Math books that fundamentally changed the way to view or learn new math?

Not a book, but learning analysis changed my view on math. In my BSc in CS I only learned discrete math and maybe some algebra also a bit of geometry, but no analysis! Analysis is in my opinion harder to start with and maybe requires some more discipline. I remember when I finally understood how to actually use the extreme value theorem or realising how useful the triangle-inequality is for proving. Another cool thing about analysis is that all machinery you develop is used in applied math. The most useful math like Fourier, DE's, Stoc. Proc, SP is all analysis under the hood.

Another thing that chanced my view on math (somewhat depressing I admit) is how important programming and systems knowledge is for using math in the real world. We had a math 'expert'/professor write an important routine at work for a year, and we ended up deleting all his code and rewriting it.

I am also biased in this regard. For me maths is just another tool, but math is not about writing algorithms it is about writing proofs.

How to Prove it by George Pólya gave me good foundations and perspective on mathematical problem solving. The Mathematical Experience by Davis and Hersh was a fascinating introduction to the way mathematicians think and their perspective on mathematical challenges.

Just be warned that classics like the Mathematical Experience and Men of Mathematics by E.T. Bell tend to give a romanticized view of mathematical research, with anecdotes mostly by high profile mathematicians doing high profile mathematics, and with up-to-20th Century perspectives that don't always hold up.

I think more creativity should be encouraged as if the student wasn't just learning material but how to be a generative mathematician of interesting new ideas and proofs.

The Moore Method is a well known pedagogical method of teaching math courses.

A text for humanities students on higher-math (it is on the lower-higher fringe I suppose) that emphasizes this perspective but is not as spartan is Mathematics A Creative Art by Julia Wells Bower.

See "The Moore Method", The Journal of Inquiry-Based Learning in Mathematics (JIBLM) has course notes http://www.jiblm.org/

Authors I have enjoyed are Spivak & Knuth but that is more about elegance and pedagogical style.

I've just started reading a volume of books by Hung-Hsi Wu[0]. They are main audience is teachers of mathematics. Two other books that seem to offer a formal rigorous approach of teaching basic math are Principles of Mathematics by Carl Allendoerfer and Modern Algebra - A logical approach by Pearson, H. R and Allen, F. B

Needless to say, these books are focussed on High School math but should be sufficient preparation for courses on a more rigorous level.

My email is in my profile if you are looking for somebody to study with.

[0]https://math.berkeley.edu/~wu/

Conceptual Mathematics: I had been exposed to "abstract nonsense" from CS papers before reading this book, but it was what made the penny drop on why we might wish to be more concerned with maps than with objects.

https://www.cambridge.org/highereducation/books/conceptual-m...

The mostly I was impressed of university’s course of “Abstract algebra” (books I’ve read was in Russian, but I believe you could take any recommended literature from MIT courses). Understanding such things as “homomorphism”, “groups”, “vector linear space” will give a fundamental base for all other math disciplines and you will start to view math in more different side than it’s studied in school.

The preface of Real & Complex Analysis by W. Rudin.

It shows that exp is periodic, the first time I read it kind of blew my mind.