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It's fine and nice and wonderful when a part of learning mathematics is chaotic, ad hoc, spontaneous, social, ...

However it would be perhaps of fundamental value to know a very central point of mathematics to which every student would travel; and those who would succeed in attaining it would have an opportunity to continue from this central point in any mathematical direction (and even in more than one).

Even a partial trip toward the point should greatly benefit the student.

Thus what in your (very subjective of course) opinion is the central point of mathematics? Quotations are most welcome too.

Try to make that central place as narrow as possible, let it be a "point". An answer like "Algebra" or "Geometry" or "Mathematical Analysis" would be weak.

REMARK   If you don't believe that there is a central point in the whole mathematics then just relax (no need to start a war)--this kind of meta-opinion would be off topic, counter-productive to this thread, one can always start a thread about no point being central. There is a better alternative for any sceptic: even if you don't believe in any center you may still answer my question conditionally, like this: if I have to choose one it WOULD be ...

Actually, I believe that selecting a central point of mathematics would immensely help mathematical education. One would get sharp focus. (The point may move over the time but not too fast).

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closed as not a real question by Qiaochu Yuan, Noah Schweber, JSE, Vidit Nanda, Vivek Shende May 25 '13 at 5:29

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

I have nothing against this question (I didn't and I won't vote to close) but it should definitely be community wiki. – Joël May 25 '13 at 2:19
@Joël: I am sure you must be right; I know only, and very little, about MO. There are some totally psychological, general, subjective, soft questions on MO, which have collected a lot of up-votes, while their authors vote to close other questions which are way less arbitrary, more objective. I just don't worry about this issue, and try to enjoy MO a bit, despite its glitches. I feel that this particular question is important for education, but also (even if much less vitally) for mathematicians. Perhaps I'll write more in a separate comment. – Włodzimierz Holsztyński May 25 '13 at 3:15
2 – Will Jagy May 25 '13 at 4:22
@Will: you have provided a very literal answer to the question in the title itself, not necessarilly in the body of the "Question". I like to have both answers :-) Thank you, @Will. – Włodzimierz Holsztyński May 25 '13 at 4:59

Pythagoras Theorem. It sounds anticlimatic, but if you think a bit, you start seeing how it has to be.

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@Rodrigo, I have a different "point" in mind but Pythagoras Theorem did occur to me, and I like your answer. – Włodzimierz Holsztyński May 25 '13 at 3:20
'tis a pity they closed the question. I would like to know your choice of "point". – Rodrigo A. Pérez Jun 11 '13 at 17:37

Once one can solve logic puzzles in Smullyan's books and understand some elementary combinatorics (up to inclusion-exclusion and Dirichlet) and number theory (up to $p|a^p-a$, say), I can teach him the rest I know in finite time. This is not the only "belly button" one can point out, but it is a fairly good one to start with. What matters, IMHO, is not the objects and statements, but the techniques and all the ones I know stem from the places I described :).

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@Fedja, I like your answer. By the way, your statement about "finite time" admits more than one interpretation :-) – Włodzimierz Holsztyński May 25 '13 at 4:02

The Schubert Calculus. The subject is over a century old--if I recall correctly, providing a suitable foundation was one of Hilbert's centennial questions. The most basic point of view requires really only a very thorough understanding of linear algebra, which is helpful for almost any direction the student might continue in (including theoretical physics). The question itself (understanding the structure of Grassmannians) is important in algebraic geometry, algebraic topology, differential topology, and group representation theory, among other places. Students can continually relearn this "belly button" using more sophisticated terminology of (co)homology rings, Chow rings, schemes, etc. (probably even derived algebraic geometry, although I've never been there myself) if their studies take them in that direction.

Note: I am providing this answer in the spirit of suggesting what the "belly button" would be, if it existed; I am not claiming it actually does exist.

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@Charles, thank you for your tolerant spirit. Your answer is perhaps quite original, certainly serious. I didn't want to overspecify my question. I think that I'd like good elementary and high school teachers to make each year a curriculum which would aim at the central point, so that at the end of the year they would believe that their students made one more significant step toward the "belly button". Do you feel that your proposed central point would serve also elementary and high school teachers? (They don't have to master The Schubert Calculus, but can they lead students toward it anyway?) – Włodzimierz Holsztyński May 25 '13 at 4:24
I was not really thinking of elementary and high school teachers when I wrote this--it's more of a "central point" for post-calculus mathematics, and the accompanying process of restructuring a student's fundamental understanding of what mathematics is about. – Charles Staats May 25 '13 at 4:57
For a focus/target point for which pre-calculus mathematics is relevant, I would suggest the rigorous definition of the complex exponential function, together with its four-dimensional graph as a real exponential revolving about the x-axis. (Fourth dimension = time.) Leading up to this could involve definitions of and rules for manipulating exponents (elementary school), non-rigorous notions of the real exponential and logarithm functions (algebra/precalculus), sine and cosine functions, rigorous definitions of the real functions via integrals, and the treatment using taylor series. – Charles Staats May 25 '13 at 5:02
@Charles, I am glad that I didn't make my q. too specific. – Włodzimierz Holsztyński May 25 '13 at 5:05
@Charles, yes, complex exponential function. Myself, I was thinking about the complex logarithm, which is very close to your choice. – Włodzimierz Holsztyński May 25 '13 at 5:08

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