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I've had a pretty poor experience going through the standardized education system in California, and now that I'm in college, I'm really fed up with how mathematics is taught (even at the college level). With this said, I thought it would be fun for me to redesign the entire math education curriculum from scratch exactly the way I would want it to be. I think this might be a fun general discussion about how math is taught in the US.

Here is what I came up with: (The wording is a little strange, but that's because I wrote this as an answer to someone on Quora who was interested in relearning mathematics from the ground up -- something I've always wanted to do).

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I feel like somewhere between steps 5 and 6 there should be a lesson on linguistics and the human limitations of semantic understanding, and how this shapes our ability to understand anything that's linguistic, like mathematics. And to highlight the difference between linguistic things and the things they represent.

Here is what I came up with: (The wording is a little strange, but that's because I wrote this as an answer to someone on Quora who was interested in relearning mathematics from the ground up -- something I've always wanted to do).

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- Go online and read the short paper “The Three Crises in Mathematics: Logicism, Intuitionism and Formalism” by Ernst Snapper.
- Learn formal logic with: Introduction to Formal Logic by Peter Smith
- Go online and read the Scientific American article “Dispute over Infinity Divides Mathematicians” by Natalie Wolchover (also in Quanta Magazine)
- Learn Set theory with Karel Hrbacek and Thomas Jech. Introduction to Set Theory. Don’t get too bogged down with this, just enjoy the read and move on when you feel ready. Go back to it when the situation arises that you need it to move forward with math.
- Read a little bit about (don’t read it all the way through; just enjoy it until you get tired of it; go back to it as you work through math and see how it all fits together) Category Theory with Lawvere, Conceptual mathematics: a first introduction to categories, 2nd Edition, 2009
- Go online to the CSUSM Spring 2009 Math 378 course website by Prof. Aitken and download all of the class lecture notes (Ch. 0 - 10). Save them before they’re taken down, and work through these
*excellent notes*as if they were a textbook. Learn it all as if you were given the Sports Alimak in Back to the Future series in 1985; it’s literally that good.**This is the most important step in this entire list; if you do nothing else, at least do this.** - Get a copy of Russell’s Principles of Mathematics on amazon: https://www.amazon.com/dp/0393314049/?tag=pfamazon01-20 Like the books on Set and Category thoery, read it, but not like your life depended on it. If you mastered Smith’s book on formal logic, you will master this book too, and it will help you clarify things that seem like magic in mathematics. But know, Russell’s work isn’t the end all be all.
- Learn about non-classical logic. Question the law of excluded middle; think for yourself — does it make sense to you? Do you believe physical reality follows this rule? All of formal mathematics from this point on, including calculus, is built on the idea that the law of excluded middle is right. In fact, even the books by Smith and Prof. Aitken, as well as all of Set Theory assume this notion. Maybe just let this question simmer in the back of your mind and continue to read about more mathematics. Don’t forget that it’s still a valid philosophical question.
- Read Principles of Mathematical Analysis by Walter Rudin; accompany this read with lecture notes and free online midterm exams from Stanford’s Math 19, 20 and 21 and Harvard’s Math 1a, 1b and 112. Just google the course websites and use what you can find. You should realize that Prof. Aitken’s lecture notes should make this transition seamless. After all, his notes could could well be called “Analysis of the Natural Numbers, Arithmetic and Algebra.” He even covers some real and complex number stuff, so when you see Rudin, you should be in a very, very solid position to blow this material out of the water. Do it. When you have, congratulations, you’ve probably surpassed the majority of college graduates understanding of mathematics. But don’t stop here, you need to understand more than just two dimensional mathematics after all.
- Read A First Course in Topology by James Munkres. This should be tons of review by this point. You should recognize things from set theory, real analysis and logic popping up everywhere. This should be an easy A, and it comes in handy as you move up to more than two dimensions.
- Read Abstract Algebra by Dummit and Foote. Steps 9–11 could probably be done in any order you like, or ever simultaneously. This book should build on set theory, Aitken’s lecture notes (it’s impossible to understate how good these are) and topology should seem relevant here as well. All the stuff taught in high school math is explained here with sets and axioms.
- Read Linear Algebra, Vector Calculus and Differential Forms, 5th edition, by Hubbard and Hubbard. Much like Rudin should have flowed seamlessly from Aitken, Hubbard and Hubbard should role off the tongue like butter to you now. You should easily grasp this material, and you should learn it because it’s important in real life. Accompany your reading with lecture videos of Math 3500/3510 by Shifrin on youtube (excellent lectures of an honors class covering multivariable math). Supplement Hubbard and Hubbard with either: 1) Linear Algebra by Levendosky, 2) Vector Calculus by Marsden and Tromba or 3) Multivariable Mathematics: Linear Algebra, Multivariable Calculus and Manifolds by Shifrin. It’s hard to say which is better. Don’t waste money buying all 3. Personally, I’d probably buy Shifrin based on his lecture videos, and also because Marsden and Tromba is on Scribd online. You can pickup Levendosky cheaply on amazon ($30 or less). If that sounds like a super good deal, buy it. It’s excellent. One last thing to add, the case could be made that Rudin will cover enough of this material to not bother with these books — that’s fair, I’d grab at least one of these just to get exposure to it though; if for no other reason than to understand physics and economics applications.
- Go ahead and solidify your linear algebra because it’s really important from now on. Have Linear Algebra done Right by Axler at hand and take Berkeley’s Math 110 midterm and final exams before opening Axler’s book (you can find them online easily enough). If they are easy for you, just scan the table of contents of Axler and read anything that sounds unfamiliar; skip the rest unless you want to read it. If you want to do the HW from 110 as well, then pick up a copy of Linear Algebra by Friedberg, Insel and Spence (optional). Don’t spend too much time on this. Just make sure you have Hubbard and Hubbard, and Math 3500/3510 down really well (youtube). Glaze through Axler to patch up anything not covered in Abstract Algebra and multivariable mathematics.
- Now you have gotten to the point where you can go online and buy any math book that interests you, and you should be able to just learn it with ease. Explore whatever you want. Algebraic topology, differential geometry, differential topology, complex analysis, physics, cryptology, computer science, statistics, anything at all. My advice, try to learn about multilinear algebra and tensors in depth. I don’t know why, but multivariable math textbooks don’t teach it, in fact the only school that I know of that teaches it is Stanford in their Math 52h class. The sky is the limit man! Have fun!

I feel like somewhere between steps 5 and 6 there should be a lesson on linguistics and the human limitations of semantic understanding, and how this shapes our ability to understand anything that's linguistic, like mathematics. And to highlight the difference between linguistic things and the things they represent.

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