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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

by 4ad 2y ago

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.

CamelCaseName 2y ago
Is there a map of this? Would love to see which topics branch of each other and which start from scratch
- plonk 2y ago
  Big schools' curricula? Look at some top school's math undergrad courses and graph them
- 4ad 2y ago
  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/ https://leanprover-community.github.io/mathlib4_docs/ [1] https://github.com/leanprover-community/mathlib4 https://github.com/leanprover-community/mathlib4
- nextos 2y ago
  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.
throwaway81523 2y ago
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.
- lupire 2y ago
  Delta epsilon is just an annoying unenlightening technicality, not the essence of real analysis. Surreal numbers (infinitesimals)solve the problem more elegantly.
  - tnh 2y ago
    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)
  - throwaway81523 2y ago
    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_Infinitesimal_Approach https://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infini...
xanderlewis 2y ago
> 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.
mhh__ 2y ago
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.
sn9 2y ago
> 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.
runald 2y ago
> 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.

by 4ad 2y ago

CamelCaseName 2y ago

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

plonk 2y ago
Big schools' curricula? Look at some top school's math undergrad courses and graph them
4ad 2y ago
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/ https://leanprover-community.github.io/mathlib4_docs/ [1] https://github.com/leanprover-community/mathlib4 https://github.com/leanprover-community/mathlib4
nextos 2y ago
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.

throwaway81523 2y ago

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.

lupire 2y ago
Delta epsilon is just an annoying unenlightening technicality, not the essence of real analysis. Surreal numbers (infinitesimals)solve the problem more elegantly.
- tnh 2y ago
  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)
- throwaway81523 2y ago
  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_Infinitesimal_Approach https://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infini...

xanderlewis 2y ago

> 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.

mhh__ 2y ago

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.

sn9 2y ago

> 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.

runald 2y ago

> 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.