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April 17, 2007

The Two Cultures of Mathematics

Posted by David Corfield

Part of what intrigues me about reading Terence Tao’s blog is that he displays there a different aesthetic to the one largely admired here. The best effort to capture this difference is, I believe, Timothy Gowers’ essay The Two Cultures of Mathematics, in which the distinction is made between ‘theory-builders’ and ‘problem-solvers’. I think we have to be very careful with these labels, as Gowers himself is.

…when I say that mathematicians can be classified into theory-builders and problem-solvers, I am talking about their priorities, rather than making the ridiculous claim that they are exclusively devoted to only one sort of mathematical activity. (p. 2)

To avoid misunderstanding, then, perhaps it is best to give straight away paradigmatic examples of work from each culture.

Theory-builders: Grothendieck’s algebraic geometry, Langlands Program, mirror symmetry, elliptic cohomology.

Problem-solvers: Combinatorial graph theory, e.g, Ramsey’s theorem, Szemerédi’s theorem, arithmetic progressions among the primes.

Gowers mentions Sir Michael Atiyah as a prime example of a theory builder, and recommends his informal essays, the ‘General papers’ of Volume 1 of his Collected Works. Indeed, they convey an aesthetic which I came to admire enormously as a PhD student in philosophy. On the other hand, Paul Erdös was a consummate problem-solver. What then of the corresponding aesthetic?

One of the attractions of problem-solving subjects, which Gowers collects under the loose mantle ‘combinatorics’, is the easy accessibility of the problems.

One of the great satisfactions of mathematics is that, by standing on giants’ shoulders, as the saying goes, we can reach heights undreamt of by earlier generations. However, most papers in combinatorics are self-contained, or demand at most a small amount of background knowledge on the part of the reader. Contrast that with a theorem in algebraic number theory, which might take years to understand if one begins with the knowledge of a typical undergraduate syllabus. (p. 12)

For someone who had recently won a Fields’ Medal, it would seem strange to feel the need to defend one’s interests, but after describing a problem involving the Ramsey numbers, Gowers writes:

I consider this to be one of the major problems in combinatorics and have devoted many months of my life unsuccessfully trying to solve it. And yet I feel almost embarrassed to write this, conscious as I am that many mathematicians would regard the question as more of a puzzle than a serious mathematical problem. (p. 11)

Two types of appeal which are commonly made to warrant the importance of one’s field are its connections to other fields and its applicability. Now,

As for connections with other subjects, there are applications of combinatorics to probability, set theory, cryptography, communication theory, the geometry of Banach spaces, harmonic analysis, number theory … the list goes on and on. However, I am aware as I write this that many of these applications would fail to impress a differential geometer, for example, who might regard all of them as belonging somehow to that rather foreign part of mathematics that can be safely disregarded. Even the applications to number theory are to the “wrong sort” of number theory. (p. 13)

The Green-Tao theorem might be a good candidate to illustrate this “wrong sort” of number theory.

Now, it’s not that, on the theory-building side, all number theoretic results emerging from the “right sort” of number theory are deemed important. Indeed, in TWF 217, John writes:

Now, personally, I think Fermat’s Last Theorem is a ridiculous thing. The last thing I’d ever want to know is whether this equation:

x n+y n=z nx^n + y^n = z^n

has nontrivial integer solutions for n>2n \gt 2.

Rather, it was the activity behind the scenes leading to the proof of the Taniyama-Shimura conjecture that is generally regarded as the major achievement. So, even were results about the existence of arithmetic progressions amongst the primes to be judged similarly as ‘ridiculous’, there might again be some general result lurking behind the scenes. However, according to Gowers, in combinatorics one deals not so much with general theorems, but rather broad principles, such as:

if one is trying to maximize the size of some structure under certain constraints, and if the constraints seem to force the extremal examples to be spread about in a uniform sort of way, then choosing an example randomly is likely to give a good answer. (p. 6)

Several similar principles are given by Tao in this talk.

