What Makes a Mathematician?
Wilhelm Magnus, in his paper "The Significance of Mathematics: The Mathematicians' Share in the General Human Condition", Amer. Math. Monthly 104 (3) (1997), 261-269, asks "What Makes a Mathematician?" We give below a version of Magnus's answer to his own question:
What Makes a Mathematician?
There exists a widespread resentment against mathematics. It is supposed to deal only with quantity (not true, since most of mathematics deals with structure and relations), or with computing (again not true, but I cannot explain that in a few words) and, on the whole, it is more worthy of a machine than of a human being. As an aid to science and technology, it does not provide values and is therefore dehumanizing. Even the claim of the mathematician to be concerned with truth is frequently answered by saying that mathematical statements are not true but merely correct. Nevertheless, it is undoubtedly true that the results of mathematics are found by human beings. Can anything be said about them?
The answer is: Not enough to enable us to recognize a mathematician if we meet one at a party. Nevertheless, there exist properties without which a mathematician cannot exist. One of them is, of course, a specific talent. But this is far from being enough. It must be supplemented by an interest in the matter, in fact by a fascination with the problems of the field. And the talent must be supported by persistence and by the willingness to spend the large amounts of time and energy needed to master a difficult craft. And the mathematician needs an exceptionally great ability to stand up under frustration. This is due to the fact (pointed out to me by a colleague) that ours is the only field with an all-or-nothing alternative. A painting or a piece of furniture may be more or less perfect. A theorem and a proof are either true or false. If either the proof or the theorem is false, we have absolutely nothing. Finally, we must be satisfied with the production of something intangible. I have found house painting to be a gratifying supplement to mathematical research. At least one can see and touch what one has done.
It follows that the mathematician needs the support of a civilization that acknowledges as valuable the products of theory, of pure thought. Although we do not set a scale of values, we would not exist without such a scale. I can be brief here, since the arguments given by the philosopher Enrico Cantore for the humanistic significance of science apply, with small modifications, to mathematics as well.
Let me conclude by pointing out one advantage that the mathematician (and, with him, the representative of the exact sciences) has. Our thoughts are eminently communicable. Not, perhaps, from person to person. But certainly from nation to nation. Mathematicians understand each other no matter where they come from. Even across many centuries we understand each other. We may not see clearly what a particular expression in Euclid means. But we are confident that, could we talk with him, we would be able to clear up the matter quickly. Nothing is more international than the community of mathematicians.