58

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)

Yemon Choi
  • 25,496
John Stillwell
  • 12,258
  • 24
    Trivial comment: this depends on how you define the word "check". – S. Carnahan May 04 '16 at 07:55
  • 13
    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. – Geoff Robinson May 04 '16 at 10:20
  • 1
    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. – user39115 May 04 '16 at 14:20
  • 2
    Is it really "checking", or is it "understanding"? – paul garrett May 05 '16 at 13:24
  • @GeoffRobinson there's a difference between "can" and "does". – Christopher King May 05 '16 at 18:47
  • @PyRulez : An example of one who does suggests to me that others can. – Geoff Robinson May 05 '16 at 18:58
  • 3
    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: http://mathoverflow.net/questions/23758/published-results-when-to-take-them-for-granted – David White May 07 '16 at 16:25

4 Answers4

49

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.

Alexandre Eremenko
  • 88,753
  • 13
    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] ..." – Yemon Choi May 04 '16 at 13:41
  • 6
    Not all endorsements are the same. A stand-alone statement that you endorse a result should not be confused with building on a result. – Douglas Zare May 05 '16 at 21:08
35

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.

Daniel Moskovich
  • 21,887
  • 4
    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. – Manuel Bärenz May 05 '16 at 18:56
  • 1
    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... – David Fernandez-Breton May 16 '16 at 12:13
11

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."

John Stillwell
  • 12,258
Mike-Doherty
  • 786
8

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.

