What is Mathematics, Really?

  • Reuben Hersh
Oxford University Press/Cape: 1997343Pp. $35, £18.99.
Facets of history: details of a wooden tarsia in the church of Santa Maria in Organo, Verona, Italy. From Polyhedra by Peter R. Cromwell (Cambridge University Press, £30, $44.95).

The title poses quite a question, and adding the word ‘really’ does not make it any easier. Empty words will not do: hard definitions are required. So what does ‘really’ really mean here?

It could mean: “What is modern mathematics, what makes it important in — or irrelevant to — our everyday life, what is it like to be an active mathematician?” There is much to say here. There has been an exponential growth in the number of scholars, books, journals, papers and theorems produced in the past century. Whole fields, such as probability theory, functional and numerical analysis, and noncommutative geometry, have been created. Older fields, such as algebraic geometry or number theory, have enjoyed tremendous progress, the solution of Fermat’s last theorem being a case in point. The use of mathematics has become pervasive in engineering for filtering noisy signals and controlling processes, as well as in economics or finance, as shown by the success of the Black-Scholes model for pricing derivative securities.

It could also mean: “What is the inner meaning of mathematics?” Is there anything more to mathematics than its outward success story? Is it an activity to be pursued for its own sake, independently of material rewards? Painters or writers are supposed to be aiming for timeless achievement in art, not for recognition among their contemporaries, which may not even come during their lifetime. Is it the same with mathematicians? Are they artists in their own way? And, if so, what kind of art is theirs?

Strangely enough, this book goes neither way. It describes nothing of modern mathematics, except for a discussion about the foundations of mathematics and its connection with logic, which is far from the preoccupations of most working mathematicians. There is nothing at all to suggest the breadth and scope of mathematics today, or how much it has changed in the past century. It is always annoying to see the same tired, age-old examples crop up again and again: how to prove that the square root of two is irrational, how to use the tools of differentiation to exploit Galileo’s law of falling bodies, and so on.

The book fails also to answer the question about the nature of mathematics. It dismisses it as “futile”, leaving to forthcoming progress in the neurosciences whatever kind of answer is possible. To quote the author: “How does mathematics come about, in a daily, down-to-earth sense? That question belongs to psychology, to the history of thought, and to other disciplines of empirical science. It can’t be answered with philosophy.”

So what does the author do? He presents his own philosophy of mathematics, which is now a fairly easy thing, considering all the questions he has decided such a philosophy should not answer. This philosophy he calls “humanism”. The basic principle is: “A world of ideas exists, created by human beings, existing in their shared consciousness. These ideas have objective properties, in the same sense that material objects have objective properties. The construction of proof and counterexample is the method of discovering the properties of these ideas. This branch of knowledge is called mathematics.”

I find it hard to understand what is meant by “shared consciousness” and to see how ideas could have “objective properties” in the “same sense” that material objects do. If I ventured such views in public, I would certainly try to clarify them, referring to such sciences as linguistics (after all, shared consciousness, whatever it is, must be mediated through language, and Chomsky has gone a long way towards a scientific analysis of the phenomenon) or to history (does a historical fact share the same objective properties as an experimental fact in physics?). Here, of course, such preoccupations are dismissed as “futile”, so the author can take his “philosophy” for granted and proceed without further ado.

In the first part of the book, the author develops the ideas that “mathematics is human. It’s part of and fits into human culture” and that “mathematics knowledge isn’t infallible. Like science, mathematics can advance by making mistakes, correcting and recorrecting them.” He contrasts these (very reasonable) ideas with a Platonist approach, which would take the stand that mathematics already ‘exists’ somewhere (written down in God’s great book), so that theorems are discovered (unveiled), and not created, by humans. Mathematicians are then divided into humanists and Platonists, and the second part of the book is devoted to a review of prominent mathematicians through the ages, classified according to the stand they have taken on this question. At the end, their political opinions are reviewed as well, and, not surprisingly, even with the help of some elementary statistics, Platonists are found to be mostly conservative-righties whereas humanists are mostly democrat-lefties.

As a source of information about mathematics in general, the book is a failure. There remains scattered information, mostly in the form of quotations, about the foundations debate in mathematics. And the author does give unexpected entertainment at times. It is difficult to keep a straight face while reading that Pythagoras, for instance, for whom no biographical data are available, except the facts that he lived in the sixth century BCand that none of his writings survive, was a ‘rightie’. There is also some humour in seeing that the author, presumably wishing to spare the critics time and trouble, concludes his book with a “self-graded report card”. The result: “Could be worse.” Of course.