How Alan Turing invented the computer, helped win World War
II and left us with one of the greatest puzzles of our time: are humans
simply computers or are we more than that? Many scientists think we
have a tenuous hold on the title, “most intelligent being on the planet”.
They think it’s just a matter of time before computers become smarter
than us, and then what? This book charts a journey through the science
of information, from the origins of language and logic, to the frontiers of
modern physics. From Lewis Carroll’s logic puzzles, through Alan Turing
and his work on Enigma and the imitation game, to John Bell’s inequality,
and finally the Conway-Kochen ‘Free Will’ Theorem. How do the laws
of physics give us our creativity, our rich experience of communication
and, especially, our free will?
Can a computer win the imitation game and pass the Turing Test?
Why do creative people make better mates than rich people?
Why are humans bad at mathematics, yet so creative?
Could an infinite number of monkeys write Hamlet?
Is our brain a quantum computer?
Is free will an illusion?
James Tagg is an inventor and entrepreneur. A pioneer of
touchscreen technology, he has founded several companies, including
Truphone, the world’s first global mobile network. He holds numerous
patents, filed in over a hundred countries. He studied Physics and
Computer Science at Manchester University, Design at Lancaster
University and Engineering at Cambridge University. He lives with his
family on a farm in Kent, England.
www.jamestagg.com
“I can’t tell you when the last time was that I had this much fun
reading and using my brain. From the very beginning, James Tagg had
me hooked with the premise; the question of whether or not humans are
the most intelligent beings on the planet....”
Janet, Netgalley
“This is a fantastic book. It seams together cutting edge
neuroscience, psychology, thought experiments, artificial intelligence/
machine learning, mathematics and even some history!...”
PFJ H., Amazon
“Hard work to read, but makes you think about the nature of human
intelligence and AI...”
Brian Clegg, Popular Science
“This is a fat book that covers a huge amount of ground. James’
topic is primarily the brain and how we think, but there is a running
theme contrasting the human brain with computers. His thesis is that
computers can never think like humans (for example, that they can never
be truly creative) and he explores many fields from philosophy and logic
to mathematics in pursuit of this proof....”
R. Hanbury, Amazon
If you have enjoyed reading this book please leave a review and if
you would like to hear more, or come to one of my talks, please join the
mailing list at: www.jamestagg.com/updates.
Are the Androids
Dreaming Yet?
Amazing Brain.
Human
Communication,
Creativity &
Free Will.

Are the Androids
Dreaming Yet?
Amazing Brain.
Human
Communication,
Creativity &
Free Will.
JAMES TAGG
Hurst Farm Books
An Imprint of
Hurst Farm Enterprises
Published by Hurst Farm Books
Hurst Farm, Dairy Lane, Crockham Hill, TN8 6RA.
+44 1732 807246
12 Williams Road, Chatham, NJ 07928
+1 646 355 1250
www.jamestagg.com
bookinfo@jamestagg.com
Copyright © James Tagg 2015
The Moral Right of the author has been asserted.
All rights reserved. Without limitation to copyright, no part of this publication may be
reproduced, stored, or transmitted in any form without the prior written permission of
the copyright owner and the publisher.
A catalogue record for this book is available from the British Library.
Publisher’s Cataloging-in-Publication Data
Tagg, James, 1964-
Are the Androids Dreaming Yet?: Amazing Brain.
Human Communication, Creativity & Free Will. /
James Tagg.
pages cm
Includes bibliographical reference and index
ISBN: 978-1-910464-03-8 (softcover)
ISBN: 978-1-910464-01-4 (ebook)
1. Creative ability. 2. Communication—Social aspects.
3. Technology—Social aspects. 4. Mind and body. 5.
Computers and civilization. I. Title.
T174 .T24 2015
303.48`34—dc23
Library of Congress Control Number: 2014945686
(hardback)
p2 220415 postc
To my family,
who have patiently listened to my interminable
ramblings about ‘Elephantine’ Equations.

PREFACE
ACPMM, Wolfson College, Cambridge
“A man may have twenty years
of experience, or one year of
experience twenty times.”
Mike Sharman
“Rules are for the obedience of
fools and the guidance of wise
men.”
Douglas Bader
I
am an inventor. I’ve always been an inventor. Ever since childhood
I’ve tinkered with electronics and computers, taking things apart
and putting them back together. There is no academic course
for inventing, so I had to choose my own path through school and
University. I studied design, physics and mathematics at secondary
school, and engineering and management at University. Part of that time
was spent in the Engineering Department of Cambridge University on a
particularly special course.
x
Are the Androids Dreaming Yet?
Mathematical Bridge, Cambridge
Every autumn about thirty graduate students arrive at the
Engineering Department in Cambridge to join the Advanced Course in
Design, Manufacturing and Management. They expect to spend the year
walking among the city’s hallowed spires, attending lectures, bumping
into Stephen Hawking and punting on the River Cam.
Instead, they get quite a shock!
In 1989, I joined the course. There were twenty-six engineers, a
psychologist and a physicist – me. There was no prescribed syllabus;
instead the course used learning-by-experience and lectures from the
experts in a given field. To study advertising, you might visit a top London
agency, for shipbuilding a shipyard on the Clyde. If you were unlucky
enough to find these two lectures scheduled for the same week, you had
to travel the length of Britain. The course runs a half dozen minibuses
to solve this transport problem. Every four weeks we would undertake a
project in a different company. I remember designing pit props for coal
mines and imaging software for a weaving company. At the end of each
project we presented our findings to each other and, with eight projects
and thirty students, this made for a great many presentations. To keep
the process manageable, the course put great store in teaching us the art
of communication.
These days I design large complex systems, and clear communication
is extremely important. My ideas are often turned into working products
and, if those products have flaws, a post-mortem usually shows the cause
Preface
xi
was a breakdown in communication. Of course, this may be a purely
personal failing, but when I talk to people in other companies they
report the same problem. It seems we all find communication difficult.
have wondered for many years why it is called the ‘art of
communication’. Surely it’s a science, governed by bits, bytes and
bandwidth. That might be true of the symbols in an email – they are
clearly encoded symbolically – but is the understanding in our brains
simply encoded by symbols? What is the physics that underlies human
understanding?
Each summer I go on holiday to escape engineering for a couple of
weeks. While away I indulge my passion for reading books by the likes
of Douglas Hofstadter, David Deutsch and Stephen Hawking. One book
that struck me years ago was Roger Penrose’s The Emperor’s New Mind.
In it, he tackles the question of what happens in the human brain when
we understand something. He extends an idea put forward by J.R. Lucas
of Oxford University that minds must be more powerful than computers
because they do something computers cannot: namely to step beyond
mere rules and see truth. Colloquially we call this ‘common sense’ or
‘stepping outside the box’.
The Lucas argument uses the theories of Gödel and Turing to
show computer algorithms have limitations. Some things are simply
not computable. Computers can do many useful things, but they cannot
discover new mathematical theorems, such as a proof of Fermat’s Last
Theorem. In 1996, Andrew Wiles succeeded in finding a solution to this
problem. This presents a paradox, solved only if we conclude Andrew
Wiles is not a computer. Indeed, since most mathematicians discover at
least one theorem during their lives, we must conclude no mathematician
is a computer! This is controversial. Most philosophers tend to the
view put forward by Daniel Dennett that the Universe is an entirely
determined place and any personal sense of free will and creativity is
an illusion. In Dennett’s worldview, Andrew Wiles is a special purpose
machine that was always destined to solve Fermat’s Last Theorem. I
believe this model is flawed. It is my aim in this book to show you why.
Indeed I am going to go further and argue all human creativity is noncomputational;
art, communication, understanding – all are based on
non-algorithmic principles.
If you consider creative thinking deeply enough you’re inevitably
drawn into the question of whether we have free will. When I get to
work each morning, the first thing I do – after a cup of coffee, obviously
– is choose which creative task to tackle first. I feel this choice is freely
made, but the determined determinists assure me I am wrong and my
xii
Are the Androids Dreaming Yet?
decision was already made. As Daniel Dennett says, “You have no free
will. Get over it!” They say I am effectively an avatar in some giant cosmic
computer game, going about my business in an entirely predefined way. I
do not agree! If they are right all the coincidences and chance actions of
my life were fixed at the time of the Big Bang. I feel this must be wrong,
but finding a chink in the determinist armor is hard work; the laws of
physics as we know them today are almost exclusively deterministic.
This book lays out the options – the chinks – that would allow free will
to enter our Universe.
To understand human thinking we would really like to look inside
a working human brain. We can’t do this yet. All we can do is observe
minds at work when they communicate with one another. If our minds
think non-computationally – as I believe – we should be able to see them
struggle when they have to translate thoughts into symbolic form. The
more symbolic, the harder it will be. This is indeed what we observe: faceto-face
communication is easy, while formal written modes are much
harder. We will explore the difference between human and computer
communication as our first step in locating the weakness in the armor
of determinism.
What do I Believe?
As a scientist, I ought not to have beliefs. I should have theories and
working assumptions. But, as a human being, I must admit believing
certain things are true. Science does not forbid beliefs. It just requires
you to be prepared to have one overturned if a better one comes along.
Richard Feynman summed this up in a lecture he delivered at Cal Tech:
“If you want to discover a theorem,” he said, “first, you guess, then you
work out some effect predicted by the theorem. Finally, you see if the
effect happens in the real world. If it does, you have a good theory. If the
effect happens a little differently, you will need to look for a better theory.”
Here are some of my overturn-able beliefs.
Preface
xiii
Beliefs
• We have true free will. We consciously decide our actions and
these decisions are in no way predetermined. We shape the
future. Allowing for free will is, therefore, a boundary condition
for any theory of our Universe.
• The world is an amazing place, but understandable. We can
understand the Universe through the application of thought and
reason.
• There is only one Universe and it appears to make sense.
• Humans think creatively, computers do not.
• The process of understanding and communication is complex,
much more complex than the digital theorems of Claude
Shannon and Harry Nyquist.
• Understanding is hard.
• The communication of understanding is even harder.
CONTENTS
Preface
ix
Introduction – Experiments, Multimedia and Puzzles 1
Chapter 1 – Mind Over Computer 3
Deep Blue 5
Man v Machine 11
Intelligence 25
The Learning Brain 35
Determinism 41
Creative Theories 49
Chapter 2 – Understanding 53
Bad Understanding Can Kill 59
The Imitation Game 65
Chapter 3 – Body Language & Banter 77
Chapter 4 – The Brain 95
Thinking 117
Chapter 5 – Knowledge 127
Chapter 6 – Kittens & Gorillas 147
Chapter 7 – Complexity & Chaos 161
Chaos 171
Chapter 8 – ∞ 177
Chapter 9 – Known Unknowns 191
The Game of Math 199
Chapter 10 – Turing’s Machine 209
The Machine 221
Chapter 11 – Software 229
Silver Bullets Can’t be Fired 233
Consequences 257
Chapter 12 – Hyper-Computing 273
Chapter 13 – Hyper-Communication 285
Chapter 14 – Creativity 295
Chapter 15 – Free Will 313
Schrödinger’s Cat 325
Twins 331
Does God have Free Will? 339
The Free Will Theorem 343
Free Will Universe 351
Chapter 16 – The Quest for Knowledge 355
Awards for Discovery 365
Chapter 17 – The Future 371
Appendix 1 – Acknowledgments 374
Appendix 2 – Bibliography 382
Appendix 3 – Puzzles and Experiments 395
Appendix 4 – Conventions in the Book 397
Appendix 5 – Index of Theorems 401
Index 405
“It is no good getting furious if
you get stuck. What I do is keep
thinking about the problem
but work on something else.
Sometimes it is years before I see
the way forward. In the case of
information loss and black holes,
it was 29 years.”
Stephen Hawking
Introduction
EXPERIMENTS,
MULTIMEDIA AND
PUZZLES
Throughout this book you will come across experiments to try,
multimedia references to track down, and puzzles to solve.
You can get additional information at www.jamestagg.com/
understanding.
If you undertake an experiment I would appreciate your leaving a
note of your results on the website and making useful comments on the
blog.
Most of the experiments and puzzles are quick and simple. The
puzzles I have set often benefit from creative thinking. I have made
finding the answers to these problems a little hard, so you are not
tempted to cheat. I want you to try to solve the problems and ‘feel’ your
brain working.
This book argues that intuitive thought solves problems in a
different way to analytical thought. The process takes time and often
benefits from putting a problem to one side while you use your mind
to process foreground tasks. I hope you read this book at a time when
the website is not available – or at least don’t peek. Give your intuitive
thought processes time to work.
Graham Wallas described the process of creative thinking in 1926
and I think it is still one of the best models we have:
First you must prepare and become fully acquainted
with the problem. It might seem impossible but don’t despair,
just commit to it. Next, you should leave the problem to
stew – incubation, he called it. After a while, you will feel
a solution is at hand. You don’t quite have it yet but you are
2 Are the Androids Dreaming Yet?
sure you will. This is intimation. Finally, some inspiration or
insight will pop into your head – this is the Eureka moment.
Now you have a solution but intuitive thinking is far from
infallible. You will need to check the solution and may find
your answer wrong the first few times. Persevere; you will
get there in the end.
As a warm-up exercise, let me give you a simple childhood riddle
to solve.
A man lives on the twentieth floor of a skyscraper with an
old elevator. Each morning he gets into the elevator and
goes down to the ground floor, but each evening he gets
into the elevator, travels up to the tenth floor, gets out, and
walks the rest of the way. Why?
ANSWER IN YOUR OWN TIME
ddd
Chapter 1
MIND OVER
COMPUTER
Computer versus Human
“I visualize a time when we will
be to robots what dogs are to
humans, and I’m rooting for the
machines.”
Claude Shannon
“The question of whether
computers can think is just
like the question of whether
submarines can swim.”
Edgar Dijkstra
“The Three Laws of Robotics:
1. A robot may not injure a
human being or, through
inaction, allow a human
being to come to harm;
2. A robot must obey the orders
given it by humanbeings
except where such orders
would conflict with the First
Law;
3. A robot must protect its own
existence as long as such
protection does not conflict
with the First or Second Law.
The Zeroth Law: A robot may
not harm humanity, or, by
inaction, allow humanity to
come to harm.”
Isaac Asimov, I, Robot
Kasparov versus Deep Blue
Deep Blue
It is 1997 and we are on the 39th story of the Equitable Center in New
York, watching a chess match. It’s no ordinary match. Two men sit
opposite each other. One, a neatly suited figure, stares intently at the
board. You can almost see the heat rising from his head as he processes
the possibilities before him. The other, sits implacably calm and, before
each turn, looks to a screen at the side of the board, reads the instruction,
and makes his move.
This is the famous match between Garry Kasparov and IBM’s Deep
Blue. Kasparov, a child prodigy, became world chess champion at the age
of fifteen and, to this day, holds the record for the highest chess ranking
ever achieved. Some consider him one of the most intelligent people on
the planet. His opponent, Deep Blue, is a massively parallel chess-playing
computer built by IBM’s Watson Research Laboratory. The machine itself
sits a few blocks north of the tournament in an air-conditioned room,
and relays the moves over a phone line to Joe Hoane, the IBM researcher
who moves the pieces.
Six months earlier, in Philadelphia, Kasparov won against Deep Blue.
This is the rematch and has generated a worldwide media frenzy. Tickets
to the event are sold out and most news organizations give a blow-byblow
report each day. On the eighth day of the tournament Kasparov and
Deep Blue are level pegging. Kasparov is playing an opening he knows
well. It’s one designed to be hard for computers to play and has been
tested extensively against Fritz, a chess computer Grand Masters use for
practice. But Deep Blue doesn’t seem fazed. Kasparov is visibly tired. On
the 16 th move he makes a dreadful blunder and sinks into despair. An
hour later, after some moments of quiet contemplation, he tips over his
6 Are the Androids Dreaming Yet?
king, gets up, and leaves the room. Kasparov has resigned, Deep Blue has
beaten him 3½ to 2½ points and is now the most powerful chess player
on the planet.
Later, when interviewed about his experience, Kasparov thought
Deep Blue must have been assisted by humans during the games because
the program appeared to play intuitively. The rules of the tournament
allowed humans to work on the program between matches, but not
during actual play. The argument has never been settled, and Deep
Blue was long ago dismantled. These days chess players avoid big public
matches against computers, arguing it is really a different sort of game.
A computer’s ability to crunch mathematically through all the many
possibilities means a chess player must play without error against a
machine, but can play a more interesting and fluid match against a fellow
human.
Chess is computer-friendly because it is a finite problem. You always
win, lose or draw. The game can’t go on forever because any position that
repeats itself more than three times is declared a draw, and if a player
makes 50 moves without moving a pawn or taking a piece, the game
is also declared a draw. In a typical game, each player makes 40 moves,
and on each turn you can choose from 30 possible moves. Although this
equates to a huge number of options, it is still a finite number.
It is possible, therefore, to create a perfect chess-playing machine.
Such a machine would project any position it encountered through
every permutation to the endgame. But, although chess is solvable using
brute force this might not be practical in our Universe. The storage
required to hold all the possible positions being analyzed would be
vast – needing most of the atoms in the Universe. You would need to
pack this information into a small enough space to allow fast retrieval in
order to play the first 40 moves in two hours. This would require storing
all the information within a sphere no larger than three light minutes.
Putting that much data in such a small space would exceed the Hawking
Bekenstein bound – a limit on the information carrying capacity of
space-time put forward by Stephen Hawking and Jacob Bekenstein
– causing the region of space-time to collapse to a black hole! Despite
these minor technical problems, an ingenious algorithm could be made
that was unbeatable: chess is essentially computable.
The term algorithm will often arise in the book, so it is worth giving
a little history. The word comes from the name of an 8 th Century Persian
mathematician, Al-Khwarizmi, and means a step-by-step procedure. We
use one whenever we do long division or look up a phone number on
Mind over Computer
7
The Music of Emily Howell
our mobile phone. It is any mechanical procedure you perform without
thinking about it. Computers are always executing an algorithm; that’s
what they do.
Fast forward to 2010 and Centaur Records releases a new classical
music CD featuring the piano music of Emily Howell. Critics are
enthusiastic about the new talent. She has composed music in a broad
range of classical and contemporary styles. You can find some examples
on my website.
But, it transpires, Emily is a computer, the brainchild of David
Cope from the University of Santa Cruz. On hearing this news critics
revise their opinion of the compositions – “repetitive and formulaic,”
“not real music,” “pastiche”. Listen again to the music and see whether
you have changed your opinion. Whatever you think, Emily has made
a good attempt at composing in the style of several great composers: J.S.
Bach and Franz Liszt, as well as modern ones such as Stockhausen and
8 Are the Androids Dreaming Yet?
Philip Glass. The compositions would get a reasonable technical score in
an exam, better than many of my attempts, but are these compositions
truly art?
There’s no question computers are gaining ground on us in certain
mathematically oriented tasks – playing chess, musical composition, and
various modeling tasks. But attempts to have them work with words and
ideas have generally produced dismal results. Until now.
In 2008, IBM unveiled Watson: a computer capable of answering
general knowledge questions. Watson has an enormous database of
human knowledge: the Encyclopedia Britannica, a billion web pages,
the entire text of Wikipedia and millions of books. It uses artificial
intelligence to trawl through this vast reservoir of knowledge and answer
questions using a statistical approach. In 2011, Watson featured as a
contestant on Jeopardy, the American quiz show, where it beat the two
record-holding contestants – the one with the highest number of wins
and the one with most consecutive wins. Let me give you a few sample
questions and see how you fare.
Question 1.
Question 2.
Question 3.
It can mean to develop
gradually in the mind or to
carry during pregnancy.
William Wilkinson’s “An
Account of the Principalities
of Wallachia and Moldavia”
inspired this author’s most
famous Novel.
Its largest airport is named
for a World War II hero; its
second largest, for a World
War II battle.
Watson answered questions one and two correctly but failed on
question three. You can probably see the final question is posed in poorly
structured English and this threw off Watson’s comprehension algorithm.
Mind over Computer
9
IBM’s Watson Plays Jeopardy
Ignoring the odd hiccup, Watson is much better at Jeopardy than a
human. Should humans be worried? First chess, then music, now general
knowledge, will all human endeavors succumb to a computer? What will
be our purpose on the planet if this happens?
Steve Wozniak
“Machines will run the world,
humans will become idle pets.”
Steve Wozniak
Man v Machine
re humans advanced computers with a temporary hold on the title,
‘most intelligent being on the planet,’ or are we fundamentally
different?
We are extraordinarily creative, but we can’t add up as
well as a cheap pocket calculator. We have poor memories, but we can
use common sense to solve problems we have never seen before. Our
communication skills are woefully imprecise, but we can tell jokes that
send our fellow humans into paroxysms of laughter. We might conclude
humans are not computers, but the scientific consensus is that brains
are ‘wet computers’. I don’t agree with this and I’m going to set out the
argument to show why man is not a computing machine.
There is an urban legend we think with only 10% of our brains. This
is not true. Science has mapped the vast majority of the human brain
using two methods. The first, an amazing set of noninvasive imaging
techniques, allows us to ‘see’ the brain as it thinks. The second is more
macabre: with seven billion humans on the planet, enough accidents
occur through sports injuries, car crashes and surgical mistakes to
provide a large enough sample to conduct research. Questioning
patients with brain-damage allows us to work out what the injured part
did before the accident.
One famous patient had an accident where the blade of a toy
sword went up his nose and damaged a small part of his amygdala and
hippocampus, the area of the brain responsible for storing memory. This
rendered the man unable to lay down permanent memories after the
accident. Events before the accident remained clear but he could not
memorize new information. You could tell a joke and he would find it
12 Are the Androids Dreaming Yet?
Turning Images to Music
funny and laugh uproariously. A few minutes later, you could tell the
same joke and he would find it just as funny as the first time. For him,
every time was the first time, because he had lost the ability to record
long-term memories. The syndrome is wonderfully depicted in the film
50 First Dates starring Adam Sandler and Drew Barrymore. Another
patient with specific stroke damage was unable to recall the names of
fruits but, oddly, could still name vegetables. Interestingly tomatoes
presented a particular problem. He had probably never known how to
catalogue them so they were partially remembered in both areas.
There are many such medical cases. In Oliver Sachs’ The Man who
Mistook his Wife for a Hat, the author relates the tale of a man with visual
agnosia who could not reliably name familiar objects, including his own
wife! He had a perfectly loving relationship with her but simply could
not name her from a picture. Sachs, Professor of Neurology at New York
University School of Medicine, provides many such fascinating stories,
along with their medical backgrounds.
The fruit and vegetable case suggests our brains are organized like a
filing cabinet. When we damage a part of the brain, it’s like losing a drawer:
All the information stored in that drawer is lost. Quite a few experiments
contradict this model and indicate many tasks are distributed around the
brain. The curious case of blindsight is one such example. People with
a damaged visual cortex can often recognize objects despite reporting
they have no sensation of vision. Show them a shape and they will report
they can see nothing. Ask them to name the shape and they might even
get a little irritated by the question; they are blind after all. But, ask them
to guess the shape and they will get it right every time. Seeing is more
Mind over Computer
13
Brain Image of Fish Hunting Prey
widely distributed in the brain than was first thought. Conscious seeing
is based in the visual cortex, but there are older pathways still active in
the brain that facilitate this unconscious seeing.
The brain is very plastic. Lose your sight through damage to the eye
or optic nerve, and the brain can repurpose the visual cortex to other
uses such as processing sound or touch. Daniel Kish has developed this
to such a high level that he can ride a bicycle despite being blind. He
clicks his tongue against the roof of his mouth and uses echolocation
to form an auditory model of the world around him. Using a similar
approach, Amir Amedi from the Hebrew University of Jerusalem has
built an audio imager that turns pictures of the world into musical sound
patterns. CAT scans of people using this system show they use the visual
cortex to convert these sound images into models of the world in similar
parts of the brain to a sighted person.
We now know roughly what each part of the brain does, but we
have no idea how it does it. The scale of an individual thought is too
small to see in a brain scan. All we can do is observe large-scale electrical
activity associated with those thoughts. A video, from a group at Tokyo
University, shows an example of electrical activity filmed in real time
as a fish hunts for its prey. Fish have transparent bodies and thin skulls
facilitating this sort of imaging. Humans are much harder subjects to
work with!
The most popular theory to explain how brains work is as some form
of computer. Computers are easy to study because we manufacture them.
They tend to crash quite frequently – usually at the most inconvenient
14 Are the Androids Dreaming Yet?
moments – so we have packed them with diagnostic monitoring systems.
These systems allow us to watch a computer think and, since they think
symbolically, we can easily read their minds.
Unfortunately computers don’t display many human-like thoughts.
They don’t laugh and cry, they don’t report consciousness and they don’t
appear to exercise free will or display creative impulses. This is frustrating
because these are the thoughts we would most like to study. It might be
that computers are not yet powerful enough, and in another few years
they will be giving Mozart a run for his money. But there may also be
a fundamental difference which renders them incapable of this sort of
thinking. This is the crux of the modern scientific debate: do humans
think differently?
Computer Brains
On the face of it, humans and computers behave very differently. Our
memories are poor, but we understand things. We are creative, but bad
at mathematics. We learn by example, computers are programmed.
We are emotional, impulsive and appear to have free will. Computers
are ordered, predictable, but lack common sense. Both humans and
computers appear to be physical, discrete systems. We both take inputs,
generate outputs and are capable of solving similar problems. Indeed,
each time we examine a problem solved by humans we usually find we
can automate it. This is known as ‘knowledge engineering’ and there are
many examples; from aerospace to finance, and architecture to medicine.
An example of where computers excel is in medical diagnosis.
ISABEL is a clinical diagnosis program designed to help ER doctors
quickly diagnose critical patients. It was created by the parents of Isabel
Maude, a little girl who presented with multiple symptoms to an ER unit.
Doctors were initially confused by the symptoms and misdiagnosed
her condition. She was later diagnosed with meningitis. Isabel suffered
multiple organ failure but survived. Her parents realized there was
something wrong with the ER triage process. They got together with
some computer scientists and built the expert system ‘ISABEL’. When
ER doctors are presented with symptoms, they must mentally scan a vast
array of literature to rule in and out possible diagnoses. The problemsolving
process is not linear; if you’ve ever watched the TV series House
it gives a great dramatization of the process. Certain symptoms might
suggest a diagnosis but are not conclusive, and there are many paths to
explore. Programmers have taken the heuristic rules from many doctors
and codified them into software. ISABEL allows a doctor to input a set
Mind over Computer
15
of symptoms and it will spit out a range of possible alternative diagnoses
with probability weightings and suggested further tests. Similar systems
are widely deployed in other fields, to build racing cars, design dams
and fight crime. Even the game consoles in our living rooms implement
artificial intelligence to make the aliens more believable and our hearts
pump faster.
Origin of Computers
Alan Turing effectively invented the modern day computer in a paper
he submitted to the London Mathematical Society in the summer
of 1936. He was not the first person to come up with the idea – that
honor probably goes to Charles Babbage – but he was the first to fully
understand its power. When we talk about computers today we mean
machines, but it is worth noting computers in Turing’s time were more
often humans using pencil and paper. The mechanical computers before
Turing were elementary at best.
Rudimentary calculating machines were developed in Greece,
Persia and China as far back as the Ming Dynasty. An astrolabe recovered
from a ship wreck off the Greek Island of Antikythera had cogs and gears
and could accurately predict the motions of the sun and planets. Many
Babbage’s Difference Engine No. 2, Computer History Museum, CA
16 Are the Androids Dreaming Yet?
of these skills were lost in the Dark Ages but, once the Renaissance was
underway in the 16 th and 17 th centuries, complex mechanical clocks were
devised that were capable of predicting the motions of the planets to a
high degree of precision. Mechanical, hand-cranked calculators appeared
in the mid-18 th century, and in 1886 Charles Babbage conceived the
first programmable computing machine, The Analytical Engine. It was
designed to read programs from cards, and used cogs and wheels to
perform the calculations. His first machine – The Difference Engine –
was designed to help the Admiralty calculate tide tables, but Babbage
realized he could generalize it to compute almost any function. He ran
out of money to complete any of his machines, but in the 20th century a
dedicated band of enthusiasts built a working model of Difference Engine
No.2. One copy sits in the London Science Museum and another in the
Computer History Museum in California. These difference machines are
not Turing complete and his Analytical Engine has never been built.
19 th Century Calculators
Mind over Computer
17
In 1935, Turing was made a Fellow
of King’s College, Cambridge, and became
interested in whether mathematical
proofs could be found automatically.
He wanted to know whether solving a
mathematical puzzle was simply a matter
of working through all the possibilities in a
methodical manner, or whether something
more subtle was required. Although chess
is a fantastically complex game, it is finite,
a big enough, fast enough computer can
play the perfect game. Is this the case
with discovering knowledge? Could a big
enough, fast enough computer calculate all
the knowledge in the Universe? Is Douglas
Adams’ fabled computer Deep Thought a
possibility, able to calculate the answer to
the ultimate question of ‘life, the Universe
and everything’, albeit with a more
enlightening answer than 42?
Turing boiled down the process
of pencil and paper computation to a
systematic program – a computer program.
Model of the Antikythera
Mechanism
He proposed a thought experiment where he would run every possible
program and see if such a procedure would yield the solution to every
imaginable mathematical problem. He was able to show this would lead
to a paradox and concluded the universal problem solver could not exist.
His discovery is one of the most important of the 20 th century – in the
same league as relativity and quantum mechanics – and I will use it as
my main tool in trying to explain the difference between brains and
computers.
Although Turing’s original paper was not intended as a blueprint
for a practical device, he was one of those rare mathematicians who also
liked to tinker with real world machines. The outbreak of the Second
World War made the practical application of his work very important,
and in Chapter 8 I will relate some of the code breaking stories that were
to make him famous and caused Churchill to credit him with shortening
the war by two years.
Calling Turing’s work an ‘invention’ is probably the wrong term;
‘discovery’ might be more appropriate. Whatever you call it, people
immediately equated human brains with computers. This is not new.
18 Are the Androids Dreaming Yet?
Each time a new advance in technology is made, people use it to explain
the working of the brain. The ancient Greeks thought the brain was a
fire consuming oxygen. When Alexander Graham Bell invented the
telephone, the nervous system resembled a maze of wires and the brain
an exchange. Brains were obviously a sophisticated telephone system.
This idea has some potentially frightening consequences, particularly in
light of the speed at which computers are improving.
The most striking feature of computer technology is the rate of
development. Cars travel faster than a person’s legs will carry them,
machines manufacture things faster than our hands are capable of
working. If brains are computers, surely it is just a matter of time before
they will think faster than humans. Turing predicted this would happen
when computers reached the level of storing around 10 billion units of
information. This happened some time in mid-2000. But today, in the
year 2014, I can report that although my computer can beat me at chess,
it still cannot fill out my expense report for me. So I am still ahead!
Maybe Turing just got the mathematics wrong. The human brain
has about 10,000 times more neurons than our most powerful computers
have logic gates. By this calculation, it’s not a billion units of storage we
need but, a trillion trillion units to put the computer on a par with a
human brain. It’s just a matter of time!
The worrying thing – especially for fans of the ‘computers taking
over the world’ science fiction genre – is that computers are improving
exponentially fast in line with Moore’s Law, and the parity point is
coming soon. Gordon Moore founded Intel with Andy Grove, and ran
the engineering department there for more than 20 years. According
to Moore’s Law, the power of a computer doubles approximately every
18 months. The next significant event in the computer versus human
competition is the gate count parity point – the moment when the
number of logic gates and the number of neurons become equal. By my
reckoning this will happen some time in 2053.
Don’t despair. There may be a few dodges yet. The gate parity point
assumes a logic gate and a neuron are equally powerful. However, some
single cell organisms with only one neuron are capable of complex
behaviors, such as hunting prey and avoiding obstacles. To perform these
simple behaviors, a computer would need as many as 10,000 logic gates,
about the complexity of my TV remote control. This gives us a bit more
breathing space. The extra four orders of magnitude push the gate parity
point out to around 2080, too late for me to see, but certainly within the
bounds of some readers of this book.
Mind over Computer
19
To give you some idea of how Moore’s Law works, the graph shows
growth in computing power over time; the y-axis is a logarithmic plot
using engineering notation. Because the growth is exponential we rapidly
end up with very large numbers. Scientists use a special notation to cope
with these large and small numbers. In scientific notation a number
is written out in a compact form. For example, three hundred can be
written as 3.0 × 10 2 . To expand it back to a regular number you move the
decimal point in 3.0 two spots to the right, making the number 300.0. A
similar technique is used for small numbers. To expand 3.0 × 10 -2 move
the decimal point 2 points to the left, giving 0.03. Why use scientific
notation? Well, once the numbers get large they would no longer fit on
a page! We can shorten the representation of numbers even further by
dropping the ‘3.0 ×’ part and just looking at the order of magnitude.
The number 10 80 , one with eighty zeroes after it, is the number of atoms
in the Earth, and 10 120 the number of particles in the known Universe.
10 -43 meters is the ‘plank number’, believed to be the smallest dimension
you can have, and 10 100 is called a googol, named by Milton Sirotta, the
Moore’s Law Extended by Ray Kurzweil
20 Are the Androids Dreaming Yet?
nephew of the famous American mathematician Edward Kasner, and
subsequently the inspiration for the name ‘Google’, the Internet search
engine.
Ray Kurzweil, the prolific inventor and futurologist, is fascinated
by this exponential growth. Exponential curves grow slowly to start with
but they pick up speed rapidly and, in the end, growth tends towards
infinity. We are all painfully acquainted with one example of exponential
growth: The common cold. Each infected cell in our body releases
virus particles into the blood which infect further cells, leading to an
exponential increase. This makes us feel rotten. Luckily our immune
system can also respond exponentially, albeit somewhat delayed, so we
survive. In the case of computer power there is no opposing immune
system fighting back, so Kurzweil thinks computers will achieve almost
limitless processing power; perhaps even within our lifetime. He thinks
this will lead to some interesting consequences, for example, allowing
people to live forever! Far-fetched? Follow his argument.
The two most important elements in keeping us alive are medical
imaging, to see what is wrong; and genetic engineering, to fix those
things. Both are improving in line with digital technology, doubling
in power every 18 months. As computers get better at seeing into our
bodies, and our ability to sequence and synthesize spare parts improves,
we will reach a point where we can fix almost any problem. Kurzweil
figures technology is improving and his body is decaying at just the right
rate to mean by the time he needs heavy duty medical intervention it will
be available. Barring a traffic accident or mad-axe-murderer, he should
live forever. Even if his calculation is slightly off, the next generation will
definitely have this option.
You might dismiss this as science fiction, but some amazing things
are already happening. Recently a female patient in the USA suffering
from bone cancer had her jaw replaced with a 3D printed component.
Doctors were able to scan her head and take an image of the good side of
her jaw, flip it right to left within the computer and repair any problems
they saw. Then they sent the image to a 3D printer. The printer made a
new jaw from tungsten powder, which was fused in a kiln. The final stage
was to cover the metal part with an inert bone-like substance to give
the human body a scaffolding on which to build real bone. They then
performed the operation to remove her old jaw and replace it with the
new one: result, brand new healthy jaw.
There are some practical limits to the power of computers on the
horizon. Currently, the wires in a silicon chip are about twenty-two
nanometers wide. That’s around a thousandth of the width of a human
Mind over Computer
21
hair, or approximately two hundred atoms wide. To match the complexity
of a brain we will need to pack an order of ten million more gates into
a silicon chip. One way to achieve this is to simply shrink the wires, but
when we get down to around ten atoms wide, quantum effects begin to
dominate. Signals in today’s chips involve tens of thousands of electrons.
We normally think of these electrons as a group, but in these tiny circuits
we need to consider the behavior of each individual electron. Problems
arise as this behavior is subject to quantum uncertainty. With only ten
electrons there is a finite probability that none of them will be where you
were expecting them to be. This causes problems for digital logic. You
can’t put a ‘1’ in a memory location and be sure when you come to read it
you will get a ‘1’ back. You have to factor in the possibility of error.
Quantum effects can be annoying – requiring us to devise all
manner of error checking hardware – but they can also be helpful.
Richard Feynman proposed using quantum bits, ‘qubits’, to perform
computation. Quantum computers can calculate many times faster than
a classical computer because a single bit can represent more than one
piece of information. Enterprising entrepreneurs are making use of this
effect to build the next generation of devices, and you can already buy a
512 qubit computer from a Canadian company called D-Wave.
The biggest problem with building more powerful conventional
chips is their area is reaching the manufacturing limit for economic
viability. Silicon wafers contain random spots of damage and, as a
chip gets larger, the chance it will have one of these spots approaches
certainty. One solution is to use the third dimension and print the logic
3D Chip, Intel
22 Are the Androids Dreaming Yet?
gates so that they communicate in the vertical direction as well. Intel
demonstrated the first three-dimensional chip in 2004, and these chips
should begin to appear in our laptops by around 2020.
Taking a chip into the third dimension solves the economic
problem but adding logic gates to a 3D chip presents a new problem
– heat. Heat is generated in proportion to the volume of the chip but
it can only be lost through the surface area. Result: the chip overheats.
Large animals have the same problem which is why elephants have huge
ears, filled with blood vessels, they can flap to cool themselves and really
big mammals, such as whales, live in the ocean. The thermal problem
is now the biggest problem in most computer designs. One data point
suggests we could solve this problem, the human brain. We pack huge
processing power into our skulls without overheating by using a variety
of techniques, including folding the surface of the brain, running each
neuron very slowly and maybe even using quantum mechanics. A very
recent discovery is that brains could be using quantum effects to transmit
signals. If true – and the research has only been recently published –
it means we may use a form of high-temperature superconductivity to
avoid overheating. More on this in Chapter 4.
Excluding exotic quantum effects, the main difference between
computer and human brains is their processing architecture. Brains
use slow, asynchronous logic to process information rather than the
fast, synchronous type used in modern day computers. Logic gates in
today’s computers work all the time, even when there is nothing to do.
For example, if I multiply 2 by 3 on my laptop the entire multiply circuit,
designed to work on 20 digit numbers will still operate, and, even worse,
it will operate on every tick of the master clock even if there is nothing
to multiply. The brain, by contrast, works only as it needs; unused gates
don’t operate. This gives a massive reduction in unnecessary power
consumption. We’d like to use this technique in modern computers but
it is very difficult to implement. Tiny changes in timing cause completely
different operation and this makes them hard to test. We accept this
sort of problem in humans, calling it ‘human error’, but we count on
computers to behave absolutely reliably, so full-blown asynchronous logic
is not likely to appear anytime soon. Some of these ideas, however, have
made their way into today’s consumer devices. For example, the chips
in the latest iPhone contain two CPUs: a small slow one operating when
the phone is dormant in your pocket, and a large fast one that switches
on when you need the power for a game or other high performance task.
All things considered there doesn’t appear to be a fundamental
engineering problem that would prevent computer hardware reaching
the complexity of the human brain. But, even with all this hardware, it’s
not clear we would have something intelligent.
Richard Branson
“I was dyslexic, I had no
understanding of schoolwork
whatsoever. I certainly would
have failed IQ tests. And it was
one of the reasons I left school
when I was 15 years old. And
if I – if I'm not interested in
something, I don't grasp it.”
Richard Branson
Intelligence
lthough my computer appears to have a mischievous sense of humor
– deleting files and crashing just before important presentations
– it does not appear to enjoy listening to Bach, nor does it ever
ask me to put it near the window to admire the view. I think this means
it’s not intelligent, but what is ‘intelligence’?
Intelligence is hard to define; I usually conclude someone is
intelligent if I have an interesting conversation with them. In the
workplace we typically hire people based on interview, rather than
practical tests. Could a computer hold a conversation with me and
persuade me it is intelligent? Meet Eliza.
Invented by Joseph Weizenbaum of the Massachusetts Institute
of Technology, and named after George Bernard Shaw’s character in
Pygmalion, Eliza runs an algorithm pretending to be a psychiatrist
offering a sympathetic ear to anyone who cares to talk. Try her out at my
website or read the sample dialogue below!
User: Men are all alike.
Eliza: In what way?
USER: Men are all alike.
ELIZA: In what way ?
USER: They’re always bugging us about something or other.
ELIZA: Can you think of a specific example ?
USER: Well, my boyfriend made me come here.
ELIZA: Your boyfriend made you come here ?
USER: He says I’m depressed much of the time.
ELIZA: I am sorry to hear that you are depressed.
USER: It’s true. I am unhappy.
26 Are the Androids Dreaming Yet?
ELIZA: Do you think coming here will help you not to be unhappy ?
USER: I need some help, that much seems certain.
ELIZA: What would it mean to you if you got some help ?
USER: Perhaps I could learn to get along with my mother.
ELIZA: Tell me more about your family.
USER: My mother takes care of me.
ELIZA: Who else in your family takes care of you ?
USER: My father.
ELIZA: Your father ?
USER: You are like my father in some ways.
ELIZA: What resemblence do you see ?
USER: You are not very aggressive but I think you don’t want me to notice that.
ELIZA: What makes you think I am not very aggressive ?
USER: You don’t argue with me.
ELIZA: Why do you think I don’t argue with you ?
USER: You are afraid of me.
ELIZA: Does it please you to believe I am afraid of you ?
USER: My father is afraid of everybody.
ELIZA: What else comes to your mind when you think of your father ?
USER: Bullies.
ELIZA: Does that have anything to do with the fact that your boyfriend made
you come here ?
If you think Eliza a bit odd, you are right. Perhaps you won’t hire
her just yet. The program looks for keywords in speech, and parrots them
back at you interspersed with a few stock phrases. If you type a sentence
containing the word “job”, the computer will say, “Tell me more about
your job.” It’s a simple program and runs to only three pages of text. It
shows how good a job you can do with very few resources and how far we
are away from a real human being. More complex programs with a larger
repertoire have been built and can hold a plausible conversation. People
are occasionally taken in by them and are quite shocked when told they
have been conversing with a machine. If you’re on your guard, they are
easy to spot. So far, no one has built a computer capable of holding an
extended human conversation and fooling a human into thinking it’s a
person. Incidentally, having a sympathetic listener is so important to
human beings these programs are used in psychotherapy and can be as
effective as drugs.
Turing proposed a test to tell whether a computer had truly achieved
human intelligence called the imitation game. His argument is as follows:
Mind over Computer
27
IQ Test
Humans are intelligent. (If you disagree with this premise then
you’re going to have a problem with this argument!) If you talk to a
computer and cannot tell it from a human, it must also be intelligent:
QED. The logic is sound but somehow feels wrong. It neatly, but
irritatingly, sidesteps the whole problem of defining intelligence.
In 1912, William Stern devised a method for measuring intelligence
in children. He named it ‘IQ’ from the German Intelligenz-Quotient.
You may have taken one of these tests at school. The tests consist of a
series of abstract reasoning problems that minimize cultural references.
For example, you might be asked to look at a set of blocks with dots on
them and identify which is the odd one out. Numerous versions of the
test have been developed over the years, but nowadays we mostly use one
of three standard tests, Wechsler being the most common.
Measuring intelligence is complicated. Culture and language play a
big part. If we take a tribe of Amazonian Indians and ask them to list the
presidents of the United States, they will fail. That does not mean they’re
stupid. Drop me into the Amazon Rainforest and I will probably starve
to death; they, on the other hand, can live off the land as hunter-gatherers
with only a few hours work per day. Who is more intelligent?
One problem with IQ is that individual candidate scores can differ
wildly from test to test, sometimes by as much as 20 points. That’s huge. At
the high end of the scale it can be the difference between being classified
as smart or as a genius; and, at the low end, between being average or
mentally subnormal. These variations don’t usually matter and most
universities and colleges take IQ with a pinch of salt, preferring more
specific tests such as SATs in America, the Baccalaureate in Europe or A
levels in the UK. IQ can be very important; and is sometimes a matter of
28 Are the Androids Dreaming Yet?
life or death. In Atkins v. Virginia, the US Supreme Court found a person
with mental disability, defined as having an IQ of less than 80, cannot be
executed.
IQ is not really a measurement, in the normal sense. Most
measurements in life are absolute, for example, distance, weight, and time.
I can prove my house is bigger than yours using a tape measure. We each
ensure our measures are the same by calibrating them against a common
reference. In the 1900s we could have walked down to the local town hall
and checked our measurements against a ‘yardstick’. As measurements
became standardized, these sticks were compared with a common central
reference. For example, the metre was a platinum-iridium bar kept at the
Pavillon de Breteuil near Paris. In the 1960s, a laser superseded the metal
reference, and today a metre is defined as 1,650,763.73 wavelengths of the
orange-red emission line in the electromagnetic spectrum of krypton-86
in a vacuum. Measurement has become very precise!
Intelligence is different. It has no yardstick. If I were to ask, “How
much intelligence does it take to design a building?” there’s no simple
answer. IQ is not an absolute measurement – it’s a relative score. Test
100 people and list their scores in order. The ones in the middle get a
score of 100; the top 5 a score of at least 130 and the top person a score
of 140. Similarly at the lower end. A person with a high IQ is probably
smarter than one with a low IQ, but it doesn’t tell you if the building
they designed will stand up. It’s rather like quoting the odds of a horse
winning the Derby. Quoting the odds does not give the speed of the
horse, nor often the winner of the race!
Despite attempts by test creators to remove cultural bias, it can
never be completely eliminated. Certain Amazonian tribes have no
concept of counting above five. For them, numbers are an alien idea and
serve no useful purpose in their habitat. In the jungle there are always
enough trees to make spears, and as a hunter-gatherer you simply need
to know where to find your prey. There is no need to count animals into
an enclosure at night. Another interesting environment is the Australian
Outback. Aboriginal Australians appear to have a remarkable aptitude for
visio-spatial memory and can remember maps or collections of objects
much better than you or I. Tests for this skill involve playing a variant
of Pelmanism. A collection of objects is placed on a tray and covered
with a cloth. The cloth is lifted for 60 seconds to reveal the location and
type of objects and then replaced. Subjects are then given a bucket full of
objects and asked to recreate the tray. You and I do a modest job. Native
Australians do this almost perfectly. Why?
Mind over Computer
29
In the vast, inhospitable Outback it is vitally important you
remember that water can be found at the two rocks near the old gnarled
tree. Forget this and you will die of thirst. It was once thought the skill
evolved through natural selection, but this might not be the correct
explanation. Recent studies show many of us can use mnemonic tricks
to significantly improve our memory. Aboriginal skills might actually be
learned and passed on from generation to generation.
IQ gives us a way to sum up intelligence using a single number
but is this too simplistic? We all have friends who would be our first
call if we met that special someone or lost our jobs. They are often not
the smartest people we know, but they are highly empathetic. These
people have ‘social intelligence’. Other friends may fail academic tests
yet demonstrate wonderful musical or artistic ability. They have creative
intelligence. As we dig deeper, more talents emerge: sporting prowess,
organizational brilliance, the ability to inspire loyalty. All these traits
appear independently of academic brilliance.
During the last century, scientists worked hard to understand these
different intelligence traits. The most influential theory came out of
studies done at the United States Army Educational testing service, by
Raymond Cattell and John Horn, and later added to by John Carroll.
Their initials give the theory its name. CHC theory breaks down the
general idea of intelligence into many different subgroups: ‘G’ factors.
If you are good at recalling all the kings and queens of England
in chronological order, or can name every member of the 1966 English
World Cup team or, perhaps, all the members of the baseball Hall of Fame,
you would have high ‘crystalized intelligence’ – ‘Gc’. It measures the sum
total of all the things you have learned and retained in your long-term
memory, your store of useful, and useless, facts. On the other hand there
is innate intelligence, the sort that allows you to solve problems where
tapping memory banks is not useful. My family often buy me puzzles for
Christmas, the sort where you manipulate bits of bent metal that appear
linked, but can be separated
with a little ingenuity. These
puzzles test our ability to work
with problems we have never
seen before and is called ‘fluid
intelligence’ – ‘Gf ’.
We can go further. A good
tennis player will have high ‘Gt’
and ‘Gv’ scores: ‘t’ for time and
‘v’ for vision, a good pub quiz
Metal Puzzle

