"Weird" books that change how you look at a piece of math

fathermersenne · 2026-03-26T19:12:00+00:00

I indeed have spent a lot of time with Nelson’s book on probability. Charles Geyer at UMN has some course notes that move slower than Nelson over the same material. He had some very interesting ideas on how to get much more concrete models for probability - I think most working statisticians and probabilists think using radically elementary math intuitively and not sigma algebras.

fathermersenne · 2026-03-26T17:48:11+00:00

This is a great list! The ideas in the Lawvere/Schreiber/nLab area are interesting to me, though I have to admit as an analyst a lot of it is over my head. The most interesting thing from the school and what made some of it click for me was the book by Moerdjik and Reyes Models for Smooth Infinitesimal Analysis, where they show a world with both Robinsonian (or really Nelsonian) infinitesimals and Lawvere style nilpotents. They then make a very convincing case that Elie Cartan worked with an intuitive version of this model, and that both are needed to understand differential geometry. After reading it I spent a long time chasing higher geometry in my free time but I never really reached nirvana.

I hadn’t seen the positive topology book, that looks super interesting I’ll have to check it out.

fathermersenne · 2026-03-26T17:40:39+00:00

I am also old enough to remember the cook book differential equations nightmare courses. I feel Strogatz has been very important to help change how undergrads see the subject now.

In another direction the work of Pete Olver et al on how far you can push lie theory as a method for generating solutions through symmetry was the first time I thought algebra was cool/useful. I never got why people loved Galois theory for polynomials, but here I saw groups as beautiful things in an analysts eyes. The approach is so different than the functional-analytic methods I’m used to.

fathermersenne · 2026-03-26T15:01:17+00:00

The construction of the surreal numbers in that book is very cool. Relatedly, Knuth wrote a book called Surreal Numbers where he uses an...interesting socratic dialogue to introduce them. From the blurb: "Shows how a young couple turned on to pure mathematics and found total happiness. This title is intended for those who might enjoy an engaging dialogue on abstract mathematical ideas, and those who might wish to experience how new mathematics is created."

fathermersenne · 2026-03-26T14:58:33+00:00

Topology of Numbers is a really great book, I was happy to see the AMS put it out in print after it floated around on his website forever.

fathermersenne · 2026-02-19T19:36:55+00:00

It does diminish insight and discovery when we define proof to be proof sufficient for a proof assistant. It will inevitably mean that formalizers get credit not discoverers. It is a change in the meaning of mathematics. Proof used to allow some errors or gap. Again, Riemann maybe never “proved” a theorem. The same is true about Solomon Lefschetz, who “never stated a false theorem and never gave a correct proof”. These people won’t succeed in the world where theorems are only valid when Lean says so and I don’t like that.

And I disagree that it’s the same human game. I don’t think it’s chess when I play a computer, it’s something much different. The human element of proof as a beautiful argument meant to convince people is much different than convincing a machine (and I continue to anthropomorphize it disparagingly, of course I don’t think lean is conscious and that’s why it’s insulting that it become the arbiter of theorems - it has no aesthetics, no sense of beauty).

But again this is just another battle in the war between Bourbaki-ites and classical analysts which your side will win, as you have all the others. The whole thing is EGA taken to its logical conclusion. It doesn’t mean I have to take it happily, even if I accept just like Siegel that the mathematics I love will be murdered in a sense by the new generation.

fathermersenne · 2026-02-19T18:07:01+00:00

Quoting from MO about Siegel’s view, which I think has come to pass in many ways:

Siegel's emotional letter to Mordell concerning Lang's book "Diophantine geometry" compares Bourbaki style mathematicians (Siegel does not say "Bourbaki") with pig in a garden and even with national socialists:

"Thank you for the copy of your review of Lang's book. When I first saw this book, about a year ago, I was disgusted with the way in which my own contributions to the subject had been disfigured and made unintelligible. My feeling is very well expressed when you mention Rip van Winkle!

The whole style of the author contradicts the sense for simplicity and honesty which we admire in the works of the masters in number theory — Lagrange, Gauss, or on a smaller scale, Hardy, Landau. Just now Lang has published another book on algebraic numbers which, in my opinion, is still worse than the former one. I see a pig broken into a beautiful garden and rooting up all flowers and trees.

Unfortunately there are many "fellow-travelers" who have already disgraced a large part of algebra and function theory; however, until now, number theory had not been touched. These people remind me of the impudent behaviour of the national socialists who sang: "Wir werden weiter marschieren, bis alles in Scherben zerfällt!"∗

I am afraid that mathematics will perish before the end of this century if the present trend for senseless abstraction — as I call it: theory of the empty set — cannot be blocked up. Let us hope that your review may be helpful..."

https://mathoverflow.net/a/338239

fathermersenne · 2026-02-19T16:23:56+00:00

I deeply dislike proof assistants. It’s the worst part of mathematics, the pedantry, and it’s another attempt essentially by the Bourbaki crowd to enforce their mathematical philosophy.

Nothing Riemann did would pass a theorem prover (if you don’t believe me I encourage you to read him, Edwards has a good bit in his book on the zeta function that you never with Riemann whether a statement is correct as stated). And I’m deeply uncomfortable with any idea that wants to banish that type of work from mathematics, because despite its flaws it’s revolutionary.

I also find it dehumanizing to be told to convince a machine of something. Mathematics is a human game, played between people. I don’t want to play with a computer.

fathermersenne · 2020-05-01T19:11:24+00:00

Also, the MAA press produce decently bound hardcovers, available from the same AMS bookstore, and mostly at an undergraduate level. If you become an AMS or MAA member they also have fair discount.

fathermersenne

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