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A colleague raised the above question with me; more precisely he said:

Suppose that a mathematician were resolved not to publish any theorems unless they had checked the proof of every theorem that they cite (and recursively the proofs of all the theorems that those rely on etc.). Can they have a career in pure mathematics?

With the obvious proviso:

Of course, there are a few well-known theorems, like the classification of finite simple groups, whose proofs are virtually impossible for any one person to check at all. But one can have a perfectly good mathematical career without ever citing any of those.

It seems to me that complete checking might be possible, though perhaps only in narrow fields of mathematics, but does anyone know of mathematicians who actually do it? (or try to)

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    $\begingroup$ Trivial comment: this depends on how you define the word "check". $\endgroup$ – S. Carnahan May 4 '16 at 7:55
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    $\begingroup$ When I was at Chicago, I remember Raghavan Narasimhan telling me that he would not use a result in his work which he could not prove himself. $\endgroup$ – Geoff Robinson May 4 '16 at 10:20
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    $\begingroup$ In Ergodic theory, there is a theorem that everybody knows, but that almost nobody has fully knowledge of its proof. The theorem is in a paper which title is exactly the theorem: Bernoulli Shifts with the Same Entropy are Isomorphic, by Donald Ornstein. User ''potentially dense'' has pointed out that ''According to Mathscinet this paper has been cited 57 times''. So, sadly, it cannot work as a criteria for filter mathematicians efficiently, as I thought. $\endgroup$ – user39115 May 4 '16 at 14:20
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    $\begingroup$ Is it really "checking", or is it "understanding"? $\endgroup$ – paul garrett May 5 '16 at 13:24
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    $\begingroup$ I just voted to reopen this question, because in general I think questions like this are important. I realize that we now have academia.stackexchange, but I still think this is the best place to get career advice from mathematicians. The question was closed as inappropriate for MO, and I disagreed. HOWEVER, I also just discovered (after voting to reopen) that there is another question on basically the same topic, so perhaps this one should be closed as a duplicate: mathoverflow.net/questions/23758/… $\endgroup$ – David White May 7 '16 at 16:25
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Possible or not, this should be a goal:-) Let me put it slightly differently: you should understand every result that you use. First of all, a theorem that you use can be wrong. So whenever you rely your proof on a theorem that you did not check, you take a risk. There are many known cases when a result was "accepted" by a mathematical community, and then turned to be either wrong or unproved. If your proof relies on a theorem that you do not understand this really means that you don't fully understand your own proof.

In the cases like finite simple group classification, you should clearly state in your publication that your proof depends on it. And in general, if you write a proof which relies on the theorem that you do not fully understand, you should make as clear as possible, where exactly and how you use this theorem.

EDIT. When you cite a result you endorse it. You are essentially saying that on your opinion it is correct. Now suppose you are simply asked to endorse some result: just to tell your opinion, whether it is correct or not. Would you endorse it publicly in print, without checking the proof ? On my opinion, citing a result in your paper without any comment is the same.

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    $\begingroup$ I like this answer very much. Personally, as a reader and referee, I would prefer people to state clearly in separate theorems the parts which they are using from the literature, and which may be hard or obscure, rather than merely saying in the middle of a proof "By [13, Theorem 2.2] ..." $\endgroup$ – Yemon Choi May 4 '16 at 13:41
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    $\begingroup$ Not all endorsements are the same. A stand-alone statement that you endorse a result should not be confused with building on a result. $\endgroup$ – Douglas Zare May 5 '16 at 21:08
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Vladimir Voevodsky comes to mind, as somebody who very much strives towards this ideal.

Of course the answer to this question depends on how you interpret the word "check". I would consider somebody's axiomatic framework to be individual and flexible. For one person, the classification of finite simple groups might be an axiom. That person's results are correct modulo this axiom. If that person then learns and understands the proof, then this axiom is replaced by whatever results were assumed during the proof, and so on.

The ideal, then, I think would be to explicitly state personal "axioms", even if these axioms are proven theorems for better-informed people. So not to check the proof of every cited theorem, but rather to explicitly state whether the proof of this cited theorem was checked, or whether it was taken as an "axiom".

It would be very difficult to live up to this ideal in the real world. But I think Voevodsky, for example, tries.