If one is according importance to mathematical activity in terms of its impact on mathematics as a whole, then rather than the transfer of theoretical results and apparatus between fields, it may be necessary to look to more subtle relationships, such as when:

Area A is sufficiently close in spirit to area B, that anybody who is good at area A is likely to be good at area B. Moreover, many mathematicians make contributions to both areas. (p. 14)

Finally, there remains the question of whether there are missed opportunities arising from the presence of a barrier between the two cultures. Gowers ends his essay by encouraging dialogue. Perhaps blogs are the right arenas in which such dialogue might take place.

Posted at April 17, 2007 9:53 AM UTC

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Re: The Two Cultures of Mathematics

Erdos was considered the Prince of Problem Solvers and the King of Problem Posers.

One of the things I like in Terrence Tao’s informal essays on his formal solutions is the narrative. He provides emotional texture to the subproblems. Numbers “want to be primes” but are prevented by “conspiracies.” This, to me, suggests that this particular genius has learned a method to use both left-brain and right-brain methodologies in optimum combination.

This tracks back, in this blog, to the threads on Story in Mathematics.

Posted by: Jonathan Vos Post on April 17, 2007 5:12 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

It looks like Tao has in fact started a little bit of this dialogue on his blog thanks to comments by Chris Hillman and Greg Kuperberg, see this post and the comments at the end.

It seems to me like an indirect invitation to the theory-builders over here to categorify the Cauchy-Schwarz inequality of extremal combinatorics…

Posted by: thomas1111 on April 18, 2007 6:35 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

Thanks for mentioning this.

How to approach the problem Tao poses of a categorical proof that if f:XYf: X \to Y is a function between finite sets, then |X× fX||X| 2/|Y||X \times_{f} X| \geq |X|^2/|Y|?

Does one look for an injection from X 2X^2 to (X× fX)×Y(X \times_{f} X) \times Y? Looking at the case X={1,2,3},Y={a,b},f(1)=a,f(2)=f(3)=bX = \{1, 2, 3\}, Y = \{a, b\}, f(1) = a, f(2) = f(3) = b, no obvious mapping from the 9 element set to the 10 element set comes to mind.

Posted by: David Corfield on April 18, 2007 1:57 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

How to approach the problem Tao poses of a categorical proof that if f:XYf : X \to Y is a function between finite sets, then |X× fX||X| 2/|Y||X \times_f X| \geq |X|^2 / |Y|

That should be a case for Tom Leinster’s theorem!

That said, I immediately take it back. While this theorem is certainly about category theory, maybe its proof is not the kind of proof that you are looking for here.

Posted by: urs on April 18, 2007 2:20 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

Nice thread! As a gesture towards dialogue, I just added the cafe to my own blogroll :-).

There is a simpler special case of the Cauchy-Schwartz inequality which may be instructive, namely the arithmetic mean-geometric mean inequality: if XX and YY are disjoint sets, then (X×X)(Y×Y)(X \times X) \cup (Y \times Y) is at least as large as (X×Y)(Y×X)(X \times Y) \cup (Y \times X). If one possesses a bijection between X and YZY \cup Z, or between YY and XZX \cup Z for some auxiliary disjoint set ZZ, then one can prove this inequality via a canonical injection or surjection, which presumably qualifies as a “categorified” proof; but the existence of such bijections for arbitrary XX, YY is essentially the axiom of choice, the use of which is presumably not in the spirit of categorification.

A more succinct way of phrasing the problem: can one categorify the assertion “the square of an integer is always a natural number”?

Finally, it may help to deal with the Cauchy-Schwarz inequality in full generality, namely if f:XYf: X \to Y and g:ZYg: Z \to Y then (X× YX)×(Z× YZ)(X \times_Y X) \times (Z \times_Y Z) is at least as large as (X× YZ)×(Z× YX)(X \times_Y Z) \times (Z \times_Y X). (Sorry, I don’t know how to use mathML markup here.) [You don’t; you type equations in TeX (more, properly, in itex) and they get automatically converted to MathML, if you choose one of the itex-enabled Text-Filters. — JD]

Cauchy-Schwarz is pretty much fundamental, not only in extremal combinatorics, but in virtually every area of analysis; it is the assertion that self-interactions control cross-interactions, regardless of how unrelated the two interacting objects are. One could almost define analysis as the branch of mathematics which uses the Cauchy-Schwarz inequality and its relatives (e.g. the triangle inequality).

