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> And guess what, you even need complex numbers.

That's not true, and that is the point. To prove the main theorems of calculus, you don't need complex numbers, but you do need irrationals.

When you first learn calculus, you do one or two epsilon-delta proofs, and then your teacher gets a little hand wavy about limits and you move on to the real work of derivatives and integrals, cause the limits stuff intuitively makes sense. When you continue on in Real Analysis or Topology of the Real Line, you discover that your intuition lied to you, and concepts like open and closed sets and intersections and accumulation points are important and are in general non-obvious.

Counterexamples In Analysis

https://www.amazon.com/Counterexamples-Analysis-Dover-Books-...

(Pdf version) https://pdfs.semanticscholar.org/a4e7/eb352e4c44bf75d8fabaf7...

I agree that having possible examples in mind is a great way to learn mathematics. There are whole books on useful counterexamples, e.g.

https://www.amazon.com/Counterexamples-Analysis-Dover-Books-...

These counterexamples are sometimes a bit involved, but I find they are often useful for understanding the purpose of the technical assumptions that accompany many theorems.

"There is in university level maths books what looks like an almost wilful disregard for how people are actually taught mathematics at high school."

The way the lower level courses are taught IS similar to high school math. Low level calculus in my undergrad institution is almost the same as AP calculus in my high school. If not, then the course picked the wrong textbook.

Anything above calculus, it is fair for textbook writers to assume mathematical maturity.

"How do students going from school to university cope? is there some secret occult ritual where all this knowledge is transmitted?"

To be able to self-teach mathematics, one would have to learn w/e mathematicians do by oneself. This is however, not impossible but difficult. Here are some disadvantages: 1. It's hard to assert one's own mathematical ability. 2. No one can give you feedback(unless, you have someone with enough mathematical maturity and also have enough time to read and correct your proofs). Programming is so much easier because you can get partial feedback from compiler/interpreter and output. In fact, anything where you can see something happens is much easier. Mathematicians need to prove what we see happen is really true and it's not a wrong intuition. 3. Math books does not try to hold hands. They leave out many details to be filled in by the reader(the notorious "The proof is left as an exercise to the reader"). Sometimes, readers without enough background could gain a wrong intuition, which will screw up everything further down. It is not easy, and it be really nice to have some professor to talk to.

Now, about this "secret occult ritual". It is basically the undergraduate mathematics scene beginning at the first introductory proof class. (depend on departments, this might be as late as the beginning of the 3rd year of study)

In UIUC, there is MATH 347. In Stony Brook there is MAT 200. Around 2/3 of the students have to retake it. Imaging this. This is a set of math majors learning these things full time, with study groups, WITH FEEDBACK and 2/3 of the students didn't get C. It's not a inherently easy thing to learn. The entire class to teach people to fight one's own intuition and mental short cuts we humans make everyday.

Once this is done, the students can further take a higher level topic(a intro to analysis or abstract algebra) to get a feel of how to apply these techniques in the intro proof class. It's a long process and there is no easy way. See this book, Counterexamples in analysis. http://www.amazon.com/Counterexamples-Analysis-Dover-Books-M... Half of the things I would believe to be true from intuitive argument turns out to be completely wrong.

Finally, one might question why one have to become a half mathematician in order to use some of the tools in mathematics. Because most math books are written for people with enough mathematical maturity that can only be gained from grinding over mathematics, and without enough maturity it doesn't make sense to learn certain things anyway.

⬐ richardjdare

Thanks for your reply, that's some useful information.

I'm writing as a self-taught programmer who has spent years trying to get a handle on the maths of game development and computer graphics; for the most part that means calculus, linear algebra and geometry.

I went to high school in the UK where I was not taught calculus, or for that matter even told that I could go to university. My experiences did lead, I must admit, to some hostility towards the educational establishment and a strong desire to succeed without their help!

I first learned of calculus when I was about 18 and I bought a book on computer graphics programming. I was all geared up to start doing some cool 3d stuff, when all of a sudden I saw this strange elongated "S" symbol, which of course was not explained in the text. This confused me, when the book had a mathematical appendix describing the simplest vector operations (which I did learn in high school) in some detail. And that was the beginning of my frustration!

There just seems to be such an enormous gulf between high school, "everyday" mathematics and anything coming after. I would like to think that in a world where programming is now so popular, that the border between everyday mathematics and the higher reaches of academia would shift a little and a kind of intermediate area would open up.

For anyone looking for more of these, you may enjoy the book Counterexamples in Analysis (http://www.amazon.com/Counterexamples-Analysis-Dover-Books-M...)