[In the following essay, Franks objects to Gleick's portrayal of mathematicians and the goals of mathematics in Chaos, and asserts that Gleick misses an opportunity to introduce the public to the rewarding creative aspects of mathematical research.]

The several responses to my review [of James Gleick's Chaos] raise some interesting questions. What does “doing mathematics” mean? Is it possible or desirable to give an honest explanation of its meaning to a general audience? How important is the role of theorem-proving in doing mathematics?

It is wrong to try, as James Gleick does in these pages, to make a dichotomy between discovery and proof in mathematics. Usually the discovery is the proof, or at least it is inextricably tied to it. Very rarely is a proof a historical afterthought. Almost always it is a proof, or the process of finding it, that creates new mathematical knowledge. A proof is not some kind of super spelling checker that merely validates mathematical facts. Most mathematicians would consider proofs to be the central content of mathematical knowledge. Who would be satisfied if God were to announce that the Riemann Hypothesis is true, but deny us the proof?

Let me paraphrase an old joke about money. In “doing mathematics” proving theorems isn't everything, but it's way ahead of whatever is in second place. Proving a theorem is one of the most creative acts of which the human mind is capable. Most mathematicians find it an exhilarating experience. It is often exciting, sometimes even thrilling. There is also pain and disappointment when a putative theorem falls through. Several years ago a well-known mathematician caused a minor flap by comparing mathematics to sex (in the page of the Bulletin of the American Mathematical Society). The propriety of his simile may have been questionable, but I prefer it to the comparison in Professor Devlin's letter in response to my review, which likens proving a theorem to a mechanic working on a Buick.

Mr. Gleick is probably right when he suggests that many mathematicians consider the concept of a “mathematician, … ostentatiously not proving much” as something of a self-contradiction. A view commonly held by mathematicians is that a mathematician is someone who creates mathematics and that, for the most part, creating mathematics means discovering and proving theorems. There is no intent here to denigrate those who use mathematics but prove no theorems, especially those who use it in creative ways. But there is a difference between a composer and a musician. And mathematicians take great pride in their craft—even to the extent of believing the theorem may outlive the application, as the musical composition outlives the performer. Mr. Gleick considers this a myopic view of the history of science; I acknowledge it is not an unbiased one. I hope, at least, he realizes it is not a petty jealousy directed at a single individual.

By and large, mathematicians have done a terrible job of communicating to the general public what we are doing (or what we think we are doing). Professor Devlin says, “Gleick very wisely steered well clear of any whiff of real ‘mathematics’ as it is perceived by most people.” I saw, instead, a missed opportunity. Mr. Gleick is a rare find for the scientific community—an exceptionally talented writer with an interest in science. I do not believe he is an antagonist of mathematics, quite the contrary. But I wish he had chosen to give his readers “a whiff of real mathematics” as it is perceived by mathematicians. Obviously this does not mean citing theorems and proofs; and it certainly does not mean reciting a list of who did what when. It does mean explaining the mathematician's perspective on mathematical creativity as a process by which new knowledge is attained. Professor [Morris] Hirsch does not suggest this process is the only legitimate topic in a history of chaos, but he is correct in saying it played a major role, which Gleick neglected in his book.

Mr. Gleick tells us that Lanford's proof wasn't needed to validate the discoveries of Feigenbaum. But, by the same token, applications to DNA weren't needed to validate the theorems of knot theorists. It is a wonderful thing when a branch of mathematics suddenly becomes relevant to new discoveries in another science; both fields benefit enormously. But many (maybe most) areas of mathematics will never be so fortunate. Yet most mathematicians feel their work is no less valid and no less important than mathematics that has found utility in other sciences. For them it is enough to experience and share the beauty of a new theorem. New mathematical knowledge, like knowledge of subatomic particles or knowledge of Mars, is an end in itself! This is part of the “whiff of real mathematics” we need to communicate to the general public.