Some time back, Kevin Buzzard (friend of the blog!) gave a series of talks in which he criticized certain aspects of the mathematical culture when it came to accepting proof. In addition to obvious targets like the classification of finite simple groups, he took aim at my paper with Boxer, Gee, and Pilloni, an in particular this passage:
It should be noted that we use Arthur’s multiplicity formula for the discrete spectrum of \(\mathrm{GSp}_4\), as announced in [Art04]. A proof of this (relying on Arthur’s work for symplectic and orthogonal groups in [Art13]) was given in [GT19], but this proof is only as unconditional as the results of [Art13] and [MW16a,MW16b]. In particular, it depends on cases of the twisted weighted fundamental lemma that were announced in [CL10], but whose proofs have not yet appeared, as well as on the references [A24], [A25], [A26] and [A27] in [Art13], which at the time of writing have not appeared publicly
Kevin asks (of this passage): “Can we honestly say that this is science?” I am certainly broadly sympathetic to Kevin’s concerns; I would say the inclusion of these remarks in the paper is some evidence of that. One interesting remark is that Kevin chose to highlight our paper rather than discuss the existential state of Arthur’s preprints.
It is now six years since our preprint was first posted, and no preprints from Arthur have been forthcoming. I have felt a growing responsibility that I should be obliged to address the issue directly on this blog. But in what form, exactly? Something of the form of this post on ABC?. In the last year I have seen talks explaining how there exist a number of genuine difficulties in carrying out Arthur’s proposed strategy. That strategy involves an inductive argument where the argument for one group might reduce to a claim for a group of much higher rank, and obviously this requires some finesse to avoid any circular argument.
What I ultimately decided was suitable was that the focus of such a post should not be of criticizing Arthur, but in emphasizing that anyone who does step up the the plate and resolves these outstanding issues really needs to be recognized for their original contributions. This is not a situation in which “the experts know how to do this” — it is a situation where the original position was “Arthur has done this” to “Arthur knows how to do this” which gradually evolved to “Arthur has explained a strategy to do this, but this strategy appears to require overcoming serious obstacles”.
But fortunately, the situation has now changed, very much for the better. In a new recent preprint by Hiraku Atobe, Wee Teck Gan, Atsushi Ichino, Tasho Kaletha, Alberto Mínguez, and Sug Woo Shin, all the promised results of Arthur’s missing papers have now been supplied. So instead, I can focus on emphasizing that this new result is a monumental achievement, and that it should be appreciated by the community as the genuine original contribution that it is. Let me add that the authors are incredibly gracious to Arthur and nobody is trying to take away from Arthur’s absolutely key fundamental contributions. But at the same time, that should not detract from our appreciation of this new work.
To return to Kevin’s question of “is this science”, there still is, unfortunately, one remaining caveat. Namely, there is another (non-existent) paper, the proof of the twisted weighted fundamental lemma as announced by Chaudouard and Laumon in 2010(!). So what is the situation here? At one point Chau told me that he was considering writing a book which would (hopefully) include this result, but that this is no longer his intention, in part because at least one graduate student (not at Chicago!) is working on this problem (I won’t say too much more to avoid adding unnecessary pressure). But the message in this case is surely the same: if the the Fundamental Lemma is worth a fields medal, a generalization of that result for which a large amount of mathematics is currently contingent should also be appreciated when it finally appears!
Science is a messy patchwork of incremental contributions and unresolved dependencies. It is hardly clean-cut. Thus, the situation described in your post qualifies 100% as such.
Note that mathematics, unlike other fields of science, upholds a higher standard of rigor. The question raised by Buzzard could be more effectively framed as whether certain practices are mathematical rather than scientific, a distinction aptly highlighted by CX’s response. If the complex status of the Langlands business is deemed “non-scientific,” then it follows that much of established physics might also not fit that definition—and may never do so.
It’s more interesting that “secret magma” has played a crucial role in significant results. Although AlphaFold3 has not been open-sourced, its creators shared a Nobel Prize, prompting important questions about the implications of proprietary research. This issue may become more pronounced as cutting-edge scientific and technological advancements increasingly intertwine with profit motives and industry sponsorship. While Buzzard may be advocating for one perspective within math, the broader landscape could very well evolve in another direction.
We may be on the brink of a bifurcated world where varying standards of rigor coexist. An alternative standard has been proposed for string theory. Are we approaching a time of multiple “paradigm shifts”? It’s certainly a fascinating time to be engaged in these discussions!
Mathematical theorems take the form “If X then Y”. As long as everyone is clear on the statement, there is nothing wrong with conditional theorems, whether we condition on the Riemann hypothesis, the twisted fundamental lemma, or any other result.
Secondly, *mathematics* is more than just the theorems — it’s a cultural practice with different approaches to “truth”, even if the ultimate truth is only rigorous theorems of the form “X implies Y”. Arthur outlined a program for proving some results. This was (1) good work; (2) mathematics. In the end the goal was reached by a different method, so what? Would the field have been better off had Arthur kept quiet about his ideas until he had a complete and rigorous proof of every statement? Should he have specifically held off on [Art04] or [Art13] because [A25] through [A27] have not been worked out yet? In a different direction is experimental mathematics “mathematics”? Observe how [BS-D]’s *numerical* work testing the (conjectural!) analytic continuation of Hasse–Weil L-functions of Elliptic curves led to a very valuable *conjecture* about their L-function in a region they could not be rigorously defined. Was this not mathematics because it wasn’t rigorous?
What is needed is *clarity* in the level of rigor of any particular claim. Authors need to distinguish theorems, conditional theorems (with clearly stated hypotheses), conjectures (with evidence and justification), proposed research programs, etc. Maybe Arthur was overly optimistic, and should not have stated the intended results of his promised work as “theorems”, but I disagree that there is a paradigm shift.
If anything, the Bourbaki-like insistence that only theorems are “mathematics” is the paradigm shift relative to historical practice.