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    $\begingroup$ I guess Voevodsky's foundations are guided by the hope of not having to understand all the proofs yourself, but let tne computer check them. You can safely trust a constructive proof verified by a computer (given no bugs in the type checker). Trouble is, very little is actually verified. $\endgroup$ – Manuel Bärenz May 5 '16 at 18:56
  • $\begingroup$ But then you'd have to verify that there are indeed no bugs in the type checker, if you really want to be completely sure... $\endgroup$ – David FernandezBreton May 16 '16 at 12:13
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I would subtitle this question "Bourbaki's dream." The dream faltered on the foundations. Bourbaki tried to give a half-baked half-formalized naive set theory resulting in an embarrassment of epic proportions (with regard to their volume Theory of Sets; of course other volumes have been extremely successful, like the Lie theory volume) that has been detailed by Adrian Mathias, an expert in the field unlike any of the Bourbaki, in a series of recent detailed critiques (not merely his essay The ignorance of Bourbaki). What I am trying to suggest is that checking all of the previous results will get you hopelessly bogged down in the foundations.


In the comments below, some of the editors requested evidence that work by Grothendieck was blocked by the Bourbaki. Here is a first sample from a 1992 paper by Corry:

Eilenberg himself was commissioned several times with the preparation of drafts on homologies and on categories, while a fascicule de résultats on categories and functors was assigned successively to Grothendieck and Cartier. However, the promised chapter on categories never appeared as part of the treatise. As we shall see in greater detail in the next section, the publication of such a chapter could have proved somewhat problematic when coupled with Bourbaki's insistence on the centrality of structures. The task of merging both concepts, i.e., categories and structures, in a sensible way, would have been arduous and unilluminating, and the adoption of categorical ideas would have probably necessitated the rewriting of several chapters of the treatise. This claim is further corroborated by the interesting fact that when the chapter on homological algebra was finally issued (1980), the categorical approach was not adopted therein.

My perception is that the Bourbaki concept of structure, while in principle similar in spirit to category theory, was hopelessly tied in with their naive set-theoretic realism, and as a consequence created an impediment of the sort detailed by Corry.

While I am in favor of systematization, the Bourbaki project to get to an alleged bottom of all of mathematics is more of a collectivization than a systematization, and is disturbingly procrustean.

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    $\begingroup$ Dear Mikhail: But the whole treatise (and much more) is rigorously based on the foundation given in the volume on set theory. Maybe it is not the foundation that some people would chose today, or not the foundation that you like, but it is enough for Bourbaki's purpose (and much more). $\endgroup$ – Fred Rohrer May 4 '16 at 8:16
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    $\begingroup$ @MikhailKatz: I am intruiged. How do you know that "Grothendieck's best work was shelved" by Bourbaki and is lying in some drawer in Paris? Can you back up your claim with references? $\endgroup$ – eins6180 May 4 '16 at 8:26
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    $\begingroup$ What? You don't need a PhD in logic to handle the amount of foundations needed for work in other fields of mathematics, just like you don't need a PhD in linear algebra to handle the amount of linear algebra you need if your field is not linear algebra. $\endgroup$ – Emil Jeřábek May 4 '16 at 8:50
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    $\begingroup$ The fact that a foundational system requires a triple negation and a double negation to express the empty set (and 4,523,659,424,929 symbols for the number 1) is of course questionable. However, I did not find all of Mathias's papers always convincing — part of the fault is clearly due to mathematicians having nothing to do with Bourbaki. Also, a recent paper by Anacona, Arboleda and Pérez-Fernández (On Bourbaki's axiomatic system for set theory, Synthese, 2014) explains how Bourbaki's system is equivalent to “Zermelo-Fraenkel + Choice - Foundation”. See especially footnote 11 in that paper. $\endgroup$ – ACL May 4 '16 at 12:33
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    $\begingroup$ I don't think that one can call this Bourbaki's Dream at all. In fact, it's rather the opposite: Bourbaki wants to write things down fully rigorously from clear foundations so that other people won't need to check the details of the proofs of the theorems they use, because they are proved in full rigorous and trusted details in Bourbaki's volumes. In my opinion, it is similar in principle to attempts to provide rigorously computer-checked libraries of mathematical theories, as people are doing more and more. In fact, see hal.inria.fr/inria-00408143v5/document $\endgroup$ – Denis Chaperon de Lauzières May 4 '16 at 13:38
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It is worth quoting what Hirsch says about J. H. C. Whitehead (footnote 23 on page 95 of Hirsch's contribution to the Smalefest volume.)

"Whitehead was very good about what he called "doing his homework," that is, reading other people's papers. "I would no more use someone's theorem without reading the proof," he once remarked, "than I would use his wallet without permission." He once published a proof relying on an announcement by Pontryagin, without proof, of the formula $\pi_4(S^2)=0$, which was later shown (also by Pontryagin) to have order 2. Whitehead was quite proud of his footnote stating he had not seen the proof. Smale, on the other hand, told me that if he respected the author, he would take a theorem on trust."

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