Download HOMEWORK / Translation From Here to Infinity by Ian Stewart

HOMEWORK / Translation
From Here to Infinity by Ian Stewart
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One of the biggest problems of mathematics is to explain to everyone else what it is all about.
The technical trappings of the subject, its symbolism and formality, its baffling terminology, its
apparent delight in lengthy calculations: these tend to obscure its real nature. A musician would
be horrified if his art were to be summed up as 'a lot of tadpoles drawn on a row of lines'; but
that's all that the untrained eye can see in a page of sheet music. The grandeur, the agony, the
flights of lyricism and the discords of despair: to discern them among the tadpoles is no mean
task. They are present, but only in coded form. In the same way, the symbolism of mathematics
is merely its coded form, not its substance. It too has its grandeur, agony, and flights of lyricism.
However, there is a difference. Even a casual listener can enjoy a piece of music. It is only the
performers who are required to understand the antics of the tadpoles. Music has an immediate
appeal to almost everybody. But the nearest thing I can think of to a mathematical performance
is the Renaissance tournament, where leading mathematicians did public battle on each other's
problems. The idea might profitably be revived; but its appeal is more that of wrestling than of
music.
Music can be appreciated from several points of view: the listener, the performer, the composer.
In mathematics there is nothing analogous to the listener; and even if there were, it would be the
composer, rather than the performer, that would interest him. It is the creation of new
mathematics, rather than its mundane practice, that is interesting. Mathematics is not about
symbols and calculations. These are just tools of the tradequavers and crotchets and five-finger
exercises. Mathematics is about ideas. In particular it is about the way that different ideas relate
to each other. If certain information is known, what else must necessarily follow? The aim of
mathematics is to understand such questions by stripping away the inessentials and penetrating to
the core of the problem. It is not just a question of getting the right answer; more a matter of
understanding why an answer is possible at all, and why it takes the form that it does. Good
mathematics has an air of economy and an element of surprise. But, above all, it has significance.
Raw Materials
I suppose the stereotype of the mathematician is an earnest, bespectacled fellow poring over an
endless arithmetical screed. A kind of super-accountant. It is surely this image that inspired a
distinguished computer scientist to remark in a major lecture that 'what can be done in
mathematics by pencil and paper alone has been done by now'. He was dead wrong, and so is the
image. Among my colleagues I count a hang-gliding enthusiast, a top-rank mountaineer, a
smallholder who can take a tractor to bits before breakfast, a poet, and a writer of detective
stories. And none of them is especially good at arithmetic.
As I said, mathematics is not about calculations but about ideas. Someone once stated a theorem
about prime numbers, claiming that it could never be proved because there was no good notation
for primes. Carl Friedrich Gauss proved it from a standing start in five minutes, saying
(somewhat sourly) 'what he needs is notions, not notations'. Calculations are merely a means to
an end. If a theorem is proved by an enormous calculation, that result is not properly understood
until the reasons why the calculation works can be isolated and seen to appear as natural and
inevitable. Not all ideas are mathematics; but all good mathematics must contain an idea.
Pythagoras and his school classified mathematics into four branches, like this:
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Pythagoras and his school classified mathematics into four branches, like this:
Three of these branches remain major sources of mathematical inspiration. The fourth, music, is
no longer given the same kind of prominence, but it can be reinterpreted as the algebraic or
combinatorial approach. (There is still a pronounced tendency for mathematicians to be
musically talented.) To these four, modern mathematics has added a fifth: Lady Luck. Thus there
are now at least five distinct sources of mathematical ideas. They are number, shape,
arrangement, movement, and chance.
The most basic and well known is number. Originally the number concept must have arisen
through counting: possessions, days, enemies. Measurement of lengths and weights led to
fractions and the 'real' numbers. A major act of mathematical imagination created 'imaginary'
numbers such as From that point on mathematics was never quite the same. Shape, or form, leads
to geometry: not just the stereotyped and pedantic style of geometry that used to be blamed on
Euclid, but its modern offspring such as topology, singularity theory, Lie groups, and gauge field
theories. Novel geometric formsfractals, catastrophes, fibre bundles, strange attractors still
inspire new developments in mathematics. Problems about ways to arrange objects according to
various rules lead to combinatorics, parts of modern algebra and number theory, and what is
becoming known as 'finite mathematics', the basis of much computer science. Movement of
cannonballs, planets, or wavesinspired the calculus, the theories of ordinary and partial
differential equations, the calculus of variations, and topological dynamics. Many of the biggest
areas of mathematical research concern the way systems evolve in time. A more recent
ingredient is chance, or randomness. Only for a couple of centuries has it been realized that
chance has its own type of pattern and regularity; only in the last fifty years has it been
possible to make this statement precise. Probability and statistics are obvious results; less well
known but equally important is the theory of stochastic differential
equations dynamics plus random interference.
