My last column explored theoretical pluralism, the philosophical stance that some scientific questions, rather than having a single solution, might generate many solutions that we choose for subjective reasons. A few days after posting the column I got an email from mathematician Ronald Graham. I hadn’t heard from him since 1997, when I wrote a profile of him for Scientific American. He also served as a source for my notorious 1993 article “The Death of Proof,” which reported that “the doubts riddling modern human thought have finally infected mathematics.” In his email Graham informed me that Timothy Chow, a mathematician at Princeton, has proposed a concept called “mathematical pluralism.” The timing of Graham’s email was purely coincidental. He had not read my post on pluralism in science. He just thought that, given my interest in the philosophical underpinnings of mathematics, I might find Chow’s view interesting. Synchronicity, man! I have also been thinking a lot about mathematics lately, because last spring I discovered that critics of “Death of Proof” named a geometric object after me. (See Further Reading.) When I emailed Chow and asked him to explain mathematical pluralism, he sent me the note below. Chow’s position implies that mathematicians, like scientists, will never converge on a single, final truth. Fortunately! Long live pluralism! –John Horgan
I want to be careful about what I say, since I'm aware of the controversy that your “Death of Proof” article (and its follow-ups) generated. I do believe that you made some valid points, and that mathematicians are sometimes unwilling to admit that mathematics is not as perfectly objective, certain, and uncontroversial as they would like to believe. On the other hand, I do believe that any claim along the lines of "proof is dead" is over-sensationalized and misleading.
In particular, I would say that virtually all professional mathematicians agree that questions of the form “Does Theorem T provably follow from Axioms A1, A2, and A3?” have objectively true answers. Admittedly, if some mathematician claims to have proved Theorem T, but the proof is complicated, then it might take some time for experts to work through the proof and check that it is correct, and some stubborn mathematicians may refuse to accept the verdict of the wider community that their proof is incomplete. For example, you may be aware of the current controversy over Shinichi Mochizuki's purported proof of the abc conjecture. Nonetheless, ultimately, mathematicians are not content to say, “Well, the question of whether this purported proof is correct and complete is just one of those debates that will never be resolved, because there's no objective fact of the matter.”
On the other hand, when it comes to the question of whether Axioms A1, A2, and A3 are true, then I think we have (what I called) “pluralism” in mathematics. The vast majority of mathematicians will assert as objective fact that there is no largest prime number, that pi is irrational, and that every differentiable function is continuous. However, there are some who agree that it is objectively true that (say) “every differentiable function is continuous” has been proved from standard axioms, but who refuse to affirm that the standard axioms are “true” and who therefore do not regard it as “true” that every differentiable function is continuous. In the other direction, if one considers axioms involving sufficiently large infinite sets, one will find that the majority of mathematicians hesitate to affirm confidently that they are “true.” I personally think that it is likely that the mathematical community will never reach consensus on exactly which axioms are “true,” and in this sense the mathematical community is (and in my opinion will remain, for the foreseeable future) pluralistic.
This kind of pluralism does not really interfere with the development of mathematics as a scholarly discipline as long as there remains a consensus that whether Theorem T follows from Axioms A1, A2, and A3 is an objective question. For comparison, note that the controversies over the interpretation of quantum mechanics do not really interfere with the operation of the physics community, because physicists all agree on what constitutes a correct quantum-mechanical calculation and on what experimental observation the calculation predicts. – Timothy Chow, Princeton
Postscript: I sent this column to a few mathematical acquaintances and they responded with these comments:
Michael Harris and Timothy Chow alerted me to an essay in the October Notices of the American Mathematical Society that defends pluralism. In “An Inclusive Philosophy of Mathematics,” John Hosack notes that mathematics consists of different systems—for example, standard, constructive and univalent mathematics--based on “incompatible” foundations, from which different results can be deduced. Hosack argues for what he calls “an inclusive attitude that does not reject other varieties as false.” He calls this philosophy “deductive pluralism.” Chow says Hosack’s outlook is similar to his.
Jim Holt: I'm sorry, did Mochizuki use any axioms beyond Peano arithmetic (or a conservative extension thereof)? More generally, are we talking about the "pluralism" that might be held to arise from Gödel incompleteness? Or about the kind that might arise from considering non-conservative extensions of Peano arithmetic (like large-cardinal axioms or axioms for transfinite induction up to large ordinals)? Is anyone a pluralist about Goodstein's Theorem?
I was under the impression that the "pluralism" that set theorists talk about doesn't touch any natural areas of classical mathematics.
Scott Aaronson: I agree with Jim and I think the point is important.
There's a totally obvious and uncontroversial "pluralism" in, e.g., the choice between Euclidean or non-Euclidean axioms for geometry. From today's standpoint, that's just pluralism in which mathematical objects we might be interested in talking about.
Also, unless we're hardcore Platonists (as Gödel was), there's "pluralism" in set theory, in the sense that there are many interesting models of ZF where statements about transfinite sets like AC or CH can be made either true or false.
But when it comes to "ordinary" parts of mathematics -- arithmetic, real and complex analysis, combinatorics, etc. -- well, Gödel taught us that no one axiom system will entail all the true statements, and sometimes more powerful axiom systems really do let us prove interesting statements that we either don't know how to prove, or provably can't prove, with weaker ones. But nothing like the "pluralism" of the set theorists has ever been observed -- e,g., there's no known interesting universe of arithmetic where Fermat's Last Theorem is false.
Personally, I'd go even further: I'm an "arithmetical Platonist," who takes it for granted that, for example, any given Turing machine objectively either halts or runs forever, regardless of whether this or that axiom system can prove it. (I'm NOT similarly a Platonist about the statements of set theory.) So even if there *were* some interesting alternative set of axioms for arithmetic that proved that Fermat's Last Theorem was false, or that there were only finitely many primes, or whatever, I would say that those axioms simply weren't talking about the same thing that I mean when I say "the positive integers."
Finally, as for Mochizuki -- my outsider's impression was that Scholze and Stix uncovered what was generally considered a fatal problem, to all but a small circle centered around Mochizuki. So, unless and until that situation changes, it's probably premature to speculate about the philosophical implications.
Mind-Body Problems (free online book)