...referenced in the article. I used a lot of his results - the basis for much of my work really. Algebraic geometry is super fucking hard abstract shit - so is number theory. The two fields are the hardest in all of math. I can see why others would likely find this new work impenetrable. The latest stuff really is backing up against the limits of human intellect. Like the article says, it would take a grad student 10 years to begin to touch this guy's paper. This guy could only do the work by isolating himself for total concentration...which is why I left academia. I wanted a more fun lifestyle.

Why spend time on it? Does it have any human use? I know I'm asking the "never use it in real life" question. Does it matter if this problem is ever solved?

Pythagoras, Hidab al-jabr wal-muqubala, Fibonacci, Gallileo, Newton, Einstein and thousands more we've never heard of who contributed to our universal understanding.

2000 years of Applied Scientists, Inventors, and Capitalists would have been lost without those contributions, often anonymous.

It is very fundamental, I can't think of anything more fundamental and yet more opaque than number theory, and yet all of our science is built on it. I am very interested to know what he is doing poking about in the foundations and set theory and the nexus between summation and multiplication, but I doubt I am up to the job, so I'll have to wait.

Our understanding of Quantum stuff is still pretty clunky, but we've been making progress on it and getting some amazing results. Getting "abstract" math sorted out will be similar- we will be able to calculate things with far more precision and efficiency.

The computational complexity of number theory results from the relationships between addition and multiplication: in general, one does not understand much about the prime divisors of a + b in relation to those of a and b considered separately. An effective proof of the conjecture might be expected to yield good algorithms, though a proof by contradiction might not. Number theory can sometimes be extremely important in other branches of mathematics and in modern physics

Beyond that, there are also aesthetic attractions

Comes out to 42, I'd be really happy.

I'm nothing like an expert in the field, but one can often tell something from style and organization

On page one, he says (for example) "For convenience, we shall use the notation l* := (l - 1)/2" -- but then does nothing with this entirely trivial definition so far as I can tell. A nonexpert like myself can only be struck by the way the material apparently flits, without plot, between apparently impenetrable jargon and rather commonplace mathematics. Unfortunately, this is really not a good sign: it is, in fact, typical of papers written by cranks purporting to have proven difficult theorems. My guess is that his huge stack of paper would be waved off without second thought, but for the fact that he has done some stellar mathematical work in the past

We know that excellent mathematical work need not be immediately comprehensible (from the story of Galois, say); that an epochal paper may be short on proof and long on suggestive conjecture (from the story of Riemann, say); that brilliant intuition itself can be valuable (from the story of Ramanujan, say); that good work can remain controversial and will not necessarily meet instant acclaim (from the story of Cantor, say); and so on. But we also know that good minds are not immune to mental breakdown (from the story of Godel, say)

He used them in the formulas. He did it because the typeset would get annoying (you will see what I mean).

A quick skim has him mentioning a lot of the mathematical constructs I did in grad school studying algebraic geometry...I have forgotten nearly all of it though. Seeing this reminds me how bloody hard it all is.

He would have picked off as a crank a long time ago had he been cranking on this. Not saying his work lacks errors, but he is legit.

the proper place for a trivial definition would be closer to the actual first use, in a form like "where l* = &c"

The same might be said of his sudden introduction of the Gaussian integral on page 4, which then seems to be dropped

There are other objective reasons for concern: ... he has declined invitations to talk about it elsewhere. He does not speak to journalists; several requests for an interview for this story went unanswered. Mochizuki has replied to e-mails from other mathematicians and been forthcoming to colleagues who have visited him, but his only public input has been sporadic posts on his website. In December 2014, he wrote that to understand his work, there was a “need for researchers to deactivate the thought patterns that they have installed in their brains and taken for granted for so many years” ... This statement about the necessity of breaking mental habits may, of course, be true -- but it is also unfortunately the well-known standard complaint of cranks

Simple statements can certainly have long proofs: one thinks of the Feit-Thompson theorem, or the four-color theorem, or the classification of finite simple groups. The latter two examples, however, both raised the question, whether one really has a proof, if nobody can survey it. That completely correct proofs may seem initially impenetrable, until the apparently tricky parts suddenly become clear, is shown by the initial reception to Apéry's theorem. But the remark, that there is no better feeling in the world that the one you have between the time you first prove a theorem and the time you find the first mistake, has a certain sagacity. And there is a certain history of good mathematicians announcing proofs that have never been accepted: ten or fifteen years ago, I heard a good number theorist discuss a purported proof of the Riemann hypothesis, which involved such heavy layers of abstraction that in the end it was really rather unclear exactly what had been proved

The interplay between addition and multiplication is very tricky, as one discerns from the effort apparently needed to prove results approaching (say) the twin primes conjecture or Goldbach's conjecture. If Mochizuki has really provided new foundations that make possible a proof of abc, someone should be able to use that environment to obtain simple proofs of many less difficult theorems

A few others, it seems, did invest done time to get through a lot if his work. There must be something there. I wouldn't be surprised if there are a couple bugs to iron out though. Still, he seems less crazy than Perleman after the Poincaré conjecture proof - which also took some years to verify.