To what extent is it true that "number theory = mathematics"? - MathOverflow [closed]

To what extent is it true that "number theory = mathematics"?

2010-05-29T03:35:08Z

In a thought-provoking answer to this MO question, Kevin Buzzard and several commentators have described a multitude of ways in which number theory is related to other parts of mathematics. It seems that, in practice, to know number theory you have to know all mathematics.

But what is "all mathematics"? The usual description is top-down -- that is, give a high-level theory, such as category theory, that includes nearly everything we currently consider to be important. Alas, there is no telling whether such a theory will continue to be a good description; category theory has only been around for a few decades.

Another way to describe "all mathematics" is from the bottom up -- give a basic form of mathematics that has always existed and which keeps growing and ramifying in all mathematical directions. Elementary number theory is very tempting bottom-up answer, because of the connections with other parts of mathematics already noted, and because it will satisfy our non-mathematical friends who think that mathematicians are people who are good with numbers.

So my basic questions are:

Is number theory a good bottom-up description of all mathematics? And if so, why?

Answers can be anything from general theories about the universality of number theory to examples of unexpected appearance of number theory in other branches of mathematics. And if you are not convinced that number theory rules:

Is there any good bottom-up description of all mathematics (one you can explain to a non-mathematical friend), and if so what?

Answer by Charles Matthews for To what extent is it true that "number theory = mathematics"?

2010-05-29T08:06:04Z

I just don't think it's true, despite my own tastes in topics. Such formulations are substantially a matter of fashion.

There is one basic axis, running from very detailed information at one end (where number theory since the time of Gauss has sat) and general theories at the other. For some perspective, there were people around 1960 who believed that everything was turning into some facet of homological algebra. If you read MO through, you'd see their point also, but it errs as an analysis by taking a single, general point of view as definitive. Round 1900 the theory of functions of a single complex variable had a similar dominant role, and Hilbert almost consciously launched an "opposition" to that hegemony

There is not so much wrong in the old division into analysis, algebra and geometry (quantitative, structural and visual/intuitive mathematics). To explain mathematics to outsiders, it will do OK. That perhaps leaves out combinatorics, seemingly, which is a rising/reviving area linked to computation; but (and I'm certainly interested in the Gowers-style debate about this) on the issue of methods I don't see that it is disjoint.

On the issue of unity of mathematics, Atiyah-style, I would also see the way of expressing the cross-links as subject to fashion. Atiyah believes Felix Klein and his Erlangen Programme to have been "premature"; one can though unify a great deal of mathematics around "Lie group", as has also been said. That's not a panacea either, of course. There is always going to be a tension between what can reasonably be seen as the technology of the subject, and what one is trying to do with it. If someone says that technical areas can be justified by breakthrough results in number theory, I'm not going to argue back too hard, but actually pluralism is a more healthy attitude.

Answer by Burhan for To what extent is it true that "number theory = mathematics"?

2010-05-29T13:20:30Z

I was wondering, in what way is this dichotomy between "top-down" and "bottom-up" mathematics related to the distinction between mathematicians who, deep down, prefer the Continuum Hypothesis to be true and those who think it is false?

(There are, of course, mathematicians who don't care at all, and this might be pertinent.)

Clarification: I was referring to Penelope Maddy's distinction that mathematicians who favored a "rich" and "large" universe of sets were against CH, while those favoring a "neat" and "controllable" universe favored CH.

Answer by Angelo for To what extent is it true that "number theory = mathematics"?

2010-05-29T15:20:31Z

I guess that the title is meant to be provocative: can anyone really believe that number theory is all of mathematics? "Number theory is all of mathematics" is equally false as "Category theory is all of mathematics". I mean, is continuum mechanics part of number theory? Or, for that matter, is algebraic geometry? I really don't think so.

Of course, "Number theory is connected with a lot of different fields", like "Category theory is connected with a lot of different fields", are statement that are both correct and interesting. And there are ideas that are influential across a large number of areas of mathematics; I am convinced that the reason why we don't perceive mathematics as one is the limitations of our brains. But this is very different from claiming that there is a single field that somehow is at the center of mathematics.

Answer by danseetea for To what extent is it true that "number theory = mathematics"?

