Editorial A Little Brief Authority

But man, proud man, Drest in a little brief authority, Most ignorant of what he' s most assured, His glassy essence, like an angry ape, Plays such fantastic tricks before high heaven As make the angels weep.

William Shakespeare Measure for Measure

A few months back, a gentleman from India sent me his work on the proof of Fermat's Last Theorem. These things come to us all from time to time, of course. Two years ago I received something which began roughly like this: "Assume FLT false. Then for all n there exists a solution to the equation x ~ + yn = zn.,, There followed a lengthy and confused, but basically correct, proof that there were no solutions for n = 4. This contradiction established FLT.

Not all attempts create such a pig's breakfast at the first hurdle. Many are infuriatingly obscure, or plausible, with the fatal error buried in the seventh sub-sub-subscript of for- mula 773B(ii)(al). But the offering from India was different. It was clearly written and tidily presented. Its author had written several times before sending the actual proof, asking very politely whether I would be willing to take a look. Pointing out that I was no expert in Number Theory, I suggested that he should have no hesitation in submitting the paper to a professional journal. They, I said in my naivet6, would be able to tell him whether his ideas had any merit. So he wrote to the Journal of Number Theory. It transpired that this prestigious organ has a long-standing editorial policy not to referee or accept papers applying elementary methods to Fermat's Last Theorem. The manuscript was duly returned, unread.

Now I can understand the editors' point of view. As a matter of practicality, they doubt- less need to do something to protect themselves against a deluge of cranks and crackpots. Their reason for the policy, in fact, is an inability to find willing referees for such material. I sympathise. After all, I wasn't prepared to take on the equivalent job myself.

But the incident did set me thinking about the role of authority in mathematics. For a start, I bet that if one of the world's top number theorists had found an elementary proof of FLT, then the editors of that journal would drop their policy like a red-hot brick. A mathematician with a sound track record is much more likely to be believed than an obscure unknown from the back of beyond. A referee will search diligently for the tiniest logical flaws in a paper from a newly qualified Ph.D.; he will almost certainly be much less wary of the "it is obvious thats" in a manuscript authored by a plausible candidate for the Fields medal. A student of a colleague of mine recently found a proof of a theorem stated, but never yet proved, by a prominent American mathematician whose depth of imagination is matched only by his tendency not to publish detailed proofs. The student would be able to publish it under the title "Proof of a Theorem of X"; but never in a million years would "Proof of a Conjecture of X" be acceptable to a journal. But in strict terms, X has never published a proof: how come he'll get the credit but the student won't? Why are we all so convinced that X's unpublished proof (if there really is one) is correct?

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Maybe all X really has is an overwhelming feeling that everything fits so neatly that the theorem has to be true? Plenty of nineteenth century mathematicians had similar feelings, and thought they had proofs: today we celebrate their ideas, but we also celebrate some- body else's solutions to the problems they left unresolved.

Every so often a case arises of a relatively obscure mathematician solving a notorious problem, and being totally disbelieved by the mathematical community. Haken's algorithm for computing the genus of a knot. Ap6ry's work on irrational values of the zeta-function. Topically, the solution of the Bieberbach conjecture by Louis de Branges. In all these cases, recognition of the true state of affairs came eventually (either sooner or la ter - - in one case very late). If the identical piece of work had come from a recognised authority in the field, it would have been taken more seriously. If the topic attacked had been more run-of-the- mill, the paper would doubtless have been accepted. Why is it that in precisely those areas where we should take really serious pains not to dismiss new ideas without good reason-- that is, the solution of central difficulties of our subject--we treat people's work so lightly that, as a profession, we simply can't be bothered to consider the mathematics on its own merits? Is it because we're all so busy talking that we've lost the ability to listen?

We mathematicians are not as rational as we like to imagine ourselves. We back our hunches. The editors of the Journal of Number Theory have a hunch that FLT can't be proved by elementary means. It's a good hunch; but it's not a fact. Topologists had a hunch that the genus of a knot would not be algorithmically computable. It looked like a good hunch, but it wasn't. That's the trouble with hunches. But none of this stops us from using our hunches to make off-the-cuff guesses as to what is, or is not, worth doing; and what is, or is not, likely to succeed. And who is or is not likely to succeed. Our hunches are tinged with the colours of authority. And in perhaps the only subject where, fundamentally, authority has no place, we defer to authority a little too willingly for our own good. The truth of a theorem should not depend on who claims to have proved it.

Ian Stewart

4 THE MATHEMATICAL INTELLIGENCER VOL. 7, NO. 2, 1985