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Comment by 4ad | Hacker News Reader

I am not sure what you are trying to say because your message and your posted link seem to be an odds with each other.

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.

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> 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.

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.

> Mathematical maturity has all to do with practice and experience and nothing to do with age.

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.

While beginning calculus students often pick up derivatives and integrals (and the associated formulas) easily, the delta-epsilon definitions of limits and continuity are a well known stumbling block for many. I've been told that the difficulty stems from that being the first place math beginners really see nested quantifiers: (forall epsilon)(exists delta)(...). In logic though, nested quantifiers are fundamental. I don't know what happens if someone tries to study logic without first having studied calculus. Maybe it's a good idea, but few people do it that way.

Delta epsilon is just an annoying unenlightening technicality, not the essence of real analysis. Surreal numbers (infinitesimals)solve the problem more elegantly.

To each his own, but epsilon-delta is my go-to example of formalizing an intuitive concept ("gets closer and closer"), which is a high-level mathematical skill.

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)

Surreal analysis is sort of a thing but it is quite far out there (e.g. you can have transfinite series instead of merely infinite ones). Maybe you meant nonstandard analysis (NSA), which is real analysis done with infinitesimals, but the machinery justifying it is way outside of what you'd see in even a theory-oriented intro calculus class. There was an intro calculus text (Keisler, 1976) that used infinitesimals and NSA. I don't know how it dealt with constructing them though.

https://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infini...

Rather than viewing delta-epsilon and infinitesimals as opposing methods, they can be seen as complementary tools I think

It’s very enlightening. Math was formalized for a reason. Now you don’t have to argue what something was supposed to mean and whether it is a correct or a false interpretation, eg: https://hsm.stackexchange.com/questions/6867/what-is-the-cor...

> I don't know what happens if someone tries to study logic without first having studied calculus.

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].

[0]: https://news.ycombinator.com/item?id=37655058

[1]: https://news.ycombinator.com/item?id=23412641

The problem is that epsilon deltas have very little practical use outside of theoretical proofs in pure mathematics. Even for cutting edge CS/statistics fields like high level machine learning, most of the calculus used are existing formalisms on multidimensional statistics and perhaps differential equations. Aside from Jensen's inequality and the mean value theorem, I have never seen any truly useful epsilon delta proofs being used in any of the ML papers with significant impact. It's perhaps mentioned once in passing when teaching gradient descent to grad students.

> Even for cutting edge CS/statistics fields like high level machine learning, most of the calculus used are existing formalisms on multidimensional statistics and perhaps differential equations.

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.

I agree, and including delta-epsilon proofs in calculus 1 seemed like a way for the curriculum authors to feel good that they were “teaching proof techniques” to these students, when in reality they are doing no such thing. I later did an MS in math, and loved the proofs, including delta-epsilon proofs…after taking a one semester intro to proofs class that focused just on practicing logic and basic proof techniques

If you want to do "exact" computation with real numbers (meaning, be able to rigorously bound your results) you just can't avoid epsilon-delta reasoning. That's quite practical, even though in most applied settings we just rely on floating point approximations and try to deal with numerical round-off errors in a rather ad-hoc way.

Is there a map of this? Would love to see which topics branch of each other and which start from scratch

Big schools' curricula? Look at some top school's math undergrad courses and graph them

I am not aware of any good one, but I realized you could probably mechanically extract such a map from Lean's mathlib[0][1].

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.

[0] https://leanprover-community.github.io/mathlib4_docs/

[1] https://github.com/leanprover-community/mathlib4

Garrity's All the Math You Missed book, mentioned elsewhere in the comments, draws a nice map of subjects, along with little introductions and book recommendations. The map is good for continuous mathematics, but IMHO fails to consider logic and type theory, which is a bit odd given the separate chapter for category theory. It also does not do proper justice to computation and clumps everything together under the label "algorithms".

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.

> Category Theory is easy because it starts from nothing, literally.

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.

It's sufficiently general as to be approachable from all angles but to actually understand why anything is being discussed I think category theory requires a certain amount of background material.

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