In what ways can you get into learning serious mathematics?

Calculus and analysis is commonly the entry point to "serious" mathematics for engineers and physicists but it is certainly not the "only true way". It's not even particularly the usual way.

I recommend the content covered in MIT 6042J, Mathematics for Computer Science (https://ocw.mit.edu/courses/6-042j-mathematics-for-computer-...). The most important part of this course is logic and proof-writing, both of which are foundational tools of serious mathematics, that sometimes get elided if you come in through the physics side (i.e. via calculus and analysis). You will also learn discrete mathematics, i.e. "a bunch of stuff that is useful for cryptography". But I think the foundational stuff about logic and proofs is even more useful. It will rewire your brain in a way that prepares you to go deeper in maths.

Mathematics has many subject areas that don't necessarily need to be learned in order. If you wanted you could start down the path of abstract algebra without almost any experience with analysis. Linear algebra is quite approachable and makes a good starting point, all while having interesting applications to help motivate the subject, and it leads directly into a huge swath of serious mathematics. I would argue that the reason analysis is often the first real course American mathematics undergraduates take is due to the fact that calculus is such a staple in high school which makes it a natural progression. Analysis is also a useful way to introduce students to mathematical abstraction by studying familiar things and showing how they can be turned into formal definitions, but this can be accomplished with other subjects as well. For example group theory has plenty of connections to real world objects that you may be familiar with and have intuition for.

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