I got a terrible education before University, I didn't even know what a theorem was, all math was just "learn how to do it, understanding will not have you pass the exams", I'm not having a bad time in college, but some of my colleagues (who got a strong math background) know things that I not even imagine that existed, and I feel offset about this.

Things Like Geometry for example, I love it, I understand what I'm doing, but some basic concepts I just accept as truth and don't know from where this result came from.

I would like to know some textbooks that would fill this gap (not only in Geometry, but in all basics), I'm not a freshman, so maybe something more rigorous will be interesting.

  • $\begingroup$ Maybe you would like reading Cohen's precalculus book. $\endgroup$ – littleO Oct 9 '17 at 6:03
  • $\begingroup$ I can't give you advice with geometry, I didn't learn that from textbooks, but the hard way (I had to: the authors of problems for mathematical olympiads love geometry). The best textbooks of mathematical analysis I met were of Russian origin, but for some obscure reason, they are translated only in the East (Iran, China, say), not in the West. $\endgroup$ – Professor Vector Oct 9 '17 at 6:13
  • $\begingroup$ Because the west has set about a different way of doing things, for better or worse I don't know. For example I have heard some American educators now refrain from teaching the epsilon delta limit definition of the derivative, instead using something like mvt or something. Why I don't know western education is a sad mess I believe were it is just monkey see monkey do. $\endgroup$ – marshal craft Oct 9 '17 at 6:30
  • $\begingroup$ Perhaps the inherent belief in limits of rigorous mathematics, now it has become strange probabilistic art field. Common problems given now have to be hard, for example finding some inverse in non commutative permutation group may involve a number of theorems, simply requiring trial and error, but a class of 30 they try different ones, some pick correct. Were they more intelligent strategies? At no point will professor ever discuss such things, leading one to conclude either there is no higher intelligent strategy, or it certainly is not taught to the class? Either way this is not moral. $\endgroup$ – marshal craft Oct 9 '17 at 6:42
  • $\begingroup$ If you edit your question to list a few of the things you "never knew existed" you might get more useful answers. In the meanwhile I suggest Courant and Robbins "What is Mathematics". It's old but good. $\endgroup$ – Ethan Bolker Oct 12 '17 at 18:01

You might take a look at some books about the history of mathematics:

A History of Greek Mathematics and Vol II - Sir Thomas Heath

Mathematical Thought from Ancient to Modern Times - Morris Kline

And then, you can always find a good text book. For example, in Calculus:

Differential & Integral Calculus and Vol II - Richard Courant

  • $\begingroup$ Thanks, this is what I was looking for. $\endgroup$ – P.A. Viana Oct 12 '17 at 18:14

"A Primer of Abstract Mathematics" by Robert B. Ash well-served the basics of formal mathematical proof. As of gemoetry, the subject of college geometry is kind of different to that in high school. Maybe more specific example will help others to give suggestion.

  • $\begingroup$ I was thinking about Euclidian Geometry and Trigonometry I know that University Geometry differs from High School, but I feel that some trigonometric identities are more like memorization than deduction $\endgroup$ – P.A. Viana Oct 12 '17 at 17:27
  • $\begingroup$ Most Tigometric identites are better understood if you know Euler's formula from complex variables, any advance engineering book will give some basic ideas. About Euclidean geometry I think "Euclidean and Transformational Geometry : Deductive Inquiry" is very suitable to your purpose. $\endgroup$ – Rikeijin Oct 12 '17 at 17:42

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