The ABC Conjecture has not been proved
As I’ve blogged about before, proof is a social construct: it does not constitute a proof if I’ve convinced only myself that something is true. It only constitutes a proof if I can readily convince my audience, i.e. other mathematicians, that something is true. Moreover, if I claim to have proved something, it is my responsibility to convince others I’ve done so; it’s not their responsibility to try to understand it (although it would be very nice of them to try).
The manuscript he wrote with the supposed proof of the ABC Conjecture is sprawling. Specifically, he wrote three papers to “set up” the proof and then the ultimate proof goes in a fourth. But even those four papers rely on various other papers he wrote, many of which haven’t been peer-reviewed.
The last four papers (see the end of the list here) are about 500 pages altogether, and the other papers put together are thousands of pages.
The issue here is that nobody understands what he’s talking about, even people who really care and are trying, and his write-ups don’t help.
For your benefit, here’s an excerpt from the very beginning of the fourth and final paper:
The present paper forms the fourth and final paper in a series of papers concerning “inter-universal Teichmuller theory”. In the first three papers of the series, we introduced and studied the theory surrounding the log-theta-lattice, a highly non-commutative two-dimensional diagram of “miniature models of conventional scheme theory”, called Θ±ell NF-Hodge theaters, that were associated, in the first paper of the series, to certain data, called initial Θ-data. This data includes an elliptic curve EF over a number field F , together with a prime number l ≥ 5. Consideration of various properties of the log-theta-lattice led naturally to the establishment, in the third paper of the series, of multiradial algorithms for constructing “splitting monoids of LGP-monoids”.
If you look at the terminology in the above paragraph, you will find many examples of mathematical objects that nobody has ever heard of: he introduces them in his tiny Mochizuki universe with one inhabitant.
When Wiles proved Fermat’s Last Theorem, he announced it to the mathematical community, and held a series of lectures at Cambridge. When he discovered a hole, he enlisted his former student, Richard Taylor, in helping him fill it, which they did. Then they explained the newer version to the world. They understood that it was new and hard and required explanation.
When Perelman proved the Poincare Conjecture, it was a bit tougher. He is a very weird guy, and he’d worked alone and really only written an outline. But he had used a well-known method, following Richard Hamilton, and he was available to answer questions from generous, hard-working experts. Ultimately, after a few months, this ended up working out as a proof.
I’m not saying Mochizuki will never prove the ABC Conjecture.
But he hasn’t yet, even if the stuff in his manuscript is correct. In order for it to be a proof, someone, preferably the entire community of experts who try, should understand it, and he should be the one explaining it. So far he hasn’t even been able to explain what the new idea is (although he did somehow fix a mistake at the prime 2, which is a good sign, maybe).
Let me say it this way. If Mochizuki died today, or stopped doing math for whatever reason, perhaps Grothendieck-style, hiding in the woods somewhere in Southern France and living off berries, and if someone (M) came along and read through all 6,000 pages of his manuscripts to understand what he was thinking, and then rewrote them in a way that uses normal language and is understandable to the expert number theorist, then I would claim that new person, M, should be given just as much credit for the proof as Mochizuki. It would be, by all rights, called the “Mochizuki and M Theorem”.
Come to think of it, whoever ends up interpreting this to the world will be responsible for the actual proof and should be given credit along with Mochizuki. It’s only fair, and it’s also the only thing that I can imagine would incentivize someone to do such a colossal task.
Update 5/13/13: I’ve closed comments on this post. I was getting annoyed with hostile comments. If you don’t agree with me feel free to start your own blog.