30 Are the Androids Dreaming Yet?
contestant a high ‘Glr’ score – ‘lr’ denoting for long-term retrieval. Carol
Vorderman, a UK game show presenter famous for mental arithmetic,
would have a good ‘Gq’ score, ‘q’ for quantitative numerical skills. With
all these types of intelligence to choose from it begs the question, “Is there
a single master intelligence from which the rest follow?”
Political correctness plays a part here. It feels rather elitist to say
smart people are good at everything. It is far nicer to think we each have
our individual talents and some just have a few more than others. But
that’s not what the science tells us. ‘Group Intelligence’ – the overall G
score – does appear to be the underlying cause of the other types of
intelligence, and smart people do tend to be good all-rounders. However,
there is one major flaw in the analysis; the studies only measure the
subjects’ ability to pass academic tests, they don’t look at our success in
real-life, nor our creativity.
Lewis Terman began the long est running study of intelligence
and its relationship to life success back in the 1920s. It continues to this
day. A group of 1500 children with high IQs were selected and tracked
throughout their lives. Terman assumed their high IQs would result in
them being very successful. They certainly did well, but studies show
they did no better than if they had been chosen randomly from the same
area (all the children came from around Stanford University). Famously
two children, William Shockley and Luis Alvarez, tested too low to be
chosen for the study but went on to win Nobel Prizes for Physics in 1956
and 1968, respectively.
There are many similar anecdotes: Apparently stupid people go on
to great things. Einstein’s teacher famously stated he would never amount
to anything and Sir John Gurdon’s school report said he was ‘too stupid’
for science. He went on to discover monoclonal antibodies for which he
was awarded a Nobel Prize! Scientists have now devised the alternative
theory of an intelligence tidemark. Once above this level – an IQ of
about 130 – you can pretty much do anything you want to. This might be
because one very important type of intelligence – creative intelligence – is
not highly correlated with the rest. Creative people tend to be sufficiently
intelligent for their field but once above that threshold the relationship
breaks down. Success in creative endeavors seems to reflect strength of
character and creative aptitude rather than raw brainpower.
Physical Basis of Intelligence
The high correlation between different sorts of academic intelligence
suggests we might find a physical process within the brain leading to
Mind over Computer
31
high IQ. Functional MRI scans show intelligent people use more neurons
when tackling a given mental task, perhaps bringing to bear greater raw
horsepower, but this is not really an explanation. It is akin to saying Usain
Bolt runs faster because he gets more power to his legs. This is obvious.
What we want to know is how.
The problem with looking at brains for a common cause is the
variation from brain to brain. We all have different genes and life
experiences. On top of this, we really only see brains post mortem and
this tends to confound comparisons of brain structure. One way to
minimize the variation is to use separated identical twins. Twins have
identical genes so their fundamental hardware is the same. We should be
able to see features of the brain that are common to smart sets of twins but
absent in less smart pairs. If a feature is not shared it can be discounted as
something accidental, caused by disease, environment, or the like.
When we examine smart twins, they appear to have greater
myelination of their neurons. Myelin is a flat protein that acts as an
insulating sheath, wrapping the nerves and the neurons in our brain.
Myelination appears to be part of the mechanism involved in laying
down long-term memory – more myelin, more memories. It may also
help sustain signals and allow them to move faster over a longer distance:
the increased insulation allowing the brain to include information from
more distant parts of the brain within a given thought. But increased
myelination may be an effect of higher intelligence rather than a cause.
The brain is responsible for a significant part of our overall energy
consumption so insulating the neurons might simply help with energy
conservation. This is an active area of research.
Evolution also gives a clue to the causes of intelligence. Humans,
nonhuman primates, and dolphins all share spindle neurons. These
spread across the brain and appear to help us coordinate complex
actions between the different parts. The high function intelligence that
characterizes these disparate species requires a great deal of cooperation
between different areas of the brain. Take playing a musical instrument.
This uses physical coordination (motor cortex), sound processing
(auditory cortex), rhythm (another part of the motor cortex), along
with emotional interpretation (amygdala). Humans have more spindle
cells than other animals so this might explain our superior ability in
performing these complex tasks.
However plausible these ideas, they are all hardware arguments. It
is like me saying my word processor is better than yours because it has
gold plated connectors. That might be true – it might allow the machine
to run a little faster without electrical errors creeping in, but we all know
32 Are the Androids Dreaming Yet?
it’s software that matters. A great computer game is great because it is
cleverly written and has beautiful graphics. The speed of the hardware
might help, but it does not define ‘great’.
Can we see these software effects in the brain?
No, unfortunately, this is where our imaging technologies fail. They
lack sufficient resolution. We would need 100,000 times more resolution
to see our thoughts, even assuming we would recognize thought if we saw
it. There is no reason to believe the brain lays out thinking in anything
resembling the computer software we are accustomed to reading.
There is one exceptional group of people that does show a software
difference on a large-scale – chess players. It seems Chess Masters use
a different part of their brain to process information about chess than
you and I. This can be clearly seen on scans of the brain and is such a
gross effect it even shows up in old-fashioned EEGs – where electrodes
are taped to your head. Interestingly the effect can be used to predict
greatness. Players likely to become Grand Masters show they use a
different part of their brain from the rest of us at an early age. Chess
players possess the only large scale wiring difference we know of, but
there is another group with a visible physical difference, London taxi
drivers. Their hippocampi are noticeably larger than the rest of ours. The
hippocampus does many things, but one of its most significant jobs is to
memorize maps. The three years it takes to acquire ‘the knowledge’ and
the subsequent years of navigating London’s complex streets give cabbies
a 30% larger hippocampus than the average London resident.
Is Intelligence Static?
We’ve all seen the headline. Every summer public examination results
come out and every year is pronounced a record breaker! Year after year,
students get better and better grades. This creates a problem. There’s is no
better grade than an A – and eventually all students get As. Welcome to
grade inflation – a problem affecting systems the world over, from British
‘A’ levels to Harvard grade point averages. Newspapers are awash with
stories bemoaning the dumbing down of today’s tests. “Examinations
aren’t what they used to be.”
Grade inflation undoubtedly exists and studies of undergraduate
grades show progressive compression into the top grades, most competent
students get ‘A’s, making it difficult to distinguish a good student from a
great one.
Mind over Computer
33
At first glance, the problem appears to be one of social engineering.
Teachers don’t want to disappoint, and academic institutions want to
improve on last year’s results. The people awarding the grades often have
a vested interest in those grades improving. Even a tiny positive bias in
the most scrupulously honest teacher is enough for grades to creep up.
However, grade inflation might not be purely a matter of over enthusiastic
teachers. IQ scores are also rising. Welcome to the Flynn Effect.
James Flynn, Emeritus Professor of Political Studies at the
University of Otago in Dunedin, New Zealand, reported in 1987 that
IQ scores rise over time throughout the world. All told the population
gains about one IQ point every three years, and approximately every
ten years IQ tests have to be re-calibrated, so the average student once
again receives the average grade. This is a mystery. It is a large effect and
cannot be explained by the rote learning of lots of sample questions. The
human race is either rapidly getting smarter or the least smart members
of society are coming up to the general average fast; either way it means
there are fewer dumb people around. The Flynn Effect has recently
slowed in western countries, suggesting it might be that intelligence is
converging rather than increasing overall. Another interesting fact is
people become more intelligent as they age, gaining about one IQ point
every ten years. Against the stereotype, it’s not all downhill after forty.
There is hope for me yet!
Until recently we thought IQ was fixed, but new research contradicts
this. Muscles get stronger with exercise, physical skills, such as playing
golf and tennis, improve with practice; why not intelligence? Scientists
used to believe brains couldn’t get smarter; you had the IQ you were
born with. You might learn more ‘stuff ’ during your life, but the G factor
stayed the same. It looks like this is wrong and we were simply not using
the right exercises.
In 2008, Susanne Jaeggi and Martin Buschkuehl, of the University
of Maryland, modified an intelligence test into a game and showed
playing the game improved ‘fluid’ intelligence and increases IQ. They
believe playing their game helps improve working memory – the shortterm
memory we use for storing sums as we do mental arithmetic –
or remembering telephone numbers. Previous attempts to improve
IQ through practice had not shown much success as the skills did not
transfer between tests, but working memory is such a useful thing it
appears to help across the board.
These factors argue against intelligence being a hardware feature of
our brain. It does not remain static but instead improves with age, time,
and education.
At the beginning of the chapter, I said Garry Kasparov was once
thought to be one of the most intelligent people on the planet. When
his IQ was eventually tested – the German magazine Der Spiegel put up
the money – he scored 135. That means, in academic terms, he is smart
but no genius. Yet, he is undoubtedly a genius by any common sense
definition: the best chess player to ever live. These days he involves himself
in politics rather than chess and is still uniquely able to concentrate for
long periods of time. Concentration seems a very important factor.
Einstein was once asked where his genius came from. He replied that
he did not consider himself a genius but instead put his success down to
his persistence and ability to concentrate on a problem for many years.
IQ tests say nothing of our ability to concentrate over extended periods
and nothing about our drive to change the world. The tests are, at best,
a useful but dangerous diagnostic tool for educators. One of the worst
things IQ can do is pigeonhole people. Would Kasparov have become
world champion if he had been given his IQ score of 135 as a teenager
rather than late in his thirties after he had conquered the world?
“Education is what is left after
what has been learnt has been
forgotten.”
B.F. Skinner
Hole-in-the-Wall Experiment
The Learning Brain
Human beings are born with an extraordinary ability to learn
through experiencing the world around them. Studies show
babies as young as three weeks understand musical ideas,
smiling as you play music to them in a major key and frowning at music
in a minor key. By six months, babies have learned to distinguish the
relationship between objects, and by two, they have a command of
language and are beginning to develop a theory of self. They understand
how to lie and become adept at playing parents off against each other!
Sugata Mitra, of Newcastle University, has run an experiment in
India to test minimally invasive education called the ‘Hole in the Wall
Project’. As the name suggests, he cut a hole in the wall of a building in
Delhi and put a computer in it. The hole opens out onto a slum district
and local children rapidly discovered the computer. Without any formal
training they picked up the necessary skills and very soon became adept
at searching the Web. Remember, in order to ‘pick up’ this skill they often
had to learn the English language as well.
Another example showing children’s innate ability to learn is Nicolas
Negroponte’s ‘One Laptop per Child’ program, which gives computers to
children in remote villages around the world. The laptops are a triumph of
cost engineering but are fully functional and can connect to the Internet.
The inspiration for the project came from an analysis of the economics
of the computer industry. Huge capital investment in the western world
is driving most costs down, but one cost that seems to have stuck fast is
the access device. Laptops tend to remain at a floor price of around $500,
far too high for much of the developing world. At $500, a computer store
makes $80 when they sell you a laptop. This is as low as is cost-effective
36 Are the Androids Dreaming Yet?
Laptops Galore
for them to stock the machine, employ someone to tell you about it, and
fix it if it goes wrong in the first year. Value for money improvements
have all focused on faster processors, more memory, sharper displays
and larger hard drives, not lower prices. These improvements are useful if
you want to shoot aliens, but overkill if you only want to surf the Internet
and learn the ‘3 Rs’. So the ‘One Laptop per Child’ project has developed
a device for $100.
Negroponte is often asked how he deals with the maintenance and
repair issues. His answer, “There aren’t any.” The computers are treasured
possessions and rarely broken or lost. Children become empowered by
the machines and can access knowledge and information far beyond the
wildest dreams of their parents’ generation. Stories abound of children
checking the spot prices for wheat or coffee on the Chicago Stock
Exchange, and advising their parents on the price to accept for their
crop. Negroponte estimates there are currently 500,000 children in South
America teaching their parents to read!
It’s interesting to speculate whether children learn spontaneously or
are somehow ‘programmed’ by the adult members of society. In both the
‘Hole in the Wall’ experiment and the ‘One Laptop per Child’ program
the children could simply be learning from adults and older children, but
there is a novel way to eliminate this influence. Negroponte and Mitra
have teamed up to run an experiment to see how children learn for
themselves. They are planning to air-drop laptops into remote villages
in the Andes. In this scenario, the children can’t possibly learn from the
Mind over Computer
37
One Laptop per Child
adults – the adults have never even seen a computer before. Instead,
they must rely entirely on their innate learning ability. At this point, the
experiment has only just started; I will put details on my website as the
experiment progresses.
The 10,000 Hour Club
Learning by experience takes
humans quite a bit of time. Anders
Ericsson, Professor of Psychology
at Florida State University, studied
musicians in the early 1990s and
found they had accumulated a
huge number of practice hours
by the time they became experts.
His research was popularized by
Malcolm Gladwell, in the book
Outliers, and by Daniel Coyle
in The Talent Code. The idea is
that humans need around ten
thousand hours of practice to
become proficient at a skill. The
more skilled players seem to have
simply accumulated even more
Dan McLaughlin
38 Are the Androids Dreaming Yet?
practice. A number of people
have wondered whether you
can take this literally, and if you
devote 10,000 hours to practicing
something you can become
world class. Dan McLaughlin
from the USA used to be a
professional photographer and
decided he might like to become
a professional golfer. He quit his
job and is now 3,500 hours in. So
far, he has achieved a 4 handicap.
I also personally got bitten by this
bug and am learning the piano. I
am about 3,000 hours in and am
making good progress.
Gladwell’s interpretation of
Ericsson’s results is not without
controversy. Ericsson stresses
‘purposeful practice’ is the
important element. Practicing
Piano Practice
the wrong thing for ten thousand
hours will just make you good at doing something wrong. Practicing
without concentration and attention will equally have little effect. One
illustrative example is the story of Edward Sanford, a supreme court
judge, who read the morning prayer aloud every day over a 25 year
period. After he retired he was asked if he could recite it from memory.
Despite reading it as many as 5000 times during his working life, he was
unable to remember it. It seems you must purposefully practice the exact
thing you want to do if you wish to learn it, in this case recall.
Computers don’t require practice to learn a skill. If their program
is right they work correctly, and if it is wrong, they are always wrong.
Computers can be programmed to learn but so far this learning has been
limited to specific problem domains, such as face recognition. They do
not have the general-purpose capability humans enjoy.

Astrological Clock at Hampton Court Palace
“The die is cast”
Shakespeare
“How does the water of the
brain turn into the wine of
consciousness?”
David Chalmers

Determinism
I
have free will.
Look…
I can choose to type any word I like.
Giotto...
Many philosophers tell me I am deluded. I was always going to type
that word and I have no free will. Everything in my life is predetermined.
I’m rather like a character in an enormous video game. The character
might think it was free to act, and its actions would appear random.
Yet from the moment the player clicked the button to start the game,
every action the character takes is determined by a preprogrammed set
of rules. This is the free will debate. How can we tell we are free? Would
there be any observable effect?
One of the big problems is that philosophers codified much of our
modern theory of free will in the 19 th century, at a time when all the
known physical laws were deterministic and reversible. They could not
see a way for free will to emerge from such physical laws. There was even
a group called the Compatibilists lead by David Hume that thought free
will could coexist with determinism. Provided you felt free it did not
matter that your actions were inevitable.
We all want free will to mean actual freedom to make conscious
choices. We would like to affect the world in which we live; not the other
way around. I dislike making definitions – I find they take away from the
core argument and only result in linguistic jousting, but it seems that two
centuries of philosophers have avoided a proper discussion of free will by
loosely defining the term. Here is my definition:
42 Are the Androids Dreaming Yet?
‘We consciously, and through the exercise of will, make decisions
between different choices without anyone or anything causing the
decision in advance. Others can influence decisions – by offering advice
or even holding a gun to our head, but we choose.’
If you can devise a better, stronger definition please email me and
I will revise my definition to your better one. I’m searching for the most
powerful definition of free will – totally free and born out of the exercise
of will.
The human mind appears to have free will. At least this is my
personal conscious experience. Computers, on the other hand, do not.
They run programs that dictate exactly how they will operate in every
situation. Could a computer be programmed to have free will? That’s
hard to do. Let’s see why.
Thinking with Clockwork
Astronomers have been predicting the motions of the heavens for
centuries and to do this they need accurate clocks. The very first clocks
were sundials. These suffered the obvious disadvantage of not working
at night, but it was also unsatisfactory to use the motion of the sun to
predict the motion of the sun. The earliest ‘heaven independent’ clocks
used water flowing through small holes in pottery vessels. They were
effective over short intervals but plagued by dust, dirt and evaporation.
It was the invention of the anchor escapement that enabled the first
accurate mechanical clocks.
By the sixteenth century clockmakers had gone to town developing
astrological clocks with more and more gears, to show all manner of
information; the phases of the moon, the motions of planets, even the
motion of moons orbiting those planets. These clocks became hugely
ornate. The astrological clock at Hampton Court Palace was built for
Henry VIII circa 1542 and, as well as showing phases of the moon
and the signs of the zodiac, it accurately calculated the time of high
tide at London Bridge, allowing Henry to travel quickly to the Tower
of London. You might also notice it shows the sun orbiting the earth!
Copernicus published his book, De revolutionibus orbium coelestium (On
the Revolutions of the Celestial Spheres) showing the earth orbited the sun
a year later in 1543, and it took centuries before it became accepted fact.
Clocks need gears. The humble gear is a simple machine. They
work because wheels of different size have different circumferences – the
distance around the edge – but one full turn is the same for all wheels.
Imagine you have a circular sweet such as a Life Saver – or Polo for
Mind over Computer
43
British readers – and you roll it once around the wheel of your car. The
small sweet will turn many times. Now put a pencil through the hole
in the sweet, jack up your car so the wheel is off the ground, hold the
sweet next to the wheel of your car and press the accelerator. The sweet
will spin round very fast and probably disintegrate in a shower of minty
sugar. This is the principle of gearing. A small circle has to do a lot of
work to keep up with a big circle. It’s very predictable. The sweet will turn
a set number of times for each rotation of the car wheel, equal to the ratio
of the circumferences of the two circles.
Gears usually have teeth to lock the wheels together, but this is really
just to make sure they can’t slip against one another when they transfer
huge forces, such as in racing cars. Some passenger cars have been built
with smooth gears; a friend of mine had one at university. If he put his
foot down too hard, the gears would slip, heat up and you would get a
terrific smell of burning rubber. If you were lucky you could leave the car
for a few hours and all would be well. But, if not, you had to replace the
rubber belt, which was very expensive. Toothed gears generally win out.
Toothed gears also have the enormous benefit of being digital. This
is quite important if you want to keep things accurate. Gears can’t move
a fraction of a tooth so if a toothed gear has ‘slipped’ forward a small
amount, it will be kicked back into position when it meshes with another
gear.
In a modern mechanical clock, a balance wheel swings back and
forth on a spring and moves the main gear one notch forward each time
it passes its central position. Gears divide this down to move the hour and
minute hands. If I put the hands of a clock at midday and let the clock
tick 86,400 times, the clock hands will come back to the same place. Once
you understand how a clock works you can play a trick. If you tell me the
number of ticks the clock has tocked, I can tell you the exact position the
hands will be in. To a small child this might be dressed up as a magicians
trick – but, of course, it is simply a matter of dividing the number of ticks
by 60 and then 60 again to calculate the amount of time elapsed. This
type of precisely predictable behavior is called deterministic behavior.
Something is deterministic if you can set it up in a particular way and
know the exact state later or, conversely, examine something and trace it
back into the past.
Modern computers scale up clockwork and make it much more
efficient; gears are translated into electronic logic gates and a quartz
crystal vibrates at 1000 million ticks per second to give us the clock tick.
On each tick, the computer can do a mathematical operation, store and
retrieve information, or branch down an avenue in its program. Using
44 Are the Androids Dreaming Yet?
these simple building blocks the computer allows us to play computer
games or process the words of this book as I write. Importantly, all these
operations are deterministic; given a set of inputs the computer will
always generate the same outputs and that means a computer has no free
will.
“Ah,” I hear you say, “but my computer plays games with me and is
not predictable, otherwise I would always beat it.” You are right, but the
computer has a clever trick to fake non-deterministic behavior: it uses
you!
Computers on their own cannot generate random numbers. All
a computer can do is generate a pseudo-random number and it does
this by working its way through a very long calculation. It could, for
example, calculate the first thousand digits of π (pi), and then start using
the subsequent digits as random numbers. The digits look jumbled up
but we know they follow an entirely predictable pattern. The computer
appears to behave randomly because when I press the button to kill
an alien the computer picks the number it had counted up to at that
moment, say the 55,678 th digit of π, and uses that. It is I, the human, who
unconsciously picks the precise moment in time and therefore provides
the random element. My choice is governed by all sorts of extraneous
quantum influences: Did I have coffee this morning? Was it a big mug
or a small cup? How hot was it? All these things will be important as
they determine the amount of caffeine absorbed across the brain blood
barrier and the exact timing of my actions.
Humans are not good at consciously generating random numbers.
We tend to choose the same numbers too often. If I ask you to pick a
number between one and ten, you are likely to choose three or seven.
This effect is called social stereotyping; magicians often use it when they
pretend to read your mind. The problem arises because we tend to over
think the problem. I asked you to pick a random number between one
and ten. You won’t pick one or ten. Five is too obviously the mid-point.
Even numbers don’t feel random. Nine is too large. That just leaves three
and seven. So the mind reading magician has you! Humans can unlearn
this social programming and become quite good random number
generators but normally we tend to conform.
There is a way two humans can generate a truly random number
without training. Find a friend for this experiment. One of you should
pick any number between one and ten and start counting under your
breath, when you get to ten just go back to one and keep repeating. The
other should wait a while and then shout stop. The number reached should
be genuinely random. Please post the results on my website and I’ll tell
Mind over Computer
45
you if this crowd-sourced random
number generator really works.
There should be no way to predict
the resulting number as both
of you are affected by quantum
randomness and, provided you
wait a little before shouting stop,
any social stereotyping should
be overcome. If you want to be
scientific, remember the random
number you started with and the
length of time before your friend
shouted stop. There should be an
improvement in randomness with
the amount of time they wait.
In the absence of human
interaction another way to give
a computer access to a random
number is from a quantum
device. A lava lamp works well!
The Lavarand, developed by
Silicon Graphics, is a hardware
random number generator which
uses images of a lava lamp to seed
a random number generator. It is
covered by U.S. Patent 5,732,138,
Lava Lamp
titled “Method for seeding a pseudo-random number generator with a
cryptographic hash of a digitization of a chaotic system.” Got that!
A computer does not acquire free will just through the injection of
randomness. You could simply put an intercept on the link from the lava
lamp to the computer and completely predict the computer’s behavior.
The system as a whole will certainly do unpredictable things, but the
computer did not make a choice; behaving randomly is not exercising
free will. Where is the will?
Consciousness
I remember my first trip to Death Valley in the United States. We were
driving along the main east-west highway at the bottom of the valley
and a sign said, “Turn off your air conditioning now.” I did as I was told
and to cool down I opened the window. When I put my hand out I felt
46 Are the Androids Dreaming Yet?
nothing; no wind chill, nothing. The air was so hot the wind carried no
heat from my hand. When I imagine hot weather it always brings back
this memory. It’s my conscious experience of the world.
Humans experience the world through a vivid lens we call
consciousness. It allows us to think about the world as we watch it and
plan actions. But, it also summons associated memories, something
scientists call ‘qualia’. Most writers describe consciousness as an internal
dialogue with themselves and see it as a consequence of human language.
That’s probably because most writers are linguists. Non-linguists, perhaps
even dyslexic engineers like me, experience consciousness as more of a
visual dialogue.
It’s hard to pin down consciousness as the difference between
humans and computers. Computers do have something that resembles
consciousness; they have watchdog functions, they plan and anticipate
actions and are aware of their own existence. But they don’t understand
or make free choices based on this consciousness. It is an entirely
mechanistic affair. A computer might know its CPU is overheating and
send a notification message to the administrator, but it does not really
appreciate what this means. It does not have our sensation of a near death
experience. This self-awareness is the ‘hard question’ of consciousness.
Why, despite the computer knowing it is overheating, does this not
translate into the intense experience we have? Philosophers, such as
Daniel Dennett, think this lack of consciousness is only a matter of time;
once computers live long enough and have sufficient internal complexity
they will begin to experience the world the way we do. We are nothing
special.
The problem with consciousness is it does not seem to have any
externally discernible effect. Anesthetics can take it away and brain
scanners can see that it has been switched off, but what is it for? I think it
comes hand in hand with our faculty of creativity. Consciousness allows
us to shape the world – not the other way round.
Steve Jobs
“We can't solve problems by using
the same kind of thinking we
used when we created them.”
Albert Einstein

Creative Theories
Once I have exercised my free will by getting out of bed in the
morning, I often decide to do something creative. Humans seem
driven to create. We compose music, draw, paint, and solve
mathematical puzzles. Computers are not naturally creative; they spend
most of their time doing exactly the opposite – following preset rules. Is
this a fundamental limitation distinguishing the computational world
from the real world?
The Conventional View
Most scientists believe pattern-matching algorithms in the brain allow
us to be creative. To see how this might work, imagine our brains are
chaotic – not hard to do – and process many competing ideas at the same
time. The neurons in our brains build millions of useful, and useless,
connections based on the patterns in the data we see and hear. Then a
selection process goes to work – something akin to natural selection – to
sift and prune the connections until something bubbles to the surface
and we get that, ‘aha’ feeling.
Douglas Hofstadter, Professor of Cognitive Science at Indiana
University, famous for the book Gödel Escher Bach, has written a computer
program using pattern matching to discover number theorems; things
like any number ending in a zero is divisible by 5. The program produces
interesting results, even perhaps generating some new theorems. He
argues the human brain is essentially a scaled up version of his program.
By the way, if you like trivia, his book Fluid Concepts & Creative Analogies
was the first book ever sold on Amazon.com.
50 Are the Androids Dreaming Yet?
The Unconventional View
Roger Penrose, Professor of Mathematics at Oxford University, holds a
completely different view. He thinks brains operate in a non-algorithmic
manner and provides a sketch of the possible mechanism in two books –
The Emperor’s New Mind and Shadows of the Mind. He suggests tubulin
molecules, which form the skeleton of our neurons, exploit quantumgravitational
effects to calculate non-computable functions. The
scientific community was initially highly skeptical that quantum effects
could survive the warm, wet environment of biological systems, but in
January of 2014, Edward O’Reilly and others at UCL discovered plants
use quantum effects to improve the efficiency of photosynthesis. No
prize has yet been awarded for this discovery but it must be a contender
for a Nobel Prize at some point. Recently Travis Craddock, now of the
Nova Institute in Florida, has submitted a paper showing a very similar
geometry of proteins exists within tubulin microtubules in the brain. He
believes this is evidence quantum effects may exist there as well.
A simple quantum effect in the brain could merely reduce the
resistance of the wiring in the brain to help conserve power and avoid
overheating. We recognize this is a major problem in building small,
powerful conventional computers. Roger Penrose suggests an altogether
more radical idea. He proposes our brains are quantum gravity computers
capable of calculating non-computable functions. We don’t yet have a
theory for quantum gravity so his idea is at the cutting edge of physics
– read highly controversial. He raises a deep mathematical question. If
the Universe is deterministic and effectively equivalent to a computation,
how does ‘creative’ knowledge emerge within it? Lots of knowledge can
be manufactured by simply mechanically rearranging data. That’s what
happens when I watch a DVD or play a computer game, but, at some
point in the past, a director or a programmer had to put in the creative
effort to make the movie or write the computer program. How did that
happen? Was it baked into the fabric of the Universe at the moment of
the Big Bang? Is what we take for a Universe really nothing more complex
than putting a DVD in the slot and hitting play?
One last piece of trivia links Hofstadter with Penrose: Roger
Penrose and his father invented the Penrose Steps, inspiring the neverending
staircase in the Escher prints featured in Hofstadter’s book. For
movie buffs, the Penrose steps appear in the film Inception, starring
Leonardo DiCaprio. The fact we get pleasure from these trivial links tells
me something is going on in our brains that is not so mechanical.
Mind over Computer
51
M. C. Escher’s Ascending and Descending (Penrose Steps)

