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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?

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closed as not constructive by Felipe Voloch, Terry Tao, Todd Trimble, Dylan Wilson, quid Oct 11 '12 at 9:59

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I think that there are three basic forms of mathematics: number theory, combinatorics and topology. They are the foundations to which we reduce something in order to turn it into something concrete. Any good bottom-up description of the subject should include the three, as they are rather distinct and not reducible to each other. –  Mariano Suárez-Alvarez May 29 '10 at 3:46
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How is category theory useful in saying what mathematics is? Of course there are entire fields of mathematics (underrepresented here) with which category theory has had no interaction. But even if we set those aside, category theory describes not only actual mathematics (e.g. semigroups) but also meaningless things which are certainly not mathematics (e.g. sets with a binary operation satisfying a(bc) = b(b((bc)(ac)))). To distinguish them you need to already know the difference. The English language describes everything I consider to be important, but that doesn't make it a top-down theory. –  Tom Church May 29 '10 at 3:56
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Tom, I don't think that your example represents something that is "not mathematics"; it's just mathematics that is not, so far as we know, interesting or useful. (I suppose I'm implicitly advocating the use of different terms for the thing defined by a top-down definition and the thing defined by a bottom-up definition. "Mathematics" is based on a top-down definition, "interesting" and especially "useful" are bottom-up notions. Regardless of which you choose to call "mathematics", you definitely do not define the same thing from top-down and bottom-up.) –  Charles Staats May 29 '10 at 4:21
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Such philosophical questions should come as wiki community. –  Wadim Zudilin May 29 '10 at 5:10
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@Mariano: Algebra, combinatorics, and topology, no? Analysis is emergent from algebra and topology, and the key ingredients to number theory are algebra, combinatorics, and analysis. Algebra is not reducible to number theory, though. Did you write that by accident? –  Harry Gindi May 29 '10 at 6:34

8 Answers 8

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).

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Apart from all the other discussion here, PDEs and wavefront sets and other such things are certainly relevant in number theory, namely, in the spectral theory of automorphic forms, and the (related) representation theory of reductive real Lie groups. –  paul garrett Oct 10 '12 at 22:08
    
I'm glad to be corrected! –  blabla Oct 10 '12 at 23:24

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.

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I don't think one should view categorical limits and colimits as analytic. They are very algebraic constructions - there are no epsilons and deltas. A limit is just a "universal cone" on a diagram in whatever category one is working in. –  BMann Jul 15 '10 at 18:26
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"Category theory should be seen as a language." To me, category theory is far, far more than "just a language" -- it points to new ways of looking at the world, uncovering cross-connections between disparate areas and speaking of them in a precise, controlled way. It is in fact a fantastically valuable branch of mathematics. And yet, it is poorly supported, and the passion with which some very intelligent people pursue it is poorly understood. Why? Maybe because of an enduring trope that it is "just a language". (Maybe you don't mean that, Colin, but other people do. It makes me sad.) –  Todd Trimble Sep 15 '10 at 0:44
    
One can make the argument that in the end it is ALL just language. –  Spice the Bird Oct 11 '12 at 18:06

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.

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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.

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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.

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I don't think they're related at all. The Continuum Hypothesis is a question about foundations. The issue at hand arises from culture: we now have foundations sufficiently powerful to include almost anything we might want to call mathematics, but that is roughly equivalent to having a natural language in which almost anything can be written. You still need criteria for determining which things are worth writing or reading, and that's where the "bottom-up" approach comes in. –  Charles Staats May 29 '10 at 14:58
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ah, i get it. i think i misunderstood John Stillwell's question, which, as you said, was more about culture than about the foundations. –  Burhan May 29 '10 at 21:48

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).

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What about Algebra? Algebra is independent of all three of those and exists on its own as a rich field of study. –  Harry Gindi May 29 '10 at 22:02
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Well, I consider algebra to be a descendant of number theory. –  danseetea May 29 '10 at 22:13
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Some of modern algebra may be inspired by the techniques used by number theorists in days long past, but saying that algebra is comprised of number theory/combinatorics/geometry is absolute nonsense. Algebraic number theory may be a special case of algebra (over Z), but algebra itself is far more than a generalization of algebraic number theory. –  Harry Gindi May 29 '10 at 23:39
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This is a subjective question, by its nature, so you shouldn't be surprised you feel so strongly against my opinion. Any way, I think that your saying "some modern algebra may be inspired by the techniques used by number theorists in days long past" is a huge understatement. I think it is a well known fact that modern algebra would not have existed at all if it wasn't for Dedekind. –  danseetea May 30 '10 at 9:16
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Well, whether that's the issue or not depends on how you interpret John Stillwell's original question. He says: "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". We just understand this in two different ways. –  danseetea May 30 '10 at 21:27

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.

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Interconnection and increase in interdisciplinary understanding of fields is hardly limited to mathematics - all the branches of science are growing together in various interesting and complicated ways. There are still parts of psychology and biology that are almost entirely unrelated, but evolutionary psychology is an important and interesting field at their intersection, for instance. Much of the 19th and 20th centuries were spent fracturing fields into different, ever-smaller subfields. Now a great deal of reconciliation and understanding is happening. It's a long project, but a great one –  DoubleJay May 29 '10 at 21:57
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@DoubleJay, this project of reconciliation was well underway since the 1930s. Nicolas Bourbaki published an essay called "The Architecture of Mathematics" about this program. People often seem to think that the mathematicians of the 20th century continued the practices of the mathematicians of the 19th century, but this is simply false. The mathematicians of the 20th century knew just as much as we know now, and there is a danger of overstating things like the above. –  Harry Gindi May 29 '10 at 23:47

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.

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One important about number theory, at least when it come to diophantine equations, is that general and universal solutions are impossible to a certain extent. So we often have to start from the "bottom up", from first principles, when solving its problems. Again, there's a Gowers-type debate here. But this also true for a lot of mathematics. –  Burhan May 29 '10 at 22:06
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Quite hard to do justice to all that: see "Why are topological ideas so important in arithmetic?" and comments. –  Charles Matthews May 30 '10 at 6:34

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