What branch of mathematics should someone study to really understand why calculus works ? I know basic calculus but for me it seems really more like apply the tools to compute stuff. I would like to know why it is true.
Any help ?
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.Sign up to join this community
As commented, the actual answer is real analysis.
Now, you say you want to know why calculus works. There are proofs in most calculus books, you know. Those proofs do explain why things work.
Finally, Spivak Calculus is an excellent calculus book, with much more emphasis on proofs than usual.
Its great that you are interested in foundations of calculus (normally even book authors and teachers are not so interested in teaching foundations of calculus to students who are learning calculus for the first time at an age of 16 years or so).
The teaching of calculus follows almost the same pattern in most countries:
In most cases a student does not have the opportunity to study both the above courses and his/her charm of calculus is limited only to the practical applications of calculus. Missing "the beauty of the simple and elegant theory of real numbers and the edifice of calculus built on top of it" is something very very unfortunate for many students.
Luckily I was fortunate enough to get hold of Hardy's A Course of Pure Mathematics when I was going through a first course in calculus. Do read this book not just for the foundations of calculus, but for the sheer joy of reading mathematics.