Chapter 2
UNDERSTANDING
Afghanistan COIN Dynamics
“Power corrupts, PowerPoint
corrupts absolutely.”
Ed Tufte
“No battle plan survives contact
with the enemy.”
Colin Powell
Originally, Helmuth von Moltke
54 Are the Androids Dreaming Yet?
John Masters stood up to address General Stanley A. McChrystal
and his military staff in Kabul. The topic, of course, the war in
Afghanistan. The main war lasted only eight weeks, but this did not
end the conflict. A level of tribal violence and insurgent warfare rumbled
on for years, killing around 30 people a week. Masters’ job was to explain
the dynamics of Afghanistan and provide politicians and military
commanders a framework to understand what was going on.
Think about your country for a moment. What maintains the fabric
of society – police, family, the local charity club, church, newspapers, the
broadcast media? All these institutions work to keep us civilized, but what
happens if a country loses them? There are institutions in Afghanistan,
good and bad: tribes, gangs, corrupt officials, families. Masters had spent
a year investigating these interactions, and questioning the returning
commanders. He and his team believed that understanding the dynamics
of the conflict was the key to bringing peace to Afghanistan.
If you live in an industrialized country, you rarely see society
without its civilizing web in place. One interesting ‘experiment’ that
shows what happens when it fails was the 1976 traffic police strike
in Finland. Finland is a fantastically law abiding country where most
people obey both the written and unwritten laws. During the strike, this
behavior changed. Many people began parking illegally but refrained
from blocking the roads. A few took advantage of the absence of police to
drive incredibly fast – twice the national limit. These would be labeled as
‘defectors’ in game theory. Without traffic police, a different automotive
General Stanley A. McChrystal
Understanding
55
morality emerged, a different structure to society. Of course all the other
parts of society remained the same. People paid their taxes and went
about their lives normally; only the traffic behavior was affected.
Afghanistan has had most of its social structures removed over the
last forty or so years. First the Soviets, and then the Taliban, took apart
much of the fabric until finally the Allied Forces swept the Taliban out,
leaving very little behind. There were no police or courts, and few laws
– or at least none enforced by the rule of law. The Allied Forces have
spent a decade rebuilding these structures. Before we examine Masters’
presentation, let’s look at the daily life of an Afghan farmer.
If you are an Afghan farmer you have a dilemma. Your most reliable
crop is opium. It grows well in the arid soil, does not require irrigation, and
is resistant to most pests. For this crop there is a financial infrastructure
to rival the Chicago Commodities Exchange. You get interest free loans
secured against the crop, and you can forward sell your product on a
futures market. Your investors can ‘add value’ by dealing with the major
pest – the US military. They do this through the simple expedience of
taking pot shots at them if they get too close to the crop. Since a field of
opium is worth $30,000 and a militia wage for the year is $350, you can
easily employ a few men to protect your investment. Of course, you are
indebted to thugs and criminals, but they are at least reliable thugs and
criminals.
On the other hand, the traditional products of the Himalayas –
walnuts, pomegranates and vines – need years to cultivate. There is no
forward market and the timescales over which you must take risks are
far greater. If you believe your American protectors will leave before the
crops mature, you will be loath to plant and care for them. But, if you
make the decision to take this risk, you have a strong incentive to foster
stability and reap the rewards of your effort. There is a feedback effect:
the balance of power between all the different parties is important to the
decisions you make, and the decisions you make affect your desire to
invest in future stability.
Masters’ team built a slide pack to demonstrate the complex
interactions between the groups: farmers, security, stability, markets,
military power, and emerging institutions. The COIN – COunter
INsurgency – dynamics slide shows just how hard it is to communicate
complex topics between human beings. The presentation is beautifully
crafted but it was a public relations disaster. At the end of the presentation
General McChrystal said jokingly, “When we understand that slide, we
will have won the war.” The slide was paraded in the press as, “the most
complicated PowerPoint slide in history.”
56 Are the Androids Dreaming Yet?
If you invest a little time on the slide you will understand it
and may even see it as a thing of beauty. But Masters’ audience was
obviously expecting something different and, presented with this level
of complexity, went into shutdown. Perhaps they wanted a simpler
presentation, a high-level summary, a few bullet points. Of course, there
is no simple presentation on Afghanistan. The lesson is that context,
timing and expectation are often as important to good communication
as the elegance of the content, and that information is a complex thing.
If you want a lighthearted poke at PowerPoint here is Peter Norvig’s
PowerPoint version of the Gettysburg Address.
Understanding
Next time you are in a business meeting, count the number of times the
word ‘understand’ is used. If you ask the people around you what it means
you’ll stump many of them. That’s because understanding has two very
different meanings. Most people don’t separate these meanings but the
distinction is important. Understanding means to decode information,
to comprehend – but, more importantly, it also means to absorb and
internalize information. That feeling you have when you ‘get it’.
If I say, “I understand” I mean I have taken in the question you
asked and decoded it into ideas so I can provide an answer. This can be
quite a mechanical process and computers routinely understand natural
language and answer questions – Apple’s digital assistant Siri being a
case in point.
When I say, “I understand a problem” or “understand a culture”
I mean something far less tangible. Somehow the information I have
gathered over my life is formed into a matrix within my brain that allows
me to ponder and run scenarios. I can predict the effects of my actions
before I do them, and often anticipate your responses. That’s clearly
a very useful evolutionary adaption, but is there more to it than that?
Roger Penrose and David Deutsch think understanding allows us to
transfer non-symbolic information from one brain to another. We don’t
run programs in our brains, nor do we store precise information such as
lists and tables. We have, therefore, had to evolve a creative approach to
communicating skills and understanding each other. One of the most
closely studied areas in the field of communication is when it breaks
down in the lead up to a disaster.
Understanding
“The human mind tends to look
for clear linear relationships, we
like solutions that are close to the
problem in time and space and
make sense when we think about
it quickly, unfortunately, those
simple solutions are usually
wrong and come from acting on
a complex system as if it was a
simple one.”
Brett Piersen
57
Gettysburg Address, Peter Norvig
Space Shuttle Columbia Crew Photo
“For a successful technology,
reality must take precedence
over public relations, for Nature
cannot be fooled.”
Richard Feynman
Bad Understanding
Can Kill
On January 16, 2003, at 3:39pm, the Columbia space shuttle took
off from Cape Canaveral. During the launch a small piece of
foam insulation broke off the fuel tank and hit the shuttlecraft.
The event was recorded on a few low-resolution video frames. They
show a tiny white object hitting the shuttle and a plume of dusty material
splattering outward. The shuttle made it safely into orbit and for two
weeks engineers on the ground debated what to do. In the end, it was
decided the risk was minimal and the shuttle could safely return to Earth.
On reentry, the shuttle disintegrated, killing seven astronauts.
NASA managers had decided the shuttle was undamaged based
on a series of presentations by the engineers. One image in particular
analyzed the potential
damage to the shuttle’s
tiles from an impact.
Read the slide, look
at the key frames, and
decide for yourself what
action you would have
taken.
Shuttle Tile
60 Are the Androids Dreaming Yet?
NASA Internal Slide
WHAT DO YOU UNDERSTAND FROM THE SLIDE?
Some images of the launch are shown on the right
ddd
Here is what you should have understood from the slide: tiles are
really tough but if the foam dislodged from the fuel tank broke through
the outer coating it would cause significant damage. The estimated speed
of the foam hitting the tile was 640 times greater than anything previously
tested. Worried?
Is this a proper understanding of the problem? You have the slide
and the images. Take another look and think hard. If you want, you can
check a video of a similar launch on YouTube to get a feel for the scale of
things, but the still frames shown all the information you need to make
your conclusion.
Understanding
61
Photographs of the Foam Impact from Video Footage
Frame Showing Foam Dislodging
62 Are the Androids Dreaming Yet?
Still from Ground Camera
LOOK AT THE IMAGES, WHAT HAPPENED?
ddd
The truth is you simply don’t know. If you are puzzling over the
strength of tiles, you have been misdirected. There is video footage of
some sort of impact on a wing mostly covered in white tiles, and a slide
describing the effect of a benign sounding ‘foam’ hitting those tiles. But
what is the evidence for an impact on a tile? The shuttle is certainly not
made entirely from tiles; I can see a window in the picture. You should
instead be asking more questions, “What happened?” “What hit what?”
and “How bad is that?”
It was bad. The foam, a very tough material, had hit the leading edge
of the wing, a weak point, punching a hole through it. The wing failed
on reentry and tore the shuttle apart. Clearly, a full discussion of the
possibilities did not occur amongst the shuttle team, or perhaps it only
happened amongst the engineers in private. Once the analysis was tidied
up and presented to ‘management’ it was a one-way communication of
the conclusions, not a discussion of the underlying ambiguous thought
process. The result: people passively listened to the information rather
Understanding
63
than interactively understanding it and agreed on the recommendation
that it was safe to return. Clearly they did not understand the ambiguity
otherwise they would have realized they did not have enough information
to form a conclusion. This is the tragedy of lack of understanding. If they
had known how little they knew, they could have deployed a spy satellite
to take pictures of the damage – one was available nearby and would
have taken a few hours to re-task – but they did not.
Ed Tufte served on the second shuttle disaster commission and
provided an analysis of the disaster. He views slides as a poor medium for
communicating complex problems and thinks documents are far better.
The danger with slides is they force you to simplify information in a way
that destroys the essence of the information. His analysis of the failure of
communication at NASA formed a major part of the final report on the
disaster. Later he coined the paraphrase “All Power corrupts; PowerPoint
corrupts absolutely.” Good communication benefits from stories and
narrative, not bullet points and graphic fluff. Instead of using bullet
points, speak! After all, we have evolved for 250,000 years to understand
language, but only 25 to read PowerPoints. ’
If you write presentations, Ed Tufte’s book The Cognitive Style
of PowerPoint is compulsory reading. He argues that much of the
information you want to communicate is complex and interconnected.
PowerPoint or any similar presentation software encourages you to
simplify it into hierarchical bullets. The format implies simple causal
relationships where none exists. This is dangerous. Communication
should convey understanding – which is very important – and not just
information. What, you ask, is the difference?
Searle’s Chinese Room
“The hardest thing to understand
is why we can understand
anything at all.”
Albert Einstein
The Imitation Game
As an experiment, I am going to ask a student to spend a week in
a locked room. The room is perfectly nice; it has a bed, a light, a
desk, some reading matter, oh, and we’ll give him some washing
facilities too! Every now and then I post some food under the door to
keep him going, Pop-tarts and pizza (thin-crust) work well.
On the first evening a note is pushed under his door with a symbol
on it. The student puzzles for a while, then opens the book sitting on
the desk. The book says, “If you get a piece of paper with symbols on it
look them up and follow the instructions.” He looks up the symbols and
the entry in the book says, “Go to page 44, write down the third symbol
on a piece of paper then post it back under the door.” He follows the
instruction and is rewarded with another piece of paper, this time with a
larger set of symbols on it. Again he follows the instructions in the book
and posts his answer back under the door. This goes on for several days.
He is somewhat bemused, but it passes the time, and he diligently looks
up the symbols and performs all the complicated actions as instructed.
Meanwhile, I meet our new Chinese graduate student and explain
to her she needs to interview a potential translator for the department.
He has just come in from Hong Kong and there is a health scare, so we
have quarantined him in the lab room. He is bored and I have some
paper for writing messages. She writes “hello” in Chinese on a piece of
paper and posts it under the door.
66 Are the Androids Dreaming Yet?
The exchange of notes goes on for a few days and the two seem to be
getting on well. There is even a little romance in the air. When the week is
over I open the door and the two meet. The graduate student says, “Hello.
It’s nice to finally meet you in person.” The man is puzzled because, of
course, she has spoken to him in Chinese. He knows no Chinese.
“I’m terribly sorry, but I don’t speak Chinese,” he says.
She is puzzled, “But I spoke with you this last week!”
“No, I really don’t speak it,” he says.
And, of course, he is telling the truth. The book he has been using
contains the rules for answering questions in Chinese, but he has
absolutely no knowledge of the language. I’ll leave to your imagination
whether the two strike up a real relationship and live happily ever after.
This is the Story of the Chinese Room. The setup is able to fool
someone into believing there is a Chinese speaking person in the room,
yet there is not. Where does the understanding of Chinese lie? The man
definitely does not understand Chinese. And the book clearly does not
understand Chinese because it is an inanimate object. Yet the person
outside the room is convinced she is communicating with a Chinese
speaker. The analogy to a computer is clear. The book is software and
the man blindly following instructions is the hardware. John Searle,
who devised the thought experiment uses it to show computers can
never understand because there is no place in a mechanistic system for
understanding to exist.
The Chinese Room has sparked huge argument in philosophical
circles; let me boil it down to its simplest form. First, let’s refute Searle’s
position with the ‘System Argument’.
The man plus the book form a system. Systems understand; their
individual components do not. My blood does not understand. My brain
without blood would not understand – it would be dead! Plug my brain
into a good supply of blood; add a dash of glucose, and it will understand
the most complex of things.
The systems argument is elegant and most scientists think this is the
definitive argument against Searle, but Searle has a neat way to counter
it. “Imagine”, he says, “that the man memorizes the book and leaves the
room. Now there is no system, there is just the man, but the man still does
not understand Chinese; he is just parroting rote-memorized words and
rules.” Computers, Searle argues, process syntax – the rules of language;
humans understand semantics – the contextual meaning of language.
Artificial Intelligence (AI) proponents hate the Searle argument.
They believe the memorization of a set of words and rules is exactly what
gives us knowledge of Chinese. That is why we go to school!
Understanding
67
A key problem posed by Searle’s Chinese Room is whether you can
know everything about a situation from just looking at the inputs and
outputs. This is very similar to the restriction posed by the Turing Test.
In that case if we were to trace the wire from our computer terminal
to the other room we would either find a human typing messages or a
large box covered in flashing lights. This would definitively answer the
question whether we were talking to a man or a machine. Similarly, if
we opened the door to the Chinese Room we would immediately know
whether there was a real Chinese speaker in there or not. But opening
the door on both tests misses the point. The question asks, “if the inputs
and outputs are the same does it matter what is really going on inside a
closed system?”
Black Boxes
Experiments involving closed systems are known as Black Box
experiments. They presume you can learn everything about the inner
workings of a box simply by probing it from the outside. Young electronic
engineers are often given black boxes as a test. Electronic components
hidden in the box are connected to three external terminals on the
outside. The student is asked to deduce what is in the box using only
an electric meter to probe those terminals. Here are a few examples of
the possible contents of a black box. They would all show up identically
on the student’s meter. Although internally different they are externally
identical. Even my ‘silly’ fourth choice with a cat in the box does not give
Black Box Equivalence
68 Are the Androids Dreaming Yet?
itself away if all you have to go on are electrical readings. (I dare say the
cat would make its displeasure know if left in there for any time.) The
contents are, therefore, said to be black box equivalent.
The reason for teaching engineers about black boxes is to help them
understand how to simplify things. We could construct option four, with
a cat and some food, but it would cost a great deal of money. Option 1 is
functionally identical from an electrical point of view, but for a fraction
of the cost. Steve Wozniak and Steve Jobs were so successful when they
started Apple because Wozniak was brilliant at simplifying logic circuits.
He could take a design with thirty chips and come back with a black box
equivalent solution using only five. It was a fraction of the cost and far
more reliable.
Scientists put great store in black box equivalence because of
a principle called Occam’s Razor. William of Occam was an English
Franciscan friar living in the fourteenth century. He proposed the idea
of minimal explanation. It states that, ‘among competing hypotheses,
the hypothesis with the fewest assumptions should be selected’. When
trying to explain the workings of a black box, the more complicated
inner workings should be discarded, as they have no externally verifiable
effect over the simpler mechanism. Our extraneous animal must be
eliminated! Sorry.
Ironically, given his calling, Occam’s Razor is sometimes wheeled
out as a disproof of the existence of God. Surely God is a complication
unnecessary to the explanation of our Universe. The argument is
illustrated beautifully in Carl Sagan’s book Contact and the film of the
same name. God gets the last laugh in Sagan’s book when the difficulty
with Occam’s Razor is brought into sharp focus. Occam’s Razor contains
an inherent paradox. At any moment in time we only have evidence to
support the simplest of explanations, yet we know many of these simple
explanations are incomplete. We regularly discover new phenomenon –
dark matter and dark energy being some recent examples. If we stopped
discovering new things, Occam’s Razor would be a good way to simplify
our thoughts. Occam’s Razor is a useful intellectual tool to prevent us
over complicating explanations, but there will often be explanations that
are correct, but for which there is not yet any observed effect.
If we go back to our black box example, we see the flaw in concluding
the boxes are identical from examining only their inputs and outputs.
Opening them would clearly show they are not identical! But, how would
this fact reveal itself if they remain closed? The answer is: over time. If
something in the box has memory or understanding, it could present
one set of results for a while and a completely different set of results later.
Understanding
69
In my trivial example, the cat could eat a wire and change the operation
of the black box. Now there is an open circuit where none existed before.
If this happened, the output would change and we would need a new
theory to explain it. If the circuit was attached to a missile control system
or a life support system, you would really want a full understanding
without waiting. It’s humans nature to try to open black boxes. This is
what MRI scans, X-rays, particle accelerators and all our other tools
of scientific investigation are for. We want to open all the black boxes
of nature and see what is going on inside: simply waiting to see what
happens is not acceptable.
In a sense, we live in a black box. We experience the world through
our senses, seeing with our eyes and feeling with our hands. The brain
never directly experiences anything; it only infers the likelihood of
Scene from The Miracle Worker. Helen Keller
pictured at the moment she understood language.
70 Are the Androids Dreaming Yet?
something from the signals it receives. This is similar to our engineer
probing the terminals of the circuit of a black box. How can we know our
experience of the world is real?
Understanding the World
The French philosopher Descartes gave us an explanation for this
paradox. He spent a long time looking skeptically at everything we
perceive. For example, when we poke a stick into a pond, the surface of
the water bends light and the stick appears to have a kink in it. Our eyes
tell us the stick is bent, but our brain ‘knows’ the stick is straight: it’s an
illusion. Descartes wondered if something so simple could be an illusion,
perhaps the whole of our experience is too.
His eventual solution underpins much of modern philosophy – ‘I
think therefore I am’, cogito ergo sum. Even if we doubt everything else,
we cannot doubt we are thinking about this doubt. At least we can rely
upon the existence of this ‘thought’ as some reality. Descartes built up
from this bedrock the real world we live in. We can be sure we experience
things and can apply logic and use thought. We can use this intellectual
faculty to tell a great deal about our Universe.
True Understanding
In the QED lecture series, The Strange Thing about Light and Matter,
Richard Feynman relates the story of the ancient Mayan astronomers.
3000 years ago they were able to predict the motion of Venus in the sky
using only pebbles. They had a simple system that could predict when
the planet would rise over the horizon. Put a stone in the jar every day,
take out a stone once a week, add a stone at every new moon. If the
number of stones in the jar is divisible by 23, Venus will rise. I’m making
up the details but you see the idea... It’s a very simple algorithm. What
should we conclude if the Mayans had perfected their calculations to
predict the motion of Venus and it proved reliable over a whole century?
Would this constitute understanding?
Feynman would say no: the Mayan understanding was not
complete. It was only black box equivalent to our modern understanding
over a limited period. We known that once the Sun begins to run out
of fuel it will swell to a red giant and explode, destroying Venus and
the Earth. Their model could not predict this catastrophic failure. Our
modern deeper understanding of the workings of the solar system allows
Understanding
71
us to predict this future even though there is no clue from the motion of
Venus today. Understanding allows us to predict discontinuous events: a
system changing its state or a star running out of fuel.
We see the same predicament in stock markets. Stock markets
normally behave in a linear fashion but, when they go wrong; they go
very wrong. Recent recessions have been made much worse by the failure
of hedging systems to handle market disruption. Some even think the
crises were caused by the automatic trading strategies of these hedging
systems.
The quants – as mathematicians in banks are called – spend
considerable effort modeling financial instruments to show that if one
stock goes down, another will go up at the same time. If the stocks
are held together your investment is safe because, on average they will
remain constant. The problem with these correlations, which often hold
reliably for many years, is that when trouble hits they fall apart. Historical
correlations don’t give us understanding of the future: something that
was only meant to happen once in a million years has happened within
six months. As they say on your investment papers, past performance is
no predictor of future results.
Do Computers Understand?
Today’s computers don’t have our general-purpose ability to understand.
Watson was thrown off by badly formatted English. The human
contestants, by contrast, had no problem with this. Just how good would
Watson have to be, to call it – or should I say ‘him’ – intelligent? How
could I judge this had happened? Alan Turing proposed an ingenious
test in his 1950 paper Computing Machinery and Intelligence using ‘The
Imitation Game.’ We now call the Turing Test.
If we ask a series of questions to a computer and we cannot tell its
responses from those a human would give, then the computer is, for all
practical purposes, the same as a human. Since we are intelligent – or at
least we hope we are – the computer must also be intelligent. QED.
That’s all there is to the Turing Test. Puzzled? Let’s pick his argument
apart.
Imagine you are chatting away on Facebook with someone you
don’t know. They may have posted a photograph so you can see what they
look like. The photo might be a fake; you have no real way to tell. What
question would you ask the other ‘person’ to prove they were human and
not a computer? There are obviously some giveaway questions. Please
multiply the numbers 342,321 and 23,294 and give me the answer. This
72 Are the Androids Dreaming Yet?
would be very hard for a human but easy for a computer. If you got a
very quick answer; the computer would have given itself away. But, the
computer has been programmed not to give itself away, and it is free to
give the answer slowly or even reply that the calculation is too hard. Our
computer can say anything it likes, including lying to pass the test! If the
computer can fool a questioner into believing it is a human then Turing
argued the computer has shown it is at least as intelligent as we are.
It used to be assumed that the field of broad general knowledge
would be hard for a computer, but Watson has shown this is not so. With
enough storage and a reasonable algorithm, winning a pub quiz is well
within the capability of a modern computer.
The really difficult questions for a computer are philosophical ones,
novel questions and things that don’t fall into a pattern. For example,
“Are you happy?”
“What do you think of Shakespeare’s Hamlet?”
“Is there life after death?”
“How went it?”
“Think Differ…”
If a computer could plausibly answer this sort of questioning for an
extended period, say fifteen minutes, should we conclude it is intelligent,
or do we need more time to be certain?
Turing’s approach to certainty was simple. Just ask lots of questions.
As you ask more and more questions, you will become increasingly
certain you are talking to an intelligent being. He characterized it
as a linear process; after 15 minutes of questioning you might be
99% certain and after a few hours 99.9% certain and after a few days
completely certain. The problem with this approach is it does not flush
out discontinuities. What if the questioning suddenly stopped without
warning or explanation? A human responder is likely to worry that the
questioner has had a heart attack and do something to find out what is
going on including leaving the room. Humans can make creative leaps,
solve non-computable puzzles or come up with a clever new joke. A
humans could even announce the test is a waste of time and walk off.
They just exercised free will! A computer cannot do these things.
Each year a group of scientists enters a competition run by
Cambridge University to win the Loebner prize, a competition to see
how close a machine can come to passing the Turing Test. If you can beat
the test you win $100,000. So far no one has come close and scientists are
beginning to realize just how hard it is.
Understanding
73
New Yorker Cartoon
With the anonymity the Internet provides we can imagine all sorts
of strange scenarios if the Turing test could be passed. You would have
no way of knowing what you were talking to. The New Yorker ran a
cartoon back in 2000. “On the Internet no one knows you are a dog.” We
come across a similar problem the other way around when we encounter
bad customer support. A few years ago, while trying to get an answer
to a computer problem, I became convinced the thing responding to
my emails was a machine. The company did use machine responder
technology so it could well have been. I asked it to prove it was human
by putting the word marmalade into an English sentence and fixing my
74 Are the Androids Dreaming Yet?
problem. The human pretending to be a machine saw the joke, fixed my
problem and replied “Marmalade is served with butter and toast.” The
test worked!
Uncannily not Human
Understanding
75
The sister test in robotics is equally hard. The goal is to simulate the
physical human form, its movements and mannerisms. It’s easy to get
close, but close is not good enough. The term ‘Uncanny Valley’ has been
coined to describe the discomfort humans have with something that tries
to simulate a human being but does not quite get there. I think it is part
of the reason Madam Tussaud’s waxworks are so fascinating. Humans
have a love-hate relationship with facsimiles of themselves. They love
the flattery but feel a sense of revulsion at anything that comes too close.
Searle and Turing
In the Turing Test, we limited our senses to the purely symbolic: using
only typed words on a screen. I could break the lock on the door and go
into the room to see what was there.
“Aha!” I would say.
“I can see you’re a computer, I, therefore, know you’ll be good at
sums and bad at creativity.”
But Turing wants us to see if the difference is given away purely
through intellect. He argues there is no way to tell. But if you follow my
argument from chapter 1, there is one way: ask the computer to find a
non-computable solution to a mathematical puzzle. This is, in practice,
a difficult test to pose because it might take a very long time. Twentyfive
billion people have lived on planet Earth during the last 350 years,
and about 5 million of them were mathematicians. None of them was
able to solve the problem posed by Pierre de Fermat until Andrew Wiles
turned up but this is a clear difference between humans and computers.
However long you give a computer it would never be able to solve the
problem.
This creativity test would take centuries to run if non-computable
thought was rare, but I think we see it often – on display even when we
tell jokes. In which case computers and humans should be easy to tell
apart: humans are the funny ones. I am not saying you can’t build a brain;
our brains are physical devices, after all. I just believe a computer or a
mechanistic machine, cannot think like a human being.
I like the Searle argument but qualitative arguments are insufficient.
We need a quantitative argument. In the forthcoming chapters, I am
going to look at the mathematical argument underlying the difference
between human intelligence and computer processing. Before we do this
let’s take one last look at a qualitative difference; the way computers and
humans communicate.

Chapter 3
BODY LANGUAGE
& BANTER
Body Language
“England and America are
two countries separated by a
common language.”
George Bernard Shaw
“I speak two languages, Body and
English.”
Mae West
“The body never lies.”
Martha Graham
In the summer of 1986 Ronald Reagan and Mikael Gorbachev met
in person for their second negotiation session, this time at the
Höfði House in Reykjavik. For five days, the leaders talked alone
except for interpreters. Reagan badly wanted to develop the Strategic
Defense Initiative; known by its nickname, ‘Star Wars’. The idea was to
put smart weaponry in space that could destroy ballistic missiles before
they reentered the atmosphere. Reagan believed this would remove the
threat of imminent destruction that had hung over the world since 1945.
Gorbachev, on the other hand, felt this was just another escalation in
the Cold War, and the Soviet Union would be forced to build yet more
weapons to overcome the American defenses. He wanted Reagan’s plans
shelved, arguing that it broke the Anti-Ballistic Missile Treaty. He was
probably right. The leaders talked back and forth, unable to overcome
the impasse. At the end of the summit there was a mad scramble to
announce some sort of deal, but this proved difficult. In the last moments
before they had to conclude a communiqué, Reagan suggested they
abolish all nuclear weapons. Reagan’s negotiating team was horrified and
shut the door.
For decades, the American strategy had been to use nuclear
weapons as a deterrent against the apparent numerical advantage of the
Soviets. In all the potential scenarios analyzed by the Pentagon, Russian
forces ended up overrunning American forward positions – otherwise
known as Western Europe! The only way to stop them was through a
release of nuclear weapons, which, inevitably escalated to all-out nuclear
Ronald Reagan and Mikael Gorbachev
80 Are the Androids Dreaming Yet?
war. It was assumed this inevitable progression deterred the aggression
in the first place, and the threat of mutually assured destruction kept the
world peaceful. Giving up this tenet of defense strategy was something
the American military just could not contemplate. Many people did not
think it a rational defense strategy; it seemed appropriate the acronym
for mutually assured destruction is MAD, but this was the status quo.
We now know our worry over Russian superiority was groundless.
The West’s technological advantage, founded on the invention of
computing and sophisticated materials technology, gave us a huge
advantage. In the only battle to be fought in the 20 th century between
Russian and Western tanks, during the first Iraq war, most of the Russian
tanks were destroyed with no losses to American tanks. We know this
now, but we are talking of a time when paranoia over the Soviet advantage
was the common view.
There is speculation that Reagan had muddled intercontinental
ballistic missiles with all nuclear weapons. I do not think this is true.
Reagan was a man of vision, quite comfortable with using his folksy way
to convey sincere belief, and I think abolishing all nuclear weapons was
in his mind. It would have been a breathtaking moment.
In the end a rather feeble communiqué was put together and the
talks declared a technical failure. But, both leaders had seen eye-to-eye;
both were prepared to make major concessions and both wanted an end
to the old strategy of mutually assured destruction. Wiping each other
out was no longer considered a successful outcome! The meeting, and
Höfði House in Reykjavik
Body Language & Banter
81
the fundamental thawing of relations between East and West, was to lead
to the Intermediate-Range Nuclear Forces Treaty and the end of the Cold
War.
Face-to-Face Communication
What really happened between these two leaders when they met and
talked? Was it a mechanical process of offer and counter-offer, as easily
executed by fax, or is human interaction more complex than this?
Reagan, as a young man, had been a liberal, sympathetic to socialist
ideals until a painful strike in California caused him to lose faith in the
politics of the left. Gorbachev, a lifelong Communist, was desperate to
reform the Soviet economy and make it more competitive. He, also, had
come to see the hypocrisies that could emerge in far left-wing ideology.
I don’t believe this common experience could have been communicated
by fax or email. Indeed, I am sure these specific points were never made,
but the nonverbal communication must have conveyed something of
their common background and purpose.
When we phone someone or exchange emails, the interaction is
factual, there is no body language, and we rarely laugh. When we travel
to meet someone, we spend a great deal of time with them. The average
length of a phone call is two and a half minutes, but meetings, especially
when one party has travelled to see the other, can be hours long. When
humans meet they greet each other, shake hands, sit in the same room,
talk at length, and laugh. Body language is important; people mirror
each other’s postures, adopt open and receptive stances, and make
eye contact. You can see this in the picture of Reagan and Gorbachev
above. Body language allows us to convey qualitatively different things,
such as trust and happiness. It is very expressive; you can see the more
guarded postures of Yasser Arafat and Shimon Pérez below, just after
they negotiated a landmark peace deal. Can you tell if the leaders smiles
are false?
Communication
Communication is one of mankind’s greatest expenditures. The US
telephone system is arguably the largest machine on the planet, while
the world’s mobile phone networks have a capital value of $2.5 trillion,
greater by an order of magnitude than all the steel plants in the world put
82 Are the Androids Dreaming Yet?
Yasser Arafat and Shimon Pérez
together. This lifeblood of our existence – long-distance communication
between human beings – turns out to be amazingly difficult, even with
all our clever technology.
In recent years the Internet has, in theory, allowed each and every
person to communicate freely with any other person on the planet.
In some of the most distant parts of the world mobile phones, and
projects such as; ‘One Laptop per Child’ are rapidly bringing unlimited
communication to all. This communication can be personal, one-to-one,
or broadcast: I can talk to people interested in a particular topic directly.
As we watch the Arab world democratize, catalyzed by the Internet, there
is no question that digital communication has now become a major force
in the world. Yet, people don’t communicate over the Internet as much
as you would expect; they often use the Internet to set up phone calls
during which they arrange meetings! This is odd. We have a fantastic
phone system and sophisticated communication technologies; email,
video and instant messaging. Yet, we still choose to travel when we want
to communicate.
On the face of it, there should be no difference between a phone
call and a meeting. In principle the same information can be conveyed.
Yet when we want to really understand someone, we always go to meet
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Smiles Fake or Real
in person. No great treaty or big industrial contract has been negotiated
without a face-to-face meeting. We see this daily: people talking on the
phone get to a certain point, give up, and arrange to meet in person.
The consequence is that we spend $550 billion annually, flying
around the globe to meet each other. Each day the world’s population
takes three million plane flights. Around 80% of these are business
flights. Some are people emigrating or going to do specific manual tasks,
but most are to have meetings. We have always assumed that this is
because the parties are unable to reach a sufficient level of trust over the
phone and need face-to-face interaction to build that trust, but it may
be that the parties are not able to convey sufficient information to fully
understand each other. Face-to-face meeting may convey much more
information than we think.
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Smiles
When we smile naturally we use a full set of facial muscles, including the
muscles around our eyes. When the smile is forced those eye muscles
remain passive and the smile, although superficially the same, is missing
something. You can’t put your finger on it, but the look is insincere. A
study of marriages in the USA analyzed smiles in wedding photographs.
The couples with false smiles divorced much earlier than the genuinely
happy couples. Similarly for high school photos; people with genuine
smiles at 18 years of age were happier later in life and in more stable
relationships. Smiling is really important. It is good to be around people
who smile, they are more successful – and nicer.
There is also a curious reverse effect. The link between our minds
and bodies is much more fundamental than we thought. If you grasp a
pencil between your teeth, it forces you to smile. Try it. The mere act
of smiling is found to make you happier, it causes the release of the
chemicals called endorphins which improve your feeling of well-being.
Micro-expression Analysis
Since the involuntary movements of the muscles around our eyes give
away genuine happiness, a whole science has evolved looking for other
biological cues to mood. The two most interested groups are the FBI,
trying to detect lies, and poker players, trying to make money! Much has
been written on the topic, including a few best sellers, but the evidence
for micro expressions is mixed. Regardless of whether involuntary
actions give away our emotions, humans voluntarily use a great deal of
body language when talking.
Body Language
A study by Albert Mehrabian is often cited to say 93% of the information
in a conversation comes through nonverbal cues. This is misquoted.
The study really stated 93% of the emotional content is nonverbal.
That’s more believable. And further studies have shown when there is
doubt, nonverbal cues win over verbal information every time. The rule
is sometimes laid out as the 7%-38%-55% rule – 7% words, 38% tone
of voice and 55% body language. Remember this is emotional content,
your conviction and sincerity. You will still have to get over the factual
information you want to convey.
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Learning Swedish with The Two Ronnies
Try this experiment on a friend. Tell them you like their shirt using
different tones of voice: sarcastic, sincere, amazed. Then see what they
understood. You will find it difficult to appear sincere because I have
told you to say you like their shirt – unless of course you really do. When
you use sarcasm they will find it hard to process your statement. It is
revealing how we use the information.
Interestingly, a piece of research described in Scientific American
shows even insincere flattery is effective. If you want a pay rise from your
boss, any form of flattery will do. Vanity appears to override skepticism!
Interaction
The normal cadence of communication between people includes a great
deal of mutual interruption. When a meeting breaks down we often see
people begin to say things like, “Please don’t interrupt me,” “Do you
mind, I was talking,” “Pleeeease, let me finish.” If the meeting is really
getting out of hand, third parties will often step in and tell one to wait for
the other. This is where the mechanics of face-to-face interaction fail, as
we need to interact in order to communicate effectively.
Because we have a lot more time in a face-to-face meeting people
can wander ‘off topic’. This is an important part of the process of
communicating. After all since most phone calls are 2-3 minutes and
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most meetings an hour, there are another 57 minutes to fill! These off
topic items bring in social experience and help us form the background
context we need to properly communicate.
What is Background Context?
Alex and Bella are both fans of the British comedy duo, the Two Ronnies,
and enjoy their learning Swedish sketch. Bella asks Alex what kind of
sandwich he wants for lunch. Alex replies ‘M’. Bella laughs. If you have
seen the sketch you will understand the background context to the joke.
If not this paragraph might as well have been in Swedish. Take a look at
the sketch on YouTube and reread this paragraph... Now you understand.
Do I think in English?
Most scientists believe we think thoughts using language, but most
scientists writing about thought are linguists or psychologists. If you are
a dyslexic engineer like me, language is a long way down the processing
chain. I think abstractly and then translate those thoughts into words.
Some ideas don’t map between languages and often, one language adopts
the words of another to fill in the gaps. Some interesting examples are:
Zeitgeist
Schadenfreude
Chutzpah
German, spirit of the times
German, enjoying others misfortune
Hebrew, audacity
All of these are fully signed up, card carrying entries in the Oxford
English Dictionary.
Some languages have fewer distinctions between ideas: truth and
law are the same word, ‘torah’, in Hebrew. Languages have different
tenses and structure. In Chinese all words are one syllable and the script
is pictographic rather than phonetic. This is unusual, even Egyptian
and linear-B, which look pictographic are mostly phonetic. With single
syllable words, Chinese uses voice inflection to change meaning; a rising
or falling tone can change the meaning of a word from ‘grey’ to ‘girl’.
In many Western languages rising voice inflection is used to indicate a
question, as in Australian English or irritation, as in English English. So
how do the Chinese show if they are annoyed or want to ask a question?
They elongate their words and accentuate the changes in intonation.
An argument in Chinese can sound quite alarming to the Western ear,
with its percussive monosyllables and extreme inflection changes. This
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degree of inflection is used in English, but only in extreme emotional
contexts: A Chinese argument over cold tea can sound like an accusation
of murder to a Western ear.
Symbolic Communication
The earliest recorded permanent human communication is cave
painting, dating to 33,000BCE. Written communication emerged in
Sumer, the southern part of Mesopotamia (now Iraq), using a script
called Cuneiform, written on clay tablets. It was used primarily for
accounting. The Sumerians are responsible for our common use of base
twelve. Twelve hours in a day, inches in a foot, and notes in the scale; all
stem from their civilization.
Although not the first to write stories, the Greeks perfected the
dramatic forms we use today: poetry, prose and plays. Watch an episode
of ‘Law and Order’ and you are seeing a direct descendant of a Greek
tragedy, complete with suffering and justice denied. All this permanent
thought art is made possible by the translation of ideas into symbols.
Scripts and Symbols
The world supports a huge variety of scripts split roughly into phonetic,
representing the component sound of words, and pictographic, stylized
pictures of the ideas.
Chinese Traditional and Simplified
Some scripts have interesting quirks. Ancient Hebrew, although
phonetic, is a script where vowels are omitted. Modern Hebrew often
leaves them out as well. This means words can be ambiguous and need
context to decipher them. A common set of Chinese characters has long
been used by Mandarin, Cantonese, and Japanese speakers even though
88 Are the Androids Dreaming Yet?
their spoken languages are entirely different. The script languages of
these people are gradually diverging and might in time become entirely
separate languages too.
The Chinese government in Beijing has moved to using simplified
Chinese for Mandarin speakers, while Hong Kong continues with the
traditional form. Japanese has developed many new characters for
modern ideas, such as computers, that differ from the Chinese, and mixes
in a great deal of Katakana, a script allowing the phonetic representation
of foreign words. If you walk around these countries their signage looks
quite different, although I am told Cantonese speakers can still read
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simplified Chinese. Take a look and normally you will find them to be
quite different. Each example in the figure is my best attempt to translate
the phrase “Hello Reader” into a script and the corresponding language.
Symbols of the World
English is one of the most irritating script languages of all. It commonly
uses etymological elements, showing the history or origin of the word
that has nothing to do with the sound of the word. A word like school
has the ‘k’ sound spelt ‘ch’, showing its historical derivation from the
Greek, but confusing for pronunciation. English has 53 sounds derived
from only 26 letters, so there are plenty of letter combinations, many of
which are irregular. Because the language favors historical convention
over simplicity, sugar is pronounced “shu-gar” whereas sand is strictly
phonetic. As for Leicestershire I’ll leave that as a test for the American
readers amongst you. If you’re British, try Mattapoisett, a town in
Massachusetts named in Native American.
Yet English is also a ‘lovely’ language. Because of its richness there
are often twenty different ways to say something, and a dozen words
to choose on any topic. One of my own favorite words is ‘jump’. It is
phonetic, but also onomatopoeic and even pictographic. Jump both
sounds like a jump and looks like a jump.
Two scripts that puzzled scholars for many years are Linear-b and
Hieroglyphics. Linear-b – found on clay tablets on the Island of Crete –
turned out to be a coded form of ancient Greek with some slight quirks,
such as dropping the letter ‘s’ from the ends of words. The ‘s’ is superfluous
in most Greek words, and dropping it saved precious clay space!
Hieroglyphics was a real puzzle. It looks so like a pictographic
language that it fooled many people for centuries. The Rosetta Stone was
discovered in 1799 and became the key to their deciphering. This stone
had the same edict written out in 3 languages – Greek, Egyptian and
Demotic. The French adventurer Jean-François Champollion decoded
hieroglyphics in 1822 and although it looks pictographic, it was found to
be predominantly phonetic. Linear-a, another script found on the Island
of Crete has yet to be decoded and remains one of the world’s greatunsolved
mysteries.
All these different ways to code ideas into symbols present the
children of the world a great learning challenge. Because written language
is so young, in evolutionary terms, our brains have not had enough time
to evolve to master it. Instead words co-opt parts of our brains originally
90 Are the Androids Dreaming Yet?
evolved for different purposes. As languages differ in their construction
they co-opt different bits of the brain. It is possible to see this using brain
imaging.
Dyslexics – and I am one – have difficulty in translating between
the realm of conceptual thought and written script. This translation is
subtly different for each language. Chinese speakers use their motor
cortex to process characters. Young children write out the characters
over and over, to memorize them, so the ‘muscle’ memory is highly
involved. French and Spanish children use the audio pathways, as most
of their language is phonetic, the motor part of writing is then an addon
and does not process meaning. English children must use portions
of their visual cortex to process the meaning of words, as many words
have spelling quirks that have nothing to do with the sound of the words.
Some studies even suggest a child dyslexic in one language, because, for
example, their audio pathway is impaired, might not suffer the condition
in another language that relied on a visual or motor skill.
Can Objects Communicate?
The process of communication has many components, starting with
something capable of communicating. Communication usually –
perhaps always – is something that occurs between sentient beings. I
don’t think of my computer as communicating with me, but rather think
of it as a medium for communication or a dumb machine. But colloquial
language around the subject is a little muddled. We all agree a lighthouse
does not communicate, even though it can signal danger, but what do
we mean when we say, “That song really spoke to me.” No one believes
the song is actually communicating, but some kind of communication
was made nonetheless. When we talk of communication do we mean the
agent or the message?
Stories
Humans enjoy communicating; we create works of art, music and
literature that transcend simple analysis. The COIN dynamic slide,
which we saw earlier detailing the strategic situation in Afghanistan,
would probably have been better communicated with a story. Humans,
unlike computers, do not cope well with large quantities of unrelated
information, and studies of memory and comprehension show we
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benefit from a narrative structure. Let me give you a basic example. One
simple trick the human brain uses is chunking. Give yourself a moment
to try to learn this string of characters.
HALTNTIBMGTATLAMATLOLPOMSGTG
TRY TO MEMORIZE THE STRING WITHOUT READING ON
Now, if I divide it into chunks, you will see it includes meaningful
information.
HAL TNT IBM GTA TLA MAT LOL POMS GTG
You probably won’t recognize all the acronyms unless you are
under 10. Even then, you will find memorizing it hard, but if you put the
sequence into the context of a story then it is much easier to learn.
HAL uses TNT to blow up the IBM building in Grand Theft Auto.
“Three Letter Acronyms are annoying,” says MAT. I’m Laughing Out
Loud; Parents Over My Shoulder. Got To Go.
We find it easier to fit new information into existing structures
within our brains rather than memorizing by rote. I’ve used quite a
bit of modern Internet slang here. You’ll find young people recall this
information better than older people for whom GTG and POMS are
nonsense.
If you want to memorize something, experts recommend you
imagine bizarre images and relate them to a story pictured in the mind’s
eye. Try it and you may very well find you can still remember my sentence
in ten years time!
Let’s try something else. The following sentences are a little different,
yet the recall scores for information in the two are dramatically different:
1. I met an old tramp on 42 nd Street wearing a dirty grey rain coat.
2. New York on a cold damp November day; as I cross the street
I bump into an old man wearing a dirty grey Macintosh. His
shuffling gait suggests some sordid intent. I think nothing of it,
but this brief meeting was to change my life.
92 Are the Androids Dreaming Yet?
The addition of contextual cues allows you to form a mental picture.
By withholding some information at the end I have used a dramatic trick
to cause your brain to free wheel and imagine what happens next. You
are involved in the story. Notice the longer story, with more data in it, is
paradoxically more comprehensible and memorable.
Ed Tufte makes the point about our ability to process information
very forcefully. He believes presentation experts are wrong when they
recommend you keep your slides to a few words! He points out the
common advice to use only six bullets per slide and six words per bullet
comes from a misconception that has blighted a generation of presenters.
Studies performed on memory in the 1960s measured unrelated word
recall. Six words are all you can remember if the words are meaningless.
But if the words have meaning we can comprehend and absorb many
pages of data. Hundreds of millions of people throughout the world read
a newspaper every morning and can recall the stories throughout the
day; the poems, songs and plays we memorize when young are usually
long, comprising thousands of words, yet we are able to remember them
verbatim for the rest of our lives.
When we tell a story, we are trying to draw the reader in so they
can to experience our imaginary world and be ‘in’ the story. When I read
a story – perhaps Harry Potter – I don’t think about the grammar and
punctuation, or even the accuracy of character portrayal. I’m transported
to a different place. I experience a piece of the reality or ‘imaginality’
the storyteller has created. I can describe the characters, the scene, the
sounds and the smells. A good author forms a complete world in our
heads corresponding with the world they have in their heads. With more
abstract information, comprehension and retention is harder. Often if
the information does not hang together in a linear narrative it can be
impossible to take in at a single sitting. However, if it forms a story and
is well told so you ‘get it’, you do not need it repeated. We experience
something of this effect when we watch a good movie. “I’ve already seen
that one,” means you have absorbed the whole story in a single sitting.
You don’t need to watch it over again to comprehend it.
Comedy
Finally, when you mix all the elements up, emotional understanding,
body language, in-person communication and empathy; you get comedy.
Humans ‘do’ comedy from a very young age and it’s vitally important to
the fabric of our lives. What purpose comedy serves in communication
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My XBox is Broken
The One Ronnie
Dead Parrot Sketch
Monty Python
Gerald the Gorilla
Not the 9 O’Clock News
Fork Handles
The Two Ronnies
Andre Previn
Morecambe and Wise
Self Defense Against Fruit
Monty Python
is not clear. In life, telling a joke will make another person smile. This
causes people to be happy and happy people release chemicals into their
bloodstream which make them healthier. Happy people then tell jokes
to others. This circular process improves the well-being of communities
and helps bond people together. But why on Earth did comedy evolve to
be the mechanism that does this?
Comedy may be an important way to avoid an argument when
context is unclear. Much of what we say can be taken the wrong way.
Simple communication of fact can sound like criticism or challenge, and
94 Are the Androids Dreaming Yet?
humans are naturally hierarchical – not unlike packs of dogs or beached
walruses. Humor allows us to test the response of others to statements,
which might otherwise be taken the wrong way. Something said in a
‘jokey’ tone of voice may not generate a negative response, even though
the raw content might be quite provocative. “Ah, late again I see…”
It is worth taking a look at some great comedy sketches because
they bring home the richness of human interaction. Here are some of my
favorite links as an antidote to the heavy-duty mathematics I am about
to inflict on you.
The World’s Funniest Joke
Two hunters are out in the woods when one of them collapses. He
doesn’t seem to be breathing and his eyes are glazed. The other guy
whips out his phone and calls the emergency services. He gasps, “My
friend is dead! What can I do?” The operator says, “Calm down. I
can help. First, let’s make sure he’s dead.” There is a silence, then a
gunshot is heard. Back on the phone, the guy says, “OK, now what?”
Spike Milligan, from The Goon Show
I think comedy is a fitness display. It demonstrates to those around
us – particularly of the opposite sex – that we can be creative and use
non- computable thought processes, just as dancing is a fitness display of
our agility and coordination. When we tell a joke we are showing others
we can ‘think outside the box’, a valuable survival skill.
At a simple level it has been proven that animals with the ability to
behave randomly escape being eaten more often than animals that follow
a pattern. Non-computability is the ultimate behavioral randomizer
since it is not an algorithm and cannot be copied. The ability to take
non-computable thinking to its logical conclusion to create and invent
has clearly taken off for humans.
Of course, another explanation might be that making people happy
is fun. People like to be around other fun people so humor encourages
crowds to form. If a saber-toothed tiger attacks you, and you are in a
crowd, you’re more likely to survive. You only have to outrun one
member of the crowd!
Chapter 4
THE BRAIN
Baby EEG
“The brain is a wonderful organ;
it starts working the moment
you get up in the morning and
does not stop until you get into
the office.”
Robert Frost
“The brain looks like nothing
more than a bowl of cold
porridge.”
Alan Turing
Physically the human brain is very boring. Alan Turing described it
as looking like a bowl of cold porridge. To get to the porridge you
must first cut through the skull, a two-millimeter thick protective
layer of bone. The adult human skull has almost no gaps in it, and the
only ways into the brain without a bone saw are through the eye sockets
or the soft area of bone at the back of the nose. Egyptian mummies had
their brains removed through the nose and preserved in a jar for the
afterlife!
Thinking with Porridge
Protecting the brain is very important and the skull does a good job
by being a tough, impenetrable barrier. But sometimes this toughness
backfires. In 2009, Richard Hammond, one of the presenters of the TV
motoring series Top Gear, suffered a crash while testing a land speed
record-breaking car. Although he was in a multipoint harness, the
crash, at over 200 miles per hour, bounced his helmeted head around
the inside of the cockpit and his brain was badly bruised. As you know
from experience, when you bruise you get swelling, and the brain is
no exception. However, the brain is encased in bone, so this swelling
has nowhere to escape. The resulting buildup of pressure is dangerous,
causing an interruption of blood supply to the un-bruised parts. Brain
damage in such accidents is often fatal; Richard Hammond was very
lucky to live through the experience.
Surgeons often need to cut into the skull to relieve pressure on
the brain, or to gain access to remove tumors. Going through the scalp
involves a great deal of blood, but once you have a clean hole in the skull
you can peel back the thin membranes, called the meninges, to reveal a
wrinkly folded whitish thing that looks a bit like a cauliflower. This is the
outer surface of the brain where much of our thinking is done. Unfolded,
this surface layer would cover the area of a football field and this intense
folding distinguishes the human brain from the brains of simpler animals.
Some animals, such as elephants and dolphins, have larger brains than
ours, but the area of their folded surface is considerably smaller. It is
thought that this efficient folding is key to giving us the ability to think
complex thoughts.
Analysis of Einstein’s brain held at Princeton University shows it
is not particularly massive, but it is strikingly more folded than average,
and has a shorter lateral sulcus – the fissure between the front and back
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Einstein’s Brain
of the brain. Whether this is related to his highly creative thinking or just
random chance is unknown, but it’s an interesting data point in our quest
to understand creativity and intelligence.
Looking through a microscope, the wrinkly grey matter is composed
of 30 trillion neurons; small whitish cells sprouting filaments that wrap
around each other like the tentacles of an octopus. The tentacles, and
there can be as many as 10,000 per cell, are known as dendrites and spread
out to nearly touch other neurons. At the other end of the neuron is a
single axon. The gaps between the end of an axon and the next neuron’s
dendrites are called synapses, about one-tenth of the width of a human
hair and varied in structure. When a nerve ‘fires’, an electrical pulse
spreads out along the axon to the end and crosses the synapses to other
brain cells. This electrical pulse is not like the flow of current in a wire:
neurons don’t conduct electricity. It is more akin to dominoes falling in
a line. Ion gates in the walls of the neuron open, letting potassium ions
flow out. As the gates open in one section, the next section is triggered
and so on. Thus, electrical signals ripple out along the axon. As the
electrical signals cross the synapses they either excite or inhibit the firing
of adjacent neurons. There is a lot more structure to a neuron than was
once thought. The textbook model is of a sequence of ion sacks stacked
end to end rather like plant cells, but neurons have a far more complex
structure. Bundles of actin and tubulin form a skeleton in the neuron and
the neuron metabolizes ATP to recharge its firing mechanism. Neurons
behave far more like small animals than inanimate plant cells.
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99
The wiring of our brain looks a bit like the logic circuits of a
computer, and our best guess is the cells in our brain form some kind of
computer. The brain cells – a specialized form of nerve cell – connect to
the rest of the body via the nerve cells that largely run down our spine.
Thoughts trigger action and, in reverse, the nerves in our extremities
sense things in the environment and relay information back to the
brain. If I think, ‘move my finger’ my finger will move, and if it touches
something I will feel the sensation. Interestingly if my finger touches
something hot a reflex will kick in. Reflexes work without involving the
brain. We don’t have to think, “that hurts.” Instead, our finger reflexively
pulls away. We may say ouch, but by the time we do, our fingers already
moved away from the heat.
Nerve cells are much slower than the electronic systems we build
with copper and silicon. This speed is quite noticeable and limits the
rate we can do certain things. It takes around 0.08 seconds for a nerve
impulse to run down to the tips of our fingers, initiate an action and
return to give us the sensation of the action. This may sound fast but if
you’re a tennis player in a rally or a pianist faced with a fast passage, the
nerves don’t have time to make a full round trip signal before the next
action must be initiated. In these instances we need to run on autopilot
and there are parts of the body where the nervous system takes action
without the brain getting involved. This is particularly the case with
things like walking and balance, which must respond fast to changes in
ground conditions. The signals just don’t have time – and don’t need – to
go all the way up to the top of the body for instructions. Rather like the
heat reflex above, the peripheral nervous system can process information
locally. After all, brain cells and nerve cells are really all one type of cell.
If you have a group of people, you can conduct a fun experiment
to show the speed of nerves. Hold hands in a big circle and squeeze the
hand of the person next to you. When they feel you squeeze, they should
squeeze the next person’s hand and so on. The rate at which people
squeeze hands around the circle is limited by the speed at which nerves
conduct the signals across our bodies.
Imaging the Brain
There are several ways to look inside the brain without recourse to a bone
saw. The methods are fascinating in their own right, even before we start
looking at the results. Each image is generated using a different physical
principle.
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X-rays
The first Nobel Prize in Physics was awarded to Wilhelm Röntgen in
1901. He had discovered ‘X’ rays; so called because he had no better
name for them. X-rays, as they became known, are just light of a very
high frequency.
Light comes in a variety of colors; at the low end of the frequency
scale we see red, higher up blue and, at the top, violet. At this point
human eyes give up and cannot see anything higher, so ultraviolet
light is invisible to us. Bees, on the other hand, can see a long way into
the ultraviolet spectrum and some flowers have beautiful ultraviolet
markings that attract bees for pollination. Daylight contains a great
deal of ultraviolet light which is wasted on us – other than to tan our
skin. But all is not lost. Clever manufacturers put fluorescent dyes
into their washing powders which stick to our clothes and convert
ultraviolet into visible light, making our T-shirts look brighter as they
reflect more visible light than fell on them. You can see this effect most
easily in a disco when ultraviolet lights are shone on the dance floor
and anyone wearing a newly washed T-shirt will glow bright white. The
other common substance that fluoresces strongly on a dance floor is
tonic water. Quinine, the active ingredient in tonic water, is a strongly
Flowers in Ultraviolet Light
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101
Pit Viper
fluorescent substance which converts ultraviolet light down into the
visible spectrum. Photoactive dyes have recently become controversial
as suggestions have been made that they are unsafe and irritate the skin.
Going to discos might not be quite as fun in the future!
Thermal Imaging
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At the bottom end of the spectrum is infrared light. Pit vipers have
evolved special organs on the sides of their heads to ‘see’ in this spectrum
and they use this sense to hunt prey in the dark. I use the word see with
some caution. We have no idea what their sensation of ‘heat-sight’
involves, but their organs are very precise, able to detect things only 0.2
degrees warmer than the background.
Infrared cues help several species of snakes, bats and insects locate
things in the dark, but the animal that excels at the task, albeit using
technology, is mankind. Special cameras allow us to use infrared to see
in the dark or detect where our houses lose heat.
X-rays are much higher in frequency – about one hundred times
that of the ultraviolet light that affects our T-shirts. The high frequency
corresponds to a small wavelength that allows the rays to pass through
our bodies. Later on in the book we will understand that frequency is not
a proper explanation for light, as it is not a wave but rather a particle that
obeys the laws of a wave. But for now we will ignore this detail.
The first use of X-ray images was to see broken bones. Bones block
the rays as they are dense, but the soft parts of our bodies are almost
completely transparent to X-rays. We can see the soft tissues if we turn
the contrast up, but there are problems when using X-rays to view the
brain. Our skull completely encases the brain and however much we turn
the contrast up, all we see is bone. The solution to this problem is to
perform sophisticated mathematical tricks using a computer to enhance
the contrast ratio and make image ‘slices’ through the living head.
The slicing technique was invented independently in the 1970s
by Sir Godfrey Hounsfield, working for EMI in England, and Allan
Cormack, of Tufts University in America, and they shared the 1979
Nobel Prize for Medicine for their work. Legend has it that EMI was
making so much money from The Beatles they could fund the enormous
development cost of the CAT scanner from the profits; true or not, it’s a
great invention.
The best way to understand the mathematics is to picture yourself
in an episode of ‘CSI’, the American television crime drama. An intruder
has attacked someone with a knife and there are blood spatters all over
the walls of the room. Enter the brilliant pathologist who reconstructs
the scene of the crime from the pattern of blood on the wall. She can map
the trajectory of the blood spatters and back-calculate that the attacker
must have been 5’ 4”, left-handed and wielding a 6” blade. In a CAT scan,
our head is hit with billions of rays that bounce and scatter over the walls
of the machine. Sensors detect the rays and a mathematical algorithm
calculates an image of the body that would produce such a pattern. To
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X-ray of Roentgen’s Wife’s Hand
simplify things we shine the X-rays onto the head as a narrow slit of light
so we only have to do the back calculation in two dimensions. Then we
stitch successive slices together in the computer to form a 3D virtual
image. Thus, doctors can ‘fly’ through the brain looking at structures
such as tumors from all angles.
104 Are the Androids Dreaming Yet?
There are two problems with X-ray imaging. Even with clever
mathematics, the dense bone in the skull blocks the rays so you don’t get
much contrast, making it hard to distinguish normal brain matter from
something like a tumor. But the bigger concern is X-rays are a form of
ionizing radiation, and ionizing radiation causes cancer.
We are told to wear sun block to protect our skin from ultraviolet
light; X-rays are 100 times more potent and can do a great deal of damage.
Fortunately, the body repairs itself quite well in the presence of low levels
of radiation. The double part of the double helix in our DNA allows a
set of proteins in our cells to go around correcting errors when they
detect a mismatch between the two strands. But, now and again an X-ray
might make an irreparable fault in both copies. If enough of these faults
accumulate, they can lead to cancer or, if the errors are in reproductive
organs, birth defects. Doctors try to minimize the radiation we receive
and give us as few CAT scans as possible during our lifetime, especially
when we are young and have not yet had children.
MRI
X-rays dominated our ability to see into the human body until the
mid-1970s when Raymond Damadian came up with the idea of using
magnetism. Magnetic fields are not absorbed by bone and present
no danger as they do not damage DNA. Ironically, the technique was
originally known as Nuclear Magnetic Resonance, ‘NMR’, which patients
thought must be dangerous because of the word nuclear. The name was
Functional MRI: Working Memory
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Diffusion Tensor Image
changed to the one we use today: Magnetic Resonance Imaging, ‘MRI’.
The system works by applying a strong magnetic field to your body to
excite the hydrogen atoms. Since we are mostly H2O there are plenty of
these.
Three magnetic fields are used. First, an extremely strong field is
applied to the whole body. This causes all the hydrogen atoms in the
water and fat to spin in line with the field of the machine. Next a gradient
field is applied to the top of your head so it is slightly more magnetized
than the bottom of your feet and, finally, a pulse of magnetism is applied
to the top of your head. The spinning hydrogen atoms line up a little
more when this pulse is applied and then randomize again when it is
switched off. As they randomize, they give off energy. The clever part is
the gradient field which causes the atoms to give off energy at slightly
different times – the top of your head first, your neck a fraction of a
106 Are the Androids Dreaming Yet?
fMRI
second later, and so on down to your feet. What you see at any one time
is a slice through a specific section of the body. You can then build up 3D
images from these slices and look at the soft watery tissue rather than the
hard bone you can see with an X-ray.
MRI scans give detailed images but today there are many more
imaging tricks you can play. Give the patient gadolinium to eat – a type
of paramagnetic material – and this contrast agent will highlight active
parts of the brain. You can ‘see’ which parts are active: the location of
emotions such as love, joy and even the effect of smells as the brain
experiences things. This is still coarse grained information; it shows only
the general area of excitation and it does not tell us what is going on at
the nerve level, but the images are fascinating.
Another recent development in imaging is the diffusion MRI. If you
remember your school physics, molecules travel with a random walk:
they diffuse along pathways just as people wander along a corridor. If the
corridor is full of people, they are jostled around and make little progress.
If the corridor is empty, they move in straight lines. This difference in
jostling affects the reading in an MRI and allows you to color code the
image according to the rate of motion of water along the pathways. You
can therefore ‘see’ the rate at which signals flow in the brain and not only
locate thoughts, but also see the links between them.
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Functional PET
PET
The last scan we will look at is functional positron emission tomography,
or f-PET. In this machine the scanner detects positrons given off by
excited oxygen atoms.
As you think, you burn glucose by combining it with oxygen.
The parts of the brain that are thinking hard use a great deal of oxygen
and this shows up in scans. Again the consecutive slice trick is used to
generate a 3D image that allows you to fly through the brain as it works
on a problem.
There is one problem common to all these methods. X-rays, MRI
and PET scans only show us the location of thoughts with an accuracy
of a few millimeters. Each pixel in the image contains around 10 million
neurons, so we can’t see the details of thought. For a scale comparison it
108 Are the Androids Dreaming Yet?
is like looking at a car factory from space. You can see cars and people
going into the factory but you can’t read the owner’s manual. We need
to be able to see at least 10 million times more detail than our current
technologies allow to see a thought.
A Quick Tour
Now that we understand how to look inside the brain, let’s take a tour
around it. The brain is a highly distributed thinking machine. Some
things, such as hearing, are located in specific places while others, like
the enjoyment of music, are spread out.
Our eyes work as an extension of the brain and use a specialized
type of nerve cell. Light falls on the retina and stimulates these cells,
causing nerve impulses to run along the optic nerve into the center of the
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Visual Processing System
brain. The impulses split and form two distinct paths, one through the
cerebral cortex, which gives us the sensation of conscious vision, and the
other into the lower brain which provides us with instinctive reactions.
The right hand side of your body is connected to the left hemisphere
of the brain and vice versa. This means each hand is controlled by the
opposite side of the brain. But, your eyes see both your hands. To resolve
this conundrum a very complex thing has to happen to the optic nerve in
the center of the brain. The optic nerve from each eye splits and crosses
over in the middle, so the left side of the left eye and the left side of the
right eye goes to the right hand side of the brain and vice versa. This
keeps the brain focused on the correct hand.
110 Are the Androids Dreaming Yet?
Frogs Eyes are Very Sensitive
The processing power of the eye is staggering. The human retina has
about 120 million rods and 7 million cones, giving it an average resolution
of 10,000 by 10,000 pixels. Each rod is sensitive to individual photons
but we register light consciously only if we see around 5-7 photons. It
is thought frogs can react to single photons because of the chemistry of
their eyes and the fact they are cold-blooded, but this is not proven.
Some animals, including some frogs and my cat, have a tapetum
lucidum. This is a reflective backing to the eye that allows each photon
two chances to react with a rod, once on the way in and, if that fails,
once on the way out. This is why you can see the eyes of some animals if
you shine a light into the forest on a dark night. Cones are less sensitive
than rods but give us color perception. In the human eye, there are three
types of cone: a red, a green and a blue, giving us trichromatic vision.
We see colors because light stimulates more than one types of cell and
we infer the color in between. A fourth type of cone is present in some
species such as birds, reptiles, and fish. This gives them tetra-chromic
vision, allowing them to see into the ultraviolet range. It is speculated
some humans might have this ability but so far none has come forward.
Some animals lack the ability to see certain colors. Most dogs can’t see
red. This gives cats a big advantage!
Many people wonder if we all see the same color as each other. Is
your red the same as mine? The brain’s perception of color is complex.
Although the color red is absolute and can be detected by a calibrated
sensor, our perception of color is relative. We perceive them in the context
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Color is Relative
of other colors – not in isolation. The two panels above contain identical
blocks of color but they look very different against the background. Check
out the website if you have a black and white book. It is an irrelevant
question to ask if my red is the same as yours, since my red against one
background is not even the same as my red against another.
People generally agree on naming colors but not all languages have
the eleven specifically named colors of modern English: black, blue,
brown, gray, green, orange, pink, purple, red, white, yellow, if you are
interested. Ancient Celtic languages, so called ‘gru’ languages, recognized
only four colors and other languages don’t distinguish purple from blue.
Color, or at least the naming of color, is a cultural thing.
Impressionist Painting, Monet Haystack
112 Are the Androids Dreaming Yet?
The resolution of the eye is not the same across the image. High
resolution is concentrated in the center, while lower resolution black
and white vision dominates the edge. This peripheral vision helps us
detect predators or play football but it is not the focus of our attention.
When we focus our attention on something, we turn our eyes to look at
it directly. The central part of our eye is called the fovea centralis and is
composed of cones. About half our cones are concentrated in this very
small section and this gives us immense visual acuity. For a computer
display to outperform this section of the eye it would need one billion by
Scintillating Dots Optical Illusion
one billion pixels. The fovea centralis is tiny, only two degrees across, so
our eyes must dart around the image to take in all the detail. Once the
basic information is encoded in our retina and sent down the optic nerve,
it goes into a production line process in the visual cortex where all the
elements are analyzed. Our brains extract information from the image
such as texture, edges and depth perception in specialized portions of
the brain. Because of this specialization it is possible to play tricks on the
brain with images that are not easy to process. Some we find pleasurable,
while others can be a little disturbing.
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Penrose Steps
Optical Illusions
This picture is an illusion that plays with your stereoscopic synthesis. The
dots appears to flip between black and white. Other illusions play with
depth perception. The Penrose Steps are a type of illusion that tries to
build an impossible physical model in our cerebral cortex. The brain sees
perspective and depth perception cues, but the resulting shape could
never exist.
Hearing
Unlike sight, hearing is an absolute sense. Our ears capture and focus
sound down to the eardrum where a set of small hairs called cilia convert
it into electrical impulses. The impulses stimulate cells corresponding to
specific pitches.
We are born with perfect pitch, yet most of us lose it early on. When
I hear Maria Carey sing a top B flat a specific set of neurons located near
the ear fires, and if she sings a top ‘A’ then a different clump of neurons
are stimulated. By the time most children come to learn music they have
edited out this absolute pitch information. One group of children who
do not lose the ability are Chinese pianists. Because Chinese is a tonal
language – where the pitch of words affects their meaning – and because
114 Are the Androids Dreaming Yet?
McGurk Effect; Go to the Website and Watch the Linked Video
Chinese children tend to learn the piano very young, they don’t lose the
absolute part of pitch. An astonishing 93% of these children develop and
retain perfect pitch throughout their lives.
There are many cross connections between the audio and video
processing systems. At parties you often can’t hear speakers clearly because
of the background noise. Watching their lips will help comprehension,
but which sense wins if there is conflict between the two? The McGurk
effect shows this.
To test the effect, go to the website, watch the video and see if you
can distinguish when a speaker talking normally and when he is making
the mouth movement of another sound. There is a winner. Try it for
yourself; check out the link on my website.
Once upon a time people imagined the brain was like a camera
forming an image of the world, but if this were the case there would be
a paradox. Who is looking at the image in our brain to make sense of it?
Modern research shows we don’t take a complete picture of the world
like a camera but rather parse the image into its constituent parts on the
fly.
If someone asks, “Which side of the house is the tree on?” your
brain parses the question and compares it with the image map in your
mind’s eye. What is the image composed of: trees, houses, sky, grass?
Your brain manipulates the linguistic question about the relationship
of elements and matches it with the visio-spatial understanding of the
image, allowing you to answer the question. You might not have to answer
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Humans’ Ability to Concentrate
the question verbally. If you hit a baseball, no language is involved; you
distinguish the ball from the background and perform quite a feat of
tracking and calculation to connect it with your bat.
Because the brain is editing the scene on the fly to keep within
its processing power, the eye only sees what it turns its attention to.
Magicians take advantage of this to play amazing tricks on us. Watch the
video on the web and then tell me what you see.
VISIT THE WEB AND VIEW THE VIDEO TO SEE WHAT HAPPENS
Tiger Woods Swing
You can see just how intensively the brain works on a given problem,
throwing away all unnecessary information.
The brain contains mirror neurons, a type of brain cell that
responds when we see another human do something. These neurons
fire as if we were performing the action ourselves even though we are
merely witnessing it. It is one of the ways we learn a skill. If I watch Tiger
Woods’s golf swing, my mirror neurons will fire as if I were practicing
his swing. Later when I practice the swing for real, my neurons will have
already been partially programmed. This effect is presumably the reason
we enjoy watching sports; our mirror neurons allow us to begin acquiring
a skill while sitting in an armchair! This is clearly a useful evolutionary
trait but you do also need to practice for real!
Mirror neurons also fire in response to witnessing emotions. When
we see an actor laugh or cry, we experience their emotion as if for real.
This helps us empathize with the person we are watching and is part of
the reason we enjoy movies and plays.
Neural Network
Thinking
“We cannot solve our problems
with the same thinking we used
when we created them.”
Albert Einstein
If you feel mentally exhausted reading this book, don’t worry. This
is normal. Mental work takes energy. Scientists estimate the brain
consumes 20% of our resting energy; around 12 watts. Physical fitness
is important for thinking. If you get out of breath running for a bus,
thinking is going to be harder for you. Studies are mixed about whether
the additional work involved in solving a difficult problem causes you to
use more energy. We certainly see an increase in the flow of glucose to
the appropriate part of the brain, but the overall energy use in the brain
is quite high in the first place, so it is hard to see the incremental effect.
Unlike muscles, which store energy locally as glycogen, brain cells
‘burn’ glucose and oxygen from the blood stream in real time. If scientists
detect glucose and oxygen flowing to a part of the brain they know it
must be working on a problem. As we know, there are several ways to
make glucose and oxygen show up in brain scanners. You can, therefore,
inject someone with the right chemical markers, wheel them into a brain
scanner, and watch them learn new skills. On a practical level, there
is limited space within a scanner and you can’t wield a golf club, for
example. Julien Doyon, a researcher at the University of Montreal, was
recounting this problem to a friend and she suggested knitting. Knitting
is a physical activity you learn just like a golf swing or a tennis stroke, with
all the initial fumbles and jerky activity, settling down to a fluid learned
skill. Most experienced knitters can engage in a full conversation while
knitting complex patterns, only needing to break off and concentrate
during a pattern change. Luckily, there are ceramic and bamboo knitting
118 Are the Androids Dreaming Yet?
needles which don’t interfere with MRI scanners, and they are small – no
golf swing problems here. Studies of knitters show that when they initially
learn a skill, several areas of their brain light up, but after a while, the
brain activity becomes concentrated in the sensorimotor striatal territory.
Glucose, the brain’s power source, is a sugar we get directly from
eating sweets or indirectly by digesting starch. Some studies show
children do slightly better at school if they eat starchy foods in the
morning for breakfast – a bowl of cereal or porridge. When you think
and work your brain consumes the glucose in your blood, and blood
glucose level drops. If there is a steady source of glucose from the starch
digesting in your gut, the glucose is constantly topped up and the level
will stay high. If there is no input of glucose from your gut, the body will
first get glucose from glycogen in your liver or generate it by converting
fat reserves. This takes more work so the body tends to avoid doing so
until it absolutely has to. You can function with slightly lower glucose
levels but the body will shut down a little. One thing that suffers as a
result is the brain’s ability to perform cognitive tasks. A quick and easy
way to fix this is to consume some raw glucose and most fridges have a
ready supply in the form of sugary drinks. Stories of kids running amok,
due to sugar highs brought on by too many sweets and sodas, appear to
be an urban legend. In tests, parents told their children have had a sugar
drink report them to be hyperactive even if they had been given a sugar
free drink. I’m not suggesting you drink lots of sugary drinks – it is bad
for your teeth and will make you fat – but the occasional soda is fine.
Memory
Scientists are just beginning to explore the mechanisms that lay down
memory in the brain. There are two main classes of theory. The first
believes memory is formed in the large scale wiring of the brain.
Neurons connect with other neurons and the number and strength of
these connections cause memory. When we learn, new connections are
formed. The electrical activity in a given part of the brain triggers the
formation of new dendrites. They grow, piloted by tubulin micro-tubes,
rather like vines growing in a slow motion nature clip. Once a microtube
guided filament is close enough to other, a synapse forms. This
gross-scale wiring growth is one method of memory formation. Another
gross-scale effect is myelination. Myelin is the insulation the body uses
on nerves cells, including nerve cells in the brain. It looks a bit like the
insulation we used in the 1930s. Before the invention of plastic, strips of
waxed canvas were wrapped around wires to provide insulation. Myelin
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Synapse
has a similar structure. It is a flat protein laid down as a spiral on the
outside of nerve cells. The theory is that cell firing causes myelination,
which permanently imprints the memory.
The alternate class of theory proposes memory is encoded at a much
smaller scale. Neurons are quite complex structures in their own right.
Inside each neuron is a lattice of proteins, which forms a skeleton. Part
of that skeleton provides structural integrity to the neuron, while other
elements provide control and motility. It is this control part of the skeleton
that people believe might encode memory. A 2012 paper by Travis Craddock
and Jack Tuszynski of the University of Alberta, and anesthesiologist Stuart
Hameroff of the University of Arizona proposes a protein called CaMKII
binds to the cytoskeleton in 32 different configurations, providing a binary
data encoding. It is an elegant idea but it also relies on your believing their
model for quantum neuron processing which is still highly controversial.
If proven, they are my top Nobel Prize tip for this decade!
Photographic Memory
Until recently conventional wisdom held that true photographic memory
was a myth and the few people claiming to have it really used some sort
of mnemonic memory technique to selectively memorize things. The
120 Are the Androids Dreaming Yet?
most famous case was a Russian journalist known as ‘S’. He habitually
memorized things using association with places. In antiquity this was
taught as ‘the method of loci’. The unusual thing was his inability to turn
the effect off, and he found it as much a curse as a blessing. He was unable
to forget useless information and found it hard to interpret complex
images, tending to see areas of color and shade rather than objects such as
trees, houses and fields.
Very recently some people have come forward, six in America and
one in the UK, who appear to have genuine photographic memories It is
well worth watching the TV documentary The Boy Who Can’t Forget to
gain a sense of what this is like. These people appear to lack the ability to
forget, and this turns our understanding of memory on its head. It seems
memory might work the opposite way we thought. We had previously
thought we only remember what we pay attention to, but perhaps we must
actively forget, and this ability is missing in these subjects. Scientists are
studying these people to see if they can understand more about memory.
The Aging Brain
We can explode a myth and encourage older readers simultaneously.
Memory does not deteriorate with age, or at least not until we are very
old. Most studies looking at memory deterioration focus on the very
old and compare them with the very young. Even then, the differences
are small. When people are asked to attempt memory problems there
is a mild drop off with age but the results are quite similar. The most
likely reason older people don’t remember so well is they don’t believe
they can. Perhaps they don’t have as much incentive to remember new
information. Why learn someone’s name if you’re unlikely to meet them
again? Since IQ actually increases with age, don’t believe people when
they say you are going downhill from the age of 40. You are not!
Computer Brains
Computers are really quite simple compared with all the evolved
baggage we humans carry around. When a computer is presented with
instructions, for example, for a program like Excel and a file such as my
expenses, it will load everything into memory and ‘run’ it. The process of
running a program is simple. Each instruction is a number. The computer
reads the number, looks it up in a table, finds a corresponding number,
and writes that down. Essentially that’s all there is to it. From a simple
mechanism like this, we get the enormous complexity of a modern
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computer. The sophistication is achieved through reading and writing
many numbers in parallel, and chaining the steps together so that if you
read a particular number it triggers another read/write process, and so
on. I’m glossing over some details such as logical functions but, if you
know how a modern computer chip is constructed, my description is
not far off. Almost all logic today is implemented in tables to achieve the
speeds we expect from modern chips.
All modern computers are clocked. A small piece of quartz rock
has been polished, coated with metal, and wired up to a control circuit in
the computer. When you apply voltage to the rock it bends and absorbs
energy. When the voltage is taken away it bends back and gives out the
energy. This is effectively a pendulum and it can be used to make an
accurate clock. I used to design these for a living. Every logic gate in a
computer is connected to this clock, and each time the clock ticks the
logic gates in a computer compute.
Most modern computers are entirely synchronous. The clock rate
is set so that the gates in the computer fully recover by the time of the
next tick, and every gate is therefore ready in its standard position when
the next instruction arrives. The human brain does not have a central
clock. Each neuron acts independently – firing regardless of whether the
neurons it is adjacent to are ready or not. It is wrong to think of the brain
as digital. Each neuron does fire and recover, but it may be triggered again
before it fully recovers. This makes for a chaotic and essentially analogue
operation. If one neuron fires when a second has only half recovered, then
it gets half an effect. If the neuron is 80% recovered, an 80% effect. Neuron
recovery time is quite long, perhaps as much as 1/1000 th of a second, and
they are wired in three dimensions to as many as 10,000 other neurons.
It is perfectly possible for a set of neurons to run one ‘program’ when
they are rested and a completely different ‘program’ when they are 50%
recovered and yet another programs if triggered from different starting
locations. I have said ‘program,’ but arguing a brain runs a ‘program’ is
misleading. It is not organized like this.
Neural Networks
A neural network is our best attempt at a computer model for the human
brain. Each neuron is represented by an entry in a table. The entry records
all the connections to it, along with the strength of each connection –
these are called ‘weights’. In some models the connections can be both
122 Are the Androids Dreaming Yet?
inhibitors and activators like in real synapses. An individual neuron will
fire if the sum of all the connections multiplied by the weights reaches a
certain pre-determined threshold.
A neural network does not run a program in the conventional
sense, and must be trained through experience rather like a human
brain. The training process allows the weights in the network table to
be adjusted to give the correct result. But, unlike the brain, you can read
the weights and even save them to a disk. The neural network tables
start with random settings. You show the network the letter ‘A’ and
adjust the weights in the tables until it gives a positive answer: ‘It’s an
A’. Repeat the process with the other letters until the network correctly
distinguishes them. As you do this a computer algorithm constantly
adjusts the weighting tables using a method called ‘back propagation’.
At the end of the training process you can show the network some
new input and see how it does. For example, a letter ‘A’ that is in a slightly
different font to anything in the training set. Trained neural networks
can perform complex tasks such as recognizing faces or making clinical
diagnoses, and they can be allowed to modify their weighting tables as
they work so they learn from experience in a similar way to a human
brain. Strong AI proponents believe making a thinking machine is just
a matter of building a really large, fast neural network and working out
how to train it efficiently.
Quantum Brains
Conventional wisdom says each brain cell is a single processing unit
making an on-off decision – fire, or don’t fire – depending on the state
of its neighbors. But, Stuart Hameroff, Professor of Anesthesiology
at the University of Arizona, thinks neurons are not the fundamental
information-processing unit in the brain. He suggests that this accolade
should go to tubulin. Tubulin is a small, versatile protein that selfassembles
into filaments rather like the way buckyballs – a magnetic
children’s toy – can be arranged. There are two types of tubulin molecule:
α and β. They slot together and wrap around to form a micro tube about
25nm in diameter.
Tubulin micro tubes do several important things in the body.
They form the skeleton of neurons and give them structure. They are
involved in guiding neurons as they grow towards each other to form
new connections, and they also operate in the nucleus of a cell to unzip
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Paramecium
DNA into its two complementary strands when a cell divides. In singlecelled
organisms, including paramecium, the ends of the tubes stick out
of the body and form the cilia that drive the organism along.
The presence of tubulin in complex, single-celled organisms
provides a clue that the smallest information processing unit might not
be the neuron. Some single cell organisms, such as paramecium, display
complex behavior: hunting for prey and escaping danger. This suggests
they can process small amounts of information without the need for a
matrix of neurons. Since we evolved from these organisms, why wouldn’t
our brain cells take advantage of this sub-cellular intelligence?
The structure of tubulin lends itself to digital processing as the
molecules forming the walls have two stable states and can flip between
them. We might recognize this as the basis of a binary computer, and
cells might have little computers within them. They would not need to
process many bits to be useful. Perhaps single-cell organisms developed
information processing capabilities in their micro tube structures that
allowed them to better survive and, as their nervous systems evolved, they
coupled these structures to form the brains we see today. This piece of
theory is not too controversial. After all, nerves have wiring within them
to carry information to the synapses and it’s likely this wiring is involved
in the thinking process. But Hameroff is not finished. He has teamed up
with Roger Penrose to bring quantum mechanics into the picture.
124 Are the Androids Dreaming Yet?
Their reasoning is straightforward but has generated a great deal
of controversy. Hameroff observes that anesthetics cause humans to
lose consciousness by binding to tubulin, but they do not halt all brain
function. He, therefore, concludes our conscious thinking is mediated
by tubulin, not the larger scale firing of the neurons. Penrose had been
looking for a mechanism in the brain that would explain how brains
solve non-computational problems. Together Penrose and Hameroff
propose tubulin micro tubes are quantum gravity computers that allow
us to think non-computationally and are the seat of consciousness. The
ideas are still being worked.
Penrose and Hameroff have a difficult task conveying their ideas
to the rest of the scientific community. Scientists don’t recognize a
need for something that can think non-computably, so they are highly
skeptical of a mechanism which performs that sort of thought. The
latest development on the Hameroff Penrose model comes in the work
of Travis Craddock, now of Nova Southeastern University, Florida, and
others. They have written a paper arguing signals propagate according
to quantum principles within microtubules through the excitation
of thiamin molecules along the length of the tube. They believe these
molecules are quantum, entangled in a similar manner to the mechanism
recently discovered in photosynthesis. The geometry of these molecules
is set out in a similar way to the active areas in chlorophyll and they
have a complementary problems to solve. Chlorophyll tries to maximize
energy conversion efficiency, while a microtubule tries to minimize the
use of energy while propagating signals along a nerve. You might wonder
Tubulin Protein
The Brain
125
Tubulin
where the light comes from since tubulin is housed deep within the
neurons inside our brains and shielded from light by our skull. It turns
out that the mitochondria which powers our bodies emit photons of UV
light as a waste product of their metabolism. The speculation is tubulin
harvests this waste energy.
Before we argue for this mechanism any further we still need to
establish that a non-computational mechanism is needed to allow human
thought. In the next chapters, we will look at the nature of knowledge
and, in particular, mathematical creativity and the Wiles Paradox.
Quantum Coupling of Tubulin in Microtubule

