Monday, August 02, 2004

Proofs and Reputations

Karl Sabbagh, the author of a recent book on the Riemann Hypothesis, recently wrote an article in the London Review of Books about Louis de Branges and his recent announcement of a proof for the RH.

There is no evidence that, so far, any mathematician has read [de Branges' proof ]: de Branges and his proof appear to have been ostracised by the profession. I have talked to a number of mathematicians about him and his work over the last few years and it seems that the profession has come to the view that nothing he does in this area will ever bear fruit and therefore his work can be safely ignored. It may be that a possible solution of one of the most important problems in mathematics is never investigated because no one likes the solution's author.

My post has nothing to say about the proof itself; I would not dare to presume even a passing familiarity with it. What caught my attention was the sense of surprise in Sabbagh's article; the unstated 'what on earth does a man's reputation have to do with his proof' ?

What indeed ?

Mathematics (and by extension theoretical CS) inhabits a Platonic world of truths and false statements (with a dark brooding Gödel lurking in the background, but that's a different story). As such, either statements are true, and become theorems/lemma/what-have-you, or they are not and fall by the wayside. The pursuit of (mathematical) truth is thus the search for these true statements. The identity (or very existence) of the searcher has no effect on the truth of the statements; there is no observational uncertainty principle.

However, mathematicians live in the real world. In this world, true and false gets a bit murkier. A theorem is no longer true or false, but almost certainly true, or definitely false. They are far closer to the falsifiable theories of natural science, although there is at least a "there" there; a scientific theory can only have overwhelming evidence in support of it, but a mathematical statement (if not too complex) can be categorically true.

The natural sciences have reproducible experiments; if I cannot reproduce the results you claim, all else being equal, the burden of proof is on you to demonstrate that your results are indeed correct. Similarly in mathematics, if a claimed theorem has a complex proof, the burden of proof does reside on the author to demonstrate that it is indeed correct. They can do this by simplifying steps, supplementing with more intuition, or whatever...

In this respect, theorem proving in the real world has a somewhat social flavor to it. And thus, there is also (it seems to me) a social compact: You demonstrate competence and capability above a certain basic threshold, and I deem your proofs worthy of study. The threshold has to be low, to prevent arbitrary exclusion of reasonable provers, but it cannot be nonzero zero, because in the real world it is hard to check a proof with absolute certainty.

This is why the many proofs that P=NP (or P != NP) that float on comp.theory don't get a fair shake: it is not because the "experts" are "elitists" who don't appreciate "intruders" poaching their beloved problems; it is because the social compact has not been met; the writers don't cross the threshold for basic reasonableness, either by choosing to disregard the many approaches to P vs NP that have been ruled out, or by refusing to accept comments from peer review as plausible criticism, and demanding that the burden of proof be shifted from them to the critical reviewer.

Such a compact could be abused mightily to create a clique; this is why the threshold must be set low and is low in mathematics. The notorious cliche that a mathematician's best work is done when young at least reinforces the idea that this club can be entered by anyone. More mundanely, there are awards for best student papers at STOC/FOCS that often go to first-year grad students (like this year's award).

Going back to de Branges' proof, I have no idea what the technical issues are with his proof, and if there are known reasons why they don't work, but going solely on the basis of Karl Sabbagh's article (and I acknowledge that he could be biased) it seems wrong to ignore his manuscripts. He for one has clearly crossed the threshold of the social compact. Reminds me of an attempt I made to read a popular exposition of Mulmuley and Sohoni's recent work on P vs NP; if this work does lead to a claimed proof, I imagine that there would be few people who could comprehend the proof, but it would deserve to be read.

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