Posted by: Terry Tao on April 18, 2007 5:40 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

A more succinct way of phrasing the problem: can one categorify the assertion “the square of an integer is always a natural number”?

Not sure if this is what you are looking for, but:

the square of a virtual vector bundle over a point is always an ordinary vector bundle over a point.

That does decategorify to your statement.

Posted by: urs on April 18, 2007 6:01 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

Thanks urs. I guess I was looking for a categorical proof of that statement and not just a categorical formulation, but I guess one could simply take this assertion as an axiom and see where it leads. For instance it seems that in the category of vector bundles, at least, one should now be able to establish the AM-GM inequality, and perhaps if Lagrange’s identity categorifies, then one also has Cauchy-Schwarz. Going back to sets, I can believe that if one accepts as an axiom that the Cartesian square of a virtual difference of sets is isomorphic to a set, then one can probably prove the original assertion about (X x_Y X) x Y being larger than X x X. I am curious though as to whether this axiom could possibly be deconstructed into something even more fundamental, or is simply an extra-categorical fact that has to be proven externally.

Posted by: Terry Tao on April 18, 2007 6:54 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

One little observation about this interesting question: at least in some cases, the explicit use of negative numbers can be avoided by assuming that:

  • for every two objects AA and BB, either there is some object DD such that AA is isomorphic to B+DB+D, or else there is some object DD' such that BB is isomorphic to A+DA+D'.

If we additionally assume that the category is extensive, then we can certainly do some things that are usually proved using negative numbers. For example, we can show that there is a monomorphism

(1)A×B+A×BA×A+B×B A\times B + A\times B \to A\times A + B\times B

for any objects AA and BB.

The property of extensivity basically says that, given any map f:XA+Bf: X \to A+B, the object XX can be (uniquely) split into X 1+X 2X_1 + X_2 in such a way that f=f 1+f 2f = f_1 + f_2 for some (unique) f 1:X 1Af_1: X_1\to A and f 2:X 2Bf_2: X_2\to B. In particular, it implies that products (if they exist) distribute over sums, and that the coprojections AA+BA \to A+B and BA+BB\to A+B are monomorphisms. Those two consequences suffice to prove the mini lemma above, but full extensivity is probably useful in other cases.

I don’t know whether these two assumptions, alone, suffice to show the existence of a monomorphism

(2)X×X(X× fX)×Y X\times X \to (X\times_f X)\times Y

for f:XYf: X\to Y. Maybe not.

Posted by: Robin on April 18, 2007 11:39 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

Nice thread! As a gesture towards dialogue, I just added the cafe to my own blogroll :-).

Thanks, nice!

Your blog has now also been included in our blogroll (thanks to Jacques Distler!), as has the Noncommutative Geometry Blog.

Posted by: urs on April 19, 2007 7:04 PM | Permalink | Reply to this

Categorifying enumerative geometry?

Hi all,

Unfortunately I will be hors de combat for an indeterminate period starting today, but I wanted to very briefly sketch how I’d like to see this develop.

As I have mentioned to some of you, several years ago I attempted to rewrite Wilf’s book Generatingfunctionology in terms of “structors” (aka “combinatorial species”), i.e functors from the restriction groupoid of Finset (toss out all the nonbijective arrows) to Finset.

As most of you know, structors “categorify” both ordinary generating functions (think “unlabeled enumeration”) and exponential generating functions (think “labeled enumeration”). In a sense this is merely a “reinterpretation” of the kind of enumerative combinatorics where one defines some kind of combinatorial structure with one or more “size” parameters and then wants to count how many (labeled or unlabeled) structures of this type exist, for each size. So why is it worth doing? Here are some reasons off the top of my head:

1. To my mind, the structor approach perfectly captures the intuitive notion of “combinatorial structure”, and in particular, how one can often naturally define such a structure by writing down some abstract “structural equation” (“categorifying” the kind of equation one classically expresses in terms of generating functions) which completely defines the desired structure, and thus (to my mind) precisely expresses what this structure “means”.