Chapter 5
KNOWLEDGE
Chimpanzee and Typewriter
“There’s an infinite number of
monkeys outside who want to
talk to us about this script for
Hamlet they’ve worked out.”
Douglas Adams
“I’m not young enough to know
everything.”
J.M. Barrie
“He has Van Gogh’s ear for music.”
Billy Wilder
Could an army of monkeys write Hamlet by bashing away randomly
on typewriters? Of course, we don’t mean this literally. We are
asking whether knowledge can be created without understanding.
Can a monkey, or perhaps some form of computerized random number
generator, accidentally type out the script for Shakespeare’s Hamlet
or write Tolstoy’s War and Peace? Is knowledge generation simply a
numbers game?
Leo Tolstoy’s War and Peace is generally assumed to be the longest
novel ever written. This is not quite true. Wikipedia reckons the longest
novel is a French book, Artamène, with over 2.1 million words. Tolstoy
comes in sixteenth, with a mere half million!
Written in 1869, War and Peace tells the story of five Russian families
during the Napoleonic wars. Originally written in a mixture of Russian
and French, and numbering over 500,000 words, it was quickly translated
to other languages. The mistress of composer Franz Liszt translated it
fully into French, where it expands to 550,000 words. Contrary to popular
myth the length of the book drops slightly in German. If you really want
to save paper Chinese is best. Because
it uses a single symbol per word, the
Chinese translation needs only 750,000
characters compared with the 3 million
for English. It is wrong to assume this
is necessarily more efficient than a
phonetic language. Although it might
save on paper, it is considerably more
laborious to write. Three strokes are
required to write ‘war’ in English
whereas the Chinese pictogram
requires ten.
War in Chinese
Computers work with numbers. It is a simple process to translate
a book into numbers because books are composed of discrete symbols.
All we need do is give each symbol a unique number and record those
numbers in digital format. Artistic works involving pictures and sound
are more difficult to represent because they are continuous in nature. We
have to digitize them first. With music or painting this inevitably means
some loss of information as we can’t cut a sound or image into an infinite
number of pieces.
The modern standard for translating text to numbers is Unicode.
Each character is represented by a five-digit number ranging from 1
to 64,000 – two bytes for those of you who know computing. This is
130 Are the Androids Dreaming Yet?
sufficient to code almost all the world’s symbols, so we can avoid any
accusation of being language-ist! Here are some examples of the
Ancient Greek, Japanese:
Kanji, Katakana, Chinese,
and Russia-Cyrillic
Symbols
characters represented by Unicode.
For our discussion, it does not matter which language War and
Peace is written in. We just treat the symbols as numbers. I am going
assume the English translation which has around 500,000 words; a nice
round number. Assuming a generous 10 characters per word, War and
Peace is approximately 10-megabytes – that’s about the same size as a
music track on iTunes. In practice, the book uses a bit more memory, as
there is some overhead for formatting information. My laptop has a 500
Gigabyte hard disk so I could fit half a million copies of War and Peace
on it!
If we take a look at the contents of the file on my computer the book
starts:

Can a computer calculate this number?
The obvious answer is YES. It is just an integer like 1, 3 or 42. Granted
it’s a large number, but the length of the number is simply the length of
all the symbols in the book coded into Unicode – about 10 million digits.
We have already determined this number can be stored on my hard disk
half a million times, so it’s not an unimaginably large number. How long
would it take to calculate the number corresponding to War and Peace?
The simplest method is to count up starting at 1 then 2, 3, 4, 5, and
so on until I try every number. Will this eventually get to the War and
Peace number? The answer is yes. Eureka! All of human knowledge is
computable. I have written this computation out as a simple computer
program below. It says, in plain English, start at zero, go round a loop
counting up one at a time and print each number as you go along.
i==0; Loop i++ Print i;
Knowledge
131
Easy!
No, unfortunately. The problem is subtler than it first appears. First
it will take a VEEEEERRRY long time. If I counted up from one, I would
print out War and Peace eventually but it would take 120 billion, billion,
billion, billion, billion… (I would need the entire length of this book
to write out all the billions) years! For the physicists amongst you, I
would need 10 30,000 years, assuming I could use every atom in the known
universe counting in parallel at the plank interval. ‘The plank interval’ is
the shortest time that can exist in the Universe as a discrete ‘tick’.
Even going at this speed using with every atom in the known
Universe would take 10 5,000 longer than the age of the Universe. This is
stupendously long. Remember scientific notation means I have a 1 with
5000 zeroes after it. It is a deceptive notation as something as innocuous
as 10 120 is equal to the number of atoms in the known universe. 10 5,000 is
an absolutely enormous number. If you hear something is going to ‘take
until the end of time’, we’re talking a lot longer than that!
You may have spotted that in the process of counting up to the War
and Peace number we also count through EVERY book ever written
shorter than 500,000 words in all the world’s languages. Interestingly we
counted through the Japanese and Chinese translations of War and Peace
quite a bit before we reached the English and finally French translations.
During the process, we also stepped through countless other wonderful
works: proofs of amazing theorems, the complete works of William
Shakespeare, and every composition ever written. Sadly, we never knew
it. The problem is my program never stopped and told me it had found
any of these wonderful things. I would have to sit staring at the screen to
spot them. If I was off doing something else – making a cup of tea, taking
the kids to school – I would miss all these wonders; the program never
tells me if it has succeeded, but quietly prints out War and Peace and
carries on. This is really annoying. It’s not a useful machine.
What I need is a machine that rings a bell when it finds something
interesting so I can break away from what I am doing and take a look.
Reading every book it writes in every language and all the nonsense
in between would take a ginormous amount of my time. (By-the-way,
contrary to statements by school teachers that ginormous is not a word
– it is!) I want a computer to come up with War and Peace without me
having to do all the work.
It’s no help if the machine writes everything down and lets me take
a look in my own good time. That only puts off the time when I have to
begin reading all the gibberish it produced. Another practical problem
is the massive storage required. Just imagine the immense piles of printer
132 Are the Androids Dreaming Yet?
paper! Stephen Hawking and Jacob Bekenstein have shown space appears
to have a limit to the quantity of information it can store. The quantity of
information we are looking at here is greater than the storage capacity of
the Universe and would collapse space-time to a black hole before I got
even a fraction of the way through. Let us try to be a bit cleverer about
the task of creating this information.
The simplest way to tie the computer down is to run a much
stricter program. Ask it to count up from one until you get to a number
representing the novel War and Peace and then print it, stop and ring a
bell.
Loop i++ until i == “War and Peace…”; Print i; ring-bell;
This program succeeds!
I am triumphant. I have calculated the War and Peace number, and
this time I did not miss the event. But, if you consider this a little more
deeply I gave the computer the answer! I told it the string “War and
Peace…” and it was able to count up, stop, and tell me it reached it. In
mathematical terminology, the program is said to have ‘halted’ when it
reached the War and Peace number and in computer science speak it
is a special purpose program designed to do only this one thing. This
program is pointless. First, it would still take a ginormous amount of
time to get there and, second, it is trivially the same as running the
program: Print War and Peace.
i = “War and Peace…”; Print i;
It’s just the same as me taking my laptop, finding War and Peace and
pressing print. In no way is this equivalent to Leo Tolstoy’s creative effort
of writing War and Peace in the first place.
What went wrong?
I wanted my computer to find an interesting string I did not already
know. War and Peace is trivially computable after Leo Tolstoy created it
but the question is whether my computer could come up with War and
Peace or some similar creative work on its own. Can it create and, more
importantly, understand it has created something? We have linked the
ideas of creativity and understanding, and this will prove to be the key
to the problem.
Knowledge
133
The Problem
One suggestion put forward by Daniel Dennett is the creative process
is a two-part task – generate ideas, then critically assess them. I can, in
principle, make a program write out every possible book less than 500,000
words long. Provided I don’t store the results this will not collapse the
Universe. This just leaves the problem of writing another program to
read all the output and ring a bell each time it finds some interesting
truth. This second program might be called an appreciation program.
Let’s examine this approach. I can write out a very simple program to do
this – provided I cheat and ignore the complexity of the term ‘something
interesting’. In plain English: Count up from one until I get an interesting
fact, write it down and stop.
Loop i++ until i == (Something Interesting), Print i
This generates two problems. We need to make a program that
can tell if something is interesting and it will need to be fast because it
is going to be handed a huge amount of junk. Clearly I have a process
running in my brain that can determine if something is interesting, but it
is quite slow. It takes me an appreciable time to open a book, leaf through
the pages and declare it either junk or interesting. Leo Tolstoy had a
process in his brain that allowed him to create something interesting but
I want to prove he did not do this by generating random junk and sifting
through it. Let’s look at the mathematics.
We know simply counting sequentially through every number
would take too much time, but why not generate random numbers and
run our critical eye over them? Surely this would give a faster result. Let
us try with a short poem. How hard would it be to come across something
as simple as a four-line poem using this technique?
This poem, by the late Spike Milligan, is only 23 words long,
including the title, and I have a powerful computer. Wouldn’t it be
possible to generate it using a computer? Unfortunately, no. We humans
don’t have a good head for large numbers and this problem is much
harder than it appears. Let’s use playing cards to get a feeling for large
numbers.
134 Are the Androids Dreaming Yet?
A Simple Poem
Rain
There are holes in the sky
Where the rain gets in
But they’re ever so small
That’s why the rain is thin.
Spike Milligan
Spike Milligan
Coming upon a poem by chance can be likened to the probability
of dealing a perfect bridge hand. Shuffle the deck thoroughly and then
deal four hands. What is the probability every player will have the ace
through king in a single suit? It’s about 1 in 1,000,000,000,000,000 hands.
Because lots of people play a lot of bridge around the world, this outcome
has been reported quite a few times. The possibility appears within the
bounds of human experience. Fifty-two playing cards seems close to the
80 characters that make up this poem and 13 choices of cards is about the
same as the 26 letters of the Latin alphabet. Wouldn’t we expect poems of
this complexity to crop up almost as often?
NO.
Knowledge
135
The 80 characters of this poem versus the 52 playing cards and the
greater choice offered by 26 letters increases the problem geometrically.
Taken together the probability of accidentally getting this poem is vastly
less than a perfect hand of bridge, 1 in 10 83 against the perfect bridge
hand of 1 in 10 20 . That’s the difference between the number of atoms
in the known universe and the number of atoms in a jug of water!
Numbers get big very quickly when we are looking at the permutation
of information. And there is another problem with our bridge analogy.
All the bridge players in the world are part of the machine finding the
perfect hand. When a human sees a perfect bridge hand they are amazed.
It is an event that usually hits the local newspapers and a couple of years
ago one reached the national papers in Britain. Each bridge player looks
at every hand, they play so there is a huge amount of processing going on
during every bridge game. To replicate this for our poem, we would need
millions of poetry classes spending hours each evening reading through
computer printouts of gibberish.
I should also add that sightings of perfect bridge hands are almost
certainly hoaxes. The probability of it happening even once would
require everyone on Earth to play bridge continuously for a thousand
years. It is reported somewhere in the world about two or three times a
year. If we are charitable, we might assume people failed to shuffle the
deck properly but I suspect some mischief is going on! The numbers
don’t stack up…
You might think the problem is one of improving the efficiency of
the filter so humans would only have to examine a smaller number of
possibilities. Surely I could improve things by writing a simple program
to ban all non-English characters, words and poor grammar; things that
don’t pass the Microsoft Word grammar checker. This would generate a
more manageable number of potential poems.
Lewis Carroll shows this does not work; my idea to use a grammar
and spelling checker to filter out gibberish just eliminated Jabberwocky,
one of the most famous verses in the English language. Take a look at
what Microsoft Word thinks of it.
136 Are the Androids Dreaming Yet?
The Jabberwocky
’Twas brillig, and the slithy toves
Did gyre and gimble in the wabe;
All mimsy were the borogoves,
And the mome raths outgrabe.
“Beware the Jabberwock, my son!
The jaws that bite, the claws that catch!
Beware the Jubjub bird, and shun
The frumious Bandersnatch!”
He took his vorpal sword in hand:
Long time the manxome foe he sought—
So rested he by the Tumtum tree,
And stood awhile in thought.
And as in uffish thought he stood,
The Jabberwock, with eyes of flame,
Came whiffling through the tulgey wood,
And burbled as it came!
One, two! One, two! and through and through
The vorpal blade went snicker-snack!
He left it dead, and with its head
He went galumphing back.
“And hast thou slain the Jabberwock?
Come to my arms, my beamish boy!
O frabjous day! Callooh! Callay!”
He chortled in his joy.
Twas brillig, and the slithy toves
Did gyre and gimble in the wabe;
All mimsy were the borogoves,
And the mome raths outgrabe.
Lewis Carroll
Lewis Carroll’s Jabberwocky
Knowledge
137
The Jabberwocky Spell Check
Microsoft Verdict on the Poem
39 of the 166 words in the poem are unknown to Word’s spelling checker
and this is an optimistic analysis of how the algorithm would fare. Many
of the words are in the spelling checker because of the poem: galumphing,
for example. Lewis Carroll’s work was sufficiently influential that part of
138 Are the Androids Dreaming Yet?
the English language was created in this poem. The same goes for much
of Shakespeare. If we used a filter method, we would have just deleted
most of Shakespeare from the English language! Indeed half the poems
in my anthology of English verse are destined for the waste paper basket
due to some minor infraction of ‘the rules’. If you want something that
completely flummoxes my spelling checker here is the Loch Ness Monster
Song by Scottish poet Edwin Morgan. I asked a Scottish friend whether
Scottish spelling checkers fared any better and he assures me, no.
The Loch Ness Monster’s Song
Sssnnnwhuffffll?
Hnwhuffl hhnnwfl hnfl hfl?
Gdroblboblhobngbl gbl gl g g g g glbgl.
Drublhaflablhaflubhafgabhaflhafl fl fl -
gm grawwwww grf grawf awfgm graw gm.
Hovoplodok - doplodovok - plovodokot
- doplodokosh? Splgraw fok fok
splgrafhatchgabrlgabrl fok splfok!
Zgra kra gka fok!
Grof grawff gahf?
Gombl mbl bl -
blm plm,
blm plm,
blm plm,
blp
Edwin Morgan
The Loch Ness Monster
Knowledge
139
The foibles of spell checkers have long been a personal pain to me
because of my dyslexia. Although I can see the red underlining Microsoft
Word kindly inserts so liberally into my text, I can’t easily see the occasions
when I use a homonym. A fine poem illustrating the problem was kindly
written by Jerrold H. Zar and published in The Journal of Irreproducible
Results. It hangs on the wall behind my computer to remind me to check
for these errors.
Candidate for a Pullet Surprise
By Jerrold H. Zar
I have a spelling checker,
It came with my PC.
It plane lee marks four my revue
Miss steaks aye can knot sea.
Eye ran this poem threw it,
Your sure reel glad two no.
Its vary polished in it’s weigh.
My checker tolled me sew.
A checker is a bless sing,
It freeze yew lodes of thyme.
It helps me right awl stiles two reed,
And aides me when eye rime.
Each frays come posed up on my screen
Eye trussed too bee a joule.
The checker pours or every word
Too cheque sum spelling rule.
Bee fore a veiling checker’s
Hour spelling mite decline,
And if we’re lacks oar have a laps,
We wood bee maid too wine.
Butt now bee cause my spelling
Is checked with such grate flare,
Their are know fault’s with in my cite,
Of nun eye am a wear.
140 Are the Androids Dreaming Yet?
Now spelling does knot phase me,
It does knot bring a tier.
My pay purrs awl due glad den
With wrapped word’s fare as hear.
Too rite with care is quite a feet
Of witch won should bee proud,
And wee mussed dew the best wee can,
Sew flaw’s are knot aloud.
Sow ewe can sea why aye dew prays
Such soft wear four pea seas,
And why eye brake in two averse
Buy righting want too pleas.
The Search for Knowledge
I hope this explanation shows you the simplest model for creativity –
working through every possibility, and examining them all – is doomed
to failure. It would take longer than until the end of time to even list all
the options, let alone analyze them.
You might wonder just how long it is until the end of time? It’s
generally assumed there are two possible ends to the Universe, a Big
Crunch or heat death. Either way the approximate estimate is our
Universe will last somewhere between one and fifty times longer than
it has lasted so far. That’s a long time, at least another 15 billion years,
but just generating War and Peace would take 5000 orders of magnitude
longer than this!
More complex models such as a three-step process have been
suggested. We could perhaps randomly create information and put
it through a mechanical filter to bring it down to a manageable set of
options and then give it to an appreciation algorithm to finally decide
whether we have created something. The real problem with this model is
the filters. If we try to reduce the effort by assembling works only from
pre-existing words, we will have filtered away many works we know and
love. Gone are Shakespeare, Lewis Carroll, Dylan Thomas and Roald
Dahl, shall I go on? Indeed, once upon a time there were no words, every
word was coined at some point. The process of creating art is continually
creative and mechanical filters can’t be applied to things they have not
seen before.
Knowledge
141
You might argue we could devise a more sophisticated mechanical
filter, something that contains an algorithm with an understanding of the
rules of language. The problem is both the size of the task and the nature
of understanding. If I devised some really good appreciation algorithm
which did not delete all the creative words of the English language, it
would still have to read and appreciate the huge quantities of input until
it hit upon something good. There is no way for any machine to read
all this information in the age of our Universe; the numbers are just too
large. And there is no way for a machine to understand all the rules of
language, they are not written down and constantly evolve.
These descriptions should give you an intuitive feel for nature of the
creative problem. If you try to deconstruct it into mechanical steps you
end up with either a mechanism that needs to be infinitely specified or
one that lets through an infinite quantity of nonsense. A human could
never sift through all that garbage to find the occasional pearl of wisdom.
Until the beginning of the 20 th century, most people thought
knowledge and creativity must be just a matter of scale. A big enough,
fast enough machine should be able to solve any problem. But early in
the 1930s two mathematicians – Kurt Gödel and Alan Turing – showed
knowledge was not so simple. Let me give you a feel for why.
Knowing When You Know
The essence of creating knowledge, is to know when you have done so.
In a sense, counting from one to infinity means I know everything, and
merely counting to 50 million creates every piece of significant symbolic
knowledge that will ever be written – all the books, plays, mathematical
theorems you could possibly want. But, if I were to list all these numbers
in an enormous imaginary book it would hardly constitute knowing
everything: I would be awash with numbers but not with knowledge.
The essential feature of ‘knowing’ is to have a small number of steps
that will definitely answer a problem. For example, if I wish to phone
someone I can look up their details on my phone. The process will tell
me their number in two or three steps. If you tell me the number is
somewhere in the phone book this is not knowledge. It could mean I
need an infinite number of steps.
If I accidentally deleted all the names in my phone – a nightmare
scenario – and just had a print out of numbers would I still ‘know’ them?
Obviously I would recognize my mother’s number, but most of them
would be useless. To know something, I need link the information to
what it is for. A number with a name allows me to predict what will
142 Are the Androids Dreaming Yet?
happen if I make a call. I will have an interesting conversation or pay
my gas bill. It’s the same with most numbers. If I have a number that
represents the design for a building or a mathematical theorem, these
numbers have purpose. If I input these numbers to a computer along
with some building design software or a copy of Mathematica they
will do something interesting; allowing a construction firm to build a
innovative building or a mathematician to check a theorem is sound.
It’s a lot harder to prove numbers representing art are functionally
useful. A work of art is in some sense not complete – it still needs to go
through the process of being appreciated by someone. We could show it
to a friend or exhibit it in a gallery but this is un unpredictable process.
Van Gogh’s paintings were so criticized in his lifetime, many people
would have denied them the label art, and Edwin Morgan’s Loch Ness
Art or Information
Monster poem is almost pure gibberish, but it’s undoubtedly art. Art is
a tricky problem but, in practice, most of us agree on what constitutes
good and bad art. We will look again at art, in Chapter 10.
Classically we assume knowledge is discovered through random
chance and iteration. To understand how this might work let’s lay out
the world’s information in a way we can visualize. Imagine every piece
of discoverable knowledge could be found in an infinitely large library.
Knowledge
143
The infinity library would contain every possible symphony, theorem,
novel, poem, and play ever written, or to be written. Its sister library
next door, the continuum library, would contain all the analogue works
of art; painting, sculpture, architecture, physical artifacts and the like.
The curators of the two libraries would constantly argue over whose
collection was the better. We’ll leave them to differ for the moment. The
infinity library is interesting enough so let’s explore it first. After all, its
sister, the continuum library, takes an infinite amount of time just to look
at the first room, and we are in a hurry!
Although the infinity library is infinite, we are probably only
concerned with entries shorter than a million symbols. All the interesting
papers, proofs and symphonies I know of are shorter than this. If I wanted
to include all computer programs, I would still only need to increase it to
100 million symbols. Looking for knowledge is not itself an infinite task.
For the sake of clarity, I will ask the infinity librarian to organize
the collection. Any book or paper will be sorted according to its title and
the contents of its pages, and similar books should be grouped together.
I also only want to look at the English section of the library for the
moment. I will still have a huge section to look through but at least every
work is titled and readable by me. Much of the information will be junk
but amongst the sea of rubbish will be islands of useful knowledge. Now,
is there a way to find knowledge in this library in an automated fashion?
Battleship
144 Are the Androids Dreaming Yet?
The best analogy I can find to illustrate iterative knowledge discovery
is the 1970s family game ‘Battleship’. The game consists of two 10 by 10
grids that you plug your ships into. All the ships are linear shapes of a
few squares in length. The players cannot see each other’s ships and must
guess where they are. A very simple way to do this would be to ask your
opponent whether they have a ship on the top left square and continue
systematically across the board, square by square, until you reach the
bottom right hand corner. This would eventually find every ship. If every
ship were a piece of knowledge we could discover all the knowledge in
the world by simply stepping through the board one cell at a time, but it
would take a long time.
A better way to play Battleship is to pick a square at random. If you
get a hit, explore linearly around the hit. This will efficiently find the
rest of the ship. The same might be true for knowledge. We could take
random shots, get lucky and move linearly to flesh out our knowledge.
Once we had exhausted an area we could take a step away at random and
again hope for another hit. This process is exactly the way some people
imagine the frontier of knowledge expands.
But, it is wrong.
The monkey moon shot story explains…
“I believe that this nation should commit itself to achieving the goal,
before this decade is out, of landing a monkey on the moon and
returning him safely to Earth.”
President Monkey
The monkey nation is asked to mount a moon shot. After a little
time a monkey is asked to report on progress.
“I can report,” says the monkey, “I have climbed a particularly tall
tree on the tallest hill on my island and have made over seven hundred
meters progress towards the moon, although this is only 0.0001% of the
way there, this has been quick so I believe we are well on the way.”
You see of course the problem. Progress in many problems is
nonlinear. Moving a bit of the way towards the goal does not provide any
actual progress: That is the problem with knowledge. It is not linear in
structure. You need to take leaps to discover new knowledge. You can not
simply look around in the general area. Such leaps are mathematically
huge. The chance of making a successful one by pure chance is virtually
zero.
Knowledge
145
But Cats Can!
As chance would have it, as I was writing this book about the impossibility
of creating great literary works at random, our new kitten, Jessie, sat
on my keyboard – she likes the warmth. To my great embarrassment
I have been proven wrong. Here is her first literary work. I managed to
capture her on camera a little later that evening, editing a spreadsheet.
My brain interprets this string as the cat thanking me for good food. I
wonder if you see the same thing? This is just a demonstration of the
strength of human pattern detection algorithms and not, sadly, of feline
communication.
Cats Creation
…. Kkkklnk gfoooooooofd0------- iiiii;;;;;;;;;;;ii…..fffffffffffffffffffffffff……
=================================================
=================================================
=================================================
============================pppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppp
pppppppppppppppppppppppppppp..oppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppppppppppppppppppppppppppppppppppppppp
ppppppppppppppppppph
Jessie Cat
Jessie, Our Creative Kitten

