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,
If a definition that introduces other variables, check the assumptions about those variables. example: why does $x_{0}$ need to both be in $X$ and be a limit point of $X$ .
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. it's also like the usual practice of breaking things off to lemmas, to make a proof more modular.
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.

Some things I sort of didn't like:

when defining the integers, rationals, and reals, he uses the equality symbol rather than the equivalence symbol, which makes checking that operations are well-defined a bit confusing; see post by gowers on gunctions.
hints too close to the exercise statement.
excessive use of numbers to refer to past results. this was actually good for exercise hints but not good for other things.
I sort of find amusing that the errata for the text is so long. Is this a fact about mathematics in general? about analysis in particular? about Tao (e.g. perhaps the way he writes in LaTeX)? But none of the errors seem very "serious" (in the sense that they can be patched up quickly/locally); this seems like an instance of the phenomenon discussed in https://www.gwern.net/The-Existential-Risk-of-Mathematical-Error

External links

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