What Mathematics Is

To students struggling with undergraduate mathematics courses, the equations of physics seem horribly complicated and opaque. What they have yet to appreciate is that mathematics is, among other things, a language. When that language has been learned, immensely complicated things can be elegantly summarized in the mathematical equivalent of a one-liner.

In this respect, mathematics differs little from other technical languages (though it is immeasurably more powerful and comprehensive). Imagine, for example, trying to explain an investment scheme to somebody in ordinary English, without being able to use the words capital, interest, or inflation. Or envisage describing the workings of a car engine without ever mentioning pistons, camshafts, gaskets, or carburettors. Perhaps the greatest scientific discovery of all time is that nature is written in mathematical code. We do not know the reason for this, but it is the single most important fact that enables us to understand, control, and predict the outcome of physical processes. Once we have cracked the code for some particular physical system, we can read nature like a book.

Beauty is a nebulous concept, yet there is no doubt that it provides a source of inspiration for professional scientists. In some cases, when the road ahead may be unclear, mathematical beauty and elegance guide the way. It is something the physicist feels intuitively, a sort of irrational faith that nature prefers the beautiful to the ugly. So far this belief has been a reliable and powerful travelling companion, in spite of its subjective quality.

And therein lies its appeal and utility. Nature is beautiful. We don't know why this is so, but experience teaches us that beauty implies utility. Successful theories are always beautiful theories. They are beautiful not because they are successful, but because of their inherent symmetry and mathematical economy. Beauty in physics is a value judgement involving professional intuition and cannot readily be communicated to the layman, because it is expressed in a language that the layman has not learned, the language of mathematics. But to one who is conversant with that language, the beauty is as apparent as poetry.

This brings me back to where I came in. Mathematics is language, the language of nature. If you can't speak a language you can't understand the beauty of its poetry. There are always sceptics who say, `What is this mysterious mathematical beauty you speak of? I don't see anything beautiful about a mess of symbols. You physicists are just deluding yourselves.' I like to reply by comparing mathematics with music. For someone who had heard only single musical notes, the beauty of a symphony would be impossible to explain. Yet who would deny that there is real beauty in a symphony, albeit of an abstract and indefinable nature? Likewise, for a person whose experience of mathematics is limited to counting numbers, how can one communicate the sense of delight, the deep and meaningful appeal, of Maxwell's equations? Nevertheless, the aesthetic quality is there sure enough. And physicists of good mathematical taste produce altogether better theories than Philistines, just as do their counterparts in musical composition.

It is one of the great tragedies of our society that from fear, poor teaching, or lack of motivation the vast majority of people have shut themselves off from the mathematical poetry and music of nature. The sweeping vista that mathematics reveals is denied to them. They may delight over the scent of a rose or the colour of a sunset, but a whole dimension of aesthetic experience is foreclosed to them.

Excerpts from Chapter 4 of
by Paul Davies
(Simon and Schuster, 1984).

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