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