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If I read a book about analysis or set theory or algebra etc I start with an introductory course (obviously) but if math, at the end is just a bunch of theorems and proofs, what is the difference between what I study and what someone who's more advanced studies? More theorems? How "deep" can you go into a subject? For example, if I learn 5 introductory books perfectly about set theory, how much am I missing from the "whole picture?

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    $\begingroup$ I don't think there's a clear distinction, it's all very relative. I've seen many books being titled ''Introduction to [some very advanced topic]'' - what may seem introductory to those who have been studying mathematics for decades, will seem very advanced for someone learning it for the first time $\endgroup$ – Mike Daas Aug 17 '20 at 21:57
  • $\begingroup$ @MikeDaas That's probably a fine answer. :) $\endgroup$ – WoolierThanThou Aug 17 '20 at 22:14
  • $\begingroup$ Here are a couple of references about how mathematics advances that I think are relevant to your question: Lakatos's *Proofs and Refutations" and Bressoud's "Proofs and Confirmations". The latter, in particular, should give you some idea about how some interesting but accessible modern mathematics came about. $\endgroup$ – Rob Arthan Aug 17 '20 at 22:47
  • $\begingroup$ It's like the difference between high school algebra and arithmetic. You can't understand algebra if you don't understand arithmetic. The more advanced math builds on the basic math. $\endgroup$ – littleO Aug 17 '20 at 22:57
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In my experience, after the introductory [usually, survey] books, you'd need to become comfortable with the material in at least three more modern textbooks before you can make sense of recent research articles in that field.

For instance, while I can't speak for set theory, if you wanted to read an article in algebra, you would read an introductory text in group/ring/field theory, then you might pick up a book in commutative ring theory, then you might pick up books in homological algebra and algebraic geometry, and depending on the research article, it might now be accessible to you.

Of course, maybe not, so then you'd need to pick up a book on number fields over elliptic curves and another book on Iwasawa theory, and then you're in trouble, because once the community has given up on giving a theory novel names and instead just names the theory after one of its developers, you know you're rather deep.

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