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I'm studying mathematics and as you all know the most important things in mathematics are proofs.

My question is, who determines if a proof that someone invents in mathematics is valid? Is there some mathematics professors who check all people's proofs in the world?

If I invented a new proof, where do I send it to? Can anyone invent their own mathematical proofs?

user642796
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bodacydo
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    In general it is a social process. If a published result is deemed important then many people will read it and find bugs in the proof. Maybe. – copper.hat Nov 18 '15 at 07:41
  • For a proof of a theorem to be validated, a searcher needs to communicate on his proof. He needs to explain it in some conferences and also to find a journal of mathematics to publish it, when he sends his proof to the journal a so-called "reviewer" will review his proof, if the reviewer decides that it is worth be published then it is published. Finally when "enough" mathematicians have seen the proof and nobody found a problem people consider the proof to be valid. – Clément Guérin Nov 18 '15 at 07:42
  • Wow @ClémentGuérin thanks for explaining. My next question is are there examples of proofs who were accepted but then 100 or 200 years later were discovered to be invalid? – bodacydo Nov 18 '15 at 07:44
  • @bodacydo, as far as I know the system is efficient, so there are no spectacular examples but for instance you have a French book called "Infirmation de l'hypothèse de Riemann" (rough translation : the Riemann hypothesis is false). The book exists (I had it in my hands when I was a student) and pretends to prove that Riemann hypothesis is false. The book is actually a proof (I don't say anything about the validity of the proof) with a lot of mathematics. The point is that the author did not follow the usual steps (publication in a journal rather than using an editor of "norrmal" books) – Clément Guérin Nov 18 '15 at 07:55
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    Usually they're not totally invalid. One example is that Euclid thought he had pinned down all the axioms he needed for Euclidean geometry, but he accidentally used his intuition to derive some results, and so technically those proofs are wrong because they do not follow from the axioms that he specified. More than 2000 years later, this was pointed out by Pasch, and the extra axiom needed is called Pasch's axiom. Euclid also did not specify clearly axioms stating the invariance of certain quantities under Euclidean transformations, so by modern standards his proofs were not completely solid. – user21820 Nov 18 '15 at 07:59
  • @bodacydo, that is why most mathematicians I know consider this book as "false". Because, if it were true, this would be published in a regular journal (I am inclined to think this too). Let me say that, if a lot of people consider the book as false, the author of the book (as far as I know) has a good reputation (at least, he is an experienced mathematicians)... – Clément Guérin Nov 18 '15 at 08:00
  • @Clément described roughly but accurately the standard process of validation. Some cases escape to this mechanism, for various reasons. One example is Perelman posting three papers on the arXiv preprint internet server. Experts of the subject understood immediately that, if correct, these papers would imply Poincaré conjecture and several groups of experts started to check the validity of Perelman"s results. Four years later the general consensus which emerged is that these are correct but the quite atypical fact is that these were never published in refereed journals by Perelman himself. ... – Did Nov 18 '15 at 08:19
  • ... To be noted is that Perelman was by no means unknown to the community at the time and that the eccentric decision to avoid publication in refereed journals (together with the fame of the claimed result) did not weaken the strength of the scrutiny these were subjected to. Another case, rather different and still in progress, is Mochizuki's development of a so called inter-universal Teichmuller theory, claimed to imply abc conjecture. Again: these are the exception rather than the rule and as such, may help to understand "the rule" ... – Did Nov 18 '15 at 08:24
  • ... WP does a decent job of presenting each of these cases and gives some references if one wants to go further. – Did Nov 18 '15 at 08:24
  • This actually happened here to prove that √2 is irrational in a new way. –  Nov 18 '15 at 08:38
  • @mistermarko What? – Did Nov 18 '15 at 08:40
  • Can you find any account of that proof before that post? –  Nov 18 '15 at 08:44
  • Proof is not a social process. Math is what it is, and a proof can be verified and understood by the writer of the proof. \ – Jacob Wakem May 16 '16 at 23:05

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That's a very interesting question. Part of the answer is already included in the question itself - for it is not evident at all that the question should not start with the word "what".

As was pointed out -- correctly I believe -- a determination of the validity of a mathematical proof is a social process. This may come somewhat as a surprise -- especially for high priests of mathematics who believe with all their heart that mathematics equals truth.

The mathematical community is first and foremost a community. It has its institutions, governments, ambassadors, pundits, enthusiasts, cults and rebellious underground movements. The current state of affairs was concisely summarized by Clement Guarin in the comments above. Observe how structured is the social process -- first the proof inventors have to believe in their proof. Then they must present it in front of other people. These other people can be members of their faculty, or readers on a site like this one. Then they ought to present it yet in front of still other people -- presumably holding more distinguished posts than the first ones -- these could be journal editors or otherwise distinguished persona in widely accepted forums, such as some mathematicians who post on the real-mathematics sites. The purpose of this all is to expose the proof to as many eyes as possible, in the hope that if there were some mistake somewhere, it surely would have been found. But this has not always been the case. See the story of the Busemann-Petty problem.

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uniquesolution
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  • The mathematical community is not first and foremost a community. it is first and foremost a set of people independently pursuing a priori truth. In this way it is possible to be a successful member of the "community" with only very occassional interaction with others! – Jacob Wakem May 17 '16 at 00:02
  • I beg to disagree. If what you say is true, then "mathematical truth" can be pursued in total isolation, e.g. in solitude on a lonely island. However, history teaches us that not a single mathematical theorem was conceived in isolation of other humans. Hence there is no innate truth in a mathematical theorem, but merely a consensus of the other living mathematicians, which romantic souls often misplace for "truth". – uniquesolution May 23 '16 at 08:12
  • uesolution So 1+2!=3 on Mars? – Jacob Wakem May 23 '16 at 20:26
  • I don't know. We should have to ask a Martian. – uniquesolution May 24 '16 at 06:45
  • That is a good point. And just because a martian may not call it 2! Doesnt mean the ideology wont be the same. Counting is counting no matter the names. – Nick Pavini Feb 17 '17 at 14:39