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?"

Everything here should be viewed from the lens of good pedagogy. This isn't directly about what the most beautiful branch is, or what's most interesting. It's also not about the elegance of the finished theory; for pedagogy, it's often a good idea to do things in a roundabout way.

I think I personally got my start at doing proofs by three routes: intro to proofs, discrete math, and real analysis. i think i pretty haphazardly jumped between various books, following my curiosity.

Intro to proofs

By "intro to proofs", i mean the kind of stuff that is in velleman's How to Prove It. so like, basic logic, proof strategies, quantifiers, sets, relations, functions, mathematical induction. maybe countable vs uncountable infinity.

Pros:

• you're learning one thing at a time. With something like real analysis, you have to learn two things at a time: how to write a proof, and the actual content of real analysis. But with intro to proofs, you just learn how to write a proof. Most of the problems are artificial; they are designed specifically to get you to write good proofs, to show you common pitfalls.
• i do like how some books here (i think rosen's discrete math book?) show how to prove things in first order logic notation. it's not literally doing derivations in FOL, but it's more robotic than how you would write a proof in prose. i think it's valuable to see this, and "intro to proofs" contexts are the main ones where i have seen this done. I can't think of why it can't be done in other contexts though (other than lack of time). An example of what i mean by this is something like writing the definition of limit like $\forall \varepsilon >0\ \exists N\ \forall n\geq N\ |a_{n}-L|<\varepsilon$, and then negating this statement by reversing all the quantifiers in a robotic way.
• since this part is so "easy" in some sense (you don't need to build up lots of complicated objects), it is possible to do a lot of this stuff in Lean. that means you get a more "gamified" feeling of doing this kind of math.
• there is something philosophically satisfying about finding out that mathematical reasoning itself can be treated formally, that there are definite "rules of the game" for how to reason in math.

Cons:

• there's not much "meat" in here? personally i really enjoyed working through basic logic, but it does feel pretty empty at the end of the day. (there is stuff like pursuing what material implication means that i spent a lot of time on as a teenager, but i doubt most people care about it.)
• in some sense i think it's better to try to intuitively reason about math first in a field like number theory or graph theory. then after a while you can start to wonder like "can i formalize how i think?" which leads to propositional and first-order logic. if you just introduce truth tables and logical connectives and stuff you might be like "well what if i don't agree with the material implication?" and stuff. seems good to get a taste of "real math" first before you start to wonder about the system with which you are reasoning.

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
• Bloch's The Real Numbers and Real Analysis (p. xxiv) mentions nested quantifiers as the differentiating factor that makes analysis harder than linear or abstract algebra for beginner students.
• Bloch also mentions the need to distinguish between scratch work and actual proof as another unique difficulty.

there is a lesswrong thread about this. Some books, like Tao's, avoid to some extent the problems mentioned there by starting with the natural numbers, integers, rationals, and basic set theory. It's only like half way through the book that the real numbers are even introduced. This gives the student some time to get used to writing proofs in a discrete domain.

Complex analysis

everybody says complex analysis is so beautiful and stuff, but i don't think i've ever seen it used as a first course in doing proofs. maybe there is a good reason? maybe because complex analysis uses results/ideas from real analysis, so real analysis always gets taught first?

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)
• almost every book sucks?

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), and talking about groups of prime order. 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. My current guess in light of this is that it's best to cover basic number theory in very rigorous detail before you start doing abstract algebra. That way, you can focus on the algebra parts without getting sidetracked into number theory.

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. it takes a rare kind of mind (or substantial experience) to even realize that the soundness theorem is important.
• 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)
  • this also means many of the topics are amenable to programmamatic treatment -- you can write toy programs to test your theories and so on.
• 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.
• are there any really interesting results that are also easy enough to understand? like a "crowning jewel" type theorem?

Number theory

I wish there was a book called something like "extremely basic number theory in extremely rigorous detail" that covered the very most basic results in number theory that appear over and over again in other areas of math. so things like gcd/lcm properties, prime number infinity theorem, uniqueness of prime factorization, and stuff like that (anything else?). The reason for this is that a few other fields (like abstract algebra) make use of a lot of these results, but the books on algebra are unwilling to treat number theory in the level of rigorous detail that i think is good for someone starting out with proofs. but existing books on number theory seem to be written pretty poorly for some reason?

Pros:

• students are already intimately familiar with the integers. this makes 'inspecting' proofs much easier.
• 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.
• number theory can be used to branch out into many different topics/fields: things like gcd can be used to cover the notion of algorithm, some stuff around fermat's little theorem can be used to talk about group theory

Cons:

• i'm not sure what a good book for this is.
• In the beginning at least, since students are so familiar with the integers, it makes it harder to know why proofs are necessary, since some things will just seem "obvious", and may make students feel like proofs are just tedious formalization of facts they are already convinced by.

Topology

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

Pros:

• i find metric spaces fun. it's a good example of abstraction. you can draw stuff a lot of times, to help you think.
• lots of counterintuitive results, which makes it fun and also teaches you to rewire your intuitions.
• non-vacuous examples of vacuous conditions, like clopen sets :)

Cons:

• i think general topology only makes much sense after going through metric spaces/point-set topology
• even metric spaces might be too hard unless you've done some real analysis in R or R^n first. i'm not sure about this. (i first went through parts of spivak and tao which do things in R, so i have no idea how i would have done if i had started with metric spaces first. I didn't like how folland's advanced calculus does things in R^n though, or maybe i just don't like folland's style. my thinking was something like "if you're gonna do the R->R^n abstraction, why not go the full way to metric spaces?")
• i think one reason topology is harder to understand than metric spaces is that the usual examples of topological spaces (except the indiscrete/trivial topology) are all metrizable, so the theory of topological spaces just reduces to the theory of metric spaces. (?)

Euclidean geometry

this is the original proof-based branch of math! i think a lot of kids (including me) got exposure to this in middle school/high school. it doesn't seem to be treated at the undergraduate level though, and i've never understood why.

there are things like 'euclid: the game' now that make this more fun.

Set theory

i'm thinking of things like halmos's naive set theory, and chapters 3 and 8 in tao's analysis I as examples, not the kind of set theory you find in books on mathematical logic.

pros:

• used everywhere
• close connection to propositional/predicate logic, such that learning how to work with sets at the same time can highlight those connections
• some good theorems like schroeder-bernstein theorem

cons:

• boring/much of it is devoid of content in a sense
• tricky philosophical issues like axiom of choice, which are difficult for a beginner

External links

• https://blog.evanchen.cc/2021/08/16/book-pitch/