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*To*: fom@math.psu.edu*Subject*: FOM: Mathematical Certainty*From*: Joe Shipman <shipman@savera.com>*Date*: Fri, 11 Dec 1998 11:30:41 -0500*CC*: sazonov@logic.botik.ru, sacook@cs.toronto.edu, ablass@math.lsa.umich.edu*Organization*: Savera systems*Sender*: owner-fom@math.psu.edu

I'd like to look at the three "big" theorems we have been discussing (FLT, 4CT, CFSG) a little more to make some points about "mathematical certainty". Wiles's proof of FLT, which may in some sense be the "deepest" of the three proofs, is also the most understandable. There must be at least a few dozen mathematicians by now who each "understand" the proof in the sense that they "know why the theorem is true" and could fill in details as far as necessary at any point in the proof, on demand. But it is a very difficult proof and in fact the original version of it had a serious gap. So Wiles, when he first announced a proof, thought he understood the proof and was "certain" that the theorem was true (or he would never have dared to announce the proof) but he was MISTAKEN. I would like to suppose this was just the myopia that comes from being too close to something you've worked on for a long time (the same reason it's much easier to catch a spelling or grammar error in someone else's writing than in your own), because in fact the gap became apparent fairly quickly to people who put in a serious effort to understand Wiles's proof, but many other theorems have had incorrect proofs that were accepted for many years. TWO separate proofs of the 4-color theorem were wrong but not shown to be so for 11 years each! (Kempe 1879 shown incorrect by Heawood 1890; Tait 1880 shown incorrect by Peterson 1891). Maybe the 4CT was not considered as important at the time because it didn't have a notorious history (after 1891 it certainly did!), and if a really serious effort had been put forth the gaps would have been found much earlier (the existence of two independent "proofs" may have reduced the urgency of checking each one really carefully). But there were false proofs of FLT before then when it was already notorious; can anyone provide information on how quickly those false proofs were discovered? I think the answer is "quite quickly", so my "myopia" theory holds up; but there will be a serious problem if someone can show that an incorrect proof of a "theorem" *considered very important at the time* was *generally accepted* for *several* years. Can anyone provide such an example from the 20th Century? Earlier examples aren't so pertinent because standards of rigor became much higher (gradually) during the 19th century. Such an example would have serious implications--it would be very hard to assert that any theorem with a difficult proof was "mathematically certain". But for now I'll stand by the following statement: For the last century or so, the following sociological criterion for "mathematical certainty" of a theorem has been sufficient: 1) Before the proof was announced, the conjecture was widely considered to be very important 2) The proof can be read by a single person (rules out computer-aided proofs like 4CT or proofs which required dozens of authors like CFSG) 3) The proof was *generally accepted* (no serious workers in the field proclaimed that the proof failed to convince them) for at least 5 years. If my statement is correct Reuben Hersh will be somewhat vindicated. Turning to 4CT, we find that the "mathematical" portion of the proof is quite well-established and many people understand it. There is no doubt, it really is "mathematically certain" that if the two algorithms given produce the desired output then 4CT is true. The "computer" portion of the proof has also been multiply verified (this took longer than most people realize, but it's been done enough by now). So 4CT is "scientifically certain" though maybe not "mathematically certain". I would actually be less surprised at this point if another gap is found in Wiles's proof than if some mistake was found invalidating the proof of 4CT. Blass and Sazonov are quite right to point out the importance of finding an "understandable" proof, but there may not be one; at some point we may just have to say "we understand this phenomenon as well as it can be understood, but it depends on so many finite combinatorial 'facts' that we still need to point to a computer output to persuade someone else that it is true". CFSG is less "mathematically certain" than FLT because the proof is not yet digestible by any one person. Depending on the word of a bunch of other mathematicians for their parts of the proof isn't any better than depending on a computer in my opinion. It is not "scientifically certain" either for the same reason. If each of the dozens of pieces has been really solidly verified, we may be justified in imputing some sort of "certainty" to the theorem, but I'm not sure exactly what to call it ("scientifically certain" is not quite right because since the proof isn't small enough for a single person to understand there is a sense in which it is "irreproducible"). Your comments, please! I'd especially like to hear from number theorists or group theorists who may have a better understanding of the proofs of FLT and CFSG than I do and can confirm or correct my characterizations (for 4CT I am confident I know what's going on). -- Joe Shipman

**Follow-Ups**:**Charles Silver**- Re: FOM: Mathematical Certainty

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