Chapter 6
KITTENS &
GORILLAS
Orangutan and Kitten
“No kitten that loves fish is
unteachable; No kitten without
a tail can play with a gorilla;
Kittens with whiskers will always
love fish; No teachable kitten
has green eyes; No kittens have
tails unless they have whisters;
hence...”
Lewis Carroll
“Once you eliminate the
impossible, whatever remains,
however improbably, must be the
truth.”
Sherlock Holmes,
Arthur Conan Doyle
s well as giving us Alice, the Jabberwocky, and the Cheshire Cat,
Lewis Carroll lectured on mathematics at Oxford University.
He wrote several books on logic, illustrated with wonderful
problems involving fish, kittens, and gorillas – much less boring than the
brown, grass-eating cows of modern textbooks. Kittens and gorillas are
not usually in much contact, but I did find one hit on Google, pictured!
The words we organize into books, poems and plays are not just
a random jumble; they have structure and a logic to them. We group
verbs, subjects and objects together to form sentences and, at a larger
scale, characters have motivations and relationships: this character
loves that character, the valet had the candlestick in the ballroom and
could not have stabbed the butler in the kitchen, and so on. We have
dictionaries to define words, but to truly understand the information
they convey we need to understand the logical rules governing how they
can be combined.
Everyday conversation is fragmented and repetitive. Fortunately,
now and again, we say something definitive. For example, “This gorilla is
brown.” The statement links a property, ‘brownness’, to a thing, ‘a gorilla’.
Logical statements are precise but often need to be put in context. If I were
standing in a forest when I made my statement you must guess I mean the
nearest gorilla. The word ‘This’ implies nearness, but nearness is not well
defined. Better to be precise. ‘The gorilla I am closest to, measured by line
of sight distance is the Pantone shade dark brown.’ However, if I talked
like this all day I would not have many friends.
Logical Beginnings
The formal study of logic began in 384BC with the publication of a treatise
called the Organon by the Greek philosopher Aristotle. A student of
Plato, Aristotle taught many of the famous leaders of his time, including
Alexander The Great. Ancient Greece was not some idyllic think tank. If
you annoyed the political establishment you might find yourself having
to leave town in a hurry. This happened to Aristotle after Plato’s death,
and he spent nearly a decade touring Europe. Eventually, he returned to
Athens where he published his study on logic.
In the Organon, Aristotle examined groups of up to four statements,
each containing up to four relationships. For example: All kittens eat fish.
Some kittens eat fish. No kittens love gorillas. No gorillas eat kittens –
luckily. It is possible to put two statements back to back and infer things.
150 Are the Androids Dreaming Yet?
I could say, “All gorillas eat leaves.” “All leaves are green.” Therefore I can
infer all gorillas eat some green things. This is a valid inference. It is not
correct to say, gorillas eat only green things.
There are 256 ways you can arrange four Aristotle statements with
four relationships but only 19 valid deductive conclusions can be drawn.
The kitten puzzle at the start of the chapter is an example of such a logical
puzzle. Can you reach the right conclusion?
ddd
TRY SOLVING THE KITTEN PUZZLE WITHOUT READING ON
Aristotle’s syllogisms are only a start. There are many other types of
logic. In antiquity, the Stoics developed a different brand of logic based
on the idea of larger and smaller. Stoic logic allows us to answer questions
of relative size. If a Mini is smaller than an Audi, and an Audi is smaller
than a Rolls Royce, then a Mini is smaller than a Rolls Royce. The Stoics
pursued their branch of logic until around 180AD when study of this sort
died out. It’s not quite clear why. Perhaps the rise of religious power and
the onset of the Dark Ages curtailed intellectual inquiry. Even after the
Enlightenment began around 1650 it took some time for the discipline of
logic to re-emerge. If you want to learn more about syllogistic logic and
how to solve Lewis Carroll’s puzzle you should read his book The Game
of Logic. The definitive book on the logic of language, in my opinion, is
Logic by Wilfrid Hodges.
Logic for Computers
Western civilization mostly survived on syllogisms and stoic logic for
nearly two thousand years before George Boole devised his theory of
binary logic in 1847. Boole developed an elegant mathematical system
for representing logical statements that allowed simple arithmetical
operations to answer logical questions. We now call this system Boolean
logic and he gave us the modern convention of using one for true, and
zero for false. Computers use his principles all the time. For example, if
it is true my bank account shows less than zero, then make it true that
someone will send me a letter warning me I am overdrawn. The best way
to get your head around Boolean logic is to solve the ancient puzzle of
the Two Guards. The puzzle featured in the 1986 movie, The Labyrinth,
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151
starring David Bowie and Jennifer Connelly. If you want to cheat watch
the film to see the answer. Here is the puzzle. I’ll put the answer on my
website.
Two guards stand barring your way and behind them are two
doors. One guard always speaks the truth, while the other always
lies. You are only allowed to ask one question of one of the guards.
Your life depends on picking the right question to ask as, based on
the answer, you must pick a door. One leads to life, the other to
certain death. Is there a question you can ask to ensure you pick
the door leading to life?
TRY SOLVING THE GUARD PUZZLE
ddd
Twin Guards - Left door or Right
If you are reading this, you picked the correct door and lived.
152 Are the Androids Dreaming Yet?
Logic for Humans
Syllogisms can be used for practical purposes. Take, for example, the
following set of statements, “I want a hot drink.” “Coffee and tea are hot
drinks.” “I always drink milk with tea,” “We have no milk.” What drink
should I choose? I’m sure you can work it out. This logical problem
follows a simple chain and results in me getting the hot drink I like.
We use Boolean logic on a day-to-day basis. The simplest form
is a checklist. Pilots use checklists all the time; do I have wings, fuel
and a copilot? If they are all there, go ahead and fly. Otherwise do not.
Mathematically speaking, a checklist is simply the product of the options.
If they are all one, then the product is one – in this case we can fly. If any is
false – represented by a zero – the product will be zero and we cannot fly.
Life is often more complicated and we have many logical tools at our
disposal. Let’s take a look at a few, starting with a famous historical one.
Benjamin Franklin invented the lightning rod and bifocal glasses,
as well as charting the Gulf Stream and all manner of other scientific
discoveries. He described his process for decision-making when there
are many pros and cons to consider.
“... my Way is, to divide half a Sheet of Paper by a Line into two
Columns, writing over the one Pro, and over the other Con. Then
during three or four Days Consideration I put down under the
different Heads short Hints of the different Motives that at different
Times occur to me for or against the Measure. When I have thus
got them all together in one View, I endeavor to estimate their
respective Weights; and where I find two, one on each side, that
seem equal, I strike them both out: If I find a Reason pro equal to
some two Reasons con, I strike out the three. If I judge some two
Reasons con equal to some three Reasons pro, I strike out the five;
and thus proceeding I find at length where the Balance lies; and
if after a Day or two of farther Consideration nothing new that is
of Importance occurs on either side, I come to a Determination
accordingly.”
Another important piece of logic is reductio ad absurdum. Reduction
to the absurd allows us to disprove something because, if it were true, it
would lead to an absurd conclusion. An alibi is a familiar form. If I was
seen in the pub when the murder occurred in the ballroom of the manor
house and you claim I committed the murder, I must have been in two
places at once. People can’t be in two places at once – that would be
absurd. Conclusion: I am innocent!
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153
Notice I not only prove I am not guilty I also prove the opposite:
I am innocent. When a mathematician uses this trick, it is called an
indirect proof and works the same way as the alibi. Assume the opposite
is true of some theory you want to prove (I am guilty). If it generates a
contradiction or paradox (can’t be in two places at once) you can deduce
the opposite must be true (innocence). Mathematicians use this all the
time. It assumes, of course, mathematics is consistent and that true and
false are opposites.
Some mathematicians argue this is too strong an assumption. Why
should we assume consistency and recognize only two logical states, true
and false? These mathematicians believe the only way to prove a theorem
is with positive argument rather than using the opposite of a negative
argument. They don’t allow indirect proofs in their mathematical
models. This type of mathematics is unsurprisingly called positivism. It’s
a pure theory but, unfortunately, if you try to follow it you lose much of
our current mathematical knowledge and understanding. Most modern
mathematicians think it a historical curiosity, but it does pop up from
time to time. Modern mathematics is founded on the axioms that
true and false are the opposite of each other and that inconsistency is
forbidden within the system. Mathematical proofs submitted to journals
are not permitted to contain inconsistencies or result in paradoxes.
Paradoxes – When Logic Fails
“I would not be a member of
any club that would admit
me.”
Groucho Marx
Paradoxes occur when a statement
makes no sense, or results
in an internal contradiction as
with Groucho Marx’s famous
quote. They are widely used in
mathematics to implement indirect
proofs. To do this, we suppose
something is true, and if
it results in a paradox then the
Groucho Marx
154 Are the Androids Dreaming Yet?
thing we thought true must be false and the opposite is true. This is a
somewhat circuitous route to prove things, but it is often the only practical
way.
Two paradoxes we are taught as children are the liar’s paradox and
Zeno’s paradox – also known as the story of the tortoise and hare. The
first is a real paradox but the second is a false paradox. The liar’s paradox
is just the simple statement:
“This sentence is false.”
It is a paradox because of the internal inconsistency: We cannot
determine if it is a true or false.
First assume it is true, but it says it is false, so it is not true. Then
try it the other way around. Assume it is false but the sentence states
it is false, so it must be true. If that were so it must be false by the first
argument and so on ad infinitum.
Either way around, the sentence contradicts itself. A paradox.
Zeno’s Paradox, on the other hand, is a false paradox. Here is the
story.
Once upon a time there was a hare. He was a very arrogant hare
and believed he could outrun any animal. A tortoise was walking along
the way and the hare jumped out in front of him. “You are so slow,”
said the hare. The tortoise replied, “You may be the fastest hare in the
kingdom but I am the most persuasive tortoise. I bet I can persuade you
of anything, including that I am faster than you.”
“I don’t believe you,” said the hare.
“OK,” said the tortoise, “let me show you. Give me 100 meters head
start since you are so fast. Then, we’ll both start to run. After 10 seconds
you will have run 100 meters and arrived where I used to be, but I will
now be ten meters ahead. After another second you will be where I am
now, but I will be 1 meter ahead again. So you can never catch me.”
The hare pondered for a while but, being a hare of little brain, could
not make out the true answer.
It is a false paradox. The time intervals are getting shorter. The
question for a mathematician is, does the problem converge to a solution.
The answer is yes, and I can reframe the problem to see how it is solved.
Let’s simply look at who would be ahead after 20 seconds: the hare!
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155
The mathematical reason for it being a false paradox is that some
series converge and some do not. If I move progressively closer and
closer to something in smaller and smaller time intervals then I may
indeed reach it. On the other hand, some series never converge. I will
never reach infinity how ever many steps I take.
The Barber Paradox
Now, for a slightly harder paradox, let’s suppose there is a town with just
one barber.
In this town, every man keeps himself clean-shaven by either
shaving himself or going to the barber; the barber shaves all the men in
town who do not shave themselves. All this seems perfectly logical, until
we pose the question: who shaves the barber?
This question results in a paradox because, according to the
statement above, he can either be shaven by himself or the barber, which
is he. However, neither of these possibilities is valid! This is because if
the barber shaves himself, then the barber must not shave himself and if
the barber does not shave himself, then the barber must shave himself.
You might think this paradox an oddity but, using this simple idea,
Bertrand Russell changed the course of mathematical history and it is
the fundamental paradox used to show computers are Turing limited.
The Russell Paradox
In the late 19 th century, mathematicians began to think about the nature
of numbers.
What is a number?
It is certainly not an object we can hold.
I can’t hold a two, unless it’s the brass number plate, for my front
door. And, in that case I am holding one number plate, so I am not
holding the idea of two, but rather the idea of one: one brass plate in the
shape of a two.
The ‘idea’ of a number is to say something about the things I have in
my hand: two apples, two oranges and two brass number plates. These
are all sets of two things and ‘two’ is the collection of all these sets.
In 1890, Gottlob Frege completed his theory of sets. The project had
taken him five years. Unfortunately, just before sending the book to the
publisher, Bertrand Russell wrote to him and pointed out the following
paradox. What about the set of sets that does not contain itself? Think
about it...
156 Are the Androids Dreaming Yet?
It is the barber paradox with the word ‘set’ substituted for ‘barber’
and ‘contains’ rather than ‘shave’. But it’s essentially the same logical
problem. You might find this rather contrived but mathematicians must
have a system totally free from paradox, otherwise there is no certainty.
Frege’s system was holed below the water line.
Eventually, after much further work, a theory of sets was worked out
that does not contain the Russell Paradox. It’s called Zermelo-Fraenkel
set theory, or ZF for short. It solves the Frege problem by forbidding sets
to refer to themselves. It’s a bit like Microsoft Excel’s solution to dividing
something by zero. It is simply forbidden and generates an error message.
Set theory was fixed and is now the basis of most mathematical thinking.
What is Logic for?
Logic is the foundation of mathematics. Applying it enables us to make
irrefutable statements about things: numbers, lines, planes, equations
and the so on – the things you learned at school – and to prove statements
about these things beyond any doubt. This is not the ‘reasonable doubt’
hurdle of our law courts, but an absolute measure: No possible doubt
whatever.
Let’s look at one of the earliest mathematical proofs: Euclid’s proof
there are an infinite number of prime numbers. Euclid created this proof
in ancient Greece around 300BC – so far back that logic was in its infancy
Euclid’s Elements 100AD
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and numbers had not yet been properly invented. Euclid used distances
rather than numbers for all his proofs but I will use the word ‘number’ in
this explanation.
First a little revision. A prime number is a number that can only
be divided by itself and one, for example three, five, seven, and eleven.
All numbers can be split into primes using a couple of tricks. First, all
numbers are divisible by a set of primes. Ten is five times two – two
primes. We are also fairly sure we can form any number by adding two
primes together. This is Goldbach’s Conjecture, set as a question in a
letter written to Euler in 1742. It is still unproven!
Euclid proved there are an infinite number of primes by using
reductio ad absurdum. Imagine we have a complete list of prime numbers
– James’ list of primes. It contains every prime number. (This is the setup.
We are proposing something we suspect is incorrect and will lead to a
paradox or contradiction. When it does, we will have proven the opposite
fact. The proof relies on the fact that a number can either be prime or not
prime. There is no middle ground.)
Let’s make a new number by multiplying all the numbers on my list
together and adding one. There are two possibilities: this new number is
either prime or not prime.
If the number is prime, it is a new prime number that was not on
my list and I have disproved the theory.
If it is not prime then it must be divisible by two prime numbers
already on my list. However, neither of these numbers could have been
on my list, because dividing by one of them would give me a remainder
of one. Remember I multiplied all prime numbers together and added
one. It must, therefore, be a new prime number, which had previously
not been on my list. Once again, I disprove the theory.
Since both routes fail, James’ list of prime numbers is not complete
and, therefore, prime numbers are infinite.
Feynman’s Proof
My favorite piece of logic is Richard Feynman’s disproof of the existence
of polywater. It’s a strange logical proof bordering on philosophy, but it
shows just how far you can take logic.
In 1969, an urban legend spread around the world that there was a
substance called polywater. It even made it into an episode of Star Trek.
Polywater was believed to be a lower energy state of water, more viscous
than ordinary water. If this substance did exist, it would be possible
to mine the oceans of the world converting water to polywater and
158 Are the Androids Dreaming Yet?
therefore generate energy. There was a concern that if the right catalyst
was accidentally introduced into the oceans they would solidify into
polywater thus dooming the human race, or at the very least making
water sports impossible!
Feynman was consulted and stated, “If there were such a substance
as polywater then there would have evolved an animal that eats water
and excretes polywater, using the liberated energy as its power source.
Since there is no such animal, polywater does not exist.”
Feynman’s proof is an elegant indirect proof coupled with a
syllogism. Polywater exists. Polywater is a lower energy form of the highenergy
substance called water. Food is a high-energy substance that can
be converted to a low energy substance by a process we shall call ‘being
good to eat.’ All things on earth that are good to eat have something that
eats them. Polywater is a food and therefore good to eat. Therefore an
animal must exist that eats polywater. No such animal exists, so either
something in our chain of logic is wrong, or the premise is unsound.
Since the chain is sound, the original premise must be wrong: Polywater
cannot exist.
In short, Feynman’s proof says: if a thing is so, then the inevitable
consequence is the evolution of something else, and since that something
else does not exist, the original thing cannot be so. QED: disproof by
nonexistent consequence.
The polywater disproof neatly demonstrates the important elements
of Feynman’s Evolutionary proof. First, life must be continuously exposed
to the thing in question, in this case water. This is clearly so as most life
on planet Earth lives in the oceans or is intimately entwined with water.
Evolution takes time, so enough time must be allowed for life to evolve.
It must be a nearly linear problem so that a solution proceeds in steps
where each step is an improvement and no step requires too high a level
of mutation or adaption. We can illustrate the boundary between a linear
problem and one requiring a step change by describing how triple drug
therapy works in the treatment of AIDS.
Until triple drug therapy entered the picture progress against AIDS
had been a depressing story of drug discovery followed by the almost
immediate evolution of the virus to evade the drug. The AIDS virus
is a retrovirus with a shell composed of sugar molecules. It is almost
trivial for an AIDS virus to mutate these outer markings to look different,
even from one day to the next. This is the way the virus continually
and nimbly evades our immune system. However, the AIDS virus does
have some components that it can’t easily mutate because they are not
merely aesthetic, they have a functional purpose. Why not target them?
Kittens & Gorillas
159
Unfortunately, it turns out the AIDS virus can even mutate its functional
parts, but this is harder. The probability of a successful functional
mutation is 1000 times less likely than a simple aesthetic mutation to the
sugar coat.
Triple drug therapy works by attacking three different functional
elements of the virus simultaneously. It is possible for the virus to modify
all these functional elements but the likelihood of it doing so is tiny.
One mutation alone does not help because the drug cocktail will still
target the other two elements and kill the virus. The AIDS virus does not
understand that it is facing a triple drug cocktail. It cannot reason like a
sentient being and random chance is not sufficient to make the big leap
necessary to overcome the cocktail of drugs. Unless you can mutate all
three elements at once your time as a virus particle on this planet is over.
Most problems we have to solve in this world require more than one
simultaneous logical step and these don’t happen by chance.

Chapter 7
COMPLEXITY &
CHAOS
Mandelbrot Set
“Life is really simple, but we insist
on making it complicated.”
Confucius
“Any darn fool can make
something complex; it takes
a genius to make something
simple.”
Pete Seeger
There was once a great King who lived in a marvelous palace. To
fend off boredom he collected all manner of interesting games
and puzzles. One day an inventor came to his palace and told the
King he had a game of such subtle complexity, yet apparent simplicity, the
King would play no other. The King learned the game and soon agreed it
was, without doubt, the best of all games. The game was, of course, ‘chess’.
The King asked the price of this game and the inventor told him it was a
mere trifle. The King should give him one grain of rice on the first square
of the board, two on the second, four on the third, and so on, doubling
each square until he filled the board.
The King called his treasurer to honor the bargain and the first bags
were brought from the storehouse. The grains were placed on the board
in each square but soon there was not enough space and the grains had
to be piled on the table next to the board. Soon this, too, was not enough
and every table and chair in the hall had to be covered. Even this was not
enough and they began to stack whole bags up in the courtyard.
When they reached the thirtieth square, the treasurer turned white.
He sat and calculated for a while before saying with a trembling voice,
“My great ruler, there is not enough rice in all the world to cover this
board.” The ruler called the inventor and told him he could not honor
the debt and the inventor should name another price. The King had
two beautiful daughters, the first knew she was beautiful and deported
herself accordingly, and the second, was bookish and shy, but perhaps
more beautiful for this. The inventor asked for the hand of the second
daughter and lived happily ever after. In the less favorable version of this
story, the King becomes very angry and has the inventor beheaded. I
prefer the romantic version.
Placing rice on a chessboard and doubling it successively
leads to wildly large numbers. Covering it completely requires
18,446,744,073,709,551,615 grains, about four hundred trillion tons and
equivalent to one thousand years of worldwide rice production. Like
the king, humans do not intuitively grasp the enormity of this problem
because we’re not good with large numbers.
Although the number of grains needed to cover a chess board
is very large, it is not hard to calculate. The treasurer is the one who
should have lost his head for not being able to do the calculation. The
equation is simply two, doubled sixty-four times, less one, 2 64 -1. A pocket
calculator can produce this number in a thousandth of a second: it’s just
long multiplication. Although calculating this number is quick, it is not
always the case. Answers to some problems have short cuts, while others
do not.
164 Are the Androids Dreaming Yet?
Mathematicians have catalogued the universe of problems into
classes rather as biologists have catalogued animals into species. Each
problem is examined and put into a genus with a name. Sadly the names
are not as readable as the Latin names for animals. For example, ‘nlogn’
is the complexity class of most sorting programs, while traversing a maze
typically sits in the class NP or P/POLY. Although the classifications look
complex the basis of cataloging is simple, a class name signifies the time
needed to solve a problem using the best possible algorithm, and the
scale this is measured in is ‘Big O’.
Big O
Every problem has a complexity. In mathematics this is expressed using
‘Big O’ notation, where ‘O’ stands for order-of-magnitude. The simplest
problems have order 1.
If I am working at my computer on a Word document and I press
print, the printer will spring to action and print the document. This
problem is of flat complexity, notated O(1). It does not matter how large
the file is; one click is all I need. I am, of course, assuming sufficient
paper in the printer and ink in the cartridges.
The next complexity class is a linear problem, O(n). For example,
walking to the store to buy a pint of milk. The farther the store, the longer
the walk. The time needed to get to the store is directly proportional to
the distance: if I am walking, a single step multiplied by the number of
steps required to cover the distance.
You might think adding two numbers together is a linear problem
– the bigger the number, the harder the problem – but there’s a clever
trick to speed it up. You can get 10 people to add each column in parallel.
They’ll need to coordinate when someone ends up with a number larger
than ten and has to carry the extra digit but this can be easily solved. A
problem gets its classification only once we’ve used the cleverest possible
trick to solve it.
Most problems we meet in mathematics are somewhere in between
flat and linear but there are some that are much harder. The most common
hard problem we come across in our daily lives is sorting. Rather than
go through a tedious written description, check out the video link on my
website. Sorting without using any spare space requires a bubble sort.
This is an example of something that needs n squared operations and,
since n squared is the simplest example of a polynomial, it is said to be in
the polynomial time, or ‘P’ time classification.
Complexity & Chaos
165
Bubble Sort Ballet
The Hardest Problems
You probably hope cracking the encryption used to secure the Internet
is one of the hardest problems known to man but I’m sorry to tell you
it is not. When you use your credit card to buy something from an
online shop, your web browser changes from http to https, the ‘s’ stands
for secure. The data you send to the Internet is coded using a system
developed in 1977 by Ron Rivest, Adi Shamir and Leonard Adleman
of MIT, which is why it is called RSA encryption. Any information you
send is raised to the power of a very large number – usually around
one hundred digits long. Raising something to the power simply means
multiplying it by itself that many times.
What makes decrypting a message hard is that division is a slow
process; it is called ‘long division’ for a reason. It turns out there is no
way to speed it up on a conventional computer so, unless you know the
right number to divide by you will have to try every number. It is this that
makes decrypting RSA messages hard.
Although RSA messages are difficult to decipher, they are nowhere
near the hardest problems. That accolade is commonly believed to go
to non-deterministic polynomial problems known as ‘NP’ problems.
NP problems are easy to describe but fiendishly difficult to solve.
Nondeterministic means each time you come to a branch in the problem
there is no way to tell which branch is the best to pursue without exploring
it all the way to the end. It’s the same as a maze; at each junction in the
maze you can decide which path to take, but the junction gives you no
166 Are the Androids Dreaming Yet?
Maze
clue which one will be better. Beware the confusing naming system, ‘N’
stands for nondeterministic in this case, whereas in normal complexity
classes ‘n’ stands for number. Sorry. That’s just the way it is. Let me give
you an example of one of these NP problems.
Let us assume we have one of those complicated recipes from the
latest celebrity chef cookbook. If all the ingredients can be bought from
one store, making the dish is straightforward, but if they come from
different stores, you will have your work cut out. What is the best order
to visit them? With 2 shops, it’s trivial. Either order will do. With 3 it is a
little harder and with 4 there is quite a bit of choice. This is known as the
‘traveling salesman’ problem because the original formulation described
a salesman wishing to find the shortest route between all the cities in
which he had customers. The complexity of this problem rises much
faster than the Rice and Chess Board problem. Try it for yourself. It
doesn’t matter if you imagine you are visiting customers or shops. I have
given you a grid to count off distances. Try to solve a problem for 3 cities,
5 and 10. What is the shortest path allowing you to visit each place?
Complexity & Chaos
167
TRY THE PUZZLE ON THE WEB
Warning: Don’t spend too long on these problems.
ddd
The reason I warned you not to spend too long is that solving the
50-city problem would take longer than the age of the known universe.
NP problems get harder very fast as the number of elements goes up. A
50-city problem is hugely larger than a five-city problem, not just ten
times harder.
The Clay Mathematics Institute has offered a $1 million prize for
anyone who can say whether NP problems are really as hard as they
appear. It may be there is a general trick or a series of tricks that allow
you to solve any NP problem in a shorter time. If you could do this, the
problem would be demoted to P, allowing fast computers to tackle it. No
one has yet found a proof of the P=NP problem. At the time of writing
several proofs are sitting with the Clay Prize judges but don’t hold your
breath. Most people assume there is no solution. If you want to have a
crack at the problem let me state it in simple terms.
Traveling Salesman
168 Are the Androids Dreaming Yet?
Imagine you wanted to find the center of a maze. Is there a way to
speed searching the maze, so you do not have to test every branch? If
you can provide a mathematical proof that there is or is not, you win the
prize.
Places Game
While it is commonly assumed NP problems are the hardest, this is not
the case. There are quite a few that are harder still. One such is called a
PSPACE problem. It’s quite difficult to explain but luckily many of you
will have played a form of it on long car trips when you were a child: My
family calls it The Places Game.
I will pick a place – ‘London,’ and you must then pick another place,
say, ‘New York’, that starts with the letter my place ends with. I’ll then
pick ‘Canterbury’ and my kids will laugh at my dyslexia and I’ll have to
switch to ‘Kansas’ and so on. Once you use a place you can’t use it again.
The mathematical question is to predict who will win given each
player has a finite list of places they know? It turns out this type of
problem is even harder to solve than an NP problem. This is because
on each turn a player gets to pick any name from their list. With the
traveling salesman problem, there is only one ‘player’ – the salesman –
so we can write out a route and check it. In the Places Game there is
no single route through the game because, after I pick my favorite town
‘London,’ you can pick any place beginning with ‘N’. I have to anticipate
an enormous table of possible paths through the game. The table takes
huge physical space – which is where PSPACE gets its name.
Remember I’m just playing the simplest mathematical games with
bits of paper and discrete ideas. I haven’t strayed into the quantum
world yet. That brings with it a whole new level of complexity to explore.
Complexity is such a diverse subject that Scott Aaronson of MIT has
created a web site called the complexity zoo to catalogue all the different
‘species’. It is much to complex to reproduce here but let me provide a
sketch.
The Complexity Hierarchy
My table below represents the hierarchical complexity of knowledge.
We start off with the problems both humans and computers find easy,
then rapidly move onto problems that even the fastest machines find
difficult: a perfect game of chess or predicting the weather. Above these
computable problems are the non-computable ones which no computer
Complexity & Chaos
169
running any algorithm can solve, and then
there are the free will problems: how do
we pick a problem in the first place? How
do inventors come up with problems no
one had ever thought to solve in the first
place, such as the invention of the Rubik’s
Cube?
Ernő Rubik’s Cube
Problem
Example
Flat
Print File (for Human)
nlogn
Searching a list
Linear
Finding the lowest number in a list
Logarithmic
Long Multiplication
Exponential
Long Division
P
Most Algorithms
Near NP
Factor Prime Number
NP-non-complete Perfect Game of Chess
NP-Complete, tractable Travelling Salesman, SAT
Chaotic
Weather
NP-Complete, Quantum Modeling a Quantum Process
NP-Complete, intractable Busy Beaver, Towers of Hanoi
PSPACE
Graph Problems, Places Game
Creativity, Finding Fermat Theorem for a
Non-computable
Turing machine, Tiling the plane with Penrose
Triangles
Non-deterministic,
Non- time divisible, Non- Free will
computable
Halting problem for a Turing Machine, some
mathematical theorems such as the Continuum
Impossible
Hypothesis in ZF+AC (Hilberts 1st). Travelling
faster than the speed of light. Understanding
the American tax code.
Known Unknowns I know that I don’t know either way.
Unknown Unknowns
I have not thought to ask that question yet.
Inventing the Rubik’s Cube
Butterfly
“Does the flap of a butterfly’s
wings in Brazil set off a tornado
in Texas?”
Philip Merilees, improving on
Edward Lorenz
Chaos
Chaos is the twin of complexity. It burst into the public psyche
in 1987 with the publication of James Gleick’s book Chaos. It’s
not a difficult concept to grasp. Complex systems can be formed
using simple rules, and very small changes in starting conditions can
profoundly affect future events. I experience this if I miss my train to
work in the morning: 30 seconds either way will change the whole
pattern of my day, the people I meet and the level of stress I experience.
I’m sure you can think of similar experiences.
Henri Poincaré, a French
mathematician, first studied the
effect back in 1880. Poincaré was
trying to solve an old mathematical
problem called the Three Body
Problem originally set by Isaac
Newton. Take the Earth, Mars and
the Sun. These three bodies orbit
each other, or strictly speaking a
point in space somewhere between
them. Is there an equation that
will tell you where the bodies will
be in, say, 100 years’ time?
The answer is surprising,
no. The three bodies will orbit in
a non-repeating way. There is no
analytical short cut, no equation
that will predict where they will be
Poincaré

172 Are the Androids Dreaming Yet?
at some point in the future. The only way to know is to build a perfect
model of the system and see what happens. Poincaré won a valuable
prize for his proof from the King of Bavaria. You can see some amazingly
complex orbits plotted below. Remember these are still deterministic and
predictable – after all, they were calculated with a computer – they are
just chaotic.
Four Body Problem
Butterflies and Sliding Doors
After Poincaré, the field of chaos remained fairly quiet until Edward
Lorenz began studying weather patterns using computers in the 1960s.
The story goes, one day his computer was misbehaving and he had to rekey
some data into the machine. Rather than using eight decimal places
he used only six to save time, and was amazed when the results of his
program came out completely different. Dropping the seventh and eighth
decimal place represents a change of only one part in a million, yet the
patterns of weather predicted by the computer were completely altered.
Complexity & Chaos
173
Lorenz went on to study the effect and created a new branch of
mathematics. His quote about the beat of a butterfly wing creating
tornados has entered the public psyche and is central to the plot of
numerous Hollywood movies. One of his functions – known as the Lorenz
Attractor – nicely illustrates the nature of chaos. A very simple equation
plots the beautiful, apparently three-dimensional, non-repeating shape.
Chaosville
Chaos, taken to its logical conclusion could explain our Universe. Stephen
Wolfram in A New Kind of Science, makes the argument that simple rules
could explain the extraordinary complexity we see in our Universe. He
applies rules to elements in a two-dimensional grid programmed on
the computer which form ‘cellular automaton’ that function a little like
simple animals, generating all manner of complex shapes and behaviors.
The inspiration for this approach is almost certainly Conway’s Game of
Life developed by John Conway in the 1960’s. In his computer game,
animals and machines seem to appear on the screen but in truth they
derive from the most simple set of rules. You can check out the website
to see a live version of Conway’s Game of Life. It’s a lot of fun. Wolfram’s
Strange Attractor
174 Are the Androids Dreaming Yet?
thesis is that we could all be living in one of these games. Perhaps our
Universe is a form of Mandelbrot diagram – albeit a 3D version with
stars and planets. If you look at the picture of a nebula and compare it to
the Mandelbrot set, you can see how this is a tempting conclusion.
In the Game of Life the rules are simple yet the behavior simulates
little animals being created and destroyed. Of course, there are no
actual animals. The things you see on the screen, ‘gliders’, ‘walkers’, and
‘cannons’, just hang together accidentally. But, Wolfram considers these
little digital creatures are animals. He argues our Universe is just like the
Game of Life: A set of simple rules leading to complex behavior. If we are
Nebula
Complexity & Chaos
175
Cellular Automaton
prepared to call ourselves animals, so should the little creatures which
emerge within the game. We simply emerged in a similar but slightly
more complex game.
This proposal would mean our Universe is entirely deterministic,
our lives the result of a gigantic computer program that we live within
and form part of. Chaos might make it impossible to predict the future
without running the program and watching what happened, but the
results would be inevitable, set in motion at the dawn of time. There is
no place for free will in such a Universe, no place for reason. The world
would simply be.
But a strange idea will come to our aid to show us the limits of
computation and allow us to question whether we live in a predetermined
world. This idea is Aleph 1 – something larger than infinity. And it is
infinity we will explore next.
Conway’s Game of Life

