Most scientists, and mathematicians in particular, must have been confronted at some point with the problem of properly quoting a theorem (or a theory, or a conjecture, …) that they do not truly understand. It may be that they have not read the proof in detail to have checked independently its correctness, or that the result involves concepts, objects and theories with which they are not familiar. Now, how should this be phrased? Surely, this is part of the information to convey to the reader of a scientific paper or book?
Well, as far as I understand (this being one of the many things that one is supposed to learn on the job), this type of doubt or indeterminacy is supposed to be hidden. The only accepted style of citing, at least judging from the typical mathematical text, is both steadfastly Olympian and touchingly Gimpelian. It implies, at the same time, a complete mastery of whatever is involved in the result mentioned, however incorrectly it may be phrased (“Wiles has proved that every modular elliptic curve is associated with a cusp form with integral coefficients”), and a childlike belief that whatever is claimed in a written source must be true (“Grothendieck has proved [SGA V, Exp. 12, Section 4, Théorème 3.1] that 27 is the only prime number which is not squarefree”). I’ve very rarely seen a quote come with a comment that the author does not claim to have understood it (which, it is true, would be somewhat embarassing for a result which is crucial to one’s proof…)
Personally, I often feel a definite awkwardness in citing a result I don’t “know” in the deepest sense of the word, and of course with the current profusion of preprint sources, things tend to become worse as more results are available for use and quotation than ever. One can try being careful (“Grothendieck has announced a proof that 27 is prime”), but what may seem reasonable when speaking of a result barely out in the arXiv, quickly becomes insulting when refering to some result which is published and is really true, but which, not to make bones about it, one has simply not checked personally. Indeed, referees may take this badly, especially if they happened to prove the theorem in question.
To give a concrete example, I have no doubt that the Riemann Hypothesis over finite fields is true, but although I have really done a lot of reading about it, and can claim to have gone in great detail in the first proof of Deligne, I can not yet claim to have mastered the second — which I’ve used much more often in my work (and even with the first, I have certainly not a full mastery of the total amount of background material, such as the complete proof of the Grothendieck-Lefschetz trace formula). This example is not academic at all: many analytic number theory results depend on estimates for exponential sums which are not accessible at all without Deligne’s work and its extensions, but very few are the analytic number theorists who understand the full proof. (My own PhD advisor said that this was essentially the only result he had ever used that he could not, if needed, reprove from scratch).
I think most uses of the Riemann Hypothesis over finite fields are fair, however: many people who use it for analytic number theory have a highly sensitive sense of what estimates should be correct, and indeed they quite often suggest precise conjectures about what they expect (which are then hopefully confirmed by people like N. Katz, using the deep algebraico-geometric framework underlying the Riemann Hypothesis): the point is that there is really an interplay between the analytic number theorists (who suggest the application of the Riemann Hypothesis, quite frequently in highly non-trivial ways) and the algebraic geometers who prove the required estimates hold. (One of the nicest example of this interaction is the beautiful paper of Fouvry and Katz on applications of the Katz-Laumon theory of “stratification of estimates” for families of exponential sums).
Another fairly common situation (at least recently), for analytic number theory, is to use the modularity of elliptic curves. Here, the gap between my understanding and what would be required to claim that I can vouch for the proof, is larger: although I’ve been exposed a lot to the general strategy, I’ve never understood in any depth the crucial final steps. But again, what I (and many others) usually want to use this theorem for is to state a corollary of a result we have proved for more general modular forms. Even if the case of elliptic curves might be the best motivation for the work, the general case is usually interesting enough that it doesn’t feel like cheating, especially if this point is explained in the introduction.
These are cases where it’s possible to phrase things professionnally enough. But sometimes, it’s much more complicated. Indeed, what should be true may be much less obvious, and the result we prove might depend completely on a truth that we can, in all honesty, only say that it’s nice that it’s true, because otherwise (and it could have been otherwise, for all we know), we would be stuck.
Thus I’ve had the occasion to use results that ultimately depend on the classification of finite simple groups (through the type of “strong approximation theorems” that say that the reduction modulo most primes of a Zariski dense subgroup of SL(n,Z), for instance, is the full group SL(n,ZpZ)) , and here it’s quite difficult for me to know how to quote this honestly. It is true that the corollaries of the classification which are involved do not really depend on its finer aspects (e.g., they do not depend on how many exceptional groups there are, provided there are only finitely many), but on the other hand, I don’t really have enough personal experience and intuition about finite groups to feel that the classification should be as it is, with a few known infinite families and finitely many exceptions.
Note that I really have no suggestion about this. Would it be good advice to suggest, say, that any analytic number theorists working on modular forms should spend a few weeks getting acquainted enough with the proof of the Ramanujan-Petersson conjecture, in order to feel better about using it? This might be what I would want to say, but how far can it get? If one starts from scratch — knowing no algebraic geometry for instance –, much more than a few weeks will be needed to feel any confidence about the correctness of the proof.
In the prevailing scientific mindset, this is an investment of time that may well be out of the question for most people (with, possibly, the exception of beginning graduate students — they, I think, should at least spend enough time acquiring the basics of many “languages”, so that they can go further later on if needed in their work; it is much harder to learn the principles of probability theory, or combinatorics, or algebraic number theory, as a post-doc or young professor, than as a student). I must confess that, at the current time, I certainly do not think I will be able to invest significant efforts in trying to understand the classification of finite simple groups…