2. To my mind, this approach greatly clarifies the relation between labeled and unlabeled enumeration, and also greatly clarifies the role of symmetry in defining and studying combinatorial structure. This helped me (at least) better understand connections with some of the most interesting topics in combinatorics.

3. In particular, it turns out that the ogf’s and egf’s arise from a more informative counting series, the Joyal cycle index, which is a generalization of the Polya cycle index. Thus, the attractive ideas of Polya counting turn out to be “secretly” controlling generatingfunctionology. Not only that, we get an apparently hard new kind of problem to chew on, decomposing an arbitrary structor into its “molecular components”. This turns out to be related to the “table of marks” used by Frobenius in studying permutation representations of groups (his work on this has since been overshadowed by his even more beautiful work on linear representations, where the character table is analogous to the table of marks but is easier to compute.

4. The centerpiece of Wilf’s book is a terminology (cards and decks) which I find singularly unappealing, plus an “exponential theorem”, which is indeed powerful, but his treatment thoroughly obscures the fact that this is a special case of a more general result in terms of “wreathed actions”, which I much prefer both because this clarifies the relationship with Polya counting and because wreath product is such an important operation that it makes no sense (to my mind) to disguise its role here.

5. Fraisse theory places the fascinating properties of the Erdos-Renyi “universal random graph” in a wider context, and relates this both to “relational theories” as studied in first-order logic, and to oligomorphic groups, a particularly tractable generalization of finite permutation groups. Here Cameron defines a cycle index which is essentially the same as the Joyal cycle index, plus a certain algebra which is somewhat analogous to the group ring. An interesting conjecture about this algebra has just been partially proven.

My interest tends to be in better understanding connections I might otherwise only vaguely sense should exist, so I am untroubled by the fact that I made no attempt to alter the classical techniques discussed by Wilf for studying asymptotics of the terms in ogfs and egfs.

Still, in the book by Bergeron et al. and in my own “zoo”, it is striking that these tools seem to apply most readily to combinatorial structures which often have the nature of “decorated graphs” of some kind, particularly “decorated trees”. But there is another kind of combinatorial structure which fascinates me. This kind has a more “geometric” flavor and arises naturally in several (related) ways:

A. In “enumerative geometry”, we seek to understand combinations of incidence relations; the best known example is Schubert calculus, where we seek to understand incidence relations between points, lines, 2-flats… in CP^n. IOW, we seek to understand a kind of geometric/combinatorial structure inside Grassmannians. This turns out to amount to computing the structure of the cohomology ring (if you think of Grassmannians as manifolds, even homogeneous spaces) or Chow ring (if you think of them as algebraic varieties), and it can be generalized (as I hope John Baez will explain in a forthcoming Week) to more general flag manifolds in reductive linear algebraic groups (I hope I said that right, John!).

B. Why did Klein not deliver the talk we know as the Erlangen Program? Some of us have speculated this might be due to his recognition that while he had a lot to say about passing from geometries to groups, and then using the group structure to organize most classically known geometries into a kind of hierarchy, he had much less to say in 1871 about going in the other direction: starting with a homogeneous space (or even just a right coset space H\G) and obtaining the corresponding geometry. JB and I have discussed different approaches to systematically obtaining the canon of “geometric elements” in such situations. When you ask how incidence relations between these elements can be defined and studied, in the context of structors, perhaps you are led to the notions John has recently been discussing involving what he is calling “spans” (in my defunct student paper “Categorical Primer” I called them objects of C^Push, were C is some category and Push is the obvious (?) category with three objects and two nonidentity arrows, since these guys “want” to fit into a pushout square).

