Monday, March 26, 2007

Math Angst

When I was at school, I hated maths. I'm still terrible at mental arithmetic, but as I always say, if god had intended us to be able to do sums, he wouldn't have invented the calculator! Recently I've been trying to understand what the heck maths is actually all about and so I pulled out a book I've had for years (but never really looked at), 'The World Treasury of Physics, Astronomy and Mathematics', and lo and behold there is a whole section on 'math angst'. Here's an extract (from an essay by Eugene Wigner):

The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories. In order to establish the first point, that mathematics plays an unreasonably important role in physics, it will be useful to say a few words on the question, "What is mathematics?", then, "What is physics?", then, how mathematics enters physical theories, and last, why the success of mathematics in its role in physics appears so baffling. Much less will be said on the second point: the uniqueness of the theories of physics. A proper answer to this question would require elaborate experimental and theoretical work which has not been undertaken to date.
And with that you might be beginning to understand why maths provokes angst. It is at the same time vital to our existence and yet bizarrely detached from it. Do we invent maths or discover it? It drives me crazy thinking about it. So what is mathematics then? Some more from Wigner:
Somebody once said that philosophy is the misuse of a terminology which was invented just for this purpose. In the same vein, I would say that mathematics is the science of skillful operations with concepts and rules invented just for this purpose. The principal emphasis is on the invention of concepts. Mathematics would soon run out of interesting theorems if these had to be formulated in terms of the concepts which already appear in the axioms. Furthermore, whereas it is unquestionably true that the concepts of elementary mathematics and particularly elementary geometry were formulated to describe entities which are directly suggested by the actual world, the same does not seem to be true of the more advanced concepts, in particular the concepts which play such an important role in physics. Thus, the rules for operations with pairs of numbers are obviously designed to give the same results as the operations with fractions which we first learned without reference to "pairs of numbers." The rules for the operations with sequences, that is, with irrational numbers, still belong to the category of rules which were determined so as to reproduce rules for the operations with quantities which were already known to us. Most more advanced mathematical concepts, such as complex numbers, algebras, linear operators, Borel sets - and this list could be continued almost indefinitely - were so devised that they are apt subjects on which the mathematician can demonstrate his ingenuity and sense of formal beauty.

In fact, the definition of these concepts, with a realization that interesting and ingenious considerations could be applied to them, is the first demonstration of the ingeniousness of the mathematician who defines them. The depth of thought which goes into the formulation of the mathematical concepts is later justified by the skill with which these concepts are used. The great mathematician fully, almost ruthlessly, exploits the domain of permissible reasoning and skirts the impermissible. That his recklessness does not lead him into a morass of contradictions is a miracle in itself: certainly it is hard to believe that our reasoning power was brought, by Darwin's process of natural selection, to the perfection which it seems to possess. However, this is not our present subject. The principal point which will have to be recalled later is that the mathematician could formulate only a handful of interesting theorems without defining concepts beyond those contained in the axioms and that the concepts outside those contained in the axioms are defined with a view of permitting ingenious logical operations which appeal to our aesthetic sense both as operations and also in their results of great generality and simplicity. The complex numbers provide a particularly striking example for the foregoing. Certainly, nothing in our experience suggests the introduction of these quantities. Indeed, if a mathematician is asked to justify his interest in complex numbers, he will point, with some indignation, to the many beautiful theorems in the theory of equations, of power series, and of analytic functions in general, which owe their origin to the introduction of complex numbers. The mathematician is not willing to give up his interest in these most beautiful accomplishments of his genius.
So maths is about beauty? Isn't that a 'crock of shit'? Perhaps it's my aversion to maths, but I just don't understand the concept of beautiful mathematics. I do at least think I understand this paragraph, from Stewart Shapiro's 'Philosophy of Mathematics':
It is surely correct to maintain that if there had never been any language (or any people), there would be trees, planets, and stars. There would also be numbers, sets of numbers, and Klein groups, if not baseball defenses. Such is the nature of ante rem structures. Once a language and a theory impose a structure and sort the universe into objects—be they abstract or concrete—one can sometimes speak objectively about those objects, and we insist (surely correctly) that at least some of the objects were not created by us. Counterfactuals about the ways the world would be must themselves be formulated in our language and form of life—for we know no other. Trivially, had there never been any language, there would be no means of discussing, say, stars, as distinguished from the particles they contain and the galaxies that contain them. However, the lack of sortals available to speakers in a given possible world has nothing to do with which objects that world contains.
So although I think no-one really knows the answers, it seems that maths is related to the structure of the universe and that's why it works...