At the very end of this 2006 interview (rm), Kontsevich says
"…many great theorems are originally proven but I think the proofs are not, kind of, "morally right." There should be better proofs…I think the Index Theorem by Atiyah and Singer…its original proof, I think it's ugly in a sense and up to now, we don't have "the right proof." Or Deligne's proof of the Weil conjectures, it's a morally wrong proof. There are three proofs now, but still not the right one."
I'm trying to understand what Kontsevich means by a proof not being "morally right." I've read this article by Eugenia Cheng on morality in the context of mathematics, but I'm not completely clear on what it means with respect to an explicit example. The general idea seems to be that a "moral proof" would be one that is well-motivated by the theory and in which each step is justified by a guiding principle, as opposed to an "immoral" one that is mathematically correct but relatively ad hoc.
To narrow the scope of this question and (hopefully) make it easier to understand for myself, I would like to focus on the second part of the comment. Why would Kontsevich says that Deligne's proof is not "morally right"? More importantly, what would a "moral proof" of the Weil Conjectures entail?
Would a morally proof have to use motivic ideas, like Grothendieck hoped for in his attempts at proving the Weil Conjectures? Have there been any attempts at "moralizing" Deligne's proof? How do do the other proofs of the Weil Conjectures measure up with respect to mathematical morality?
Best Answer
I would guess that Grothendieck's envisaged proof, via the standard conjectures, would be "morally right" in Kontsevich's sense. (Although there is the question of how the standard conjectures would be proved; since they remain conjectures, this question is open for now!)
The objection to Deligne's proof is that it relies on various techniques (passing to symmetric powers and Rankin--Selberg inspired ideas, analytic arguments related to the positivity of the coefficients of the zeta-function, and other such things) that don't seem to be naturally related to the question at hand. I believe that Grothendieck had a similar objection to Deligne's argment.
As a number-theorist, I think Deligne's proof is fantastic. One of the appeals (at least to me) of number theory is that none of the proofs are "morally right" in Kontsevich's sense. Obviously, this is a very personal feeling.
(Of course, a proof of the standard conjectures --- any proof, to my mind --- would also be fantastic!)
[Edit, for clarification; this is purely an aside, though:] Some arguments in number theory, for example the primitive root theorem discussed in the comments, are pure algebra when viewed appropriately, and here there are very natural and direct arguments. (For example, in the case of primitive roots, there is basic field theory combined with Hensel's lemma/Newton approximation; this style of argument extends, in some form, to the very general setting of complete local rings.) When I wrote that none of the proof in number theory are "morally right", I had in mind largely the proofs in modern algebraic number theory, such as the modularity of elliptic curves, Serre's conjecture, Sato--Tate, and so on. The proofs use (almost) everything under the sun, and follow no dogma. Tate wrote of abelian class field theory that "it is true because it could not be otherwise" (if I remember the quote correctly), which I took to mean (given the context) that the proofs in the end are unenlightening as to the real reason it is true; they are simply logically correct proofs. This seems to be even more the case with the proofs of results in non-abelian class field theory such as those mentioned above. Despite this, I personally find the arguments wonderful; it is one of the appeals of the subject for me.