In the approach I advocated in my defunct student paper “Symmetry and Information”, one exploits the Galois correspondence between (pointwise) stabilizers and fixsets to obtain some basic information. In the case of finite groups this involves in effect computing the table of marks restricted to the lattice pointwise stabilizers (a sublattice of the lattice of subgroups), but it can sometimes yield some kinds of primitive incidence data. However, I suspect that this only scratches the surface. Still, I think it’s interesting to see how the “complexions” and “conditional complexions” specifying motions of fixsets under the symmetry group (possibly subject to knowing how some other fixsets moved) obey the same formal laws laid down by Shannon. This is not a coincidence, since we can recover Boltzmann’s notion of combinatorial entropy in a simple and natural way.

C. In the study of reflection groups, incidence relations and their study in terms of parabolic subgroups plays a key role. (John can probably explain how this is very closely related to A.)

So, I’d like to see someone (presumably JB) develop a “categorified” approach to enumerative geometry. In recent Weeks he seems to toss out some strong hints that this is one place he intends to go. I can’t help hoping it will be possible to explicate this in terms of structors, which I think are easy to understand, without appealing much to n-categories, since I’ve never felt comfortable with worrying about “coherence relations” which I typically feel I don’t understand.

I’d be particularly curious to see if a notion of “complexity” arises. Put very informally, an “entropy” should be subadditive while a “complexity” should be superadditive (both should satisfy some other formal properties). That is, the entropy of the whole is at most the sum of the entropy of its parts, while the complexity of the whole is at least the sum of its parts.

As some of you know, I’ve also been intrigued by the idea of “sizing” solution spaces to a system of PDEs, or a more “geometric” equation, without trying to actually characterize “the general solution”. Here one idea— probably a very crude one— which was discussed by Einstein (who may have gotten the idea from Hilbert or Noether) is to count the number of partial derivatives which remain after using a PDE to eliminate as many as possible. In simple cases this amounts to computing the Hilbert series of an ideal in a polynomial ring, but the treatments I’ve seen appear oddly limited. Does anyone have a better idea about what to “count”? For all I know, a really insightful categorification of these notions might also relate them to some of the other areas which have been mentioned in this thread, such as scene reconstruction.

Having explained (sort of) what most intrigues me about what John may or may not be up to, I have a question regarding Terry’s challenge: “categorify” his Cauchy-Schwarz argument bounding the size of a finite “fiber product”. My question is this: does anyone know of an impressive example in which categorifying an inequality useful in analysis (C-S qualifies!) results in a more general statement which can be “decategorified” to give a new but useful inequality?

Oh yes, one last thing: in my “rewrite” I failed (as I recall) to fit Dirichlet generating functions neatly into the picture. As most of you know, these are useful for many things involving multiplicative structure of the integers, including the prime number theorem. Anyway, this reminds me of a silly thought: the dual of fiber product (a subset of the Cartesian product) is the fiber sum (a quotient of the disjoint union), so one might very naively expect that “dualizing” categorifications inspired by multiplicative phenomena might sometimes lead to “dual” additive phenomena. So one vaguely wonders if something like “CS” has some kind of crazy “dual”. This crazy idea might be worth keeping in mind, if only to explain why it is naive! Certainly this naive expectation doesn’t seem to agree very well with the distinction between multiplicative and additive as characterized by Terry in various writings at his website. More generally, I suppose, one might demand to know why specific categories drastically “break” naive symmetries expected from very general notions in category theory, such as duality (reversal of arrows).

Posted by: Chris Hillman on April 21, 2007 10:59 PM | Permalink | Reply to this

Re: Categorifying enumerative geometry?

I quote Lawvere here, saying

Not every statement will be taken into its formal dual by the process of dualizing with respect to VV, and indeed a large part of the study of mathematics

space vs. quantity

and of logic

theory vs. example

may be considered as the detailed study of the extent to which formal duality and concrete duality into a favorite VV correspond or fail to correspond.

Posted by: David Corfield on April 23, 2007 4:02 PM | Permalink | Reply to this

Re: Categorifying enumerative geometry?

Yay! Chris Hillman is posting comments to our blog!

There’s too much to respond too here, so for now I’ll just say that Dirichlet series don’t seem to fit in the family as ordinary and exponential generating functions.