2010-05-29T20:16:49Z

To answer in the same style the question was asked at, my equation would be that "mathematics = number theory + geometry". I think it all stems from these two; the fact that this was historically so supports my view.

In an insightful comment to the original question, Mariano Suárez-Alvarez wrote: "I think that there are three basic forms of mathematics: number theory, combinatorics and topology". Our answers would be almost the same if we could classify combinatorics as part of number theory. I am certainly not competent enough to give an opinion as to whether that would be a fair classification (and I have a feeling that such a classification might hurt the feelings of some combinatorialists).

Answer by Tran Chieu Minh for To what extent is it true that "number theory = mathematics"?

2010-06-10T15:45:33Z

This post offer some argument in favor of the post by danseetea, yet it might be a bit too long for a comment.

Let me try to propose some candidate for the interpretation of what does it means by "top down" and "bottom up".

I think a vague definition of mathematics = study of interesting structures. I believe we generally agree that mathematics is the study of structures, the difficulty lies in trying to make clear what is interesting.

A. top down approach: Give a suggestion of what is interesting with the risk of overkilling. Candidates:

Set theory: Offer none explanation what is interesting. (Failed)

Category theory: Interesting properties are properties invariant under certain class of function. That seems better but possibly still leave some junks inside.

B. bottom up approach : Try to find some generator of the set of interesting structure with the risk of leaving out a few things. Candidates:

Number theory (Failed) It seems at least we need something like geometry.

Number theory + Geometry

Number theory + Combinatorics + Topology

Analysis +Algebra + Geometry

I want to fail the the last two candidates since it appears to me that Topology and Algebra seem to contain certain elements of the top down (mid air?) approach rather than bottom up. That is because not all algebraic structures and all topological structures are interesting. Certain topological results are interesting perhaps because they describe properties of interesting mathematical structures. There are also pathological results but am I right to say that mathematician generally has limited interest in them?

Number theory, geometry seems to be good since they come from time and space which are the two basic elements of our perception of the external world.

Answer by Bruce Westbury for To what extent is it true that "number theory = mathematics"?

2010-06-10T18:08:21Z

This is a somewhat oblique answer. My recollection is that I read an obituary for Brauer. I don't remember the details except it may have been written by Alperin. Anyway the obituary said Brauer was simultaneously a specialist and a generalist. He was a specialist in the sense that he was only interested in the modular representation theory of finite groups. He was a generalist in that he was prepared to learn any area of mathematics that was relevant to this area and consequently he had learnt a broad range of mathematics.

Answer by Colin Tan for To what extent is it true that "number theory = mathematics"?

2010-07-02T11:59:46Z

I think one needs to distinguish between phenomena, expression and techniques.

Number theory and geometry are phenomena. We observe that all techniques: combinatorics (first principles), algebra (recursion), analysis (approximation) are employed to investigate number theory and geometry.

Regarding the descriptor "interesting", one could call those portions of these technical areas that are employed to investigate the phenomena of number theory and geometry as interesting.

So where then is set theory and category theory? These are expression. Set theory is a language consisting only of nouns. All functions, relations are articulated as sets, a noun. Category theory is a language consisting of verbs and nouns. Functions are modelled as morphisms, a verb. Spaces are modelled as objects, a noun.

In fact before set theory was formalized, there was still mathematics. This primitive language was used in word problems (there are 12 chickens and rabbits in a cage. There are 30 legs in total. How many chickens are there?) before set theory came and bijected the animals to some finite cardinal. It was at this primitive stage that there was no (set-theoretic) definition of natural numbers or real numbers. Yet there was arthmetic and geometry. But all constructive.

As such, one needs to be careful when saying that category theory is a technique. Category theory should be seen as a language, and there is still analysis (limits and colimits), algebra (obvious), and combinatorics (higher categories) in category theory.

Answer by blabla for To what extent is it true that "number theory = mathematics"?

2012-10-10T19:31:07Z

You really don't need PDE's and propagation of singularities (just to name an example) in order to do number theory. It simply never comes up. However PDE's are a part of mathematics, a big one at that. Therefore this disproves your equation (I'm a number theorist and in honesty I do find the equation mathematics = number theory borderline offensive).