Chapter 8
∞
Hilbert’s Hotel
“All infinities are equal, but some
are more equal than others.”
George Orwell, paraphrase
“Only two things are infinite, the
universe and human stupidity,
and I’m not sure about the
former.”
Albert Einstein
“God gave us the integers, all else
is the work of man.”
Kronecker
Health warning! The man who discovered infinity had a mental
breakdown. This subject may tax your brain.
Georg Cantor didn’t really ‘discover’ infinity but he was the first
mathematician to put it on a firm theoretical footing. In the late 19 th
century, most mathematicians thought infinity was a curious idea with
no proper place in mathematics. They treated Cantor’s attempts to make
it into a real mathematical object with contempt. This affected Cantor’s
morale and caused him to suffer several bouts of deep depression,
retreating to a sanatorium from time to time.
Infinity is a difficult idea to grasp but it is vital to our study of
information. It behaves counter-intuitively but is not impossible to grasp.
The reason it is important is that information can always be translated
into numbers and numbers go on to infinity. If you want to know all
about information, you must understand infinite numbers.
History
Indian scholars began studying infinity in the 4 th Century BC. It turns
up naturally in all manner of places. In geometry, parallel lines extend
forever in either direction without ever meeting. To define a parallel line
you must contemplate infinity. In arithmetic, even if you pick the largest
number you can imagine, there is always a larger one; just add one. In the
physical world if you look up at the night sky it appears to go on forever.
Again you have infinity.
Historically there were two interpretations of infinity. The first,
favored by Plato, was a journey. When you embark upon a journey, you
can always take another step. Infinity is the idea of ‘one more’ or neverending.
It can never be reached. The second definition is more radical.
Infinity is a thing, a number so big you could not imagine anything bigger,
but it is one number. Plato thought this second definition tantamount to
madness. Today we embrace this madness and go a whole lot further. Let
me show you how.
If infinity were a number, you should be able to perform mathematics
with it; add it, multiply it, and even raise it to a power. This is not as
radical as it might first seem. Until comparatively recently, zero was not
accepted as a number – if you consider recent to be one thousand years!
Nowadays it is.
At the end of the first millennium Indian scholars found, against
their intuition, that you can use zero as a number without generating
contradictions. Take addition. I can have zero cakes, add one, and I
have one cake, add another, and have two cakes and so on. In this way,
180 Are the Androids Dreaming Yet?
the number zero behaves just like any other counting number. It also
works with multiplication. If I have zero lots of 4 cakes, I have no cakes.
Zero times four is zero, so multiplication with zero works. There is one
embarrassing exception, if I divide by zero I seem to get infinity. When I
was a child this was a definition for infinity, but nowadays mathematicians
simply forbid the operation. Division by zero is not allowed and if you
try it on your computer, you will get the not terribly useful, #DIV/0!
Error. That’s progress I guess!
Zero had been tamed. What about infinity?
Cantor showed that while you could think of infinity as a number,
it might not be just one number. He proposed there are many infinities.
In fact, there are a greater than infinite number of them! He did this
through a rigorous analysis of a new branch of mathematics called set
theory.
Set theory is now the cornerstone of modern mathematics, but it
was treated with suspicion in Cantor’s time. Rather than embrace the new
thinking, many mathematicians ridiculed it; Poincaré wrote that Cantor’s
ideas were a grave disease infecting the discipline of mathematics! This
seems odd given our modern propensity to embrace innovation, but the
tone of science back then was different: innovation was not necessarily
considered a good thing.
At the turn of the 20 th century, scientists were on a mission to tidy
things up. Lord Kelvin announced in 1890 that mankind had discovered
everything there was to know and the role of future scientists was simply
to catalogue and observe the consequences of these laws, and to improve
the accuracy of measurement. The last thing scientists wanted was a
completely new set of numbers that behaved in strange ways. Cantor was
upsetting the apple cart, but he was in good company. Just a few miles
away in Berlin, a young Albert Einstein was beginning to study physics
in his spare time. Those studies would culminate in his four papers of
1905, two on Quantum Mechanics and two on Relativity, ushering in the
modern age of physics.
How to Count
To understand infinity you need to count in a particular way. You’re
probably used to counting with numbers. You count apples: one, two,
three, and say, “I have three apples.” You can do the same with oranges.
If you have three apples and three oranges, the totals are the same and
you can declare you have the same number of fruits. This is the first way
to count.
∞
181
But there is a second way of counting. Take your apples and put each
next to an orange. If they match up, you can easily see they are equal in
number. “Look,” I say, “I have the same number of apples as oranges.” This
method is more primitive and does not require the concept of numbers,
but it is very useful. If I’m a shepherd I can hold a set of counters in a bag,
one for each sheep. To ensure all my flock are gathered in for the night I
drop one counter into the bag as each sheep enters the enclosure. I don’t
need to give the counters number names.
The Munduruku tribe, from the Amazon rainforest, have no concept
of number names beyond five. Their counting system simply goes one,
two, three, four, five, many. Yet this second way of counting allows them
to function successfully, deciding whether two groups of things have
the same number of elements, even if there are more than five of them.
For example, if they need to determine if they have enough spears for a
hunt, each person simply stands next to their spear. If everyone has one,
they’re ready. If not, then the empty handed Munduruku simply make
one. No need for pesky numbers or mathematics lessons.
This second way of counting is particularly useful when tackling
infinity because we are not sure what infinity is. Treating it the same way
the Munduruku treat the number ‘many’ is the safest thing to do. The
first question we would like to answer is whether all infinite things are
the same.
Spears and Hunters
182 Are the Androids Dreaming Yet?
We know from our childhood that infinity plus one is still infinity.
Is there anything we can do to make infinity bigger? Perhaps multiplying
infinity by infinity will do the trick.
Infinity times infinity can be visualized as a square with edges of
infinite length. We can show that this square is the same size as a onedimensional
infinity through a clever trick – the zigzag method. Mark
the infinity square into a grid. Start in the corner square, go across,
diagonally down, then across, diagonally up, and so on. I’ll draw you
a picture. We visit every square in our grid using a single line. We can
then lay down our infinite zigzag line next to the infinite line of one
of the edges. The lines are the same length as they are both infinitely
long! So infinity, times infinity, can be matched to infinity, they are the
same. Cantor thought this a very strange result and wrote to a fellow
mathematician, Dedekind, “Je le vois, mais je ne le crois pas!”, “I see it,
but I don’t believe it!”
If you are struggling with this, don’t worry. We just jumped forward
to quite a complex concept. Let’s take it more slowly. One way to get a
better grip on infinity is through the stories of David Hilbert and the
Infinity Hotel.
Infinity for Dummies
Hilbert’s Hotel is a mythical building with an infinite number of rooms.
Other than this strange feature it is a regular hotel complete with
minibar, dodgy TV, and slightly mad manager. The rooms are numbered
sequential starting at one, then two, three, four, and so on. The hotel
allows you to play a series of mathematical games to see how infinity
behaves.
Are there the same number of minibars as there are rooms? That’s
easy. I said every room has a minibar. We can use the matching technique
to match minibars with rooms. Go to the first room. There is a number
on the door and a minibar inside. The same goes for room 2 and 3 and
this goes on forever. I’ve just proven two infinite things are the same –
rooms and bars, but I still have not shown you why the zigzag line is the
same length as the edge line.
When you first explain infinity to a child they immediately ask
“What’s infinity plus one.” A particularly smart kid I met, Dermot, asked,
“What’s infinity plus three?” Hilbert’s Hotel allows us to answer this
problem in a way we can visualize.
∞
183
Traversing an Infinite Plane with a Line
The infinite hotel is full. A man comes to the front desk and asks for
a room. The hotel manager says, “I’m terribly sorry, but we are full… But
I may be able to help you. Let me think.” He ponders for a moment and
then says, “OK – I’ve found you a room.” He calls the people in room 1
and asks them to move into room 2. He calls the people in room 2 and
explains that due to a double booking they must move out of their room
to let the people from room 1 in. But it’s OK; they can move into room
3. Everyone moves up a room and the new guest gets the checks into the
now vacant room 1.
This is a little harder to understand. We did not have a perfect oneto-one
match as with the rooms and mini-bars. We had a mismatch of
guests to rooms. But, we were able to show it is possible to re-establish
a one-to-one match by doing something to every guest, having them
move up a room. There is no problem with the last guest because it is an
infinite hotel, there is no last guest! Another way to visualize the problem
is to ask ever hunter to pass their spear to the right in the picture below.
184 Are the Androids Dreaming Yet?
Hunter with Spears
Provided there are an infinite number of hunters there is always someone
to hand the spear to and the person at the front of the line now has space
for another spear.
You can probably see how to answer Dermot’s question. The hotel
manager calls the guest in the first room and asks him to move 3 rooms
up rather than one. He then calls the remaining guests and tells them
the same thing. Thus, he has managed to fit three more people into
the infinite hotel. Infinity plus 3 is infinity. You may worry that it takes
the manager an infinite time to call all the rooms, but it’s OK; he lives
infinitely long so it all works out.
What about fitting an infinite number of new guests into the already
full hotel? Surely then we will get stuck.
No, Hilbert’s Hotel can fit an infinite number of extra guests. Here’s
the trick: ask all the people currently in the hotel to move to the room
with double the number they are currently in – 1 goes to 2, 2 goes to 4,
3 goes to 6, and so on. Now all the odd numbers are empty and you can
fit an infinite number of people into the empty odd rooms. Infinity plus
infinity is infinity. Voila.
∞
185
Now, a very clever or annoying student asks, “What happens if an
infinite number of infinitely large buses arrive at the hotel. Can they all
fit in?” The mathematical question is “does infinity times infinity, equal
infinity?” Let us ask all the guests to get out of the bus and line up in the
parking lot in neat rows. Passengers from bus one in line 1, those from
bus 2 in line 2, and so on. All the guests now form a two-dimensional
grid. We already know how to map a two-dimensional grid to onedimension
using the zigzag method. We can fit them all in the hotel and
we are done!
Is Anything Larger than Infinity?
Is there any bus or combination of buses that would cause the manager
of Hilbert’s Hotel a problem.
The answer is yes and it involves a subtle change to the contents of
the bus.
An infinite number of buses turn up but this time the buses are
filled with men and women. The hotel manager is asked to put everyone
in a room and once again he obliges using the zigzag method.
At the end of the process the tour guide comes to him. “I think you
have missed some people,” he says. “Since I am just one person, I know
you can fit me in. But, I have a whole bus in the car park you completely
missed.”
“No,” says the manager. “I did every bus.”
Infinity Plus Infinity Equals Infinity
186 Are the Androids Dreaming Yet?
“Ah, no,” says the tour guide. “The first bus you accommodated
had a man in the first seat but this has a woman. The second bus had
a woman in the second seat but this one has a man and so on. This
bus has a different gender in at least one seat to every bus you so far
accommodated. It is a new bus.”
The manager finds room for the passengers from the new bus but
the tour guide comes back a moment later.
“You have missed another bus. This one has a different gender
in at least one seat to every previous bus, including the one you just
accommodated. It looks like there are an infinite number of buses you
missed, all lined up to get into the infinite hotel.”
What is it about these buses that make them so difficult to
accommodate? They are all just filled with people after all.
The manager is defeated by the more complex information held in
the contents of the buses. An infinitely large bus full of binary information
has more information in it than an infinitely large bus specified only by its
size. This is a larger infinity than the counting infinity. The permutation
of all the possible options for the occupants of the bus is larger than
infinity.
Real Numbers
What about the real world we live in? Is the larger infinity we failed to
fit into Hilbert’s Hotel present, or was it just a mathematical fiction?
Hold up your thumb and index finger for a moment. The gap between
them is a distance. Most likely this is a whole number with an infinite
decimal digits after it – say 2.2320394386…. centimeters. The infinite set
of decimal digits in this measurement is the larger type of infinity: called
the continuum. Distances in space form a continuous unbroken line of
points, with no gaps in between. The counting numbers, on the other
hand, form a broken line. We take discrete steps from one number to the
next. This is a hard distinction to grasp but it is the same distinction we
used in Hilbert’s Hotel. Imagine you believe you have a list of all the real
numbers in the world. You can take the first decimal digit from the first
number and add one, the second digit from the second number add one
and so on generating new numbers not on the original list. Therefore,
you cannot have a list all the real numbers; they are not countable. Let’s
take a closer look at these real numbers.
Here’s a quick test. Which is the larger number, the first or the second?
∞
187
Holding a Real Number in your Hand
ddd
First: 3.1233249837583462136421472374
Second: 3.1233249837583462134421472374
You have 2 seconds to answer!
TRY ANSWERING WITHOUT READING ON
The first is larger. I changed one digit. Can you see?
Notice, you need time to read each digit and process the information.
If you were an obedient reader and attempted it in two seconds you either
guessed or gave up. Two seconds is too short to take in all the digits.
Let me give you another test. Again, I’ll ask you the question, “Is the
first number larger than the second?”
ddd
First
3.12332498375834621364214723751646464646464636…
Second
3.12332498375834621364214723751646464646464636…
188 Are the Androids Dreaming Yet?
I know you’re looking for the difference but you won’t find one, as
I did not have time to write the numbers out in full. The 10 20000th digit is
different, but even if I took the whole age of the universe and counted
as fast as possible I would not reach this digit. Any number greater
than, 10 120 /10 -43 digits cannot be distinguished from another in the age
of the observable universe. Real numbers are in practice subject to an
uncertainty principle. Some mathematicians even wonder whether they
really exist. But, they do exist in our minds and our thought experiments.
In my view, any model of the Universe that ignores them is likely to be
wrong.
Random Numbers
Which of the following numbers is random?
ddd



THINK ABOUT YOUR ANSWER THEN READ ON
Each of the numbers could be random. There is no reason any set
of 10 digits is more likely than another, but it feels very unlikely that if
I tried to generate a random number I would get 15 consecutive digits.
What a human means by random is a jumbled up number: one with
varying digits that have no real pattern. An American mathematician,
George Chaitin has been able to explain this by saying that a random
number is uncompressible. This means there is no way to describe the
number more efficiently than writing it out in full. A string of ones
can be compressed. “Write a million 1s” takes only 18 characters, yet
accurately describes a number that is a million digits long. By contrast
8988376132 can’t be compressed very much at all, its information
is just a jumble. There are many interesting numbers around. Some
numbers are Hamlet; some numbers are pi. One interesting number is
the following: 17733173332032037377. It is the genetic sequence for the
virus smallpox, or at least the first 20 digits. Copies of the full sequence
sit under lock and key in the Pasteur Institute in France and the CDC in
Atlanta. This number is a candidate for an ‘evil’ number. You might think
there are many numbers that could represent smallpox because there are
∞
189
Smallpox Virus
many languages in the world and many ways you could code the genetic
sequence of GATC. But, there will be one most efficient binary coding
for smallpox and that number is the nearest we have to an evil number.
The other important element of random numbers is the process
by which they are created. Computers can’t genuinely generate random
numbers. The numbers they generate are predictable and eventually
Child Survivor of Small Pox
190 Are the Androids Dreaming Yet?
repeat. To create the random number in my example above I went to www.
random.org, a website that uses fluctuations in atmospheric quantum
noise to generate random numbers. As far as we know quantum effects
are truly random and have neither rhyme nor reason.
Numbers are more complex than they first appear. They are infinite,
yet there are different infinities, and they have meaning. The smallpox
example above and the Turing numbers we will discover shortly suggest
numbers do have meaning independent of culture and language. The
next two chapters will show us what happens when we think about the
meaning of numbers. We will also explain one more ‘super infinity’ and
this will be the key to understanding creativity.
“There are known knowns; there
are things we know we know.
We also know there are known
unknowns; that is to say we
know there are some things we
do not know. But there are also
unknown unknowns - the ones
we don’t know we don’t know.”
Donald Rumsfeld
United States Secretary of Defense
(2001-2006, 1975-1977)
Chapter 9
KNOWN
UNKNOWNS
Donald Rumsfeld
In the spring of 1981, London staged its first marathon. The field of
runners included 1200 international athletes and 20,000 amateurs.
An estimated 20 million viewers watched from around the world.
The top international runners stayed together for the first twenty miles
and then two runners, American Dick Beardsley and Norwegian Inge
Simonsen, made a push for the finish. They were long-standing rivals
and, as they ran the final mile each man challenged the other to see if
they could get ahead and gain the advantage. Because of the fine balance
human muscles maintain between anaerobic and aerobic metabolism,
the small set advantage could prove insurmountable. The other runner
would need to sprint to catch up and the resultant lactic acid generated
would turn their legs to jelly. As the two runners neared the finish line
they glanced at each other, smiled, reached out and held hands as they
crossed the line. Who won?
We all instinctively know the answer. The race was a draw, but the
rules of the International Athletics Federation are clear. Read rule 164.
RULE 164
The Finish
1. The finish of a race shall be denoted by a white line 5 cm wide.
2. The athletes shall be placed in the order in which any part of
their bodies (i.e. torso, as distinguished from the head, neck,
arms, legs, hands or feet) reaches the vertical plane of the finish
line.
The organizing committee held a brief conference and the result
declared a draw. They had interpreted the rules in the same way 20
million TV viewers already ‘knew’ to be true.
This story should set your minds thinking about the nature of rules
and truth and how the two are often different. According to the rules,
one person crossed the line a little ahead of the other. The truth, as we
all instinctively know, is that the race was a draw. Maybe the rulebook
is missing a rule – ‘The contact draw rule’. Clearly you could amend the
rulebook to add this one rule. I checked the current athletics rules and
they don’t contain this amendment. If the rules were amended the mischievous
amongst you will realize an unsporting athlete could grab the
hand of their opponent as they crossed the line to force a draw. The rules
would have to stipulate that holding hands must be voluntary for both
parties, and refinements could go on for some time. What if I held your
hand but you tripped and let go? What if my attempt to hold your hand
Known Unknowns
193
caused you to trip? You could go on
forever, generating rules to cover every
eventuality.
Clearly, in the fuzzy world of
human endeavor, truth and rules often
part company. Yet, we all assume mathematics
is free of such uncertainty. Let
me tell you this is not so. The brilliant
mathematician Kurt Gödel proved
this when he was just 22, and his proof
says something fundamental about the
nature of knowledge.
The story of his discovery involves
Kurt Gödel
some of the greatest mathematical
thinkers in history. My introduction to it came about from a chance
accident. I became ill in my first year at University (mononucleosis,
otherwise know as glandular fever, if you’re curious) and was eventually
sent home to recover. Lying in bed for two months is boring. So
to pass the time my mother suggested I read Bertrand Russell’s, The
History of Western Philosophy. I think she figured I had plenty of time, so
picked a thick book. This nearly 800-page tome charts the entire history
of philosophy from the time of the ancient Greeks. I presumed Russell
was a philosophy professor, but he was originally a mathematician. He
was a mathematician. And because he lived and worked productively for
almost all of his 97 years, spanning much of the 19 th and 20 th centuries,
he crops up repeatedly as a central figure in many areas of intellectual life.
Russell the politician, Russell the philosopher, Russell the mathematician
and Russell the peace campaigner are all the same man – not, as I had
incorrectly first guessed, a prolific family. In his early career, Bertrand
Russell was a Fellow of Trinity College, Cambridge, working on a broad
range of mathematical problems. Meanwhile, in Germany, his contemporary
David Hilbert, also a polymath, held the chair of mathematics at
Göttingen University. Both men shared a common objective: to tidy up
the loose ends in mathematics and set down the rules once and for all.
This movement was called Formalism.
Formalism
David Hilbert and Bertrand Russell believed you should be able to set
out all the rules of mathematics even though it might be a complicated
affair. Without contradiction or inconsistency you should be able to
194 Are the Androids Dreaming Yet?
write down the rules and then play the ‘game of mathematics’ to derive
every possible truth. Hilbert despised the idea that there could be
unknowable things and was a forthright speaker. His battle cry was: Wir
müssen wissen — wir werden wissen! “We must know — we will know!”
He believed there were no fundamental unknowns in the world.
Donald Rumsfeld famously summed up the problem of unknowns
in an attempt to clarify a question from a journalist at a Whitehouse
press conference:
“There are known knowns; there are things we know we know. We
also know there are known unknowns; that is to say we know
there are some things we do not know. But there are also unknown
unknowns – the ones we don’t know we don’t know.”
Interestingly Donald Rumsfeld, like Bertrand Russell, is another
person to span a huge swath of time in the public eye. He was both
the youngest and the oldest serving U.S. Secretary of Defense, serving
under both Richard Nixon and George W. Bush. We will shortly discover
Rumsfeld’s convoluted view of the world turns out to be closer to the
truth than Hilbert’s tidy mathematical aspiration.
As well as believing there were no unknowable unknowns Hilbert
thought mathematics was completely abstract. You did not need to know
what you were talking about. Whether the symbols meant dogs, cats or
numbers all you needed to do was apply the rules and all would be well.
His belief is captured in his quote below.
“It must be
possible to
replace in all
geometric
statements the
words point,
line, plane by
table, chair,
beer mug.”
David Hilbert
Geometry with Beer and Furniture
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195
Newton’s Principia
PM
In 1890, the Cambridge mathematicians Alfred North Whitehead and
Bertrand Russell embarked on the mammoth task of writing out all the
rules of mathematics and publishing them in a set of books called Principia
Mathematica. Every rule is written down in meticulous detail. The books
are heavy going and look like more like computer programs than text.
They set out precisely what you can, and cannot, do with numbers, and
are the most impenetrable textbook you will ever read. Just to give you a
flavor here is one line where Russell proves 1+1=2. It has taken about 100
pages of densely packed equations to get to this point!
One Plus One Equals Two, PM
PM is a 3-volume set of books. Volume One costs £480 on
Amazon. This is a significant work and a collector’s item. The last time
a first edition volume came up at auction in 2007 it went for over £800.
Cambridge University Press printed only 750 copies and I suspect they
196 Are the Androids Dreaming Yet?
Amazon Listing for Principia Mathematica
are undervalued. When mathematicians use the letters ‘PM’, they are
usually referring to Russell and Whitehead’s Principia Mathematica
rather than the afternoon.
Hilbert’s Problems
In 1900, while Russell and Whitehead were in full flow writing out
their rules, David Hilbert was invited to deliver the annual lecture at
the International Congress of Mathematicians in Paris. He asked a
mathematician friend what subject he should pick for the talk and, in
a moment of inspiration, the friend suggested laying out a vision for
the future of mathematics. Rather than tell people how wonderful
mathematicians were, and why their discipline was the pinnacle of
human scientific endeavor, why not try modesty and list all the problems
on which they were stumped? Hilbert liked the idea and devoted his
talk to all the problems he thought mathematicians would solve in the
20 th century. Hilbert’s Problems were simply an intellectual challenge.
He offered no prizes. At the turn of the 21 st century, the Clay Institute
created the Millennium Prizes for solving the most important modern
mathematical problems. Each solution wins a prize of a million dollars!
There are 23 numbered Hilbert Problems in all: ten in the original
lecture and a further 13 in the written transcript. In 1928, he clarified
the 2 nd and 10 th problems, refining them into three distinct questions: Is
mathematics consistent, complete and decidable? Ironically this means
that Hilbert’s 23 problems actually number 24! The most important
Hilbert questions where these last three. They ask whether Russell
and Whitehead would be successful – can you write out all the rules
of mathematics and then simply calculate the answer to any problem
or derive any proof. This is known as the Decision Problem. Can you
mechanically decide any mathematical question without doubt? To
explain Hilbert’s Problems, I need to define mathematics properly.
Giuseppe Peano, Mathematician
“A mathematician is a blind
man in a dark room looking for
a black cat which isn’t there.”
Charles Darwin
The Game of Math
One of my most vivid childhood memories is driving my mother
distraction by asking the ‘why’ question. Most children go
through this phase:
Me: “Why is a sponge wet?”
My mother: “Because it has soaked up water.”
Me: “Why has it soaked up water?”
My mother: “Because it has small holes in it.”
Me: “But what makes water wet?”
My mother: “Because it is made of wet stuff.” a bit weak now.
Me: “What is wet stuff?”
…
You can ramble on indefinitely unpeeling a never-ending onion.
Sometimes, if you are unlucky, you may get stuck in a loop. For example,
“where did the chicken come from?” “An egg,” “and where did the egg
come from?”...
Mathematics breaks this cycle!
In mathematics, there is no danger of an infinite number of ‘why’
questions because at its core are a clearly defined set of absolute rules
called axioms. You cannot ask the ‘why’ question of an axiom. It is a
RULE!
Starting from an absolute minimum of fundamental rules
everything else is built up so that no step requires any leap of faith nor
generates any contradiction. Let me give you a concrete example and, in
the process, show you how numbers are defined.
Known Unknowns
199
Numbers
It was not until the late 18 th century that numbers were properly codified.
The mathematician Giuseppe Peano gave us the rules, so they are called
Peano axioms. Here are his ‘axioms’ in natural language.
Peano Axioms
1. The first number is named zero.
2. Every number has a next number (called its successor). Example:
the next number after one is two.
3. Numbers are singular. Every number with the same name is the
same thing.
4. If something is true of a number, it should be true of the next
number (the successor number).
From this we can prove some very simple things.
1+1=2. Because the next number after 1 is 2 and ‘+1’ means take
the successor. (You can see I cheated here a little and did not take 100
pages for the proof.)
Back to my poor mother: “Why is the lowest number zero,
Mummy?” “Because I say so!” Or, at least “…because Mr. Peano said
so.” That’s what an axiom is.
“OK, but why is 3 greater than 2.”
“Because I said that each number has a thing that comes after it.
“But, why can’t 3 come after zero!”
“It can!”
“But then, if 3 is the thing after zero, I could count 0, 3, 2, 4…”
“Yes, if you want to…”
“I’m sort of lost. Now, you are saying that 3 doesn’t really ‘mean’
anything. It just comes after 0.”
“Yes. You can make up any symbols you like. You just have to
remember what you said and be consistent.”
The dialogue shows the importance of definition in mathematics.
I could define my counting numbers as 0, 1, 2, 3, 4 or as ο, π, ρ, σ, ς, or
享 , 仇 , 仕 , 仝 or to be really annoying and confusing 0, 3, 1, 2, 4; they
are only arbitrary symbols. It helps us to learn the numbers because 1
is a single line, 2 is two lines joined, three is basically three lines looped
together, and four is four lines, but we could have used any symbols
we cared for. It is the rules for manipulating these symbols that are the
important part and give mathematics its meaning.
200 Are the Androids Dreaming Yet?
The Game of Mathematics
When I was a child, our living room carpet had a square pattern. You
could use boiled sweets to play checkers on it. Even though there was
no board and no pieces, it was clearly a game of checkers because we
followed the right rules (with the one exception that if you jumped over
a sweet you got to eat it). Mathematics is like a game with a set of rules.
If you follow the rules, you are doing mathematics.
Consider the simple mathematical theory that if A equals B, then B
equals A. This seems clear-cut, but you can get into trouble if you’re not
careful when defining the word ‘equals’. ‘My dog equals naughty’ does
not imply ‘naughty equals my dog. Here I have used ‘equals’ to mean
‘has the property of.’ My dog has the property of being naughty. This is
an attribute, not equivalence. You must be careful with mathematics. A
equals B implying B equals A is a property of numbers when the equals
sign is used to mean equivalence.
Here are the rules of the game that provide a proof for this theory.
Let us start with the position in which we don’t know whether A
equals B implies B equals A. We have these three axioms, call them rules
for now since we are using the game analogy.
Rule 1: If I have no minus sign in front of a letter I can assume there
is an invisible + sign there.
Rule 2: If I have a positive letter (or a letter with no symbol in front
of it) I can put a minus in front of it and put it on the other
side of the equals sign.
Rule 3: I can swap the plus and minus signs of all the letters in my
equation if I do it to all of them.
Now I am ready to prove my theorem.
A = B is the same as +A = + B. (rule 1)
+A = + B is the same as -B = - A (rule 2 done twice)
-B = -A is the same as B = A (rule 3)
Success.
So A = B is the same as B = A.
I have my proof. It might be glaringly obvious, but that’s not the
point. The point is you can apply rules to symbols and derive new rules.
It does not matter what the symbols are or how obvious it is. Here’s the
same proof with dingbats.
Known Unknowns
201
Rule 1: If I have no glyph in front of a symbol I can assume there is
an invisible Ψ there.
Rule 2: If I have a positive letter (or a letter with no symbol in front
of it) I can put a � in front of it and put it on the other side
of the →
Rule 3: I can swap the Ψ and �symbols of all the symbols in my
equation if I do it to all of them.
The proof in symbols
� → ß is the same as Ψ � → Ψ ß. (rule 1)
Ψ � → Ψ ß is the same as �ß → �� (rule 2 twice)
�ß → �� is the same as ß → � (rule 3)
Any collection of symbols will do. The symbols have no meaning
in themselves other than the meaning we have given them. A tribe in
the Amazon jungle could demonstrate a proof without knowing any
mathematics. All I need say is, “Hey, I want to play a game with you. Can
anyone make this into that, in the fewest possible steps, while obeying
these rules?”
But, is it true we can ignore the meaning behind the symbols. Does
it matter that we were talking of numbers rather than spears, counters, or
crocodiles? If we look at the marathon winning analogy again, we know
the nature of a game is important. In a running race we can interpret
holding hands to mean the two athletes are treated as one, the existing
rules can then be applied as normal and the pair become a single winner.
But, in tennis, there would be a problem. I wouldn’t want to come on court
and find I’m playing against two opponents! On consideration though
I’d be happy if they had to hold hands while they played so that they
constituted a single player. When we examine the actual circumstances,
we can add a rule and show the rule works, but we have to see something
about the specific sport that makes the rule fair and workable.
Hilbert was convinced mathematical truth is not like this and
that proofs follow from the rulebook without any knowledge of the
circumstances, i.e., the sport being played or any other analogous thing.
He was to be proven wrong by Kurt Gödel.
202 Are the Androids Dreaming Yet?
Königsberg Bridges
Gödel
Gödel studied mathematics at Königsberg University, Hilbert’s
hometown. Königsberg is famous for having a mathematical problem
related to the seven bridges that link the city together. It’s quite fun to try
to solve. Find a route across the city that crosses each bridge once and
once only. You can start anywhere, but no walking halfway over a bridge
and no swimming!
Euler discovered a rigorous mathematical proof that there can
be no solution in 1735 after five hundred years of failure by other
mathematicians. The answer is you cannot.
In 1931 Kurt Gödel, then working at the University of Vienna, proved
mathematics is like our sporting analogy. There are true statements in
mathematics that cannot be proven by the rules of the system. Someone
outside the system, with common sense, can see a statement is true, but
it’s impossible to prove this if you constrain yourself inside the system. It
is the equivalent of all the members of the London Marathon Committee
wondering what to do about the race while all of us watching the TV are
shouting, “It’s a draw!” Looking at the rulebook ‘really hard’ doesn’t help.
Known Unknowns
203
You have to step back and think about the problem in the round and then
devise some additional rules to handle the circumstances. Mathematics
is like this also.
Here is how Gödel proved his result.
It is easy to turn logic or any text into numbers. That’s how this
book is stored on my laptop. All we need do is translate sentences into
ASCII or Unicode. In this way, any theory can be reduced to a string of
numbers.
Since Gödel’s proof predates the invention of the computer, he had
to come up with a novel way to store information. He deployed an old
Roman invention; a substitution code. The number one was represented
by 1, two by 2 and the symbols by larger numbers, for example, ‘=’ was
coded as 15 and so on. He then raised a sequence of prime numbers to
the power of each of these codes and multiplied all the results together.
This generated a single enormous but unique number that he could later
factor back into its constituent parts to recover the information. This is
a truly complicated solution to a very simple problem. Today we would
solve it by storing each number in the memory of a computer as an array.
Let’s use the easier table method to store things and code as follows:
000 will stand for ‘start of proof ’. Each step in the proof will start with 00
and each symbol in the proof starts and ends with a zero. This way we
can code one plus one equals two as follows.

I think this is simple enough for you to guess the coding scheme.
Hint: 111 stands for 1. The scheme is on my website if you can’t work
it out. Using this technique, any series of mathematical statements can
be turned into a number. As a series of mathematical statements is a
proof, we can generate proof numbers. They are just the sequential list of
all the instructions. These numbers are sometimes referred to as Gödel
numbers.
Gödel’s next step was to say one number demonstrates the proof of
another number. For example, the number 000820962 might demonstrate
the proof of another number 000398... This is the mathematical equivalent
of my saying a Word file demonstrates the truth of your mathematical
theorem. Any statement can be represented by numbers, provided
you have a consistent coding scheme that allows you to get back to the
meaning.
Now Gödel set up his paradox:
204 Are the Androids Dreaming Yet?
Every correctly formed theorem number has another number,
which demonstrates the proof of that number.
If this is universally true there should be no contradiction.
Unfortunately if you apply the theorem to itself you get something
similar to the liar’s paradox.
“This proof number is not a proof of the truth of this theorem
number.”
The proof number proves the theorem number is true, but the truth
of the statement is that it can’t be a proof of the statement… Paradox.
The only way to resolve the paradox is to go back one step and
realize that not every correctly formed theorem number has a proof
number using only the rules of that system.
Concisely, Gödel’s theorem says, “Within any formal system of
mathematics there can be statements that are true but are not provable
using only the rules of that system.”
When Hilbert heard of Gödel’s proof, his first reaction was anger.
After all, he had spent 30 years of his life trying to prove mathematics was
tidy and complete. Gödel had just shown it was not. Hilbert never worked
on formalism again, but the rest of the mathematical establishment
largely ignored the result. Gödel’s proof did not stop mathematicians
proving new theorems nor doing useful mathematics. They went on
much as before, using a mixture of intuition and analysis. The only
difference was someone had told them analysis alone would not succeed.
The repercussions of Gödel’s theory have more to do with understanding
our place in the Universe and the nature of knowledge discovery. These
are ‘big’ philosophical questions, which don’t greatly affect the day-today
ability of a mathematician to do their job. However, it is important
to understand that knowledge discovery is not simply analysis. Knowing
this helps us understand human creativity.
Inconsistency
In the proof above, I said the only way to resolve the paradox is by saying
there cannot be a proof number for every mathematical statement and
therefore mathematics is incomplete. There is one other way to solve the
paradox, and that is by allowing inconsistency into the system. Gödel’s
proof assumes you can prove something true or false, but what if you
could prove it true and false? In this case, the system is complete but you
can prove truths and untruths within it! This may seem an acceptable
solution, but inconsistency in a mathematical model is a cancer that will
Known Unknowns
205
spread through the entire body. Think about it. If I am allowed to prove
anything either way, of course, my system is complete. It can say anything
it wants, but the proofs I make are worthless.
Let us imagine, for a moment, we created a new system of
mathematics where all the numbers in our new theory behave as we
expect, except for the numbers 5 and 6. You may use them to count, but
they are also equal to each other! This feels bad and it certainly breaks the
Peano axioms. In my new system 1 plus 5 and 0 plus 5 are the same, so I
can equate 0 to 1. Because 0 and 1 are the basis of binary arithmetic, all
numbers can be equated. Numbers now have no guaranteed meaning in
my system and, what is worse, since logic uses 1 and 0 to represents true
and false, all of logic falls apart as well. Whenever we allow inconsistency
into mathematics it rapidly brings the whole pack of cards down.
The example I gave was glaring; an inconsistency right in the
middle of the counting numbers! Maybe I was too aggressive and a
subtle and less damaging inconsistency might be tolerable. However,
any inconsistency allows me to make zero equal one somewhere in my
system and, therefore, any theorem based on proof by counterexample
will be suspect.
There might be systems where inconsistency could be a legitimate
part of a mathematical system, but I would always need positive
corroboration for each proof. If I tried hard enough, I could always prove
something either way. I would need to formulate a new mathematical
rule – something like “I will believe short, sensible-looking proofs to be
right and circuitous proofs to be wrong.” Mathematics would be a bit like
a court of law. You would have to weigh up the evidence from a variety
of sources and the verdict would be a matter of subjective opinion rather
than objective fact. Inconsistency is very bad in mathematics.
The Lucas Argument
J.R. Lucas of Oxford University believes Gödel’s theorem says something
fundamental about the nature of the human mind. In 1959, he wrote a
paper, Minds, Machines and Gödel, where he argued humans must be able
to think outside a fixed set of formal rules. The paper has been causing
arguments ever since. Strong AI proponents have a visceral reaction to
it. Forty years later, in 1989 Roger Penrose picked up the baton and put
the Lucas argument on a stronger theoretical footing. The Lucas-Penrose
argument is this:
206 Are the Androids Dreaming Yet?
If humans used a formal system to think, they would be limited by
the incompleteness theorem and unable to discover new theorems that
required them to extend the formal rules. Humans do not appear to have
such a limitation and regularly extend their appreciation of mathematics
by expanding the rules, and seeing through to the truth.
Many scientists dislike this argument and think it farfetched, saying
there is no evidence to show people see past the limitation. Our brains
could be following a formal system capable of discovering everything we
have discovered to date or, indeed, might encounter in the future. Why
should we assume human minds are constrained in the same way as the
mathematical systems they discover? There is no evidence to suggest a
human thinking about Peano arithmetic is running a Peano based model
in their head. When Peano discovered his theorem he was certainly
extending our mathematical knowledge, but this does not imply he was
extending the capability of his brain.
The critics of Lucas and Penrose have one big problem to deal with.
The formal system in our head would need to be able to see the truth in
everything we could ever encounter. But, our formal system appears to
be small. As infants, it is almost nonexistent. Where does this enormous
system come from? It can’t come from our parents because they have
the same problem; they were once children. You might argue that the
capability of the human brain is huge and we can learn from all the other
humans on earth, but let me remind you what Gödel said. However large
Two Giants
Known Unknowns
207
a system you have and however much you extend it, the system will
always be incomplete. And we really do mean; however large. Even an
infinitely large formal system would be incomplete.
The only way to avoid this problem is with some sort of conspiracy
theory where we only come across problems our formal system can
already solve. Such a theory is a determined Universe. In a determined
Universe, all the mathematical problems we ever solve must be expressed
by the formal systems existing in the Universe. We must never encounter
a problem where we need to extend the system and break the Gödel limit
because we are pre-determined not to do so.
The Inconsistency Defense
An argument put forward by opponents of the Lucas-Penrose position
is that humans are inconsistent formal systems. Inconsistent formal
systems are not subject to the incompleteness limit. Humans certainly
behave inconsistently with remarkable regularity but simply making
inconsistent statements is not sufficient to show the underlying formal
system is, itself, inconsistent. Inconsistent beliefs can come simply from
making mistakes or reading the same story in two different newspapers!
We need a fundamentally inconsistent thinking mechanism inside our
brains to break the constraint. The very machinery itself would have
to be inconsistent. But this is exactly Penrose’s point. Constructing a
machine capable of reasoning in an inconsistent but useful manner would
need exotic technology, some sort of non-deterministic, rationalizing
computer. The components to make it could not be computer logic as we
know it today. All such logic is entirely computationally deterministic.
Let me see if I can reframe the Lucas argument. Imagine IBM’s
Watson computer was let loose on mathematical reasoning. Watson could
scan every mathematical theorem ever written down. It would know
every programming language created. It would have its enormous bank
of general knowledge to call upon and it could answer many questions.
It would sometimes appear inconsistent because the information it had
trawled from the Internet would be wrong. But Watson would still be a
consistent formal system and Gödel’s theorem says there would be truths
Watson could never see. Lucas argues humans can see such truths where
a machine cannot, and these truths would allow a human to discover a
proof to a mathematical problem that would forever elude Watson.
The Lucas argument runs into a brick wall because it asserts we see
truths a machine cannot. For each alleged creative step, his opponents
simply assert your brain was already sufficiently powerful to perform
208 Are the Androids Dreaming Yet?
that creative step. Lucas’s argument is largely a philosophical one. Surely
all this creativity can’t all be pre-coded within the brain. Surely we must
be extending our model in order to extend mankind’s mathematical
model. “Prove it,” say the detractor, and he cannot. We need something
more practical if we’re going to show a difference between humans and
machines - something an engineer, or even a physicist, could grasp! That
thing is a Turing Machine. We will examine this next.
Chapter 10
TURING’S
MACHINE
Alan Turing
“A computer would deserve to
be called intelligent if it could
deceive a human into believing
that it was human.”
Alan Turing
“The only real valuable thing is
intuition.”
Albert Einstein
“Mathematical reasoning may be
regarded rather schematically as
the exercise of a combination of
two facilities, which we may call
intuition and ingenuity.”
Alan Turing
It is 1943 and a small group of Polish mathematicians sit, ears glued
to their wireless set, waiting to hear whether the German army will
advance on Warsaw. The Polish Intelligence Bureau badly needed
to know what the German army was planning and had recruited this
group of young mathematicians as code breakers. Up to this point, codebreaking
had been the domain of linguists able to see word patterns
in apparently random sets of letters. The arrival of electro-mechanical
machines made this method redundant, and code-breaking had become
the domain of mathematical minds. The British, French, and American
intelligence agencies were all hard at work deciphering the German
codes, but only the Polish group, motivated by the imminent threat of
invasion, had made real progress. The code they were breaking: ‘Enigma’.
As with many inventions, Enigma got off to a difficult start. The
inventor, Arthur Scherbius, tried to sell it to the army but they rejected it
saying it did not provide any real military benefit. Instead, the machine
went into service transmitting commercial shipping manifests. However,
some senior figures in the German military had not forgotten the lesson
of the First World War. During that war, the German army suffered
major setbacks because the British broke all their codes early on. With
the onset of World War II, Rommel ordered the German Army and Navy
to deploy modern coding machines. The previously rejected Enigma was
rapidly pressed into service and, all of a sudden, Europe went dark to
Allied Intelligence. The man to lead the task of breaking Enigma for the
English was Alan Turing.
Alan Turing
Alan Turing was conceived in India but born in London in early 1912.
He was precocious from an early age and an extraordinarily determined
character. His first day at Public School, Sherborne in Dorset, coincided
with the British General Strike of 1926. With no public transport available,
the thirteen-year-old Turing cycled the 60 miles to school, staying in a
guesthouse on the way and earning a write-up in his local newspaper.
Turing went on to study Mathematics at King’s College, Cambridge and
was made a Fellow at only 22. In 1936 Turing, aged 24, published On
Computable Numbers and their Application to the Entscheidungsproblem,
not a snappy title, but one of the most influential mathematical works of
the 20 th century. The paper described the new the science of computing
and solved Hilbert’s ‘Entscheidungsproblem’, a mathematical puzzle
212 Are the Androids Dreaming Yet?
simply translated as ‘the Decision Problem’ – could you decide the truth
of a mathematical statement using some sort of automatic computation
– an ‘algorithm’ as we now call it?
It is difficult to imagine, but Turing worked on ‘computing’ before
the invention of the computer. When he talked of computing, he
meant the abstract idea of doing something mechanically. The nearest
thing he had to a ‘computer’ at the time was a human mindlessly but
methodically calculating something with pencil and paper! The scientific
paper he submitted to the London Mathematical Society described both
the theoretical basis of computing, and the design of a general-purpose
computing machine: the forerunner of all modern computers.
At the time, only a handful people in the world could assess
Turing’s paper. One of them, Alonzo Church, was based at the Institute
of Advanced Mathematics in the USA on the Princeton University
campus, next door to the Institute for Advanced Study that housed
Einstein. Turing travelled to America in 1937 and completed his doctoral
thesis at Princeton. He might have stayed, but Europe was heating up
and war seemed inevitable, so Turing returned to England to take up
a part-time job in the government code-breaking branch. Here he was
able to indulge his passion for hands-on engineering, experimenting
with the newly invented valve technologies. When war finally broke out
Turing was ordered to report to Bletchley Park, just north of London.
This was to be the home of the top-secret British code-breaking group
tasked with cracking Enigma. Turing’s first task was to debrief the
Polish mathematicians and see what they had discovered. The Polish
mathematicians had seen there were flaws in Enigma that made it repeat
itself. They had made a copy of the machine to test different coding
configurations and had been routinely cracking Enigma for 6 years,
but the Germans had been getting smarter and it was taking longer and
longer to crack the codes. Turing realized he could apply the Polish ideas
in a more general way and break the codes on an industrial scale. He was
installed at Bletchley Park to lead the project.
Initially he was successful but as the war continued, Enigma
developed subtleties making it harder to break. At one point, it was
taking a whole month to break a single day’s messages. Turing realized
the only solution was to use computer technology to fully automate the
decryption. He built a computing machine that could simulate thousands
of Enigma machines and try out all the possible settings in a short space
of time. The machine acquired the nickname ‘a bombe’, perhaps because
of the ominous ticking sound it made as it calculated (or maybe as a
reference to the smaller Polish machines).
Turing’s Machine
213
Thanks to Turing’s insight into coding schemes and the machines
he designed, the British were soon able to read almost every coded
message the Germans sent during the war, giving the Allies an enormous
advantage. The D-Day invasion involved convincing Hitler that the Allies
had a huge army of nearly 400,000 men, massed around Dover preparing
an attack on Calais head on, with a second army in Scotland poised to
attack Norway. In truth, they had only 150,000 men planning an assault
on the Normandy Beaches in the South. Just before the landings messages
were decoded showing Hitler had fallen for the Allied subterfuge. Even
as the Normandy landings began, Hitler still thought this a bluff and
kept his 28 divisions at Calais waiting for the imagined attack. Without
this intelligence advantage, the Allies would have needed a much larger
invasion force, and Churchill believed Turing’s work shortened the war
by as much as two years.
The cracking of Enigma remained a secret after the war and
Turing’s story remained untold for many years. When Churchill wrote
his history, The Second World War, a massive work in six volumes, all
sorts of sensitive information featured, but Turing’s work was omitted.
One sentence hints that Churchill might write something about it in the
future, but he never did. Churchill considered the work at Bletchley Park
so sensitive he had it put in the highest classification – extending the
30-year secrecy rule. We must presume the decoding schemes were still
being deployed during the Cold War. The papers were finally released in
2010.
In one of those sad turns in history Turing was found guilty of gross
indecency for homosexuality in 1954, a criminal act at the time, and was
prescribed hormone treatment. This affected his mental state and he took
his life by eating an apple laced with cyanide. He was eventually honored
posthumously as a war hero and one of the most significant thinkers of
the 20 th Century. A Turing Award is the equivalent of the Nobel Prize for
Computing. He was given a royal pardon in 2013.
To see how Turing came up with the idea for the Turing machine
and solved the decision problem, we need to get a feel for theoretical
mathematics. That might sound a little heavy going but don’t worry, I will
use a simple piece of mathematics to explain, one we have all played with
as children, secret codes.
214 Are the Androids Dreaming Yet?
Codes
Everyone has played with some sort of
secret code as a child – the Aggy Waggy
game, passing notes written in invisible
ink made from lemon juice, or perhaps
a simple cypher. If I want to send you a
secret message, I can use a substitution
code. Let’s see how good a code breaker
you are. Can you decode this?
Gdkkn Qdzcdq
It’s really easy. You might guess the
Enigma Machine
message from the pattern of letters and
your knowledge of my writing style. There are a couple of interesting
patterns to note: the 3 rd and 4 th letter of the first word are the same and
the first and last letter of the second word are the same. As a test I gave
this code to my wife and my eight-year-old daughter to see how long it
took them to decode… Less than a minute for my wife – a linguist. We
will come back to my daughter shortly!
Roman Emperors used this sort of simple code to secure their
messages, but modern codes have to be a great deal more sophisticated.
Let us use a progressive cipher where we vary the substitution using a
secret word. Take the name of my dog and write it down repeatedly next
to the letters of the message you want to keep secret. Now translate all
the letters in the message and the code into numbers ‘a’ = 1, ‘b’ = 2 and
so on. Then add the letters of my dog’s name to the letters of the message
one at a time. If I get to 26 (‘z’) just wrap around to ‘a’ and carry on. This
is called modulo arithmetic. This coding scheme will translate ‘l’ to ‘a’ the
first time but ‘l’ to ‘c’ the second making it much harder for a linguist to
see any pattern.
hello reader can you read this code
georgegeorgegeorgegeorgegeorgegeorge
Gives
ojacveyjpvlwghpegcvzoilfkehzpxghcvle

