Is it possible to have a research career while checking the proof of every theorem that you cite?

Question

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)

Answer 1

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.

Answer 2

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.

Answer 3

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."

Answer 4

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.

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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.