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This felt like it was written by a physicist or engineer. Too much emphasis on differential equations and not enough on things like topology, functional analysis and/or non-introductory parts of algebra like say representation theory.
> My goal here is to provide a roadmap for anyone interested in understanding mathematics at an advanced level. Anyone that follows and completes this curriculum will walk away with the knowledge equivalent to an undergraduate degree in mathematics. NO, NO, NO. There is no real way to go up to the real deal without having understood elementary Functional Analysis, which the article doesn't even mention. FA is roughly…
Earlier quoted context omitted.
Second Axler. "Linear Algebra Done Right" is probably the pure mathematics textbook I've most enjoyed reading ever (but be warned you will learn very little about applied methods from it if that's what you care about). Also enjoyed Artin's Algebra.
My university course in linear algebra taught me how to manipulate matrices. It was super uninteresting, and easy. I aced every test, but got a B in the course, because the professor assigned an asinine amount of homework (that I either aced or didn't do), perhaps holding the article author's view that: > solving problems is the only way to understand mathematics. There's no way around it. ...without also understandi…
I think this is a common first experience when first hitting pure mathematics. Mathematics often feels like very rote applications of rules drilled into one's mind, and then you hit a pure mathematics textbook and the questions become a step change in difficulty where you're expected to derive novel insights on your own that the text doesn't hold your hand in showing. A single problem can easily occupy days of your time before the "aha!" moment, but as you say, once you get the "aha!" you realize your understanding is quite profound as opposed to a shallower understanding of just how to apply a given set of rules.
> [Mathematics] is the purest and most beautiful of all the intellectual disciplines. It is the universal language, both of human beings and of the universe itself. [...] That doesn’t mean it’s easy — no, mathematics is an incredibly challenging discipline, and there is nothing easy or straightforward about it
I am always, always going to condemn this unnecessary mystification and idealization of mathematics. It's exclusive and misleading.
For real analysis it recommends as essential Abbott's "Understanding Analysis" and Rudin's "Principles of Mathematical Analysis". If you "haven't gotten your fill of real analysis" from those it recommends Spivak's "Calculus".
I'd consider promoting Spivak to essential, but using it for calculus rather than real analysis, replacing their recommendation of Stewart's "Calculus: Early Transcendentals".
By doing calculus with a more rigorous, proof-oriented introductory calculus book like Spivak, there is a good chance you won't need a separate introduction to proofs book so can drop the recommended Vellemen's "How to Prove It: A Structured Approach".
IME (as a math-degree-haver) the value of mathematics is in improving one's ability to mentally model and reason about complicated real-world phenomena . A lot of folks lose sight of the reality and get lost in the mysticism, especially within the academic regime. > [Mathematics] is the purest and most beautiful of all the intellectual disciplines. It is the universal language, both of human beings and of the univers…
"Sadly, there is all sorts of baggage around learning it (at least in the US educational system) that is completely unnecessary and awful and prevents many people from experiencing the pure joy of mathematics. One of the lies I have heard so many people repeat is that everyone is either a “math person” or a "language person” — such a profoundly ignorant and damaging statement. Here is the truth: if you can understand the structure of literature, if you can understand the basic grammar of the English language or any other language, then you can understand the basics of the language of the universe."
:)
> My goal here is to provide a roadmap for anyone interested in understanding mathematics at an advanced level. Anyone that follows and completes this curriculum will walk away with the knowledge equivalent to an undergraduate degree in mathematics. NO, NO, NO. There is no real way to go up to the real deal without having understood elementary Functional Analysis, which the article doesn't even mention. FA is roughly…
I'm a fan of functional analysis, but even in my (very competitive) undergraduate curriculum, it wasn't required for a bachelor's in mathematics. I think Susan's guide covers most of what the undergraduate programs I've seen require.
I am bit surprised there is nothing about graph theory in there. Also nothing about combinatorics or knot theory to mention two other subjects. If you want to make people dive into mathematics, it might be a good idea to show a broad range of subjects instead of focusing on the traditional subjects.
You listed some of my favorite stuff. Weirdly, when I was 11, my math tutor told me I’d probably really like finite mathematics. She turned out to be right.