0

I had a thought about how one might fake a proof or indeed fool yourself that you have proved something.

First take the proof and convert it into an equivalent question in a field of mathematics that is very obscure and not many people work on. Preferably a field that is less rigourous such as physics or string theory or loop quantum gravity or category theory.

Then use lemmas cited from many different sources that prove various steps. (But don't check yourself if these lemmas are actually true! Just assume they are because they were published.)

Convert it back to into everyday mathematical language. QED.

Now, you have limited the number of people who will understand some steps. Ideally, some steps will only be understood by one person (who will not like to admit it if they made a mistake in their own lemma).

This is why sometimes I doubt whether very long proofs such as Wile's Fermat proof, the classication of groups proof, or the Poincare conjecture proof might have some mistakes in them. And why I think we will only know for sure once these proofs have been converted into computer language to be checked.

I wonder, do you know of any cases where in fact someone has deliberately used this tactic to fake a proof?

zooby
  • 4,343
  • 1
    You may enjoy reading https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-3-strong-induction/MIT6_042JF10_proof.pdf – JMoravitz Oct 08 '20 at 17:04
  • lol. I saw a video of a conference once of someone who said "X has not been discovered anywhere in the literature so it can't exist." First time I saw booing at a physics conference. Another time someone said "All you physicists think this is true but it has never been proved" They also go booed. So goes both ways. Another time someone said "We all agree about X so..". And someone stood up and said "Physics is not about consensus!" Seems to happen a lot in physics... – zooby Oct 08 '20 at 17:11
  • 1
    This is definitely a thing which can happen in principle. That said, I don't know of any case of it occurring deliberately, and there's a good control against it: in general, if you see someone use a "magic technique" to solve a big problem you have a huge incentive to learn that technique yourself before it becomes common parlance. – Noah Schweber Oct 08 '20 at 17:43
  • 1
    Actually this seems like a terrible way to whip up a fake proof: drowning the reader in a million lines of standard reasoning strikes me as much more feasible. Almost all of the big claimed proofs (by respected mathematicians) I can think of which are generally thought of as unsalvagable suffer not from field-switching but rather just too long arguments peppered with tiny mistakes or ambiguities, which are individually solvable but collectively wind up losing the reader. A "we solved X by going to the Y literature" argument is just too easy to check. – Noah Schweber Oct 08 '20 at 17:43
  • @Noah, but probably a million lines of standard reasoning could easily be computer checked. Whereas an abstract field might not be so easily converted into computer undertandable language. And the more different fields you use the harder it is to check it by computer. – zooby Oct 08 '20 at 20:06
  • @zooby "a million lines of standard reasoning could easily be computer checked." Only after somebody writes them in the appropriate formal language as opposed to the natural language proofs are actually written in by humans, and that would be a gigantic undertaking. By contrast, multiple human experts - who together cover the relevant areas - would pretty quickly be able to tell that something was suspicious about the kind of trickery you're asking about, and there would be strong incentive (per my original comment) for such a group to do this. – Noah Schweber Oct 08 '20 at 20:10
  • I really don't think that the particular thing you're worried about here is a serious concern: barring some evidence to the contrary I think it's really a non-issue. (Of course in the broader picture of mathematical errors/frauds in general, computer verification is absolutely the sine qua non - although it's very hard to do in practice, even when the proof being verified is actually correct, unless the original proof was specifically written to be computer checkable.) – Noah Schweber Oct 08 '20 at 20:12
  • That is: if you're worried about FLT or CFSG (personally I think a certain amount of worry is always reasonable barring a computer verification, but it's rather small for FLT and more significant for CFSG), the real sources of worry should be the sheer number of small boring steps which have to be right rather than any sort of "misuse of magic." Again, it's just too easy for the community to uncover an error of magic, and there's too much incentive for it to do so. – Noah Schweber Oct 08 '20 at 20:15
  • Also: "Preferably a field that is less rigourous such as [...] category theory." Wait, what? How is category theory less rigorous? Also, an argument within the confines of a non-rigorous field would not be considered a proof at all. This whole thing seems like a real non-issue. – Noah Schweber Oct 10 '20 at 23:46

0 Answers0