The Driving Force
The driving force of mathematics is problems. A good problem is one whose solution, rather
than merely tidying up a dead end, opens up entirely new vistas. Most good problems are hard:
in mathematics, as in all walks of life, one seldom gets something for nothing. But not all hard
problems are good: intellectual weight-lifting may build mental muscles, but who wants a
muscle-bound brain? Another important source of mathematical inspiration is examples. A really
nice self-contained piece of mathematics, concentrating on one judiciously chosen example,
often contains within it the germ of a general theory, in which the example becomes a mere
detail, to be embellished at will. I'll combine the two, by giving some examples of mathematical
problems. They are drawn from all periods, to emphasize the continuity of mathematical thought.
Any technical terms will be explained later.
(1) Is there a fraction whose square is exactly equal to 2?
(2) Can the general equation of the fifth degree be solved using radicals?
(3) What shape is the curve of quickest descent?
(4) Is the standard smooth structure on 4-dimensional space the only one possible?
(5) Is there an efficient computer program to find the prime factors of a given number?
Each problem requires some comment and preliminary explanation, and I'll consider them in
turn.
Studenti sa brojem indeksa 31The Historical Thread
One feature illustrated by my selection of problems is the unusually long lifetime of
mathematical ideas. The Babylonian solution of quadratic equations is as fresh and as useful now
as it was 4,000 years ago. The calculus of variations first bore fruit in classical mechanics, yet
survived the quantum revolution unscathed. The way it was used changed, but the mathematical
basis did not. Galois's ideas remain at the forefront of mathematical research. Who knows where
Donaldson's may lead? Mathematicians are generally more aware of the historical origins of
their ideas than many other scientists are. This is not because nothing important has happened in
mathematics recently quite the reverse is true. It is because mathematical ideas have a
permanence that physical theories lack.
Really good mathematical ideas are hard to come by. They result from the combined work of
many people over long periods of time. Their discovery involves wrong turnings and intellectual
dead ends. They cannot be produced at will: truly novel mathematics is not amenable to an
industrial 'Research and Development' approach.
But they pay for all that effort by their durability and versatility. Ptolemy's theory of the solar
system is of historical interest to a modern cosmologist, but he doesn't use it in serious research.
In contrast, mathematical ideas thousands of years old are used every day in the most modern
mathematics, indeed in all branches of science.
The cycloid was a fascinating curiosity to the Greeks, but they couldn't do anything with it. As
the brachistochrone, it sparked off the calculus of variations. Christian Huygens used it to design
an accurate clock. Engineers today use it to design gears. It shows up in celestial mechanics and
particle accelerators. That's quite a career for one of such a humble upbringing.
Yes, But Look At It This Way
As well as problems and examples, we mustn't ignore a third source of mathematical inspiration:
the search for the 'right context' for a theorem or an idea.
Mathematicians don't just want to get 'the answer' to a problem. They want a method that makes
that answer seem inevitable, something that tells them what's really going on. This is something
of an infuriating habit, and can easily be misunderstood; but it has proved its worth time and time
again.
For example, Descartes showed that by introducing coordinates into the plane, every problem in
geometry can be reformulated as an equivalent problem in algebra.
Instead of finding where curves meet, you solve simultaneous equations.
It's a magnificent idea, one of the biggest mathematical advances ever made. But let me 'prove' to
you that it's totally worthless. The reason is simple: there is a routine procedure whereby any
algebraic calculation can be transformed into an equivalent geometric theorem, so anything you
can do by algebra, you could have done by good old geometry. Descartes's reformulation of
geometry adds precisely nothing. See? Like I said, it's useless.
In point of fact, however, the reformulation adds a great deal namely, a new viewpoint, one in
which certain ideas are much more natural than they would have been in the original context.
Manipulations that make excellent sense to an algebraist can appear very odd to a geometer
when recast as a series of geometric constructions; and the same goes for geometrical ideas
reformulated as algebra. It's not just the existence of a proof that is important to mathematicians:
it's the flow of ideas that lets you think the proof up in the first place. A new viewpoint can have
a profound psychological effect, opening up entirely new lines of attack. Yes, after the event the
new ideas can be reconstructed in terms of the old ones; but if we'd stuck to the old approach,
we'd never have thought of them at all, so there'd be nothing to reconstruct from.