User:IssaRice/Subfield of math that is best for introducing proofs

From Machinelearning

This page compares subfields of math by how good they are as an introduction to proofs and rigor. In other words, the question to ask is "What is the first proof-based math course one should take?"

Real analysis

Pros:

  • students are already familiar with the basic objects of study (assuming they have taken calculus), including the real numbers, continuous functions, etc. so less motivation is needed for explaining why these objects are interesting to study
  • lots of crazy things happen in real analysis (see Abbott's book or chapter 1 of Tao's book) that motivate the need for rigor

Cons:

  • lots of subtle stuff happen, which might make it particularly challenging as an introduction to proofs

there is a lesswrong thread about this

Linear algebra

Pros:

  • objects of study (linear maps) are simple
  • there aren't a lot of paradoxical/unintuitive things
  • knowledge of linear algebra helps in many places

Cons:

  • boring? i think SVD might be the only interesting theorem.
  • everything is isomorphic to R^n so vector spaces are not a good example of abstraction (Tim Gowers makes this point somewhere)

Abstract algebra

Pros:

  • unsolvability of the quintic might be a good target to work toward
  • although algebraic structures are "abstract", it's possible to give many concrete finite examples that build intuition. when examples are finite, you can "see everything"/specify things completely in a way you can't e.g. specify a continuous function completely (by giving a list of where inputs map).

Cons:

  • boring? i think many of the introductory stuff feels like "what's the point of this?" why care about groups, subgroups, normal subgroups?
  • something that's kinda funky about intro group theory: many of the non-trivial proofs are actually about number theory. like, because things like lagrange's theorem are phrased in terms of multiples of numbers, the practice problems are also phrased that way. same thing with stuff like x^n=e implies n is a multiple of ord(x). also of course, cyclic groups <=> modular arithmetic connection. so yeah. it's like, you thought you were in here to learn algebra, but in fact, you're just being forced to work through a bunch of basic number theory. and since the algebra book isn't a number theory book, it's not exactly gentle/pedantic/rigorous/self-contained about the number theory it teaches. So you get an "intro to number theory" aspect but it's actually sort of a mickey mouse/dumbed down/not-the-real-thing experience.

Computability and logic

Pros:

  • proofs in analysis and linear algebra (and probably other places too) often make use of "algorithmic" ideas, e.g. the bisection proof of bolzano-weierstrass theorem. there is a sense in which we like our proofs to be computable, but without learning computability it's hard to express what we even mean by this. I think Stillwell's Reverse Mathematics talks about this issue?
  • several interesting theorems, including equivalence of semi-decidable and recursively enumerable sets, the existence of non-r.e. sets, various examples of diagonalization.

Cons:

  • this isn't usually taken to be an intro-to-proofs subject, so the textbooks might not assume a level of innocence. In other words, the teaching material might be better for other subfields.
  • some of the material seems like unhelpful pedantry, like how interpretations are defined, and the proof of the soundness theorem.
  • might be too meta as an introduction to proofs, e.g. always distinguishing between object level and meta level

Discrete math

Pros:

  • many topics to pick and choose from
  • many interesting topics that can be covered that have wide applicability
  • topics tend to be concrete, so you can easily play with them (e.g. automata, finite graphs)
  • no philosophical issues (?)

Cons:

  • there's something about proofs in discrete math/abstract algebra where when you can see the whole thing (because it's finite), it becomes really tempting to say that it's "obvious" and handwave through a proof.

Number theory

Pros:

  • students are already intimately familiar with the integers.
  • the problems in number theory are easy to state.
  • good concepts like gcd that "feel obvious" but require careful treatment. (e.g. what is gcd(0,0)?)
  • non-trivial results like bezout's lemma, euclid's algorithm for finding gcd.

Cons:

  • i'm not sure what a good book for this is.

Topology

i think i should split this up into metric spaces vs general topology, because i have different opinions about them.

Pros:

Cons:

  • i think general topology only makes much sense after going through metric spaces/point-set topology
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