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Im not a math student and thus I've never studied mathematics at a high level (i just know some calculus, linear algebra and abstract algebra), most of the exercises I've done were all about computations (I'm an engineering student) and never really about proving things. Recently I've started to self study math and I've noticed that most of the exercises in textbooks are about proving things (when I studied math in college I didn't use a textbook, so this was new to me), and even googling "unsolved math problems with prizes" I noticed that many problems are about proofs. So my question is, are higher level mathematics exercises all about proofs and not really about computations (or exercises like, for what $k$ is the function differentiable? Find the solution of the equation ...)? Can you guys tell me "what is like" to study higher level mathematics? What kind of problems students have to solve?

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    You could check out this thread: https://math.stackexchange.com/questions/616595/difference-between-school-mathematics-and-university-real-mathematics – Manan Aug 01 '20 at 15:50
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    Even a problem "For what $k$ is the function differentiable?" would require an answer of the form "It is differentiable if and only if $k$ has property $X$. Proof: Let $k$ with property $X$. Then ... so that $f$ is differentiable. On the other hand, let $k$ with $\neg X$. Then ... and $f$ is not differentiable" – Hagen von Eitzen Aug 01 '20 at 15:51
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    @Hagen von Eitzen well actually you're right they are proofs anyway, I didn't think about it, but if it's about proving theorems and sentences you know what I mean? Usually in calculus you have to calculate integrals limits etc, in linear algebra solve system of equations etc, but never really proving that the set of natural numbers is bigger that the one of the reals.. I hope what I mean is clear, I'm like a kid whose studying arithmetic in elementary school asking a question to high school students because they use letters instead of numbers‍♂️ –  Aug 01 '20 at 15:57
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    And similarly for seemingly computational problem: you actually have to prove you really have a solution (or the complete set of solutions) or that no solution exists. For example, solve $Ax=b$ where $A$ is a matrix and $x,b$ vectors. Sometimes there are general theory that you can rely on, but at other times you really need to work hard to prove it. – user10354138 Aug 01 '20 at 15:59
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    Some people might say mathematics is only about proofs. – Klaus Aug 01 '20 at 16:15
  • Even a computation is just a proof that $A=B$. –  Aug 01 '20 at 17:15
  • A lot of "higher" mathematics does consist of using previous theorems to prove other theorems. However, applications of these theorems to solve problems in mathematics and other areas is also important. This is called "applied" mathematics – Somos Aug 01 '20 at 18:17
  • @Somos I think "applied mathematics" is just a fancy name for "inductive sciences" (opposed to deductive science, which is math) like physics etc –  Aug 01 '20 at 18:40

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