Economist and mathematician Steve Landsburg writes:
The (really really) big news in the math world today is that Shin Mochizuki has (plausibly) claimed to have solved the ABC problem, which in turn suffices to settle many of the most vexing outstanding problems in arithmetic. ...No, this is contrary to two millennia of mathematical wisdom. Mathematicians only care about what is provable. For them, the truth is what is provable. Nobody changes the rules of the game.
No mathematician would consider rejecting Mochizuki’s proof just because it relies on new axiomatic foundations. That’s because mathematicians (or at least the sort of mathematicians who study arithmetic) don’t particularly care about axioms; they care about truth. ...
Mathematicians care about what’s true, not about what’s provable; if a truth isn’t provable, we’re fine with changing the rules of the game to make it provable.
That in turn implies that there is such a thing as mathematical truth, independent of what we can prove.
Since Euclid axiomatized geometry, a geometry truth meant a theorem with a proof from the axioms. Sure, people have invented other geometries with other axioms, but that has not changed fundamental ideas about mathematical truth or the necessity of proof.
Mochizuki's proposed proof of the abc conjecture is being taken seriously here, here, and here.
In the following discussion, we shall work with various models — consisting of “sets” and a relation “∈” — of the standard ZFC axioms of axiomatic set theory [i.e., the nine axioms of Zermelo-Fraenkel, together with the axiom of choice — cf., e.g., [Drk], Chapter 1, §3]. We shall refer to such models as ZFC-models. ...So in fact Mochizuki cares very much about proving his theorems in ZFC, even if he likes to use tools that go outside ZFC. Mathematical truth is what is provable in ZFC.
We shall refer to a ZFC-model that also satisfies this additional axiom of the Grothendieck school as a ZFCG-model. ...
Although we shall not discuss in detail here the quite difficult issue of whether or not there actually exist ZFCG-models, we remark in passing that one may justify the stance of ignoring such issues — at least from the point of view of establishing the validity of various “final results” that may be formulated in ZFC-models — by invoking a result of Feferman [cf. [Ffmn], §2.3] concerning the “conservative exten- sionality” of ZFCG relative to ZFC, i.e., roughly speaking, that “any proposition that may be formulated in a ZFC-model and, moreover, holds in a ZFCG-model in fact holds in the original ZFC-model”.
Update: In the comments on his blog, Landsburg says that algebraic geometers cheerfully cite the work of Grothendieck without worrying about foundational issues. Sure, mathematicians like to specialize, and most of them do not study foundations. Algebraic geometers especially have a cult-like reputation of ignoring others. The Italian school of algebraic geometry is famous for making sloppy assumptions that are sometimes wrong. But even still, today's algebraic geometers do not condone changing the rules of the game because some (alleged) truth is not provable.
Saying that algebraic geometers cite Grothendieck without worrying about foundations is about like saying that they don't worry about differential equations. Yeah, they would rather let someone else worry about the differential equations. But that does not mean that they don't care about what is provable.
Update: Wikipedia says:
Grothendieck originally developed étale cohomology in an extremely general setting, working with concepts such as Grothendieck toposes and Grothendieck universes. With hindsight, much of this machinery proved unnecessary for most practical applications of the étale theory, and Deligne (1977) gave a simplified exposition of étale cohomology theory. Grothendieck's use of these universes (whose existence cannot be proved in ZFC) led to some uninformed speculation that étale cohomology and its applications (such as the proof of Fermat's last theorem) needed axioms beyond ZFC. In practice étale cohomology is used mainly for constructible sheaves over schemes of finite type over the integers, and this needs no deep axioms of set theory: with a little care it can be constructed in this case without using any uncountable sets, and this can easily be done in ZFC (and even in much weaker theories).So ZFC is adequate for mathematicians.
Update: For another contrary view, John Horgan once wrote a SciAm article on The Death of Proof. It was very unpopular among mathematicians.
Update: The above explanation is not entirely correct, according to these matheticians:
Ulrik Buchholtz on September 7, 2012 said:The point remains that all these mathematicians are interested in proving the abc conjecture in ZFC. If mathematicians were really willing to lower their standards of proof, then they would accept this heuristic argument. More info on the proposed proof is here.
It seems to me that Mochizuki defines species, mutation and morphism-of-mutation as being whatever defines a category, functor or natural transformation in any model of ZFC. Thus, by the completeness theorem, these are nothing but formalizable-in-ZFC categories, functors, and natural transformations. ...
On another note, on p. 43 Mochizuki seems to claim that Grothendieck set theory is a conservative extension of ZFC, which it certainly is not. What the Feferman-paper he refers to shows, is that an extension of ZFC with a constant for a universe satisfying schematic reflection, ZFC/s, is a conservative extension of ZFC, and that’s obviously because every model of ZFC can be extended to a model of ZFC/s (using a Löwenheim-Skolem construction), but a model of ZFC/s need not be a model of Grothendieck set theory.
Terence Tao on September 8, 2012 said:
On reading the Mochizuki paper, it seems to me (if I am not mistaken) that all the set theoretic and model theoretic stuff is really only used in Section 3, whereas the ABC conjecture is proven (at least allegedly) by Section 2. As far as I can tell, there are no forward references to Section 3 in previous sections other than side remarks. So perhaps all the set and model theory here is in fact something of a red herring as far as the application to ABC is concerned, and are primarily relevant for further development of Mochizuki’s inter-universal geometry instead? (Among other things, this would render the issue of the non-conservative nature of Grothendieck set theory somewhat moot.)