The Math Gene: How Mathematical Thinking Evolved And Why Numbers Are Like Gossip

305 Pages·2001·2.29 MB·English

by  Keith Devlin

Table of Contents

Title Page

Table of Figures

Acknowledgements
PROLOGUE

Chapter 1 - A MIND FOR MATHEMATICS
NOT ENOUGH TIME
IT’S NOT JUST NUMBERS
WHAT IS MATHEMATICS?
THE SCIENCE OF PATTERNS
WHAT DOES IT TAKE TO MAKE A MATHEMATICAL MIND?
THE PUZZLE OF BRAIN SIZE

Chapter 2 - IN THE BEGINNING IS NUMBER
THE NUMBER SENSE
THE HORSE THAT DIDN’T KNOW 2 + 2 = 4 AND THE RAT THAT
DID
WHAT ABOUT THE CHIMPANZEES?
THE RISE AND FALL OF PIAGET
CAN BABIES COUNT?
BUT CAN THEY ADD?
BUTTERWORTH’S CASEBOOK

Chapter 3 - EVERYBODY COUNTS
THE COUNTER CULTURE
SYMBOLS OF SUCCESS
NUMBERS IN THE MIND
THE SOUND OF NUMBER
THE CHINESE ADVANTAGE
WHY MATHEMATICS SEEMS ILLOGICAL

Chapter 4 - WHAT IS THIS THING CALLED MATHEMATICS?
THE NATURE OF THE BEAST
HOW DO MATHEMATICIANS GET INTO SHAPE?
THE GEOMETRY OF ANIMAL COAT PATTERNS
THE GEOMETRY OF FLOWERS
THE PATTERNS OF BEAUTY THE EYE CANNOT SEE
WHY EVEN MATHEMATICAL LONERS LIKE TO WORK IN GROUPS
NOW FOR THE REALLY HEAVY STUFF

Chapter 5 - DO MATHEMATICIANS HAVE DIFFERENT BRAINS?
THE STRANGE CASE OF EMILY X
HOW DO MATHEMATICIANS THINK?
THE FOUR LEVELS OF ABSTRACTION
LIVING AMONG SYMBOLS
SOME CAN, SOME CAN’T—OR CAN THEY?
INSIDE THE HOUSE, OCCASIONALLY LOOKING OUT
HERE’S LOOKING AT EUCLID
BACK INSIDE THE HOUSE
INVENTION OR DISCOVERY?

Chapter 6 - BORN TO SPEAK
DID SHAKESPEARE KNOW THE DIFFERENCE?
VARIATIONS ON A SINGLE THEME
WORDS IN BOXES
THE KEY TO SYNTAX
HOW DO TWO-YEAR-OLDS MANAGE TO LEARN ALL THIS
STUFF?

Chapter 7 - THE BRAIN THAT GREW AND LEARNED TO TALK
TWO PUZZLES ABOUT THE EVOLUTION OF LANGUAGE
HOW A LARGE METEOR LED TO SMARTER APES
TO SURVIVE IN THE WIDE OPEN SPACES—ACT SMART!
WALK TALL, THINK SMART, AND COMMUNICATE EFFECTIVELY
THE DOG THAT CAN LEARN NEW TRICKS
FOLLOWING THE LEARNING CURVE
Chapter 8 - OUT OF OUR MINDS
WHAT’S YOUR TYPE?
TELL ME ABOUT IT
SYMBOLS OF PROGRESS
SYMBOLS FOR SYMBOLS
CREATURES THAT CHANGE THE FACE OF THE EARTH
SEWING THE COMMON THREAD OF ALL LANGUAGES
THE ATTRACTOR THEORY
LINGUISTIC EVE
SMALL CHANGE, BIG EFFECT
CROSSING THE SYMBOLIC RUBICON
IN A WORLD OF OUR OWN
THE SYMBOLIC MIND LIGHTS UP

Chapter 9 - WHERE DEMONS LURK AND MATHEMATICIANS WORK
HE SAID, SHE SAID
OH, THAT PI—HE’S SO IRRATIONAL, YOU KNOW!
BARBIE WAS RIGHT
WHY DO SO MANY PEOPLE SAY THEY CAN’T DO MATH?
RIGHT ANSWER, WRONG REASON

