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I study electronic engineering at university, 3rd course. I had to use mathematics a lot, from basic algebra to analysis. Yesterday, after watching some mathematics-related videos and reading some posts I made a conclusion that I don't have a general understanding of this beautiful field of science. I don't know if this kind of question is appropriate here but I want to ask a basic but nontrivial (for me) question:

What is mathematics?

That is, what branches it has, what is the foundation, the starting point, etc. This question emerged when I decided to get a more profound insight of the matter and start learning mathematics from scratch to form a systematized knowledge and after all I love using and studying it.

P.S.: I apologize for my English: it isn't my first language.

Shaun
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Bran Tran
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  • Welcome to stackexchange. That said, I'm voting to close this question because it is much too broad and vague. You might enjoy this Map of Mathematics https://www.youtube.com/watch?v=OmJ-4B-mS-Y (Your English is just fine. Ask mathematical questions whenever you wish.) – Ethan Bolker Jul 12 '18 at 17:10
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    Math in a giant subject. The foundations of math is in itself a giant subject. You probably won't get any good comprehensive answers. – JuliusL33t Jul 12 '18 at 17:12
  • Richard Courant & Herbert Robbins & Ian Stewart, What is mathematics : An Elementary Approach to Ideas and Methods (2nd ed 1996). – Mauro ALLEGRANZA Jul 12 '18 at 17:12
  • Basic definition that I use: mathematics is the art of recognizing numerical patterns in the universe. For branches, see the wiki article. The basic foundation is number, and you can't do better than Landau's Foundations of Analysis for a rigorous, but accessible treatment of the fundamental number systems. – Adrian Keister Jul 12 '18 at 17:13
  • Thank you for your responses. I will close this question but before that could you provide some links where I can read what questions are "right" and which ones are not. If you could also recommend me where I can ask this kind of questions I would be very happy. Thanks. – Bran Tran Jul 12 '18 at 17:14
  • @Bran Tran: Check out What Is Mathematics? by Richard Courant (ISBN-10: 0195105192) – Moritz Jul 12 '18 at 17:15
  • That's mathematics. – rafa11111 Jul 12 '18 at 17:17

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One encyclopaedic definition I've seen, which I think is very good, is that mathematics is the study of the logical consequences of axioms. (Technically, the consequences depend also on the rules of inference.) Such study cares not for what mathematical objects are, but what they do. For example, any construction of the real numbers will prove the same facts about them, so it's meaningless to ask whether they "are" Dedekind cuts or something else isomorphic. See also this David Hilbert comment.

J.G.
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    I've also seen it described as "the logical analysis of patterns", which I quite like. – zhw. Jul 12 '18 at 17:33
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    Since this was downvoted, I'd appreciate advice on how I can improve it. – J.G. Jul 12 '18 at 17:39
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    So would, say, analysis predating the rigorous construction of the reals (or indeed any comprehensive set of axioms for analysis) not count as mathematics? I don't think this works. At best, we can say that mathematics is somehow captured by the study of the logical consequences of axioms, or perhaps "proper" mathematics is the study of the logical consequences of axioms, but I don't think it is that if we accept that all mathematicians do mathematics. (cont'd) – Noah Schweber Jul 12 '18 at 18:16
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    Additionally, while I'm not an expert in the history I think there have been serious mathematicians who were actively skeptical (or more) of the ability of axiomatic systems to faithfully capture mathematics; I think Bishop, for example, fell into this category. – Noah Schweber Jul 12 '18 at 18:19
  • Does this make mathematics a rigorous philosophy? – Mathematics enjoyer Dec 28 '23 at 01:17
  • @Mathematicsenjoyer Anyone who agrees with the first rhyming couplet here will think that's an understatement. – J.G. Dec 28 '23 at 08:23