Stewart - Natures numbers : the unreal reality of mathematics

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Copyright 1995 by Ian Stewart.

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Stewart, Ian.

Natures numbers : the unreal reality of mathematics / Ian Stewart.

p. cm. (Science masters series)

ISBN 978-0-78672-392-8

1. MathematicsPopular works. I. Title. II. Series.

9 8 7

CONTENTS

I have a dream.

I am surrounded bynothing. Not empty space, for there is no space to be empty. Not blackness, for there is nothing to be black. Simply an absence, waiting to become a presence. I think commands: let there be space. But what kind of space? I have a choice: three-dimensional space, multidimensional space, even curved space.

I choose.

Another command, and the space is filled with an all-pervading fluid, which swirls in waves and vortices, here a placid swell, there a frothing, turbulent maelstrom.

I paint space blue, draw white streamlines in the fluid to bring out the flow patterns.

I place a small red sphere in the fluid. It hovers, unsupported, ignorant of the chaos around it, until I give the word. Then it slides off along a streamline. I compress myself to one hundredth of my size and will myself onto the surface of the sphere, to get a birds-eye view of unfolding events. Every few seconds, I place a green marker in the flow to record the spheres passing. If I touch a marker, it blossoms like a time-lapse film of a desert cactus when the rains comeand on every petal there are pictures, numbers, symbols. The sphere can also be made to blossom, and when it does, those pictures, numbers, and symbols change as it moves.

Dissatisfied with the march of its symbols, I nudge the sphere onto a different streamline, fine-tuning its position until I see the unmistakable traces of the singularity I am seeking. I snap my fingers, and the sphere extrapolates itself into its own future and reports back what it finds. Promising... Suddenly there is a whole cloud of red spheres, all being carried along by the fluid, like a shoal of fish that quickly spreads, swirling, putting out tendrils, flattening into sheets. Then more shoals of spheres join the gamegold, purple, brown, silver, pink.... I am in danger of running out of colors. Multicolored sheets intersect in a complex geometric form. I freeze it, smooth it, paint it in stripes. I banish the spheres with a gesture. I call up markers, inspect their unfolded petals, pull some off and attach them to a translucent grid that has materialized like a landscape from thinning mist.

Yes!

I issue a new command. Save. Title: A new chaotic phenomenon in the three-body problem. Date: today.

Space collapses back to nonexistent void. Then, the mornings research completed, I disengage from my Virtual Unreality Machine and head off in search of lunch.

This particular dream is very nearly fact. We already have Virtual Reality systems that simulate events in normal space. I call my dream Virtual Unreality because it simulates anything that can be created by the mathematicians fertile imagination. Most of the bits and pieces of the Virtual Unreality Machine exist already. There is computer-graphics software that can fly you through any chosen geometrical object, dynamical-systems software that can track the evolving state of any chosen equation, symbolic-algebra software that can take the pain out of the most horrendous calculationsand get them right. It is only a matter of time before mathematicians will be able to get inside their own creations.

But, wonderful though such technology may be, we do not need it to bring my dream to life. The dream is a reality now, present inside every mathematicians head. This is what mathematical creation feels like when youre doing it. Ive resorted to a little poetic license: the objects that are found in the mathematicians world are generally distinguished by symbolic labels or names rather than colors. But those labels are as vivid as colors to those who inhabit that world. In fact, despite its colorful images, my dream is a pale shadow of the world of imagination that every mathematican inhabitsa world in which curved space, or space with more than three dimensions, is not only commonplace but inevitable. You probably find the images alien and strange, far removed from the algebraic symbolism that the word mathematics conjures up. Mathematicians are forced to resort to written symbols and pictures to describe their worldeven to each other. But the symbols are no more that world than musical notation is music.

Over the centuries, the collective minds of mathematicians have created their own universe. I dont know where it is situatedI dont think that there is a where in any normal sense of the wordbut I assure you that this mathematical universe seems real enough when youre in it. And, not despite its peculiarities but because of them, the mental universe of mathematics has provided human beings with many of their deepest insights into the world around them.

I am going to take you sightseeing in that mathematical universe. I am going to try to equip you with a mathematicians eyes. And by so doing, I shall do my best to change the way you view your own world.