The rig of ordinary generating functions is a decategorification of the 2-rig of ‘structures on finite sets’:

hom(FinSet 0,FinSet)hom(FinSet_0, FinSet)

Similarly, the rig of exponential generating functions is a decategorification of the 2-rig of ‘structures on totally ordered finite sets’:

hom(Δ 0,FinSet)hom(\Delta_0, FinSet)

(Here FinSetFinSet is the category of finite sets, Δ\Delta is the category of totally ordered finite sets — also known as ‘the algebraist’s category of simplicesΔ alg\Delta_{alg} — and the subscript 00 means we’re considering the underlying groupoid, where the only morphisms we keep are isomorphisms).

Dirichlet series don’t seem to arise this way — though god knows, I’ve tried. It seems Dirichlet series are best seen as elements of the monoid algebra of the multiplicative monoid of natural numbers. (This monoid is the free commutative monoid on countably many generators, one for each prime!)

The concept of ‘monoid algebra’ is a slight generalization of the more familiar concept of ‘group algebra’, and a special case of the still more general ‘category algebra’ described in week244. So, Dirichlet series can be seen as elements of a category algebra. And, in fact, the Riemann zeta function is the ‘zeta element’ of this particular category algebra, as defined in week244.

I’ll categorify some enumerative geometry later in This Week’s Finds. In particular, Jim, Todd and I have been putting a lot of energy into trying to understand the Schubert calculus and Littlewood–Richardson rules more deeply. So, I have a lot to say about that… too much, in fact.

Posted by: John Baez on April 24, 2007 2:42 PM | Permalink | Reply to this

Re: Categorifying enumerative geometry?

Yay! Chris Hillman is posting comments to our blog!

Yay, indeed. But judging from this, we’ll have to work hard to overcome his problem with MathML.

If you do venture back to this thread, Chris, it would be great if you could make accessible those papers on information and entropy. Were there others not cached by Citeseer?

Posted by: David Corfield on April 24, 2007 5:35 PM | Permalink | Reply to this

Re: Categorifying enumerative geometry?

The question is: which web browser could Chris Hillman be using, so archaic and/or obscure that it doesn’t do MathML? It couldn’t be Firefox!

More likely Chris didn’t take the time to set up MathML on his browser.

Posted by: John Baez on April 24, 2007 8:45 PM | Permalink | Reply to this

Re: Categorifying enumerative geometry?

Actually, when using a not-too-old version of Firefox, there is nothing one needs to do in order to be able to read almost everything on this blog.

Extra fonts need to be installed, for Firefox, only if a small subset of characters that might appear here once in a while should also be readable.

The only situation where one really has to do something is when one is using Internet Explorer. Then one has to spend \sim40 seconds for downloading this plugin and then restarting the browser.

Posted by: urs on April 24, 2007 8:51 PM | Permalink | Reply to this

Re: Categorifying enumerative geometry?

Oh, you could never say too much about Schubert calculus! I know what you meant, though: you have too much to say about this to easily sum it up in a few Weeks. But please keeping trying anyway!

One thing I’d like to see you explain (on the way to “more interesting” examples than Grassmannians) is intersection numbers and how these are related to integrating products of special Schubert cycles over a Grassmannian.

(“Must have javascript enabled”?! Isn’t that like announcing “I’m vulnerable— phish me!”? Needless to say, I only enabled javascript for this one session at this one site. I AM using Firefox, with the NoScript plugin. Perhaps my MathML problem is due to something running afoul of my semi-paranoid “safer browsing” precautions, details of which I decline to discuss here for the obvious reasons.)

Posted by: Chris Hillman on April 28, 2007 2:56 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

Gowers’ essay is an important one. I think the view that favors theory-builders is a short-sighted one (though my taste tends towards theory-building rather than problem-solving). The problem-solving of the nineteenth century led to the theory-building of the twentieth.

Posted by: Walt on April 18, 2007 6:27 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

Here is a quick comment on the Green-Tao theorem and the `other culture’ to which I perhaps belong. At issue are two theorems that mirror each other:

(1) The theorem about primes in arithmetic progressions.

(2) The theorem about arithmetic progressions in primes.

