User:IssaRice/Reflections on working through Tao's Analysis

Some thoughts I've had while working through Tao's Analysis I (and probably soon Analysis II).

Notational pedantry: I like how Tao almost always introduces variables by explicitly naming their type. Even after having used $X$ to denote a subset of the reals throughout a chapter, he still begins each definition (that uses some subset of the reals) with "Let $X$ be a subset of the real line" (or similar). I had seen some other mathematicians like Tim Gowers say "Always introduce your variables to the reader before going on to talk about them", but in my experience most writers don't actually follow through on this. Vipul complains about the usual mathematical practice here.

Similarly, I like how Tao generalizes notation like that of limits to

\lim _{x\to x_{0};\,x\in E}f(x)

, where the "

x\in E

" allows us to pass in a separate set

E

. This allows us to define e.g. left and right limits separately.

Definition, example, theorem, proof: I had previously thought that I didn't like this style of writing mathematical texts, but somehow I like it in this text, so that tells me I need to introspect on this practice more, to see what specifically I like/don't like about it.
At each theorem, check necessity of hypotheses.
At each definition,
For each definition of an operation, check that it is well-defined.
For each "superseding" definition, check consistency with old definition.
For each definition that "clobbers up" a notation, check consistency with old definition (See inverse image for functions).
Disambiguate similar-seeming concepts: example of distinct vs disjoint set.
"(Why?)": this feels similar to a "not implemented" error in programming.
rolling up concepts to make them more processable: epsilon-close, epsilon-adherent, continually epsilon-close, etc. see gowers blog post as well, where he does this for the definition of limit of a sequence.

External links

I've written about this sort of thing in some other places and I should consolidate/organize these all at some point.