View Single Post
P: 800
What is high level mathematics really like?

 Quote by Nano-Passion I've been self-studying calculus and I got to say I really love it (part of the reason why I'm leaning toward theoretical physics). I fell in love with the way they state extremely powerful mathematical concepts through rationale and logic and just because I find it pretty darn cool. But do I have an overly romanticized view of mathematics? I've been seeing a lot of whining and complaining about higher level mathematics and that its proof based (true?). Anyhow, what is higher level mathematics really like?
Math is proof-based because they need to know that each step of reasoning is absolutely correct. For example in calculus they tell you what a limit is. But given some function or sequence, how do you know that the limit even exists? So in a course called Real Analysis, which you take after the first two years of calculus, they go back to square one. They give a logically rigorous construction of the real numbers, and they prove the least upper bound property: any nonempty set of real numbers that is bounded above, has a least upper bound.

With that principle in hand, you can be certain that functions and sequences that "should" converge to a limit, actually do. Then they can make rigorous definitions of continuity and differentiability, give a rigorous definition of the integral, etc. So it becomes totally proof-based.

But you don't need to worry about that right now. However it's true that once you get past the first two years of calculus, the nature of math classes changes substantially. It's all definition/theorem/proof. But the concepts you study are very interesting in themselves, so don't let the idea of proof put you off.