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I am a high-schooler who still has not learnt calculus completely. I have good introduction to proofs, propositional logic,all tenth grade math,functions,set theory and introductory combinatorics.
I want to learn a large number of topics in math. But do not know from which one to start if I can follow only one book at a time. Here are the topics:

  1. Proof based Calculus-I am following Introduction to Real analysis-By Robert Bartle and Donald Sherbert.
    2.Graph Theory(non-calculus type;kind of discrete)-I am following Graph Theory by Ronald Gould
    3.Naive set theory(Paul Halmos' book)
    4.Raymond Smullyan's Beginner's guide to mathematical logic-Two volumes
    5.Elementary Introduction to Number Theory-by Calvin T.Long(no calculus)
    So here is my first question:
    (1) Where will it be correct to start? I did not find any other stack exchange site which will fit the second question. This question is least fit for this site,but I am desperate for help. Thus,I had to put it here.
    For someone who is in high school, one has to give a significant amount of time to assignments from other subjects,projects and exams. The same is happening with me as a result of which I am not able to give sufficient time to give to mathematics.
    Also,I have very important exams this year which is essential for my future and thus,I am forced to ignore math.
    Since a lot of academics in maths are present here, I just wanted to ask: (2)"How did you manage your time for mathematics when you were in school and how did you give time to other subjects?"
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    I don't think this question is suitable for this site, but I would suggest reading an analysis book. Michael Spivak's Calculus is a brilliant introduction to rigorous mathematics, but you might find it too difficult. (Despite its name, I would describe it as an analysis book rather than a calculus book). However, I would suggest getting to grips with calculus first. Also, don't neglect your other school subjects—I would presume that many academics only started studying mathematics very thoroughly at University. – Joe Aug 28 '21 at 15:27
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    There are two modes of learning math: 1) grokking the details, growing a foundation of deeply understood material; 2) “Big picture first” learning. In “big picture first” learning you look to the horizon, skimming or reading parts of books that you don’t understand yet, but getting the big picture and the main high level ideas and seeing how everything fits together. Big picture first is like looking at a globe before learning your way around a particular city. You can use both of these modes of learning. It will help you navigate through the vast amount of knowledge there is to learn. – littleO Aug 28 '21 at 17:08
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    In other words, you don’t have to read just one book at a a time. It’s ok also to look ahead, skip ahead to the most interesting or useful material, then backtrack and fill in gaps as necessary. You are like a bird flying over the ocean of knowledge, occasionally diving down to catch some prey. – littleO Aug 28 '21 at 17:11
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    That said, looking at your list, I would recommend doing elementary number theory first and proof-based calculus next. – littleO Aug 28 '21 at 17:14
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    I would recommend reading up on the history of mathematics and on the lives of famous mathematicians. That offers you lots of overlap with interesting non-mathematical subjects. – Rob Arthan Aug 28 '21 at 22:30

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(1) Strictly speaking all "higher" math is based on logic and set theory. However, going into these topics seriously is not only quite challenging, but also unnecessary for most purposes. The basics on logic and sets are covered in almost all undergraduate books (say on analysis or linear algebra). Even Halmos "Naive set theory" is perhaps overkill. Additionally, research in logic and set theory is not very active nowadays.

Although I'm not US-based, I believe the biggest step between high school math and college math is the amount of abstract objects and formalism. In this respect, linear algebra might look a bit more scary to you than analysis, but if you want to gain a head start I would definitely add linear algebra to your list (I can't recommend a book since I learnt in my native language). Elementary number theory and graph theory is certainly accessible and quite rewarding as well.

(2) During my school time I was busy doing homework and didn't think about time management at all. I spent time with silly elementary number theory puzzles (often my own creations) and tried to read books with limited success. In college there were many better educated fellow students, but some got bored and others distracted. I won't overestimate the influence of "pre-college" effort. In the end, all what counts is staying interested and fascinated by math.

Brauer Suzuki
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