Examples First highlights

Note: these posts are copied over from the ‘mathbucket’ section of my old tumblr blog and I haven’t put much effort into this, so there is likely to be context or formatting missing.

Rereading a bit, that blog post comment section is probably the original source of my minor obsession with the role-of-intuition-in-maths literature.

(This isn’t actually a particularly major real-life obsession, just a secondary one that I like writing about/have worked out how to write about/have bored everyone I actually know with so I need to go talk about it somewhere else where people can opt out easily)

The only thing I can’t find in there is the Vladimir Arnold rant, which I must have picked up somewhere else. And actually Rota doesn’t make an appearance either. But there’s a load of the standard lore, the usual quotes from Thurston and Poincaré (it’s always the same quotes because this subfield is tiny). 

I’d kind of forgotten that, because I think of it more as my starting point in figuring out how to learn differential geometry. Following the references in the comments taught me more than any maths course I took in undergrad. 

Anyway, some highlights:

  • “I have a colleague (in CS, not math) who reads papers as follows: First he skims the paper by skipping all English and reading only formulas, then he reads the introduction, and then he reads again forcing himself to read some of the English too.”

  • that story about Grothendieck thinking that 57 was prime

  • “One day I realized it was all a lot clearer if I specialized the arguments. As a simple example, a theorem about differentiable real-valued functions on an interval might reduce to the case of the behavior, at 0, of a differentiable function f satisfying f(0) = 0 and f'(0) = 0. Cosmetic assumptions like these simplify the difference quotient and make the key issues clearer (to a novice anyway). The “general case” of such a theorem is often the result of composing the specific proof with an affine transformation. The symbols implementing this transformation play no essential role in the argument.”

  • Fields Medallist admits they never really understood what all that Sylow subgroup stuff was on about

  • ”My undergraduate days left me afraid of many subjects: complex analysis, measure theory, most of algebra and almost all geometry, for example”

  • link to John Baez on normal subgroups. I gave up with trying to understand group theory when I didn’t understand what the definition of a normal subgroup was on about. Unfortunately this is like week 3 of an intro to group theory course, so that was kind of it for me and abstract algebra. I clicked on the link but never read it properly and still don’t really get what a normal subgroup is. Even so, after reading this I felt better for realising it wasn’t some completely obvious thing I ought to ‘just get’ and didn’t.

  • “Dualization is a rather simple idea but I think it is perhaps one of, if not the, most powerful tools in mathematics, especially in the modern era. There is, I’m sure, a good story about why. Perhaps someone can explain or tell me where to find an explanation?”

  • someone asks how to start learning differential geometry and a student of Chern turns up to answer

  • ‘I like to call differential geometry “nonlinear linear algebra”.’

  • a long involved interesting argument about whether you should identify the tangent and cotangent space when you can to save bothering to keep track of a distinction you don’t need right now, or whether you’ll confuse yourself more in the long run

  • anecdote about helping a six-year-old who “could do 3+2 with no problem whatsoever. In fact, she had no trouble with addition. She just couldn’t get her head around all these wretched apples, cakes, monkeys etc that were being used to “explain” the concept of addition to her.”

  • “I was talking to two students about conjugation and talked about how gfg^{-1} is the function that takes g(x) to g(y) if f takes x to y. I then asked them to come up with a function from the reals to the reals that takes x^3 to (x+1)^3 for every x. After a while, one of them had the idea of taking the cube root, adding one, and cubing. But it was clear that he did that by forgetting all about my discussion of conjugation and just looking at the example. Only afterwards, when I pointed it out, did he realize that he had just done a conjugation.”

  • “Kazhdan’s advice to my friend: You should know everything in this book but don’t read it.”

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