Turing’s Machine
215
The advantage of this cipher is that I can easily remember the name
George. I don’t need to write it down. And the circular application makes
the message sufficiently obscure you can’t easily work it out…
Is this, therefore, a good code?
No.
This cipher is easy to break. Once you have guessed that I have
applied a repeated short code word, you can write out ALL the possibilities
and decrypt my message! This may be tedious, but if you are fighting a
war and your life depends on it, you can employ a thousand people to
write them all out. The British government employed 10,000 people at
Bletchley Park, many of them doing exactly this. You might think that
applying ALL the possibilities is too time consuming in practice but
there are many shortcuts. If I suspect the message contains the name
of a German town all I need do is try keys until I find a German town
somewhere in the message then work my way outwards from there. Or
perhaps I suspect the key is something easy to remember like the name
of the Commandant’s dog. I can try ALL German dog names until I get
lucky. If I’ve 10,000 people working for me this is easy.
The Enigma machine and the coding process set up to operate it
was designed to remove these loopholes. For a start, the keys were all
random numbers taken from a code book – no dog names allowed – and
the machine took the idea of a simple progressive cipher and made it
much more complex.
Imagine I took my GeorgeGeorgeGeorge pattern but then every
3 rd character added one, every 14 th character subtracted 15 and every
40 th character added the 3 rd letter of the First Mate’s mother’s maiden
name. Now this would be a VERY hard code to break. I would need a
machine to code messages because if I tried to do it by hand I would
make so many mistakes that the messages I send would be unintelligible.
The Enigma machine made these coding schemes a practical possibility.
But, although Enigma is hard to break it is not impossible with enough
computing power. Is there any code that is impossible to break?
An Unbreakable code
Is there a way of coding a message so you can never break it?
The answer is there are two ways to code a message so it is
PERFECTLY safe. The first is to use a one-time pad and the second is
quantum cryptography.
216 Are the Androids Dreaming Yet?
One perfect way to encode a message is to use a one-time pad. On
a sheet of paper I write a completely random set of numbers or letters
– since we are going to translate numbers to letters it does not matter
which. I make a copy and give it to a person I later want to send a coded
message. Because I will only use these two paired sheets once it helps to
make a few of them – a pad in fact. By convention, we refer to a single
sheet or a whole book as a one-time pad code. Here is the one-time pad I
created earlier. It is just a random sequence of letters and spaces.
kaleygnqaloiuebldlan dlkawoqyevbax gmlsosuebal
To code a message, I substitute numbers for letters as with the
progressive cypher earlier again using modulo arithmetic to wrap around
if I reach the letter ‘z’. I have applied my one-time pad to the hello reader
message below to get ‘sfacngfvbpta’.
hello reader
sfacngfvbpta
This code is unbreakable – almost! Notice there are very few clues
for anyone wanting to decode it without holding a copy of the pad. Spaces
do not necessarily indicate breaks between words, and letter patterns are
absent. It has only one flaw. The total number of characters and spaces
could have some meaning. This is a problem because if I routinely
communicated bombing targets and my message was “Bomb Bath”. You
could figure out the sender was not going to bomb Bristol if the message
were shorter than 11 letters and spaces. To avoid this problem, messages
are extended with nonsense at beginning and end to make sure no
information can be gleaned from the length. The convention is to code
messages to the full length of the pad. You must never reuse a pad. Each
time you code a message, rip off that page rather like a calendar. Destroy
it and use the next page for the next message. At the other end, the
recipient uses his copy of the pad to run the process in reverse. Decode
the message by swapping each letter according to the modulo method,
rip the page from the pad, and burn it. Because each key is only used
once you can’t use any sort of statistical method to work out the message,
making the one-time pad perfectly secure. Claude Shannon proved this
in 1945 while working for Bell Corporation but, due to wartime secrecy,
his proof was not published until 1948.
Turing’s Machine
217
The Perfect Code
The proof that a one-time pad is perfectly secret is straightforward.
Imagine I take a coin and flip it 1000 times. I’ll write down some of the
results as follows:
HHTHHHTTHTTTHTTTTTHTH…
I give you a copy of my results and keep one for myself. Now we
each have the same random set of Heads and Tails recorded on a piece
of paper. I can convert any message from letters to binary numbers: ‘a’
= 00000001, ‘b’ = 00000010, ‘c’ = 00000011 and so on. If you are not
familiar with binary just assume I have a code where we only ever use
combinations of 0s or 1s. To encrypt the message we flip each bit – 0 goes
to 1 or 1 goes to 0 – using my random list of heads and tails according
to the following rule: If I have a head flip the bit, otherwise leave it the
same. I now have a randomized message, and it really is truly random. To
convince yourself, imagine answering the question, do you like coffee or
tea? Think of your answer and flip a coin. If the coin lands heads change
your answer otherwise leave it the same. Now write your answer down.
Try it out a few times. Do you see you end up with a totally random set
of decisions – tea, coffee, coffee, tea, tea, tea. If you don’t record the coin
toss there is no way to determine your true answer.
Similarly, the message I encoded above now looks like a completely
random stream of 1s and 0s and the only person who can decode it is the
party with the other record of the coin tosses. Apply this to the message
and, as if by magic, the message reappears. Any other random sequence
will yield gibberish. It has to be the SAME random sequence I used in
the first place.
Mathematically, the proof involves working out that the probability
of getting the right answer by applying a random sequence is 1 in 2 n and
the probability I could guess the answer is also 1 in 2 n ? The same! So the
chance of decrypting the message knowing the encryption method is the
same as simply guessing the message and getting lucky. Therefore, the
message is perfectly encrypted.
Quantum Cryptography
It turns out there is one other perfect encryption method that involves
thinking about the nature of secrets. Normally we consider the primary
problem with sending a secret message is coding it so that it can’t be read
218 Are the Androids Dreaming Yet?
by anyone but the intended recipient. However, wouldn’t it be equally
valuable to know if someone other than the recipient had intercepted
and read the message? This is the trick quantum cryptography gives us.
Taking a measurement with a quantum device disturbs the system
so measurements can be taken only once with the same results. By the
same logic, I could send you a message and if someone else has read it in
the meantime, you will know. I could arrange to meet with you in Berlin
and if you detect the message has been intercepted, you could simply not
show up.
I could use this same technique to send you a one-time pad. If you
receive it without it being overheard, I could then safely send you an
encrypted message. In 2007, this technique was used to transmit the
results of a Swiss election from the polling booths to the central counting
center.
Enigma
World War II accelerated the evolution of encryption from simple
substitutions a human could perform to complex ciphers only a machine
could calculate. You might wonder why everyone does not use a onetime-pad
since it is a perfect code. The problem is distributing and
maintaining the pads while keeping them secret. My daughter cracked
my earlier code because she knows my laptop password, broke in, and
read the answer. That’s the problem with codes – security. The pads
could be sent out in sealed envelopes but it would be easy to intercept an
envelope, copy the pad and reseal it. You would then have a perfect and
undetectable way to break the code. Also, if I were an Admiral wanting
to communicate with my fleet of submarines I would need a huge pad
– one page for every message I want to send – and either a pad for each
submarine or one pad for all submarines. If I use only one pad, then I
cannot talk to a submarine privately, and if any pad were lost all security
would be breached. One-time-pads were used by both sides during
World War Two, and often printed on nitrocellulose – a chemical similar
to the explosive nitroglycerine. This allowed users to burn the codebooks
quickly if an enemy threatened to capture them.
Both the Americans and British captured Enigma machines and
codebooks during the war. A Navy Enigma machine was a sought-after
prize, as it was more complex than the Army version, with extra dials
and plug settings. To crack the more sophisticated codes Bletchley Park
Turing’s Machine
219
needed to get hold of Enigma machines, ideally without the Germans’
knowledge. The film U-571 merges two such capture stories into one,
taking a few dramatic liberties along the way, but it’s well worth watching.
Even with a captured machine, the codes were hard to break. You
needed a starting point – a crib to give you a clue what the machine
settings were. Helpfully, the German Army often began their messages
with a weather report. Everyone knows the German word for weather –
‘Wetter’. Decode the first 20 letters of a message until you found ‘Wetter’
and the message is unlocked. The German Navy, however, was less chatty
and avoided obvious words in their messages. One way the Allies could
find a crib was to blow something up. They would sail to some point in
the Atlantic, fill an old boat with oil drums, and set it alight. The German
Navy would get wind of this and go to investigate. The first thing they
would do is to radio a message back to base with the coordinates of
the wreckage, which, of course, the British already knew. This gave the
British a crib, and once they were in, they could decode messages for
several days in a row because the Enigma machines often cycled through
a repeating pattern.
Throughout the War, the German military never suspected the
British had cracked their codes and thought they must have traitors giving
away their secrets. The Enigma machine was an elegant compromise
between a truly unbreakable code and a simple cipher. Unfortunately for
the Germans, Turing was on the side of the Allies.
In the 1930s almost all mathematics, accounting, and code-breaking
were performed by humans using pencil and paper. It was the science
behind this process Turing sought to understand. We’ll take a step back
in time again to 1935 and Turing’s discovery of a solution to the Decision
Problem – the Entscheidungsproblem.
Lego Turing Machine
“Machines take me by surprise
with great frequency.”
Alan Turing
The Machine
Turing probably learned of the Entscheidungsproblem in a lecture
given at Cambridge University by Max Newman. Newman
described a new proof by Gödel showing mathematics was
incomplete. The proof solved the completeness and consistency problems
by turning mathematical statements into numbers and showing you
could generate a logical paradox if you tried to argue for completeness
and consistency at the same time. Thus, of the three original Hilbert
problems, completeness, consistency and decidability, only decidability
remained unanswered.
Turing spent all of 1935 and much of 1936 thinking about
this question: Is mathematics intuitive, or could a machine decide
mathematical questions automatically? Eventually, cycling through
the Cambridge countryside one day, he stopped to rest in a field near
Grantchester and in a flash of inspiration envisioned his mathematical
machine. The machine was entirely imaginary but made as if from
mechanical parts common in the 1930s.
The idea was to reduce the process of computing with pen and paper
to its most basic level. Turing hit upon the idea of using a long ribbon of
paper tape similar to the ones used in telegraph machines. A paper tape
is simpler than rectangular paper as it can be handled mathematically as
a single sequence of numbers – we don’t have to worry about turning the
page or working in two dimensions. If you are worried that a tape is less
powerful than a sheet of paper remember Cantor’s theorem: an infinite
plane is the same as an infinite line. The use of a tape massively simplified
the mathematics, and subsequently many early computers used tapes, as
they were easy to handle in practice as well as in theory.
222 Are the Androids Dreaming Yet?
The eye, hand and pencil of a human mathematician was modeled
as the read-write head of a teletype. It allowed the machine to read
input from the tape and write information back so as to keeping track of
intermediate calculations or provide the final output. The operation of
the machine was straightforward. At each moment in time the machine
could read a symbol on the tape, move the tape forward or backwards,
and write or erase a symbol. That’s all he needed to model a human doing
something like long multiplication. Turing argued his model was exactly
analogous to a human performing a computation.
Turing’s imaginary machine was now able to perform computations
just like a human. You could write down the rules for a given procedure
and the machine could, for example, do long multiplication. At each step
of the calculation, the computer would examine the state machine, look
up the state in the instruction book and put the machine into its new
state. If you recall Searle’s Chinese Room, this is the same process the
man in the room followed: get a symbol, look it up in a book, and reply
with the corresponding symbol.
Universal Turing Machine
We have missed one important step from our explanation of the modern
computer: the ability to run programs. Nowadays, we take for granted
you can download a program from the Internet or buy one from a shop.
In the 1930s adapting a single machine to multiple purposes was a radical
idea. Machines were built to do one thing, and one thing only, and there
was no concept of a general-purpose machine. Nowadays this is hard to
comprehend, but there is a similar revolution going on in manufacturing
today with the widespread adoption of 3D printing. Today most factories
use tools – lathes, drills and saws – to fashion objects. Each machine does
a specific job and is not ‘general purpose’. But innovative new machines
can now be purchased relatively inexpensively called 3D fabricators,
which print entire objects. The same happened for electronic logic in
Turing’s time.
Before computers, logical tasks were performed by banks of relays.
How these banks work can be illustrated by the workings of an oldfashioned
elevator. If you pressed a button to call an elevator, you closed
a switch coupled to a relay in the basement sending power to the car.
Another switch was tripped automatically when the elevator reached the
desired floor. All the functioning of the elevator system was fixed. Once
you pressed a button to go up you could not change your mind and press
Turing’s Machine
223
Old Fashioned Relay Mechanism
the button to go down. That logic did not exist in the relay banks. If you
wanted to improve the logic of the elevator you would need to rip out all
the relays and rewire everything from scratch.
Turing’s first imaginary machine was set up in the same way. It
had a fixed set of hard-wired logic, a rule book. In order to perform
different tasks – say addition or multiplication you had to use a different
rule book. His revolutionary idea was to write a rule book that told
the machine to read a soft-wired set of instructions from the tape and
execute those instead. He called this a Universal machine since it could
perform any procedure written on the tape. Today we call this software.
It is fair to say Turing was not the first to use this idea. Charles Babbage’s
analytical engine could read instructions from cards and execute
different procedures, but Turing thought through all the ramifications
of the idea and made it general purpose, giving us the modern science
of computing. It is easy to build a real Turing machine, but by today’s
standards it is a little clumsy; a team in Denmark has built one using
Lego. You can see a link on my website.
Very soon after Turing’s paper was published, a number of people
proposed better practical implementations. In 1943, John von Neumann
of Princeton University created the architecture for ENIAC, the first
stored program computer, developed for the United States Army’s Ballistic
224 Are the Androids Dreaming Yet?
3D Printing Machine
Research Laboratory. The laptop I am writing on uses the von Neumann
architecture, and most modern computers evolved from it. By contrast,
mobile phones are descended from the Harvard architecture developed
by IBM and first supplied to Harvard University in 1944, hence its name.
The distinction in architectures has blurred over the years. The world
supports two main computer chip technologies, one built for desktop
and laptop computers, designed by Intel in Santa Clara, California, and
the other, designed for mobile devices by ARM, in Cambridge, England.
All these computers can, in principle, run any piece of software.
Programs
Software is just a series of numbers. When you click an icon on your
desktop, the computer reads the number and interprets it as a series
of instructions. There is a decoder inside the computer that knows the
number ‘1’ means add the next two digits and the number 5493 means
display them on screen and so on. On my computer the operating system,
Apple’s OSX, takes the number, decodes it and passes it to the CPU for
Turing’s Machine
225
execution. You might ask what runs the operating system and that is a
smaller program called the BIOS. What runs BIOS? An even smaller
program called the Bootstrap. Once all this is up and running you have a
working computer, which can run any program you throw at it.
The problem with programs is they tend to crash – usually at
the most inconvenient times. It is often not clear whether a program
has truly crashed. It might be stuck in an infinite loop, or it could be
calculating the answer to a complex question, such as the answer to life,
the Universe, and everything. How would we know? If only I had waited
a little longer before rebooting, the program would have run to its end
and given me the answer to Douglas Adams’ question.
It would be very useful, and save a great deal of time, if I had a way
of telling whether a program will ever stop. An elegant solution would be
to have a second program called ‘Halt’, which would test the program and
output ‘will halt’ or ‘will crash’ as appropriate. It turns out this program
would be more than just useful. It could be used as an oracle, capable of
answering almost any question imaginable.
I could, for example, write a program that says: for every index in
Fermat’s puzzle try every number and halt if you find a solution greater
than 2. Now if I run my halt program on this program and it states ‘will
crash’, I will have solved Fermat’s Last Theorem! Do you see why?
If we give ‘Halt’ an input: a program we are interested in, along with
some data, it will tell us if the program finds an answer. If I am trying to
solve Fermat’s Last Theorem, we will ask it to try every possible index for
the equation 3 x +4 x =5 x and halt when it finds a true result greater than 2.
If the halt program says yes and halts, you can trace through the program
and work out how it did it. The theory would be proved. If the program
says no, the theory is disproved. This gives us a way to discover proofs of
many mathematical theorems.
I could try almost any puzzle using a program with this form. All I
need do is put a problem in the following decision format: try all possible
options, and then stop and ring a bell if a solution is found. The Halt
program would then give the result leading to untold riches, winning all
the remaining Clay Mathematics prizes at the very least and earning me
$6m.
Does such a magical program exist? The answer, sadly, is no. There
is no Halt program and the final part of Turing’s paper proved there can
never be.
226 Are the Androids Dreaming Yet?
The Proof
Let’s write a list of all the possible programs my laptop could ever run. A
comprehensive way to do this is to start at one and try every number.As
I count up I am simply generating numbers, for example, 5,433,232, then
turning each number into a program file and running it. For a bit of fun,
I created a couple and tried them out on my laptop. They did nothing, so
it was not very edifying. Most numbers are just junk because programs
have to be in the right format for the computer you are working on. It’s
just like words. If you randomly take a handful of scrabble tiles out of a
bag, most of the time you will have nonsense, but every now and they
you will have an actual word. Be careful with this; you could accidentally
write, “delete every item on my hard disk.” Of course, the probability is
astronomically low, but Murphy’s Law says it will happen, so back up
your data!
As you count up, you will generate every possible program along the
way. A mathematician would say programs are recursively enumerable.
The word recursive means there is an algorithm and enumerable means
to count. Therefore, there is a counting algorithm that would run every
imaginable program. Here is a list of them, or at least a some of the
highlights:
0 (probably doesn’t run)
1 (ditto)
00 (ditto)
01 (ditto)
011001001001000100 (makes the computer beep once)
… (from here on I’ll give the program names since the numbers are
too large to print)
Does Nothing (there are many of these)
Is Gibberish (there are an infinite number of these)
Junk (an infinite number of these)
Print Something (again an infinite number of these)
More Gibberish
Excel
Word
PowerPoint
Mathematica...
Fermat’s Last Theorem enumerator (runs for ever)
A nonworking version of the Halting Program
A nonworking version of the Crashing Program
Really big programs that don’t fit on my hard drive
Turing’s Machine
227
and so on.
You can see that every program imaginable is generated in our
list. If you are wondering which version of Word or Excel, the answer
is every version and every bug ridden unreleased version as well. We
are enumerating every program that could ever be run in the known
universe!
Perhaps you can see a problem looming. I can pose any mathematical
puzzle in a clever way so that a program only stops if there is a solution.
I am about to list every possible program that could ever be created. If
halt exists this will automatically prove every mathematical theorem
imaginable.
Let us see if this is so.
For our thought experiment, we will assume every program takes
an input. Historical convention in computing means this is generally the
case. If you type a program into the command line of a computer with
some words listed afterwards, the computer will usually run the program
with the words as input. For example, if you type, “Print ‘Hello World’”,
most computers will print ‘Hello World’.
We now imagine there is a Halt program that can run on an infinity
of inputs. Will it work for every input? We are looking for a paradox
caused by the existence of the Halt program. If Halt causes a paradox
then Halt cannot exist.
Here goes...
If there is a Halt program, we can write a Crash program. That’s a
program that goes into an infinite loop if it detects a program will halt.
Now what happens when we feed Crash into itself? Does Crash halt if it
runs with the input Crash?
This creates a paradox; there is no solution which makes sense. It’s
similar to the Barber Paradox of earlier. Since a paradox is created there
must be a fault in our original theory. The error is the existence of Crash.
Since Crash cannot exist and it was created as the logical opposite of
Halt, Halt cannot exist either. QED. There is no general program that will
tell if another program will halt because such a program could not run
with the negative of itself as input.
This places a limit on the power of computers to automatically
solve problems. There is certainly no general purpose algorithm which
will solve every problem. Slightly more subtly there is no general
purpose program that is guaranteed to solve one arbitrary problem.
228 Are the Androids Dreaming Yet?
If there were, you could just write a program to sequentially present
every problem to the arbitrary problem solver and you would have
solved everything.
This presents us with a puzzle. A huge software industry has
grown up based on Turing’s ideas, employing tens of millions of people
worldwide. This industry regularly solves all manner of problems.
The proof from Turing’s original 1936 paper suggests there should be
quite strict limits on the power of computers. In the next chapter, we
will examine this industry and take a look at Turing’s theorem from a
modern view point. The chapter can be read as a stand alone article but
was originally written as an integral part of this book.
Chapter 11
SOFTWARE
Fred Brooks
Medieval Block Print from ‘No Silver Bullet’
“The bearing of a child takes nine
months, no matter how many
women are assigned.”
Fred Brooks
“Adding manpower to a late
software project makes it later.”
Brooks’ Law
In No Silver Bullet – Essence and Accidents of Software Engineering,
Fred Brooks explains why writing software is hard, and why machines
are not going to do it for us anytime soon. The original article
appeared in the proceedings of the Tenth World Software Conference.
It was subsequently expanded into the, now famous, book, The Mythical
Man Month.
Brooks believed solving real world problems involves understanding
the essential complexity of life. ‘Accidental Complexity’ – the simple type
– is the time-consuming part of writing software, for example, listing all
220 countries of the world in a website, or making sure all the buttons in
an interface line up correctly. These tasks are tedious – you have to look
up all the countries in Wikipedia and make decisions, such as whether
the United Kingdom will be denoted ‘UK’ or ‘GB’. They don’t need any
real ingenuity. ‘Essential Complexity’ is altogether different. It involves
understanding the world and setting out the rules in meticulous detail.
Brooks argued essential complexity is not susceptible to being sped up
by machine processes. Navigating these architectural decisions cannot
be automated. He gives us an analogy by comparing writing software to
building a house.
When you build a house, an architect designs it, an engineer makes
the calculations to ensure it is safe, and a construction firm builds it. The
construction process dominates the cost and time. In software projects,
an engineer writes a program that precisely defines the design and the
construction and calculation is done by a compiler – software that
takes the design and makes it machine-readable. Compilers operate in
a fraction of a second. Making software is, therefore, dominated by the
design time, and design is all about capturing the essential complexity
of a task.
This chapter will try to show where essential complexity comes
from, why computers can’t tackle this sort of complexity and, therefore,
why they can’t write software. Good news for programmers as this means
job security!
For a more thorough treatment of the mathematics read my paper
The Free Will Universe at www.jamestagg.com/freewillpaper.
James Tagg’s Home Page
“Computers are stupid. They can
only give you answers.”
Pablo Picasso
“Software is like sex: it’s better
when it’s free.”
Linus Torvalds
Silver Bullets
Can’t be Fired
Human brains are wonderfully creative things. We can compose
music, play golf, write novels, and turn our hands to all manner
of problems. Many people use their brains to write software. In
our modern-day lives we use software all the time: when we access the
web, type on a word processor or play a computer game. Software also
inhabits many apparently dumb devices. Modern cars contain dozens
of computers quietly working away; providing entertainment and
navigation, controlling the engine, and helping the car brake safely. In
my living room I count over a hundred computers. Many are tiny, like
the one in my TV remote control, while others are hidden as parts of
larger machines. The laptop on which I write has over twenty computers
inside it, besides the main Intel processor.
One thing all these computers have in common is that a human
being sat for many hours writing their software. Software is formal logic
written in something resembling English.
If I go to my ATM and try to withdraw cash, a programmer will
have written out the logic for the transaction as a set of rules.
When I put my bankcard in the slot, and type in my PIN, a line of
software will ask: If the bank balance of ‘James Tagg’ is less than twenty
dollars and I have pressed ‘withdraw’ for an amount in excess of twenty
dollars, then display, “We are sorry we cannot process the transaction at
this time.” and return the card. There seems to be an unwritten rule that
the things a computer says should be accurate but unhelpful!
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Alice, Ted and Software Specification
It would have been much more helpful if the computer had said,
“You do not have enough balance in your account.” And, it would have
been more helpful still if it had asked whether I needed a temporary
overdraft. However, such a feature needs many more lines of software
and this is time-consuming to write.
Software takes time and is expensive, because it has to be written
in a general-purpose way. Any name could substitute for James Tagg,
and any amount could be used. After all, it would be useless if an ATM
machine could only give out $20 to one person. The generalization of
software makes use of variables instead of fixed values and this renders
it hard to understand. Wherever we meet an idea that needs to be
generalized, a letter must be used instead of a fixed value. Computer
programs tend to look like this: if ‘a’ wants to do ‘b’ with ‘c’ then allow
it only if ‘d’ is greater than ‘c’. The software programmer has to keep
track of all the possible values that could be inserted into each of the
variables and make sure each and every combination would make sense.
My ATM scenario gets complex quickly. It needs to be able to answer a
range of questions for all the bank’s customers, deal with any amount of
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money and handle security when communicating with foreign banks is
necessary. A human being must write lines of code for all the rules and
every exception, making provision for any gibberish that might be typed
in by the customer.
Many people ask, “Wouldn’t it be great if my computer could write
software for me? Humans could sit back and put their feet up.” While
most people don’t actually believe this could happen, they will often ask
why we can’t specify software exactly and use unskilled people to write
it. Both proposals fundamentally misunderstand the nature of writing
software.
What do Programmers Do?
A human software programmer can write up to 1000 lines of code per
day. At the beginning of a project, when the work is unconstrained,
programmers write fast. Things slow down once programmers encounter
the enemy: the real world. By the time the code is complete and selling
in shops, the productivity of a programmer can be as low as one line
of code per day. This is staggeringly low and luckily only applies to big
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commercial software, equating to about 10 words per day. A good typist
types at 80 words per minute and most programmers are reasonable
typists. So software writers in a big project spend only a minute or so
per day in the act of writing. The rest is taken up by meetings, process
discussions, email, reporting and so on. In projects that avoid much
of this administrative overhead, good software programmers reach a
long-run average of about 225 lines per day. This has been the level of
productivity on the products I have developed in the past. These projects
were lucky. They had a single team on the task from beginning to end and,
in general, the projects took few wrong turns. Still these programmers
were spending only 10-20 minutes of each day on actual programming.
What were they doing the rest of the time?
In the early days of programming you might have a great idea,
but the process of turning this idea into software was immensely longwinded.
I learned to program at Manchester University in the 1980s. The
enormous machines in the basement of the computer building provided
heat for both our building and the mathematics tower next door. We were
not permitted to play with these basement monsters but were ‘privileged’
to submit instructions to a mini computer in the undergraduate section
– a PDP11-34.
For those of you not acquainted with computers I can tell you the
process of writing software in the 1980s was immensely tedious. To
add two numbers and display them on a screen took a month of lab
time, using detailed instructions written in machine code. Everything
was manual, including writing your code out in pencil on special paper
with little numbered squares and then giving it to someone to type in
overnight! You would return the next day to discover whether you had a
usable program or a something riddled with errors. If you found an error,
it would require editing. This was nothing like using a modern word
processor. The online editors of the day were the ultimate in annoying
software. If you misspelled a word, you would need to count up the letters
and spaces manually on a printout and enter a command – replace letter
27 of line 40 with the character ‘r’. Each and every typo would take five
minutes to correct. I managed to finish the simple program required for
course credit – I think it displayed an eight-digit decimal number – and
ran for the hills. In my second year I bought a PC and decamped to
the physics department next door where I remained for the rest of my
undergraduate life.
The PC revolution provided programmers with a new and intuitive
software creation environment where almost all the tedium was removed.
A wealth of tools for creating software was pioneered by Bill Gates of
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Microsoft and Philip Kahn of Borland, along with intuitive applications
such as the spreadsheet invented by Dan Bricklin and Bob Frankston and
made popular by Lotus Corporation. Today all computers have elegant
WYSIWYG, ‘What You See Is What You Get’ interfaces, where you drag
and drop elements into place on the screen. Over the last 25 years writing
software has sped up and stopped being tedious – becoming almost a joy!
In No Silver Bullet, Brooks explains that writing software can’t be
accelerated any further because all the tedious mechanical tasks have
already been removed. Remember his analogy: Writing software is like
building a house, but with some important differences. With a house,
an architect handles the design and then turns over construction to
a building company. Construction takes an appreciable time, more
time than the design and quite a bit more effort. But in software the
construction is totally automated. When we complete the design for a
piece of software we press compile on the computer and the software
is built and tested automatically in a matter of seconds. Speeding this
process up any further would make only a tiny improvement in the
overall software creation time, since the process is already 99% design
and 1% building. For the most part, the creative process of writing
software cannot be improved through mechanical means.
This is not always the case. I recently upgraded the machines for
some developers I work with. We added solid state hard drives. Compiling
a program now takes only 10 seconds, compared with 6 minutes before.
Because programmers nowadays tend to compile their programs very
regularly we estimate this saves them as much as an hour a day. This is
the only real innovation I have seen in the build phase of software in the
last 5 years, and it’s arguably not an innovation at all. We just forgot to
keep on top of the build time and allowed it to get out of hand.
You might argue some counter examples. Modern software design
suites let you drag and drop things on the screen to make applications
or build a website. Two hundred million people have managed to put
together WordPress websites using this technique. These are mechanical
procedures for solving a programming task and seem to contradict my
argument. They allow us to lay out graphics, press a button and turn the
design into software. But they perform very simple tasks. The computer
simply notes the coordinates of each box on the screen and places those
numbers into a file. The process is entirely mechanical and could be
performed by a clerk with no programming knowledge following a set
of rules. The computer just does it faster. I did the clever work; I had the
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idea for the software, I came up with the idea for the interface, I decided
where to place the boxes, and I chose all the colors, fonts and graphics. I
did all the creative bits!
So, now we know what programmers do all day. They create!
Origins of Software
Alan Turing first described the modern day computer in a paper presented
to the London Mathematical Society in 1936. He was not trying to invent
the computer. That was a by-product. He was trying to solve a puzzle that
had been troubling mathematicians for 30 years: The Decision Problem.
David Hilbert set out the challenge during a public lecture to the
French Academy of Science in 1901, marking the turn of the century.
Rather than give a boring lecture extolling the virtues of scientists, he
decided to give his audience a list of all the puzzles mathematicians were
stumped on.
Rather like the XPRIZE of today, he presented the problems as
a series of challenges. Sadly for the mathematicians of his time, there
were no million dollar prizes on offer, just a moment of fame and the
adulation of their colleagues. Each challenge was given a number. The
list included many famous puzzles; the Riemann Hypothesis, the puzzle
of Diophantine Equations and the Navier Stokes Hypothesis, to name
only three. A group of these questions were to coalesce into what we now
know as the Decision Problem.
The Decision Problem is very important to computer science
because it asks whether an algorithm can be written to automatically
discover other algorithms. Since all software is itself algorithmic you
could rephrase the question: Can software write software? This might
seem esoteric. But, if you are a computer scientist, it is an important
question. If we could solve all mathematical problems automatically
we would not need mathematicians anymore. And, since programs are
applied mathematics, the same goes for computer programmers.
Before you breathe a sigh of relief because you are neither a
mathematician nor a computer scientist, you should remember it is
possible to describe all knowledge using numbers. That’s what your
iPhone does when it stores music. If everything can be represented by
numbers, then a fast-enough computer could use an algorithm to create
everything! You really could set Douglas Adams’ Ultimate Question of
Life the Universe and Everything before a computer and it would come
up with the answer – presumably extrapolating the existence of rice
pudding and income tax along the way.
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Algorithms
Back in the 1930s no mechanical system could perform a calculation
with any speed. People still used pencil and paper for most things; the
newly-invented mechanical cash registers were slow and could perform
only one calculation for each crank of the handle. If you wanted to
calculate something complex, you had to employ a computer: a person
who could do mental arithmetic enormously fast. Richard Feynman’s
first job was computing for the Manhattan Project. The question was:
Could a computer, either mechanical or human, blindly follow known
rules to decide all mathematical questions? Hilbert’s 10 th Problem asked
this question of a particular type of mathematical expression – called a
Diophantine equation.
Hilbert’s 10 th Problem
“Given a Diophantine equation with any number of unknown
quantities, devise a finite process to determine whether the
equation is solvable in rational integers.”
David Hilbert
Diophantus lived in ancient Persia – now Iran. His son died young
and Diophantus was so consumed by grief he retreated into mathematics.
He left us seven books of mathematical puzzles – some he devised himself
and some of them taken from antiquity. The puzzles look deceptively
simple and are all based on equations using whole numbers. His most
famous puzzle is set in a poem which tells how old Diophantus was when
he died. Can you solve it?
“Here lies Diophantus,’ the wonder behold. Through art algebraic,
the stone tells how old: ‘God gave him his boyhood one-sixth of
his life, One twelfth more as youth while whiskers grew rife; And
then yet one-seventh ere marriage begun; In five years there came
a bouncing new son. Alas, the dear child of master and sage, after
attaining half the measure of his father’s age, life chill fate took him.
After consoling his fate by the science of numbers for four years, he
ended his life.”
Diophantine puzzles look straightforward. Hilbert asked if these
problems could be solved by a mechanical procedure, in modern terms,
by an algorithm. To show you what is meant by this, allow me to take you
240 Are the Androids Dreaming Yet?
Long Multiplication
back to your childhood. Do you recall being taught long multiplication
at school? Take a look at the next illustration and it will all come flooding
back. Once you learn the process of long multiplication you can follow
the rules and get the right answer for any similar problem every time. To
do this, you lay out the calculation in a particular format and apply the
logic. Multiply each number by a single digit of the other number and
then add the results together.
Diophantine problems are a little more complex than long
multiplication and some of them are a bit abstruse. But there is one
very famous Diophantine problem we can all recite. “The square on the
hypotenuse is equal to the sum of the squares of the other two sides.” The
equation for a Pythagorean triangle.
The theorem applies to right-angled triangles and there are sixteen
whole number solutions, known as Pythagorean triples; three, four, five;
is one example.
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Purists may protest that Fermat’s Last Theorem isn’t strictly
Diophantine because it refers to a variable exponent – the x to the n
part. This is hair splitting. But, of course, the splitting of hairs is bread
and butter to a mathematician. We will see later that Fermat’s Theorem
can be made Diophantine, but we are jumping ahead of ourselves a little.
A question that taxed mathematicians for many centuries was
whether there are triples for higher powers, such as cubes. In other words,
would the cube of the hypotenuse be equal to the sum of the cubes of the
other two sides for some set of numbers? After much work, it was proven
no triple exists which can solve the cubic equation. But what happens if
we substitute higher indices?
The next shape to consider is the hypercube – a four-dimensional
cube. That may stretch your visual imagination but the equation is simple,
3 4 +4 4 ≠5 4 . Again the challenge is to find a whole number solution for:
Hypercube
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“The hypercube of the hypotenuse is equal to the sum of the hypercubes
of the other two sides.” A picture of the hypercube might help you
visualize things.
It’s quite difficult to get your head around this shape because it is
hard to think in four dimensions. This seems strange because we have no
problem seeing in three dimensions on flat, two-dimensional paper – it’s
called a picture, but four dimensions on flat paper appears to stump us.
Again there is no solution for a hypercube: no Pythagorean triple exists.
Fermat’s Last Theorem asked whether this inequality for the cube
and the hypercube is true for all higher dimensions – for the hyperhypercube,
the hyper-hyper-hypercube and so on. Tantalizingly, he
claimed to have found a proof but wrote that it was too large to fit in
the margin of his book. It’s partly due to this arrogant annotation
that it became the most famous puzzle in mathematics, frustrating
mathematicians for nearly 400 years.
Hilbert’s question back at the turn of the 20 th century was whether a
machine could find a proof of this conjecture by following a mechanical
procedure, similar to our long multiplication example above.
The puzzle was eventually solved in 1995 by Andrew Wiles, a mere
358 years after Fermat claimed to have solved it. Wiles’ proof runs to
eighty pages of densely typed mathematical notation – considerably
larger than the margin in which Fermat claimed his proof did not quite
fit! There is an excellent book by Simon Singh – Fermat’s Last Theorem –
that tells the whole story.
We now know for certain, thanks to Wiles, that the answer is ‘no’.
There are sixteen answers to the two-dimensional triangle puzzle but
there is none for any higher dimension all the way up to infinity. How
might a computer tackle this problem and find a proof?
A computer could apply brute force and try many solutions; every
combination up to 100 million has already been tried and no exception
found. But, mathematicians are haunted by big mistakes of the past.
There were theories they imagined to be true until someone discovered
a counterexample. This sort of thing dogged prime number theorems.
Mathematicians don’t like to look foolish and are suspicious of
practical answers, “Well, I’ve tried it and I can’t seem to find an exception.”
This sort of argument does not wash with them. That’s what engineers
and physicists do. Mathematicians are better than that!
Mathematicians want definitive answers; “It is certain no solution
can exist”, and these sorts of answers require an understanding of the
problem to see why no solution could exist. That’s a very high bar. What
we need is a program that, rather than mechanically trying every possible
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combination, takes our problem and definitively says, “Yes, there is a
solution,” or, “No, there is not.” There are plenty of man-made proofs of
this nature. Pythagoras’s proof there are an infinite number of primes
is an example. Pythagoras did not have to try every prime number. He
simply understood the nature of prime numbers and gave us a logical
reason why it is so.
Mathematicians love a general solution. One way to solve Hilbert’s
10 th Problem would be to find a single mechanical way to solve every
problem. If you could solve every possible problem, you could certainly
solve Hilbert’s 10 th Problem. It turns out there is a way to test whether
every problem has a mechanical solution – pose the Halting Question.
The Halting Question
I should say for a little historical color that the Halting Problem was not
called that by Turing. The name was coined much later, in the sixties, by
Martin Davis. Turing knew the problem by the less catchy name of the
“not crashing” problem, or as he preferred, “Being circle free”, meaning
the program did not get caught in an infinite loop.
To understand halting we should imagine a brute force program
stepping through all the possible solutions to Fermat’s problem. If there
is a solution this stepping program will eventually halt and answer ‘true’.
If there is not, the program will run forever. Can we predict a program
will not run forever? At first pass this is hard. We can’t watch it forever
and say, “It never halted.” So is there a clever way to do this? An algorithm
perhaps?
The Answer to the Ultimate Question
The answer is ‘No!’ In 1936, Alan Turing proved there is no generalpurpose
mechanical way to tell whether a program is going to find an
answer at all, much less what the answer is. This means Hilbert’s Decision
Problem has no solution; there is no general purpose algorithm which
will discover all mathematical theorems.
Turing succeeded in proving this by turning the problem on its
head. He proved that a crash detection program is unable to see whether
it will crash itself. Since you cannot tell whether a program will crash
– and by this I mean go into an infinite loop – you cannot tell if it will
halt. He used the simple argument that since you can’t tell if the crashing
program will halt, you have already proved you can’t predict if every
program will halt.
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Impossible Shape
That is Turing’s argument in a nutshell. But if that was too large a
step, let’s take the argument a little more slowly and prove it a couple of
different ways. First, we will use a proof by counterexample, known by
mathematicians as an ‘indirect proof ’. These may tax your brain. If you
want a visual image to help with the idea of an indirect proof, take a look
at the impossible shape. It is paradoxical, which means it does not exist.
QED.
The Proofs
There are several ways to prove the non-existence of the Halting Program.
I am going to present a few in the hope one of them will hit the mark and
allow you to see why. The first proof uses a software flowchart. I have
laid this out on the assumption the program exists and then attempted
to apply it to itself. Unfortunately, the flowchart contains a paradox
and thus there can be no Halting Program. The paradox is at once
straightforward and confusing. It is a more elaborate version of the liar’s
paradox: “This sentence is a lie.” If the sentence is true it must be false,
and if the sentence is false then it must be true.
The Halting Program
Let us suppose there is a Halting Program. Remember that a Halting
Program simply takes another program as input and predicts if it will
halt or not. It follows there must also be a program called Haltcrash.
Haltcrash goes into an infinite loop if it examines a program with input
that halts, otherwise it halts itself.
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Halting Flowchart
Now we create a third program called RunMe. RunMe runs
Haltcrash on itself. Still following this? Now execute RunMe with RunMe
as its own input. What happens? The analysis is as follows:
1. RUNME started on input RUNME halts. If RUNME started on
RUNME halts, then Haltcrash started on RUNME with input
RUNME halts. If Haltcrash started on RUNME with input
RUNME halts, then HALT decided that RUNME started on
RUNME does not halt!
Therefore,
RUNME started on input RUNME halts implies that RUNME
started on input RUNME does not halt. (contradiction)
2. RUNME started on input RUNME does not halt. If RUNME
started on RUNME does not halt, then Haltcrash started on
RUNME with input RUNME does not halt. If Haltcrash started
on RUNME with input RUNME does not halt, then Halt decided
that RUNME started on RUNME halts!
Therefore,
RUNME started on input RUNME does not halt implies that
RUNME started on input RUNME halts. (contradiction)
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Both analyses lead to a paradox! There is only one way out. There
can be no halting procedure. I’m sorry if this is quite convoluted.
Philosophical Proof
If you find these technical proofs difficult to follow, it may be easier to
examine the problem philosophically. Consider the consequence of
the existence of a Halting procedure. A Universal Turing Machine is a
relatively small program. Roger Penrose gives a three-page example in
The Emperor’s New Mind, and Stephen Wolfram has implemented one
using a cellular automaton with as few as five component parts.
A Halting Program running on such a machine should be able
to compute all the knowledge in the Universe. Every structure, every
work of literature, every galaxy could be the output of this single, simple
program. My pocket calculator could, theoretically, paint like Picasso
and compose like Mozart. All art, knowledge and science would be
entirely determined in our Universe and we would have no free will. If
you philosophically rebel against this then the Halting Problem must
have no solution.
Gödel’s Insight
Another way to understand this conundrum is through the earlier work
of Gödel. Solutions to mathematical puzzles are neat, orderly sequences
of statements where the problem is solved step by step. Computers are
good at step by step processes. Surely a computer could simply proceed
in a painstaking fashion to check all the possible combinations of words
and symbols to discover a proof.
An analogy might be trying to find your hotel room if you have
forgotten the number. You could simply find it by trying every room.
As you progressed through each floor, you would try every corridor
and retrace your steps to the main hallway before attempting the next.
Eventually you would succeed.
Finding proofs of theorems is often understood to be the same sort
of task: search systematically through all the numbers and you will find
the solution. But this is not so: There is a hidden problem.
Although it is true to say problems and proofs can be described by
numbers, they are not simply related like a lock and key. We need the
first number to translate into a set of symbols meaning something about
mathematics: for example, that x squared plus y squared equals z squared
but for higher powers there is no equality, and the second number to
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denotes a set of sequential steps we can apply to demonstrate this fact.
These steps must have meaning and obey the rules of mathematics, but
what are these rules? Are they written down in a text book?
It turns out there is no way to find this set of rules; it is a superinfinite
task. We would need to reach into our infinite bag of numbers
and pull out rule after rule, turning each into a mathematical model
that explains numbers and logic and what can be done with them to
form mathematical statements. The number of ways to do this is not just
infinity, but two to the power of infinity. This is the number of ways to
permute all possible mathematical rules.
Your mind may be rebelling at this. Surely, if I have an infinite
set of numbers I can just pluck all the numbers from my bag and then
I am certain to have the solution. Unfortunately, it turns out there is
no complete, consistent set of rules; no valid dictionary that maps all
numbers to all of mathematics. That is Gödel incompleteness theorem.
Despite a fundamental limit on mapping all numbers to all of
mathematics, there might still have been an algorithm which could
practically find solutions for a given arbitrary problem. Turing proved
this is not the case.
The Wiles Paradox
Turing showed us there can be no general purpose, mechanical procedure
capable of finding solutions to arbitrary problems. A computer program
cannot discover mathematical theorems nor write programs to do so. Yet
computers regularly solve problems and generate programs. That’s what
software compilers do. This seems to be contradiction.
The solution to this apparent contradiction is to propose a boundary:
a ‘logic limit’ above which computers may not solve problems. With a
high boundary a general-purpose machine could solve most problems
in the real world, though some esoteric mathematical puzzles would be
beyond it. But if the boundary were low, many activities in our daily life
would need some sort of alternative, creative thinking. It is crucial to
know where the logic limit lies.
The Logic Limit
Amazingly, in many branches of science it is possible to pinpoint the exact
location of the logic limit, but finding that boundary in mathematics has
taken forty years work from some of the greatest mathematicians of the
20 th century.
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The story starts back in the 1940s at Berkeley University with a
young Julia Robinson, one of the first women to succeed in the previously
male-dominated profession of mathematics. By all accounts, she had a
wry sense of humor. When asked by her personnel department for a job
description she replied: “Monday—tried to prove theorem, Tuesday—
tried to prove theorem, Wednesday—tried to prove theorem, Thursday—
tried to prove theorem, Friday—theorem false.” Like Andrew Wiles, she
fell in love with one of the great mathematical puzzles, and although she
made great strides, the problem passed from her to Martin Davis for the
next steps.
The final elements were put in place in the 1970s with the work of
another young mathematician, this time a Russian – Yuri Matiyasevich.
Robinson wrote to him when she heard of his proof, “To think all I had
to do was to wait for you to be born and grow up so I could fill in the
missing piece.” The complete result is the Robinson Davis Matiyasevich
theory which sets out the limits of logic and algebra. What, you may ask,
do we mean by logic and algebra?
Mathematicians like to turn everything into logical statements, even
ordering a round of drinks! The discipline of logic emerged from ancient
Greece as the study of language. The starting point was the syllogism:
Statements such as, “All cows eat grass.” or Lewis Carroll’s assertion,
“There are no teachable gorillas.” Over time the study of logic became
ever more precise with, for example, the introduction of variables and
equations; a=all cows, b=some grass. The formula “a eats b” translates by
substitution into, “The cows eat the grass.” This doesn’t look much like a
step forward but, trust me, it is.
The modern way to represent logic is using prenex normal form.
This mouthful simply means separating relationships between things
from the things themselves. The following four statements say the same
thing, each in a more formalized way.
Speech: Harry loves Sally
Logical: x loves y (substitute Harry for x and Sally for y)
Formal: There exists an x, there exists a y (x loves y)
Prenex:
∃x∃y (x R y), Where R, the relationship, is ‘loves’
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The final example is in prenex normal form. The symbol ‘∃’ means
‘there exists’ and R stands for relationship in this equation. All logical
statements can be translated into this form using a purely mechanical
process. There is even a website that will do this for you. It’s useful but I
don’t recommend it as entertainment!
In the example above, something exists in relation to the existence
of something else: one person who loves another. Give me a name and I
can look up the person they love. This is simple. A computer can easily
solve such problems. Indeed there are hundreds of websites doing this
every day. Once you’ve solved one problem of this type, you have solved
them all.
We can rearrange Diophantine equations into many different
prenex forms. The simplest form might be, ‘there exists an x which solves
the following equation, x equals three.’ This would be written out as ∃x,
x=3 and is of the ∃ class – ‘there exists’. There are slightly more complex
classes than our simple ∃ relationship: ∀∃∀ ‘for all, there exists for all’ or
the class ∀ 2 ∃∀ ‘for all, for all, there exists, for all’. Each of these groups of
equation is called a ‘reduction class’.
One way to think about a reduction class is as a problem in topology,
‘knots’, to non-mathematicians. Imagine someone handed you a bunch
of tangled cables – the sort of mess you get when they are thrown
haphazardly into a drawer. You can tease them apart and rearrange
them but you must not cut them or break any connection. Once you
have done this you will be left with a series of cables on the desk. They
are all separate, looped or in someway knotted together. Each cable has
a fundamental topological arrangement: straight cables, granny knots,
figure eight, and so on. You have reduced them to their simplest form,
their logical classes. The same goes for logical statements. Once you
have rearranged logical statements into their simplest form you can lay
them out and group them together according to their complexity. Each
group makes up a reduction class and you can ask whether that class as a
whole is automatically decidable. It is a huge task to untangle and classify
mathematical problems, and it took Robinson and her colleagues nearly
forty years to succeed.
It turns out problems with a form as simple as ∀∃∀ (for all,
there exists, for all) have no general purpose algorithm. Each must be
examined individually and solved by something that is not a computer.
This is a remarkable result as the logic boundary is set quite low. An ∃∃,
(exists, exists), class of problem is automatically solvable by a general
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algorithm, but a ∀∃∀, (for all, there exists, for all), is not. Each individual
type of problem within the class must be examined with insight and
understanding.
Our lives are full of problems – playing chess, finding a mate,
designing space ships and simply getting to work in the morning.
Imagine we expressed everyday problems as logical problems. Where is
the logic limit for life? We have no answer for this yet, but we do know
the logic limit for computing; it is given by Rice’s Theorem.
Named after Henry Rice, and proven in 1951 as part of his doctoral
thesis at Syracuse University, Rice’s Theorem states: “No nontrivial feature
of a computer program can be automatically derived.” You cannot tell if
a program will halt with a given input. You cannot tell if one program
will generate the same output as another. You cannot tell if a simpler
program could be written to do the same task as a more complex one.
In fact, no nontrivial thing can be proven. This means the logic limit in
computers is low, and computer programmers have job security.
For Programmers
For the programmers amongst you, here are some of the things that
cannot be done automatically even given infinite time:
• Self-halting Problem. Given a program that takes one input,
does it terminate when given itself as input?
• Totality Problem. Given a program that takes one input, does it
halt on all inputs?
• Program Equivalence Problem. Given two programs that take
one input each, do they produce the same result on every input?
• Dead Code Elimination. Will a particular piece of code ever be
executed?
• Variable Initialization. Is a variable initialized before it is first
referenced?
• Memory Management. Will a variable ever be referenced again?
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Can humans solve ‘unsolvable’ problems?
The question of whether Fermat’s Last Theorem could be solved
mechanically remained unanswered until 1970 when Yuri Matiyasevich
filled in the missing piece in Julia Robinson’s proof. Matiyasevich used
an ingenious reduction method to match up sequences in Robinson’s
theorem with a set of Turing machines. This showed that if Robinson’s
theorem was false you could solve the halting problem and since you
can’t solve the halting problem, then Robinson’s theorem must be true.
All this effort proved Diophantine equations have no general algorithmic
solution. This was a hugely important result but, as we noted earlier,
Fermat’s Last Theorem is not, strictly speaking, a Diophantine. It is an
exponential Diophantine equation. We still had no definitive answer to
Fermat.
In 1972 Keijo Ruohonen and again in 1993, Christoph Baxa
demonstrated that Diophantine equations with exponential terms could
be rewritten as regular Diophantine equations with one additional
complication – the necessity of adding an infinite set of terms to the end
of the equation. In 1993, J.P. Jones of the University of Calgary showed the
logic limit for regular Diophantine equations lies at thirteen unknowns.
Matiyasevich had already pointed this out but never completed his proof.
Since infinity is greater than thirteen, all exponential Diophantine
equations are above the logic limit and, therefore, undecidable. Finally,
we have a proof that Fermat’s Last Theorem is unsolvable by a computer
– or at least by a general purpose algorithm running on a computer.
Matiyasevich went on to show many mathematical problems can be
rewritten as exponential Diophantine equations and that much of
mathematics is undecidable. For example, the Four Color Conjecture:
“Given an arbitrary map on a Euclidean plane, show the map can
be colored in a maximum of four colors such that no adjacent area
shares the same color.”
Meanwhile, Andrew Wiles, an English mathematics Professor at
Princeton had been secretly working on Fermat’s Last Theorem. When
I say secretly, he had not told anyone in his department, and only told
his wife late in 1993 when he suspected he might have a solution. He
had been working on the problem a long time, having fallen in love with
it at the age of 8! In 1995, after nearly 30 years work, he announced he
252 Are the Androids Dreaming Yet?
Four Colors is All You Need
had found a proof. He had solved an unsolvable problem, a problem that
could not be answered by using a computer. Therefore, Andrew Wiles
cannot be a computer!
As with all real-life stories, it was not quite as neat as this. It turned
out Wiles’ initial proof had an error in it, identified by one of his referees.
Wiles had made an assumption about a particular number theory that
had not been proven: it was still a conjecture. Working with another
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mathematician, he managed to prove this conjecture and so, two years
after first announcing that he had solved Fermat’s Last Theorem he could
finally lay it to rest.
The Special Purpose Objection
Before I declare mankind’s outright victory over computers, the Special
Purpose Objection must be overcome. The objectors would argue that
Wiles is a Special Purpose computer. Special Purpose computers are at
no risk of breaking the Turing limit when they solve problems they have
Theorem (Undecidability of Hilbert’s tenth problem)
There is no algorithm which, for a given arbitrary Diophantine
equation, would tell whether the equation has a solution or not.
been programmed to answer. The objection misses the key point. I am
not arguing having a solution to a given mathematical puzzle presents a
difficulty to a computer; I am arguing a computer cannot discover one.
Take, for example, the search engine Google. If I type “where can
I find the proof of Fermat’s Last Theorem?” into the search box, it will
retrieve a PDF of the proof as the third result. It appears this special
purpose computer solved the problem. But you immediately see the
difficulty. Google search already knew the answer, or more precisely had
indexed the answer. The computer was not tackling a random problem
from scratch. It was tackling a problem for which it knew the answer, or
at least where an answer could be found. There is no sense in which the
search engine discovered the proof.
To really understand this objection we need to examine exactly
what Turing and Matiyasevich proved.
An arbitrary problem is one you do not already know the solution
to when you write the algorithm. You can think of it as a variable. Is
there an algorithm that can solve problem ‘X’? The alternative is a special
program. It can solve problem Y. Y is a problem it knows. It must have
the solution coded somewhere within it in a computably expandable way.
You might think of this as a table of constants; problem Y has solution
1, problem Z has solution 2, and so on. But it could be more subtle than
that. Problem Y might have a solution which is encrypted so you cannot
recognize it within the program, or it might even be the result of some
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chaotic equation so complex that the only way to see it is to run the
program and watch the output: no form of program analysis will give
you any clue as to what it produces. There is only one stipulation. The
answer to problem Y MUST be held within the program as a computable
algorithm. Put another way, the computer must already be ‘programmed’
to answer the question.
Could a human mathematician be pre-programmed from birth?
Yes, there is no fundamental objection to this. Mathematicians could be
born to solve the problems they solve. But this would present a couple of
issues. Where is this program stored? And who, or what, programmed
the mathematician? Could we perhaps find an experiment to determine
whether mathematicians are pre-programmed?
One view held by philosophers is that the Universe programmed
the mathematician. They believe we live in an entirely determined
Universe with no free will. There is then no mystery as to how Andrew
Wiles came up with his proof. He was destined to do it from the dawn of
time. The ink that fell from his pen to the paper was always going to fall
in just that way. We live in a clockwork Universe and although we might
feel we have free will, this is an illusion. I simply don’t believe this. If I
am right and humans do exercise free will, Andrew Wiles cannot be a
computer. And because Andrew is not alone in discovering proofs, those
mathematicians cannot be computers either. Humans are, therefore, not
computers.
The Chance Objection
I said there was no automatic way to solve any problem above the
logic limit, but this is not quite true. There is one automatic method
you could deploy to generate a non-computable proof, the infamous
‘monkeys and typewriters’ idea where we use random chance to generate
information. Many people have suggested it is possible to write a play
such as Shakespeare’s Hamlet by simply typing random characters until
we happened upon the play. The argument is flawed.
The first flaw is the process would take a super-astronomically
long time. Even if every atom in the Universe were a monkey with a
typewriter, it would take orders of magnitude longer than the age of the
known Universe to come up with the script to a play or a mathematical
proof.
The probability of finding a solution to Fermat’s Last Theorem
by chance is about 1 in 10 50,000 . That’s 1 with 50,000 zeros after it. For a
comparison, there are only 10 120 atoms in the known Universe. To be, or
not to be, certain of finding the proof, you would need to run a computer
long enough to calculate all the possible proofs up to the length of Wiles’
solution. Currently, a computer using every particle in the Universe
clocked at the Plank interval – the fastest conceivable computer running
at 10 34 operations per second – would take 10 500 times the age of the
known Universe to do this. If someone tells you this is astronomically
unlikely they are making a huge understatement. A computer running
until the end-of-time would only scratch the surface.
The second flaw is even more damning. Even if the monkeys
succeeded in generating something interesting, something else needs to
spot this. If an algorithm stumbled upon a proof of Fermat’s Last Theorem,
what would recognize it as such? There are no ways to systematically
analyze proofs. There are no mechanical methods that understand these.
Dalek Trouble
“All non-trivial abstractions, to
some degree, are leaky.”
Spolsky’s Law
of Leaky Abstractions
Consequences
Machines cannot discover theorems using algorithms, yet
mathematicians do it all the time. Do the rest of us break the
logic limit? It seems we do. People appear creative – painting,
composing, sculpting and so forth. But, are these endeavors creative
in the mathematical sense. To prove this, ironically we need to find
something outside mathematics that is definitely non-computable. This
is tricky. Most artistic things are fuzzily defined and there are no written
rules we can apply. How can we prove a work of art could not have been
generated by a computer?
Trivial proofs exist but they are rather contrived. For example, it
would not be possible to make a film with a solution to the still unproven
Riemann Hypothesis on the blackboard in the background of a movie
scene. All the mathematics Good Will Hunting had been already
discovered before the movie was made. New mathematics cannot be
accidentally generated by a set designer – unless, of course, they also
happened to be a world class mathematician.
These trivial proofs might lead a mathematician to argue the theory
is proven. There are some artworks which cannot be computed. QED. But
these are not very satisfactory proofs. I could create almost any movie I
wanted without tripping over this rule. What I really wanted to know is
whether Good Will Hunting as a whole could have been generated by a
computer. Not that some weird version with a particular mathematical
proof on the blackboard is forbidden. Movies are a difficult subject for
258 Are the Androids Dreaming Yet?
this argument, but music is much easier to analyze. It is linear, highly
mathematical and largely uniform by culture and language. Yet it is
universally appreciated. Is music a computational or a creative endeavor?
Is Music Computable
To prove a piece of music is non-computable requires two tests. First to
show we can ‘reduce’ it to a problem that is already non-computable and,
second, to demonstrate it ‘looks like’ or ‘sounds like’ a piece of music. An
accountant would say it needs to pass ‘the smell test’.
The first non-computable problem to be studied in depth was
Emil Post’s Word Problem. Post was a contemporary of Alan Turing
and studied at the Institute of Advanced Mathematics in Princeton. He
solved the Halting Problem six months before Turing, but his proof used
a complex recursive method called the lambda calculus. Turing’s method
was far more practical, which is why we now refer to Turing machines
rather than Post machines. Later in his career, Post came up with a
branch of non-computable mathematics called ‘Post Problems’. They
look like a puzzle you might find in a newspaper. Imagine starting with
the word ‘camel’ and being asked to turn it into ‘aardvark’, using only a
few simple rules. We’ll make the problem very easy to start with: cam
↔ aard and el ↔vark. This solution is obvious; just do the substitutions
and you are there. But what if the rules were a little more complex?
Gennadií Makanin, a Russian mathematician based at the University of
Moscow, found a set of extremely simple puzzles that are nevertheless
non-computable. Here is one:
{“CCBB” ↔ “BBCC”, “BCCCBB” ↔
“CBBBCC”, “ACCBB” ↔ “BBA”, “ABCCCBB”
↔ “CBBA”, “BBCCBBBBCC” ↔
“BBCCBBBBCCA”}
Word Problem
Can a computer tell us which word problems have a solution and
which do not? The answer is ‘no’. Word substitution puzzles are a class
of non-computable problem. Martin Davis proved this in 1948. Using
a reduction argument we can use these word problems to prove some
music is also non-computable.
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Let us start by substituting the notes of the musical scale for the
letters of the alphabet to create a piece of ‘music’. Since it is a direct
analogue of the word problem, we have created a non-computable piece
of music. It is definitely non-computable, but is it music? If it just looked
like a random jumble of notes it would be unconvincing, but luckily there
are many forms of music that look exactly like a word substitution puzzle.
Bach’s Art of Fugue, the canons of Tudor composers such as William Byrd
and Thomas Tallis, and the works of Grieg all use sequences of chords
that move from one to the next using substitution rules. If you were to
listen to the steps in our word substitution music, they would definitely
sound musical. I think they should pass the main artistic criticism – that
they should not sound formulaic.
But is any actual human composition non-computable?
Unfortunately, we cannot prove whether a particular piece of Bach, Tallis
or Grieg is non-computable because we don’t know the specific rules
used to compose it. All we know are the general musical principles of
harmony and counterpoint that applied at the time. We don’t have these
composers personal rule sets because they were held in their brain and
they are, of course, long since dead. It is statistically likely that most pieces
are non-computable because there are an uncountably infinite number
of them, whereas computable pieces are merely countably infinite. But
that’s just probability; it is no proof.
I puzzled for some time whether there is a way to prove it but had to
conclude it is impossible. However, and this is how creativity works, once
I had given up on the problem, my brain continued to work on it. I was
not conscious of this, I was only aware that failing to solve the problem
annoyed me. I then had a Eureka moment. Although I couldn’t prove
a piece of music was non-computational, I could make one! – a piece
that could not have been created
using computation alone. This
requires me to inoculate your
brain.
Take either Andrew
Wiles proof of Fermat’s Last
Theorem or Alan Turing’s proof
of the Halting Problem; both
proofs are non-computable.
Each document is made up of
symbols, the Roman alphabet
and some special Greek symbols
such as α, β, ζ, and so on. Let us
Creative Inoculation