Chapter 10 - ROADS NOT TAKEN

EPILOGUE
APPENDIX - The Hidden Structure of Everyday Language
REFERENCES
INDEX
A NOTE ABOUT THE TYPE
Copyright Page
Table of Figures

FIGURE 4.1 Regular Polygons. A polygon is regular if all its sides are equal
and all its interior angles are equal. There are regular polygons of any number of
sides greater than two. The first four are shown: the equilateral triangle, the
square, the regular pentagon, and the regular hexagon.
FIGURE 4.2 Regular Solids. A polyhedron is regular if all its faces are identical
and all its interior angles are equal. There are exactly five regular polyhedra.
FIGURE 4.3 Wallpaper patterns. There are exactly seventeen different ways to
repeat a fixed pattern indefinitely to cover the whole plane. The illustrations
show one wallpaper pattern that repeats in each of the seventeen different ways.
FIGURE 4.4 Animal coat patterns produced mathematically on a computer
screen, by solving the equations written down by mathematical biologist James
Murray. By changing the value of a single parameter in his equations, Murray
could change a spotted tail into a striped one.
FIGURE 4.5 The Koch snowflake starts to take shape.
FIGURE 4.6 Repetition of a simple reproduction rule rapidly generates a tree-
like structure. The rule is to add two new branches two-thirds of the way up each
topmost branch, each one equal to one-third the length of that branch.
FIGURE 4.7 An electronic lilac generated on a computer by the iteration of
some simple growth rules. Examples such as this demonstrate that much of the
seeming complexity of the natural world can be produced by some very simple
rules.
FIGURE 4.8 The circle looks exactly the same if it is rotated about the center
through any angle or reflected in any diameter.
FIGURE 4.9 Reflecting a clock face in the diameter from 12 down to 6 swaps
the 3 and the 9.
FIGURE 4.10 The square looks exactly the same after a rotation about the
center through one or more right angles or a reflection in any of the dashed lines.
FIGURE 4.11 The triangle looks the same if it is rotated about the center
through 120° in either direction or if it is reflected in any of the dashed lines
marked X, Y, Z.
FIGURE 4.12 Multiplication table for the symmetry group of an equilateral
triangle.
FIGURE 6.1 ASimple Parse Tree
FIGURE 6.2 A Very Basic Parse Tree
FIGURE 6.3 Two Depictions of the Same Parse Tree
FIGURE 6.4 Another Parse Tree
FIGURE 6.5 The Fundamental Language Tree
FIGURE 6.6 Construction of the Noun Phrase the red door
FIGURE 6.7 Construction of the Noun Phrase the house with the red door
FIGURE 7.1 The Protolanguage Tree
FIGURE 8.1 The Syllabic Structure Tree
FIGURE 8.2 Part of a simple neural net that reads three-letter words and
associates with them various properties of the objects they denote. The inputs
(bottom layer) are individual letters in particular positions in the word. The
middle layer represent units that the network recognizes as complete words. The
outputs (top layer) are properties of the objects depicted by the words. The
network is constructed so that three signals must reach a node in the middle layer
in order for it to activate and send a signal to the output layer.
FIGURE 8.3 A top view of mental activity in a thinking human brain. Such a
picture, called an electroencephalogram (EEG), is obtained by attaching
electrodes to the subject’s skull. It provides a map of the amount of electrical
activity in each part of the brain. The lighter shaded regions are where the
activity was most intense at the time the measurement was taken. In this
example, there was considerable activity in the left side of the brain. The activity
at the rear of the brain (bottom of picture) is the processing of visual signals
received from the eyes.
FIGURE 8.4 An impossible figure by the artist M. C. Escher, Ascending and
Descending (1960). Our minds can imagine a figure that is physically impossible
to achieve.
FIGURE A.1 The Fundamental Language Tree: The Generic Structure for X-
bar Theory
FIGURE A.2 Construction of a Noun Phrase
FIGURE A.3 Construction of a Prepositional Phrase
FIGURE A.4 Construction of the Noun Phrase the red door
FIGURE A.5 Construction of the Adjectival Phrase red
FIGURE A.6 Structure of a Noun Phrase with Two Complements
FIGURE A.7 The Generic Structure for X-bar Theory
FIGURE A.8 Disambiguation of an Ambiguous Noun Phrase
FIGURE A.9 The X-bar Structure of a Verb Phrase