Monday, October 23, 2023

When did Math Get So Hard- Part 2

Click here for When did Math Get so Hard-Part 1, though it was not called Part 1 at the time. 

This post is not so much about WHEN math got so hard but an example of math BEING hard. The main issue is that so much is known that the PREREQUISITE knowledge can be overwhelming.

My interest in Hilbert's tenth problem (see here) and an email from Timothy Chow (reproduced in that article) lead me to the book

                               Rational Points on Varieties 

                                  by Bjorn Poonen

(see here for amazon link. Disclosure: Lance and I are amazon affiliates).

Here is the prerequisite for the book as stated in the preface: 


A person interesting in reading this book should have the following background:

1) Algebraic Geometry (e.g. [Har77]: up to Chapter II, Section 8 as a minimum, but familiarity with later chapters is also needed at time)--- this is not needed so much in our Chapter 1. 

2) Algebraic Number Theory (e.g., [Cas67], Fro67] or [Lan94, Part One] or [Neu99 Chapters I and II).

3) Some Group Co-homology (e.g. [AW67] or [Mil13], Chaper 2]). 

[AW67] M.F. Atiyah and I.G. Macdonald. Introduction to Commutative  Algebra, Addison-Wesley, 1969

[Cas67] J.W.S Cassels. Global Fields, Algebraic Number Theory (Proc. Instructional  Conf, Brighton), 1965), 1967, 42-84

[Fro67] A. Frolich, Local Fields, Algebraic Number Theory ((Proc. Instructional Conf, Brighton, 1965), 1967, 1-41.

[Har77] Robin Hartshore, Algebraic Geometry, Springer-Verlag, 1977, Graduate Texts in Mathematics, No. 52

[Lan94] Serge Lang, Algebraic Number Theory, 2nd ed. Grad Texts in Mathematics, Springer-Verlag. , 1994.

[Mil13] J.S. Milne, Class field theory (v4.02), March 23, 2013. Available at here

[Neu99] Jurgen Neukirch. Algebraic Number Theory,  Fundamental Principles of Mathematical Sciences Vol 332. 1999.


This seems like quite steep prerequisites. I don't have them so perhaps they are easier than they look. 

But in any case, Some parts of math are hard because, over time, so much math is known that builds on earlier math, that just getting through the background material is hard. Comp Sci hasn't been around as long, but its been around in the 20th and 21st century when more was being produces, so its also gotten hard, as I discussed here. Note also that computer science uses some of that hard math, and is also an inspiration for some hard math.


  1. Math is hard because ALL of the "structure" is to be funneled through human mind (as your post well exemplifies) and there is no full formalization which could use processing power to tame the volume.
    And don't tell me about proof assistants, they are ridiculously complex and limited and add to the problem instead of alleviating it.

  2. Real things are really hard, super-duper hard; in fact its supposed to be exactly that way!
    Imagine creating living things including humans from bare-bones organic compounds on a primordial earth.
    It's not like chemistry back then was any different from what we get to work with today.
    It took nature ~4 billion years of constant toil to get us from there to here; to figure out how to reliably retain what works and relentlessly discard the stuff that does not work over this whole period.
    While we might hope that math should be more tractable because its an abstraction that we humans created, the reality points in the opposite direction.

    1. "While we might hope that math should be more tractable because its an abstraction that we humans created, the reality points in the opposite direction."

      Actually there are good reasons to believe that the so-called "reality" is also a creation of the human mind:

      Idealist Implications of Contemporary Science. (PDF)

    2. The preprint has been scrapped after publication on September 1, it is now behind a paywall.
      Idealist Implications of Contemporary Science.

  3. In graduate school on the first day of one class, the professor listed the prerequisites, then noted that no one had all of these prerequisites. One of the things that you learn how to do in graduate school is to learn stuff when you don't have all the prerequisites. Basically, you fill in the ones you need as you need them. Of course, it helps to already have some of the prerequisites.

    1. I feel that the distinction between a good mathematician and a random graduate student is that mathematicians somehow manage to read and comprehend all of the prerequisites precisely. It is possible to have some ideas without a solid foundation, but it is impossible to make a real contribution.

  4. The landscape of math (and math theorems) is like a Mandelbrot set ... you can keep zooming-in and you'll always find new nice places.

  5. Alexander Grothendieck8:54 AM, October 26, 2023

    I assume an early year PhD studying number theory and arithmetic geometry is suppose to know these materials.The list does not include more advanced stuff such as etale cohomology.

    1. Ha! I sort of guessed that. I can't tell what's "more advanced" in this rarified air universe, but I did notice that the main three prereqs were 1967, 1967, and 1977, which is an age ago in math terms.

      More generally though (for us beginners), math builds on things. One can't do Lie Algebras without linear algebra and modern algebra; you just can't. It very much sounds like all of modern math is like that. Classical number theory may have been different: my impression is that Ramanujan's work was more about brilliance in ability to manipulate stuff than in depth of structure. Maybe.

      But the bottom line is that modern math is about building structures on (and between (e.g. Langlands)) structures that came before. And one isn't going to play in the fast lanes without doing the work. For we amateurs, there's a lot of beauty in even the simple stuff, so there's pleasure to be had, but if you need it for work in another field, e.g. Comp. Sci., life is going to be harder...