260 Are the Androids Dreaming Yet?
write out the symbols in a table and assign a musical note to each. It is
straightforward to put these notes into a synthesizer and play the piece
of music. I have provided a link to such a piece. Warning: once you listen
to this you will have been ‘creatively inoculated’.
This resulting piece of music, based on the transliteration of a proof,
is non-computable. You might immediately argue with this, “The piece
of music was translated from proof text to music file using a computer. It
is clearly computed.”, but this is not my point. The music could not have
come into existence in our Universe as a result of a computation. It is a
computable translation of a non-computable string. It could not have
been generated solely by a computer: It was done in two steps, the first of
which could not have been computed.
If, up to this time, our Universe has never contained a piece of
music that was generated non-computationally, it does now. If you listen
to this piece, you will find it impossible not to be somewhat inspired by
it. You cannot erase the experience from your memory. And once you
have heard it you will have been creatively inoculated. I have defeated
Daniel Dennett and his like, and given you creative freedom!
www.jamestagg.com/noncompmusic
Having made at least some music above the Turing limit I could
declare victory but I want to go further. Using the same reduction method,
I believe we can show all art is above the limit. First let’s attempt novels
and plays. Do you enjoy those crime novels by Agatha Christie and Colin
Dexter? It must be possible to construct a plot sufficiently complex, and
a murder sufficiently baffling that it exceeds the logic limit. I could keep
extending this idea to provide any number of examples and, therefore,
prove all art and creative output is above the logic limit.
There are many other arts we could apply this argument too. In
the visual domain there are non-computable images. In principle, it is
possible, to draw or paint things beyond the capability of a computer.
Roger Penrose has created non-computable visual puzzles such as tiling
an infinite plain with special jigsaw pieces. Creating an image containing
a solution to his visual puzzle is non-computable.
This extension argument also applies to me. There is an argument
that I am a finite being and therefore can be simulated by a computer.
Since I can be simulated by a computer, I am the same as a computer
and therefore incapable of non-computable thought. The argument is as
follows: James Tagg will have during his life a finite number of inputs and,
equally, a finite set of outputs. This means you could model me using a
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Jackson Pollock
computer. You could simply create a table of all the possible inputs and all
the possible outputs I would make and this would be a perfect facsimile
of me. A number of people have posed this as an argument to refute
Roger Penrose’s assertion that humans are capable of non-computable
thought.
But this analysis misses a key point. There is no way to calculate all
the contents of this table. My past could be tabulated. It is the history of
all the things I ever did, but my future cannot. I might yet discover some
great theorem that could not be computably generated. This would be
a part of my output which could not be generated by an algorithm or
any mechanical process. This forms a non-computational arrow of time;
we can write down the past, we cannot write out the future. If a creative
person such as Andrew Wiles could be simulated in advance, we would
have an automatic way to find a solution to Fermat’s Last Theorem. Since
this is not possible, it follows that creative people cannot be simulated.
This also means the Turing test is not passable by a machine. Humans
can create; machines cannot. That is the difference.
Will Computers Take over the World?
Ray Kurzweil, the American inventor and futurologist, has suggested
computers are getting exponentially faster and will soon reach such
immense power they became effectively infinitely powerful. They could
instantly answer any question posed and solve all our engineering
problems. He dubs this point ‘the singularity’: a point of near infinite
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Watson and Our Future?
computing power and therefore universal knowledge. This could herald
a Utopian future; global warming, cancer, all things of the past. But
computers might just as easily become bored and determine we humans
are the real problem. If we are lucky, they may treat us as amusing pets.
If we are unlucky...
These consequences might have come to pass if the answer to the
Halting Problem were ‘yes’, but as the answer is ‘no’! This is not the future
we face.
Mummy, where do Bugs Come From?
One consequence of the logic limit provides a theoretical basis for the
origin of computer bugs. The mention of ‘bug’ conjures up stories of
dead creepy crawlies stuck in early computer circuits, but the term had
been in use for over 150 years before the computer was even invented.
Bugs are not simply annoying mistakes.If you misspell my name as Stagg
instead of Tagg that’s just carelessness. Real flaws creep into a computer
program when you fail to understand Brooks’ essential complexity, or by
my terminology, you stray above the logic limit without realizing it.
Imagine we have created a piece of software. The software goes
into test and is subjected to a range of use cases. Some of these will fail
because we did not take into account all the real world possibilities.
Then a strange thing happens. We get trapped in a loop of patching the
errors in the program in a rather mechanical way. Find an error, patch
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it. Find another, create a work-around, and so on. By doing this, we are
effectively mechanically generalizing our solution. This is forbidden as
it breaks the Turing limit, so we can’t mechanically solve a general logic
problem above the logic limit. We need instead to use intuitive or creative
thought. In our panic we did not stop, take a step back and engage our
brain. Instead, we attempted, unsuccessfully, to blindly hack our way
through the problem.
If we eventually succeeded in perfecting the code this way, we
would have broken a fundamental law of the Universe. Something nasty
would have to happen to prevent it, such as rupturing the space-time
continuum or an event equally horrible! Luckily something prevents this
and keeps our Universe intact – BUGS! Bugs stop us breaking Turing’s
limit.
The next time you curse a bug, remember if they didn’t exist you’d be
in danger of causing a logical paradox. There is no problem in redefining
the domain and then creatively producing an all-encompassing design,
but, you can’t patch and hack your way there. This theory of bugs leads to
an explanation for some modern programming rules of thumb.
Written specifications are valuable because they force you to lay out
the whole problem. You don’t need to be detailed regarding the depth,
but should be expansive about the breadth, covering all the logical
complexity. This might result in many details as a by-product, but a
specification needs to delineate the edges of the problem space and not
simply focus on a few key points.
Writing the tests for the software in advance is helpful as it is likely
to tell you early whether your design encompasses the whole problem
space.
Also, building a prototype, throwing it away, and then building the
real thing can help greatly. It may be the only way to examine the edges
of the problem space in detail. Armed with a full understanding, you
can then imagine solutions to the complete problem in a single creative
sitting. Whatever techniques you use to improve the quality of your
software, remember you are engaged in a creative process that is not,
itself, open to automation.
The Art of Programming
Programming is an art: a creative endeavor. It is also, of course, highly
scientific. When you work with a good programmer – and I have been
fortunate to work with some of the best in the world – they all follow
a similar process. First they talk with you at length about your needs
264 Are the Androids Dreaming Yet?
Geek Humor
and examine the full scope of the problem space. Even if you say, “Oh
don’t worry about that bit,” they always will. They want to know about
everything. Then, they write a high-level list of features, some simple
block diagrams, and occasionally a flow chart, only then do they begin to
code, ticking off the list as they go. Sometimes, they will check to see if
their list is the same as your list but more often they will come back and
just check the high-level purpose. “If I give you something that achieves
this, will that do it for you?” They test as they code so you end up with is
something that meets your high-level purpose, and can prove it does so
in its own right. At the end of the coding they write out the specification
for the project so that they can remember what they did, or a colleague
can pick it up in the future.
This is not how students are taught. Students are told to write a
detailed specification at the start and then simply implement it. If you’ve
been following my argument, they are being taught to do something
impossible! There is no ‘just’ to programming. Sometimes teams are
even split up so that one person writes the specification and another the
code – again an impossible task. If the specification was the answer to
the problem, it must have required creative thought to develop and so
would be as complex as the program itself. Since it is not yet a program
you cannot test it, so it becomes an untestable solution to a creative
problem. Since the specification is not the answer but rather a general
list of tasks, the great danger is to give it to a separate programmer and
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they implement it mechanically. You see, of course, the problem. It
will be riddled with bugs because they have missed the creative step of
imagining the whole problem and solving it in the round.
This fundamental misconception of software is common in many
organizations. “Ah,” says the finance director, “I’ll write a very detailed
spec and then we can get someone cheap to just program it.” This does
not work. If the finance director has done the creative work of taking a
problem and turning it into a detailed specification for the programmer
to ‘just program’ – removing any ambiguity and therefore the creative
overhead – he will have all but written software himself, albeit in
a computer language of his own making. On the other hand, if the
specification is a linear list of issues with no creative thought, he will not
have reduced the time needed to program. He may have improved the
quality by effectively getting a second pair of eyes onto the requirements
gathering stage, but this does not help the programming effort itself.
Ideally, you should never split up specification and coding. It is a
creative process best handled by very small numbers of people working
intensively on it. Of course, there is one big problem with this: some
software tasks are huge. Before we look at the science of splitting up a
software project, it is worth pointing out that many of the most famous
projects were written by one man. I have met many of these people and
they are all exceptional – Linus Torvalds, Linux; Anthony Minessale,
FreeSWITCH; Daniel-Constantin Mierla, Kamailio; Eric Allman,
,SendMail. Before splitting a project between many people, it is worth
considering whether you can give it to just one individual. To do this you
266 Are the Androids Dreaming Yet?
will need to unload this person of ALL interruptions and administrative
burdens. This is the most effective way to solve a creative programming
task. Practically, once your task is over the limit for a single human, a
software project must be split up. This requires great care. Dividing a
problem efficiently means specifying the interfaces between them and
decoupling the components. This is the art of an architect or a producer
in the creative arts. The creative process operates similarly in other walks
of life. There are many examples of successful creative duos – Rogers
and Hammerstein (The Sound of Music), Ben Elton and Richard Curtis
(Blackadder).
Good managers, therefore, find ways to break projects into
manageable sub-projects that can be worked by pairs or rely on single
super-programmers with support around them. If you are lucky enough
to gather together a group of super-programmers and can divide a
problem efficiently amongst them, you can achieve great things. You
see this pipeline in movie production. A script writer generates a script
creatively. The casting director finds the actors, a director is in charge of
filming, and an editor puts it together. In very great movies you will often
find a great director or producer who had a hand in almost everything
holding it all together. They are often accused of micro-managing but
you can see that’s what they must do. They are the super-programmer
with the whole creative work in their head, and an eye on the audience
and financial backers.
If you talk with great programmers you will be amazed by their
breadth of technical, commercial and product knowledge. Why do they
need all this commercial information to do their job in the round?
Rules and Tips
I began writing some rules on how to split up a project, and almost
immediately ran into exceptions and special cases. The job of dividing
things into sub-tasks is, itself, a creative problem and must not be done
mechanically. Any ‘one size fits all’ rule will fail and you must apply
domain knowledge and careful thought to the process.
It is the job of architects or a senior engineer to split projects into
smaller chunks. To do this they must accurately ‘guess’ boundaries
between subtasks to create self-contained, creatively solvable problems.
This can be done by either vertical or horizontal abstraction. Both have
their problems.
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Horizontal abstraction is the simpler of the two to understand,
and the more common. Computer systems are built ‘on the shoulders of
giants’. That is to say we no longer need to place individual pixels onto
the computer screen. We can assume a computer will draw a square if we
specify the dimension and coordinates of the center. That’s abstraction.
Today’s computers are even more helpful. We can ask them to draw a
rotating cube lit from a certain angle and the computer will do the whole
job for us. But, there are always practical limitations to this.
I want my cubes to move around the screen naturally but I am not
sure what physics model has been implemented. What will happen when
they bump into each other? If the abstraction is not thoroughly thought
through they pass through each other in a very odd way, breaking up
and showing me they are really made of triangles, the illusion of three
dimensions is lost. Whenever we work at an abstract level, we risk being
exposed to its inner guts at some point. Joel Spolsky, a computer scientist
who worked on Microsoft Excel, proposed the Law of Leaky Abstractions
to explain this. An example of his law in action is the TCP/IP protocol
stack that transports data over the Internet. The stack is hugely reliable,
yet I have to debug one of these stacks at least four times a year!
The problem is that the TCP (Transmission Control Protocol) is
designed to provide reliable delivery of information: internet pages,
my bank account and the like. But, the internet protocol ‘IP’ on which
it relies is only designed for best-efforts. When a link loses a packet of
information, the TCP has to retransmit it. This takes additional time. TCP
provides an abstraction of a reliable connection, but the implementation
is not as robust as it may seem, and the details leak through as variable
latency and throughput. This explains why your web pages sometimes
do not completely render. You are told it is reliable, but often it is not!
Experience is so valuable to a programmer because they know which of
these specifications to take with a pinch of salt and when they are likely
to leak. They are battle scarred by previous naivety.
I think Spolsky’s Law follows from Rice’s Theorem and ultimately
from Turing’s no halting proof. If leak-less abstraction was possible you
could, in principle, write a recursive partial halting solution. By layering
abstraction on top of abstraction you would be able to solve some very
complex problems, eventually including the Halting Problem. We know
this is impossible, so non-leaky abstraction cannot exist.
The other method of splitting software is vertically. This is often
done following the natural boundaries of an organization: functional or
geographic. Again there will be leakage between the systems; the data
you get from the finance department might not be detailed enough for
268 Are the Androids Dreaming Yet?
Specification Cartoon
the engineers or vice versa, and so groups have to interact. The main
problem with vertically divided software is each group tends to reinvent
the wheel, so you end up with multiple similar implementations of the
same thing.
All said, the architectural job in software is a dynamic one. You can
split up software into separate elements but you must take into account
the leakage between them. When you detect a leak you must bring people
together to collaboratively solve the problem, rather than insisting on
the original partitioning. While doing all this you must keep track of
the overall aim and all the irritating small details contained in the many
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lists that form the project specification. I should confess that I am no
great fan of specifications, because they can mislead you into thinking
you’ve solved the problem, but I concede a good specification is helpful.
Spolsky’s Second Law is ‘Always write a specification.’ Engineers should
collaboratively write the specification as a response to the desires of the
project creators. But they must not blindly implement the specification
they’ve been handed. They must not forget the creative element.
270 Are the Androids Dreaming Yet?
The Role of ‘Process’ in Creativity
We hear a lot about ‘process’ when developing software and other
creative tasks. The first thing to realize is process does not write software
and every moment spent on process is a moment not writing software.
Excessive process can bring the productivity of the average programmer
down from a thousand lines per day to one. On the other hand, we all
know that using no process to write software results in useless software.
Good solo programmers, playwrights or composers are surrounded by
lists and post-it notes full of process. Where is the balance to be struck?
In my view ‘process’ is there to help humans with the tasks we find
naturally difficult. Humans, as we know, are dreadful at remembering
lists of symbolic information. Give a human ten numbers to memorize
and they will quickly forget them. Give Microsoft Excel ten numbers and
it will remember them forever, or, at least, until your next upgrade! So
the first job of process is to collect lists of things and sometimes even lists
of those lists.
Another significant affliction affecting humans is procrastination.
We tend to put off decisions. Process can set waypoints; when will the
job of splitting a project occur, when will we begin the test, and so on.
The third job of process is to keep track of the division of labor – if
the project has to be divided. Who will do what? Essentially we are back
to lists again.
The most important job of process, in my view, is to keep track
of scope. ‘Logical scope creep’ when unrecognized destroys software
projects. Scope creep is fine if it just adds more linear work. “Could we
add three more product types?” “Could you do another language?” “Can
you make this interface prettier, less cluttered?” It may cause a busy team
to groan, but it does not damage the integrity of the design. To put it back
in Brooks’ language, accidental creep is fine – provided you add some
resource. Essential creep is not. Adding the french language prompts to
a project in English might be fine, putting language translation into a
project may be a step too far. The project may have strayed into a different
logical class. Increases in logical scope often require redesign, you must
stop and re-architect if you are to avoid bugs in plague like quantities.
If programming software is a creative task, how can we help improve
productivity? The most important factor is to provide uninterrupted
peace and quiet. Programming is a task where people need to hold many
ideas in their head at the same time, and this requires deep concentration.
To get some idea of the creative process at work, listen to the excellent
TED lecture by John Cleese.
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A common and costly mistake is to put off thinking about a class of
things you are going to need in the next release because of time pressure.
‘Time out, that’s for the next release’ and similar statements spell disaster
for the future of a project as when you come to the next release, you may
have to rewrite much of it from scratch. This is why good architects are
so valuable. They anticipate the future even when they are told to ignore
it and ship now!
Just as there are artistic geniuses, there are programming geniuses.
Hold onto them if you get one. They are rare. We don’t know if they
can be made or they are lucky accidents, but statistics shows that some
people are 1000 times more productive at writing code than the average.
If you can find lots of them and make them work together you will build
the next Google or Facebook. If you have a tight deadline, a superprogrammer
may get you out of a hole, producing in a week what might
otherwise take a year. Remember your great programmers will most
prolific if you can get process and distraction out of their way. Just make
sure they have a clear idea of purpose.
Laws
A programmer interrupted eight times a day does no work.
A creative person interrupted eight times a day does no work.
Programming is a creative endeavor.
There are creative geniuses. Hold onto them.
Bugs save us from collapsing space-time when we are lazy and try
to use mechanical means rather than creative thought to write software.

Chapter 12
HYPER-COMPUTING
What’s in a Brain
Perpetual Motion from the 1600s
“If you are in a spaceship that is
travelling at the speed of light,
and you turn on the headlights,
does anything happen?”
Stephen Wright
If you believe humans outthink computers, be warned; you are in
controversial territory. This would need a hyper-computer and many
scientists speak of these in the same breath as perpetual motion
machines.
I’m not sure it’s an entirely fair analogy. We understand machines,
and the physical laws of our Universe forbid perpetual motion. We
don’t understand brains, so we can’t reasonably dismiss human hypercomputing.
Humans commonly demonstrate one clear example of
thinking which appears to break the Turing limit, namely finding
solutions to mathematical puzzles. We need an explanation for this.
Let me take you on a whistle-stop tour of all the schemes people have
imagined that might lead to a hyper-computer.
A hyper-computer is a machine that can calculate a function which
a Turing machine can not. For example, when given a number denoting
a problem such as Fermat’s Last Theorem, it can give me in return a
number representing a valid proof. We are not concerned here with
speed. We are talking about fundamental ‘do-ability’. Such machines are
often dubbed ‘super-Turing’.
Epic Fails
Let us first look at some proposals that blatantly fail. My children call
these ‘epic fails’, and they are the perpetual motion machines of the
hyper-computing world.
Could we run many Turing machines at the same time, perhaps
even an infinite number? Then we would have a much more powerful
machine that must beat the Turing limit.
The answer is no.
Turing machines are already infinitely powerful and we know from
our chapter on infinity that all countable infinities are the same. Infinity
plus infinity, infinity times infinity, infinity to any power; all are equal.
One single, fast, one-dimensional machine can simulate them all. We get
no greater power with an infinite number of similar machines.
The next technique which might realize a hyper-computer is to
have a machine which simultaneously runs every possible branch in a
program. Each time the machine gets to a point where there is a binary
decision, it can take the ‘yes’ branch, spawn a copy of itself, and run the
‘no’ branch as well. Logically this machine should be able to calculate
anything since it tries every conceivable option. The process is called
non-determinism. This doesn’t mean the computer has free will. It just
means the computer never chooses one option over another. It just
276 Are the Androids Dreaming Yet?
assumes each could be correct and travels down both. Solving a problem
using a machine like this can be fast. The problem is this machine has
no greater power than a regular Turing machine. Let me show you why.
A non-deterministic machine is essentially the same as a single
Turing machine; each time there is a branch in the program you would
start running two processes. The first process works on every even tick
of the computer clock and the other on every odd tick. Now we have a
single machine running two branches at the same time. Using this trick
over and over again, a single machine can run a program exploring every
possible branch. Although it generates an enormous number of branches
and takes a huge time to run, it is still a single machine and we have an
infinity of time on our hands. Therefore, the machine is limited as before.
We are not doing well so far and we have already exhausted an
infinite number of options! Let’s try a different tack. We know true
randomness is non-computable, the sort of randomness generated by
the Lavarand we examined earlier in the book. Might this help? Truly
random processes can’t be simulated by a computer. If we throw this into
the pot might it let us compute something a Turing machine cannot?
Again, no.
This idea still only generates a machine as powerful as the nondeterministic
machine above. A non-deterministic Turing machine runs
every possible program. All a random one does is choose some of the
same paths at random. It, therefore, can’t be any more powerful. The one
difference is that it can generate non-computable numbers. However, the
only interesting characteristic of these numbers is they are truly random
and this randomness was an input. Their presence does not make the
machine any more powerful.
There are quite a few proposals for hyper-computers that are just
cleverly dressed up versions of the machines we have already met and
dismissed. For example, it has been proposed the Internet could form
a super-Turing machine. This is known as a site machine because the
processing is distributed across many sites linked together through the
Internet. It is proposed each site could act as an oracle to the others. This
is quite an elegant idea, and some proofs have been offered that show
such a machine is capable of generating non-computable functions. The
problem with this idea is that you can simply draw an imaginary line
around the whole site machine and it looks exactly like a big Turing
machine. There is no conceptual difference between such a machine
and a regular computer with subroutines. After all, that’s in Turing’s
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original proof. Again we have reached a dead end. We need something
qualitatively different to a traditional computer in order to break the
Turing limit. The obvious place to turn is the quantum world.
Quantum Computers
Quantum computers have had an extraordinary run in the press recently.
It has been variously claimed they offer limitless computing power and
can break all known security schemes; cracking, for example, the prime
factors that form the basis of public key cryptography. This is big news.
These codes are used to protect all the financial transactions we make on
the web.
In a regular computer, bits of information are processed by switches
that make simple ‘yes’ or ‘no’ decisions. In a quantum computer each
switch can take both the yes and no branches, at least for a short time called
the decoherence interval. The calculations are said to be superposed.
This allows a quantum computer to calculate exponentially, rather than
linearly, as the number of logic gates increases. Grover’s algorithm and
Shor’s algorithm use this superposition to speed up factoring numbers
and looking things up in databases, respectively.
Grover’s algorithm gives us the ability to find something stored in a
random place without having to look in every box. If you think about a
standard search, say for your lost car keys, you must look everywhere to
guarantee finding them. It does not much matter in what order you do it.
When you are halfway through the search, you will be 50% likely to have
found your keys. But, with a quantum computer, you can be fuzzy and
look in many places at once. A quarter of the way through a quantum
search, you are 50% likely to have found your keys. That might sound
like a small improvement, but when working with very big numbers, it
makes an enormous difference.
Shor’s algorithm works a little differently and, yes, it does allow
a quantum computer to break Internet encryption, so the newspaper
headlines are true up to a point. Some time in the future we will need to
move to a more secure type of encryption.
The largest quantum computers today can process 300 qubits at
a time or remain ‘coherent’ for about two seconds. These results are
pitifully low. The largest prime number factored so far is 143, a mere 7
bits long! By way of comparison, internet security routinely uses 1024
bits. But, quantum computers are improving exponentially faster than
classical computers: They really do change the rules of the game. If you
remember our discussion of chess, the quantity of space needed for a
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calculation can be the limiting factor. A quantum computer is very space
efficient. When the computer branches and makes a copy of itself, it does
so without needing more space. There are two theories for how it does
this, (well, three, but the third is highly controversial). The first theory is
the computer doesn’t need the space because it hasn’t made its mind up
yet; somehow the

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