Algebraic number theory

What Hacker News says

1. impendiaJul 2015
   It would take an extraordinary effort. I doubt there is anyone who could learn this material given anything less than a year of solid study -- and I am thinking of the leading researchers in the world, who are already experts in everything related. I am a professional mathematician myself, and I doubt I could manage it even within a year.

   This is dictated by the difficulty of the subject, and it is even more so since Mochizuki's papers are notoriously difficult to read. What I have heard from other research mathematicians is that Mochizuki doesn't make much of an effort to make himself comprehensible, to answer questions, or in general to explain his results to the community.

   Also please be warned that the consensus is generally that Mochizuki's theory is probably incorrect.

   That said, if you want to learn the stuff, there is no reference other than Mochizuki's papers themselves. If you would like to learn some general background theory I would recommend learning some algebraic geometry, for which see Ravi Vakil's book:

   http://math.stanford.edu/~vakil/216blog/FOAGapr2915public.pd...

   and maybe Milne on etale cohmology:

   http://www.jmilne.org/math/CourseNotes/LEC210.pdf

   If you find that incomprehensible, start with Dummit and Foote's Abstract Algebra (read it cover to cover), Neukirch's Algebraic Number Theory, Atiyah and Macdonald's Commutative Algebra, and Fulton and Harris's Representation Theory.

   If you indeed want to read all that, you may as well enroll in a Ph.D. program in math....

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2. mebassettMay 2017
   The top-rated answer states that "the goal is to learn how to do mathematics, not to 'know' it". I don't think that's a defeatist sentiment or a rationalization for the state-of-things. It's certainly true that one would gain more insight that shows how modern-maths-as-we-know-it evolved from a centuries' long struggle to understand basic patterns (exempli gratia, Cox's "Primes of the Form x^2 + ny^2"). But it is also true that, when you are doing mathematics, such a book is not nearly as helpful as an impenetrable tome like Neukirch's Algebraic Number Theory.

   From my own experience (which, I'll admit, is not much to extrapolate from), the former was great for its "insight" and the latter just had too much unnecessary "information". Until I got stuck in an actual maths problem. Then suddenly my opinion switched, and the latter became a trusted map through strange lands.

   You're right about the amount of information that's encoded in a definition or theorem. And it is very difficult to portray why that information is important (never mind how to access it, see other's comments about scaffolding and cathedrals and the like). I fear that trying to describe why groups are defined like they are would just lead to more impenetrable tomes - but instead of being maps to the lost traveler, they'd be more like tourist adverts. But the questions the source was asking is "why are books like this?" and "how do others [learn maths] in this situation?"

   Well, maths books are like this because they are guides to help people doing maths. And you learn maths by doing it. I don't think its defeatist to say that no book or lecture about riding a bike would compare to the experience of actually riding a bike. To paraphrase the top-rated answer, why would you expect anything different from maths?

   But to answer the underlying question of both yourself and the source: how do we improve this? I think pedagogy should focus on getting to "do maths" faster. You really learn by getting stuck in a problem.

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