Author of All the Math You Missed: But Need to Know for Graduate School
Mathematical maturity has all to do with practice and experience and nothing to do with age.
Category Theory is easy because it starts from nothing, literally. You can learn it at any age and with no almost no prior education. Same with various formal logics.
You can't study or use in any way the theory of Calabi–Yau manifolds unless you have mastered all of its prerequisites.
Certainly the advice of not choosing textbooks you don't understand is spot on, however. Unfortunately (?) most textbooks assume quite a bit of background, so you don't often have much choice in this regard.
The intuition and the formalism are presented together (at least, they should be!). To learn the role of epsilon and delta, the student needs to jump back and forth, finding the correspondences between equations and the motivation. This is a skill that needs practice; this was one of the first places I found the equations dense enough that I couldn't just "swallow them whole".
(The earlier I remember is the quadratic formula, which I first painfully memorized as technical trivia. It took me a couple of years to grasp that it was completing-the-square in general form. Switching between the general and the specific is another skill that you develop)
https://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infini...
When I was in college, the Philosophy department offered this course. It was considered an easy way to get a general education math credit without needing to be good at math. It was a really enjoyable course[0] that put me on the path to becoming a computer programmer. It occasionally comes in handy[1].
If you mean experimental work, then sure, that's like laboratory chemistry. You run code and write up what you observe happens. If you're trying to prove theorems, you have to understand the epsilon delta stuff even if your proofs don't actually use it. It can be somewhat abstracted away by the statistics and differential equations theorems that you mention, but it is still there. Anyway, the difficulty melts away once you have seen enough math to deal with the statistics, differential equations, have some grasp of high dimensional geometry, etc. It's all part of "how to think mathematically" rather than some particular weird device that one studies and forgets.
Since Lean builds everything from scratch, this should be doable, albeit Lean builds everything on top of type theory which is not the only choice possible. Different foundations will result in a different graph.
Also the best way to learn math is probably not by following this sort of graph, it would be far too abstract and disconnected from both the real world and usual practical applications.
Good alternatives are The Princeton Companion to Mathematics by Gowers and Mathematics: Its content, Methods and Meaning by Alekxandrov, Kolmogorov, et al. Those present much more detailed maps so YMMV.
It has virtually no prerequisites, at least in classical mathematics. But I wouldn’t call it ‘easy’ (indeed, many proficient in elementary calculus and so on find it very hard). If you study category theory with no knowledge of any of the concepts it’s designed to abstract it’s not going to make any sense and the whole exercise is pointless. You may be able to follow it and complete exercises, but you won’t actually grok it.
That's OP's point.
"Age-appropriate" was put in quotes because it was referring to a metaphorical "mathematical age" to tie it to the concept of mathematical maturity.
And, indeed, certain books require more mathematical maturity to get through than others, even though the prerequisites may be minimal. You'll see this often explicitly described in the preface of textbooks and reviews of those textbooks.
They mentioned studying certain books to develop mathematical maturity because that's where the practice and experience happen. Calculus is one such course used as part of this process, as are many courses intended as a first exposure to proofs like linear algebra and discrete math. Some might use the Moore Method with point-set topology.
I disagree, you need to have capacity for abstract thought for abstract topics, most especially with category theory. Normal people have quite difficulty understanding abstractness, you either have the aptitude for it, or work your brain hard enough that it becomes somewhat easier. Children and younger people especially have difficulty understanding non-concrete topics.
Proving theorems is the easy part. Getting an intuition on which theorems you need to prove and which you can assume are true is the hard part.
For research math, it is "immature", incomplete, .... But I was aiming the remark at students plus/minus a few years of being a college freshman.
For what theorems to prove:
(1) "Alexander Grothendieck, who spoke of tackling hard problems by creating a gradually rising sea of ideas around them."
https://www.quantamagazine.org/monumental-proof-settles-geom...