Mikhail Katz
  • 15,081
  • 1
  • 50
  • 119
  • 2
    If you have a look at the theories that were rigorously built on Bourbaki's treatise, especially some parts of Grothendieck's work, I do not think "embarrassment of epic proportion" is appropriate here. – Fred Rohrer May 04 '16 at 07:59
  • I was referring to Theory of sets. I clarified this point in my answer. – Mikhail Katz May 04 '16 at 08:00
  • @FredRohrer, in fact some of Grothendieck's best work was shelved by the Bourbaki and is still languishing in some drawer in Paris. – Mikhail Katz May 04 '16 at 08:02
  • 6
    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). – Fred Rohrer May 04 '16 at 08:16
  • 13
    @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? – eins6180 May 04 '16 at 08:26
  • 1
    @Fred, Mathias shows that the foundation was not rigorous and on the contrary he documents numerous errors. I suggest you read Mathias. It is a real eye-opener. – Mikhail Katz May 04 '16 at 08:26
  • 1
    @eins6180, I suggest you read Mathias. I also saw elsewhere that Grothedieck proposed a volume on infinitesimals that was never published by Bourbaki. – Mikhail Katz May 04 '16 at 08:27
  • 1
    Dear Mikhail, may I ask you to provide a precise reference? – Fred Rohrer May 04 '16 at 08:32
  • 1
    Mathias, A. R. D. Hilbert, Bourbaki and the scorning of logic. Infinity and truth, 47–156, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 25, World Sci. Publ., Hackensack, NJ, 2014. @Fred – Mikhail Katz May 04 '16 at 08:38
  • So, the actual answer to the question is that the dream is possible, just use a decent book for foundations rather than Bourbaki? – Emil Jeřábek May 04 '16 at 08:39
  • 1
    Rather, first get a phd in logic :-) – Mikhail Katz May 04 '16 at 08:39
  • 2
    @MikhailKatz: I am sorry, but both references (Ignorance of Bourbaki; Hilbert, Bourbaki and the scorning of logic) do not back up your claim. And that you've read it somewhere else just isn't convincing enough. However, both papers seem to be an interesting read. So thanks for that. – eins6180 May 04 '16 at 08:42
  • 10
    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. – Emil Jeřábek May 04 '16 at 08:50
  • 1
    @EmilJeřábek, I agree with you. However if you want to trace every result you use all the way back, as John's colleague suggested, you are in for a tall task that some famous people failed at. – Mikhail Katz May 04 '16 at 08:56
  • 2
    @eins6180, consider the following passage from Mathias: "A possible and regrettable consequence of the uniformising tendency of Bourbaki is Grothendieck's withdrawal from the group and subsequently from contact with other mathematicians: one wonders whether he felt threatened by Bourbaki in the same totalitarian way that Chevalley may have been shouted down by Dieudonn'e, Cantor was blocked by Kronecker, and Nikolai Lusin was menaced by, and Giordano Bruno actually suffered, a death sentence." I am not sure I would endorse every detail here, but those who wish to believe in perfect harmony... – Mikhail Katz May 04 '16 at 09:10
  • 1
    ...between Bourbaki and Grothendieck are kidding themselves. – Mikhail Katz May 04 '16 at 09:11
  • 1
    @eins6180, here's a quote from Mathias that proves my point in an even more poignant fashion: "Eilenberg acknowledged that Bourbaki hadn't thought through their position on foundations clearly and that what they had provided was a mess. But then he felt foundations was always a work in progress, an outlook shared by several members of Bourbaki: but when, later, after the impact of category theory had become evident--especially after the adjoint functor theorem and Grothendieck's work-- – Mikhail Katz May 04 '16 at 09:14
  • 1
    ...it was suggested they go back and do a fresh treatment of foundations from scratch, Weil vetoed the idea as a mis-application of energy and resources. Thus Weil, the tyrant, imposed his static view of logic on his colleagues." – Mikhail Katz May 04 '16 at 09:14
  • 5
    @MikhailKatz: I read these passages before; however, they don't hint at unpublished material. They just speculate what could have been possible. And that Grothendieck and Bourbaki had a troubled relationship is well known. – eins6180 May 04 '16 at 09:27
  • 1
    I remember reading that a manuscript on infinitesimals was rejected by Bourbaki. Perhaps this was in Corry rather than Mathias. @eins6180 – Mikhail Katz May 04 '16 at 09:40
  • 3
    No, not a tall task at all. The trick is to do it all along while learning the results. I pretty much did check (even was forced to) everything I learned as part of my undergraduate curriculum, which was more or less systematically built from foundations. I definitely cited papers I didn’t check, but mostly as pointers to related literature, only a small fraction were results I actually used, and I believe it would be feasible for me to track the proofs down now if I wanted (I can’t remember ever using anything of epic proportions like the classification of finite simple groups). – Emil Jeřábek May 04 '16 at 11:47
  • 2
    @EmilJeřábek, interesting. I assumed that anybody claiming that he checked "everything" is simply not aware of the complexity of the foundations, but now that a professional logician (unlike myself) is making such a claim I have to concede. Still, I think there are a lot more of fundamental results (of the finite simple group classification type) than people realize. – Mikhail Katz May 04 '16 at 11:54
  • 10
    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. – ACL May 04 '16 at 12:33
  • @ACL, interesting. Are you sure this is choice or perhaps global choice? – Mikhail Katz May 04 '16 at 12:56
  • 18
    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 https://hal.inria.fr/inria-00408143v5/document – Denis Chaperon de Lauzières May 04 '16 at 13:38
  • 1
    @Denis, I think computer-checked libraries are fabulous. As far as Nicolas is concerned, I think he wanted to check this both for himself and for others, just like Vladimir Voevodsky mentioned in Daniel's answer. The fact is that Nicolas failed, as documented by Mathias, but Vladimir possibly succeeded. – Mikhail Katz May 04 '16 at 13:52
  • @DenisChaperondeLauzières : I'm confused about your comment. Are you saying that computer-checked libraries are similar to Bourbaki or that they're the opposite? They seem similar to me because they both do all the checking for you, but it sounds like you're saying the opposite. – Timothy Chow May 04 '16 at 18:10
  • To Timothy Chow: sorry if my formulation was unclear; I was indeed saying that I view Bourbaki's work as similar to computer-checked libraries. (Bourbaki having the advantage of readability and exercises...) – Denis Chaperon de Lauzières May 04 '16 at 18:43
  • 1
    Mikhail, does your irony towards Bourbaki mean that you don't approve any activity on systematization of something? – Sergei Akbarov May 04 '16 at 19:39
  • 5
    @MikhailKatz No, the founders of Bourbaki did absolutely not begin to write his treatise to have every detail checked, but to have a usable textbook at their disposal that would fit their taste. This is clearly documented by the notes of the first meeting (December 12, 1934; http://archives-bourbaki.ahp-numerique.fr/archive/files/1ddd9c24f785342286f51d403482afca.pdf) – ACL May 04 '16 at 22:43
  • @SergeiAkbarov, I am strongly in favor of systematization. However, 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. – Mikhail Katz May 08 '16 at 07:08
  • 1
    Mikhail, in my opinion, you are too hard on them. I also don't like their "Theory of sets", but my impression is that it is not difficult to rewrite this concrete book, and everything will be acceptable. – Sergei Akbarov May 08 '16 at 07:55
  • @Sergei, I have not written anything on the Bourbaki so my personal opinion is less relevant than that of Corry, McLarty, and Mathias, who have been rather critical (Mathias more explicitly than the other two). – Mikhail Katz May 08 '16 at 08:07
  • 1
    Do they have complaints against other books, apart from "Theory of sets"? – Sergei Akbarov May 08 '16 at 08:15
  • I haven't studied Corry and McLarty in detail so it is hard for me to say but Mathias is a logician and is concerned with their covering not merely of set theory but of logic as a whole. I do remember reading in either Corry or McLarty that there is an odd aspect of the Theory of Sets: while other books are cited in higher-level volumes in accordance with the philosophy of "the great tree of mathematics", the volume Theory of Sets apparently is not, even when the real numbers (constructed here) are used in other volumes. @Sergei – Mikhail Katz May 08 '16 at 11:42
  • 1
    Dear Mikhail: The real numbers are constructed in TG. And there are several citations of E in most of the following volumes, basically everywhere where it is logically needed. – Fred Rohrer May 08 '16 at 15:48
  • I will have to check my sources. Probably the claim concerns the natural numbers rather than the real numbers. The natural numbers are constructed in Theory of Sets but apparently the construction is not referred to elsewhere, even though obviously you are going to need them :-) @FredRohrer – Mikhail Katz May 09 '16 at 14:40
  • 1
    Dear Mikhail: Indeed, checking your sources seems to be a good (and necessary) idea! The first occurence of the set of natural numbers after E is in A.I.1.1 Exemple 2, and there is a reference to E.III, as it should be. – Fred Rohrer May 09 '16 at 18:17