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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 ?

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    $\begingroup$ Basic real analysis $\endgroup$ – user258700 Jul 23 '16 at 17:03
  • $\begingroup$ Some topology and linear algebra wouldn't hurt either. $\endgroup$ – got it--thanks Jul 23 '16 at 17:04
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    $\begingroup$ Yeah as mentioned, analysis is exactly what you're looking for. It starts calculus from the beginning again, but proceeds by explaining and proving everything $\endgroup$ – Ovi Jul 23 '16 at 17:05
  • $\begingroup$ There is even a book with almost that title: Elementary Analysis - The Theory of Calculus by Kenneth A. Ross. $\endgroup$ – lhf Jul 23 '16 at 17:13
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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.

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  • $\begingroup$ Apostol's Calculus is also excellent. $\endgroup$ – lhf Jul 23 '16 at 17:15
  • $\begingroup$ Yeah there are proofs but in my case we didn't spend too much time on them in my class. We just learned how to use the tools more. $\endgroup$ – copper Jul 23 '16 at 17:52
  • $\begingroup$ @copper I know you didn't spend much time on them in class, nobody ever does. But you asked for a reference. Whatever reference you use you're going to have to read it, outside of class. Nothing's stopping you from simply reading those proofs you skipped in class. $\endgroup$ – David C. Ullrich Jul 23 '16 at 18:05
  • $\begingroup$ You're right but math aren't my only classes so .... I was more worried about having my other works done and didn't want to drag. $\endgroup$ – copper Jul 24 '16 at 18:17
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    $\begingroup$ For what it's worth, going through the proofs in YOUR calculus text is going to take much, much less time than trying to plow through Spivak or Apostol, to say nothing of a typical real analysis text. Even if you do decide to plow through one of these other texts, it would be worth while to read along the proofs in your calculus text at the appropriate places (because the treatment is almost certainly more elementary in your text and because you are already familiar with the notation and other aspects of your text). $\endgroup$ – Dave L. Renfro Jul 25 '16 at 14:59
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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:

  1. First one is taught calculus as a weird complicated tool which has many many interesting applications within and outside of mathematics. The focus here is on learning techniques and tactics of calculus to apply it for various practical problems. Proofs are almost never provided by using the banal excuse that "proof of this theorem is beyond the scope of the book/syllabus" and some book authors commit intellectual fraud by giving incorrect/non-rigorous/intuitive proofs and students think that those proofs are real.
  2. Next the student is taught the same concepts of calculus (plus a few more abstract ones) and this time the focus is on proofs and foundations and applications within mathematics. And then it is no longer called calculus but rather real-analysis.

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.

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