(2) Something missing: As a grad student studying the Kuhn-Tucker (KT) conditions and the constraint qualifications (CQ), there was interest in implications among the CQs. Two of the famous CQ were (a) the Zangwill and the (b) KT, but the implications between them were "missing". So, that was a problem, a theorem "need to prove". My approach was to look for a counterexample among wildly goofy sets, e.g., the Mandelbrot set or Brownian motion. As appropriate for the KT work, both sets were closed. Hmm .... So, needed an optimization objective function to be minimized. So, ..., soon enough, for each closed set there is a real valued function zero on the closed set, strictly positive otherwise, and infinitely differentiable. Then I had a counter example. Two weeks of work in a reading course. Published it.
Was doing some AI for monitoring but wanted a better approach, the "need". Used the probabilistic concept tightness to get another approach, the basis of the "proof", widely applicable because still distribution free (i.e., made no assumptions about probability distributions, e.g., Gaussian). Published it.
(3) A problem from outside math: As in
https://news.ycombinator.com/item?id=40893566
the FedEx BoD wanted some revenue projections, wanted so much it nearly killed FedEx. So, as in that Hacker News URL, got some "intuition", ..., got a simple differential equation (the theorem), solved that (the proof).
Currently my startup needed some progress, and I formulated a suitable theorem and proved it.
A concern about such theorems and proofs, for the published ones, the check has yet to arrive.
Was eating lunch with some well known mathematicians, and they asked what I was working on. I explained, "scheduling the fleet at FedEx, which airplanes go to what cities in what order". Immediately one of the mathematicians with contempt scoffed and said "the traveling salesman problem" as if that was the "theorem" to be proved, i.e., P = NP.
Nope: I was just trying to save FedEx some money. So, my approach was 0-1 integer linear programming (ILP) set covering; that this is in NP-Complete was to me next to irrelevant; I just wanted feasible solutions that would save money. Maybe over a year the savings would be some $millions, but each feasible solution might be $1000 above an optimal solution. At 365 days a year, I'd leave $365,000 on the table. Fine with me!!! To the mathematician, all that consideration of money was irrelevant -- he wanted to focus on P = NP and regarded that as too difficult (it still is) and I was foolish for working on it (I wasn't working on it). In short I was counting the millions to be saved, not the thousands of saving to be missed.
Later there was a 0-1 ILP with 40,000 constraints and 600,000 variables. I used the IBM OSL (Optimization Subroutine Library) and in 900 primal-dual iteration for Lagrangian relaxation got a feasible solution within 0.025% of optimality.
Lesson: O-1 ILP can be a good tool in practice, sometimes can save a lot of money.
So the well-known mathematician and I disagreed on your "which theorems you need to prove"!!!
Of course there is the now famous
Garey and Johnson, Computers, and Intractability, Bell Labs, 1979.
The authors were trying to find a least cost design for some Bell network.
On pages 2-3 we see some cartoons with
"I can't find an efficient algorithm, I guess I'm just too dumb."
and
"I can't find an efficient algorithm, but neither can all these famous people."
It turns out by "an efficient algorithm" they meant (a) getting least cost solutions, least down to the last tiny fraction of a penny, (b) to worst case problems, (c) guaranteed, (d) with computer time growing no faster than some polynomial in the size of the problem.
I just wanted to save FedEx some $millions a year.
The famous mathematician insisted on (a)-(d) or no savings at all.
Previously, I thought certain math topics were "hard" (e.g. category theory) while others were supposed to be "easy" (e.g. Calc I). I beat myself up for struggling with the "easy" topics and believe this precluded me from ever tackling "hard" topics.
I was thirty-something years old when I finally realized math has a well-documented maturity model, just like emotional maturity or financial maturity. This realization inspired me to go back and take a few math classes that I had previously labeled as "too hard," with the mindset that I was progressing my math maturity.
My point is that choosing an "age-appropriate" (in terms of math maturity, not actual calendar age) textbook is important. I also find it extremely helpful to chat with people who are more mathematically mature than I am, in the same way it's helpful to seek advice from an older sibling.