The link between (1) and `fancy’ mathematics proceeds through a *Galois-theoretic* interpretation. It is *possible* that the question of whether (2) can be linked to Galois theory is a very deep one, and one that points in a good direction.

Posted by: Minhyong Kim on April 18, 2007 8:08 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

Dear Minhyong,

There is a famous distinction in prime number theory between the number theorists who like to multiply primes, and the number theorists who like to add primes. As the primes are very heavily multiplicatively structured, the mathematics of multiplying primes is very algebraic in nature, in particular involving field extensions, Galois representations, etc. But the primes are very additively unstructured, and so for adding primes we see the tools of analysis used instead (circle method, sieve theory, etc.).

“Counting primes in arithmetic progressions” can be easily converted (via the multiplicative Fourier transform) to “counting primes twisted by characters” - an eminently multiplicative question, and one which is handled by primarily algebraic means. “Counting arithmetic progressions in primes” however seems to be a stubbornly additive question and thus not really amenable to algebraic attacks. (On the other hand, “counting geometric progressions in primes” is very easy. :-) ).

Posted by: Terry Tao on April 19, 2007 11:19 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

[Reposted with itex-enabled setting - DC]

Dear Terry,

Thanks for your comment. Even if what follows is very vague, I thought I’d indulge in some further explanation since this forum seems to tolerate such ramblings.

You see, the reason I thought about connection between (1) and (2) is rather related to your comment about the additive nature of your theorem (whose proof I’m not claiming to understand).

The background `theory’ that my speculation comes from goes under the name of anabelian geometry, a program of Grothendieck from the 80’s according to which certain classes of schemes should be determined by their arithmetic fundamental groups. An outstanding example where this works out is a theorem of Neukirch and Uchida that says that a number field FF is determined by its absolute Galois group G F=Gal(F¯/F)G_F=Gal(\bar{F}/F). More precisely, given an isomorphism G FG KG_F \simeq G_K, its conjugacy class comes from an isomorphism FKF \simeq K. A nice special case of this theorem says that all automorphisms of the Galois group of Q are inner.

I don’t really know how Grothendieck thought about his program, but my friend Shin Mochizuki has the striking view that anabelian geometry is primarily concerned with the relation between the additive and multiplicative structures of rings. It’s not so easy to explain this point of view since it arises from the actual proofs of theorems rather than their statements. To get a sense of it, think of the N-U theorem as giving a way to

*recover F from G FG_F*

Now, consider the fact that given G FG_F, it is rather easy to recover the multiplicative group F *F^*. This involves only the abelianization G F abG_F^{ab} and standard class field theory. But it’s the additive structure on F=F *{0}F=F^*\cup \{0\} that’s the real challenge. To this day, one can’t seem to do this directly. But in many other theorems of anabelian geometry, for example, Mochizuki’s theorem on hyperbolic curves over pp-adic fields, some sort of a direct reconstruction is carried out. So within the general picture, it turns out to be quite natural to break down the reconstruction problem into two parts, multiplicative and additive.

What then is necessary to get the field FF? One deep ingredient is a theorem of Neukirch that says that a subgroup of G FG_F that looks like a decomposition group of a prime is a decomposition group. This allows us to recover the primes of FF. The other part, regarded as more standard, is the Cebotarev density theorem, which is of course the non-abelian extension of the theorem on arithmetic progressions. So from the point of view of the previous paragraph, the non-abelian theorem on arithmetic progression plays a key role in recovering the additive structure of FF, albeit indirectly. Note, however, that the Cebotarev theorem by itself is only weakly non-abelian, since it follows rather easily from the abelian case.

If my nonsensical thought of linking your theorem to Galois theory could ever be made coherent, what I’m suggesting therefore is that the full non-abelian Galois group should be the relevant object. The lack of any obvious link to algebraic number theory could be an artifact of thinking primarily about G F abG_F^{ab}. In any case, since the Galois group G FG_F contains the data of the primes of FF and its additive structure, perhaps its not entirely far-fetched to ask what your theorem says about this group. Incidentally, is there a number field analogue of your theorem that makes any sense?



Posted by: Minhyong Kim on April 20, 2007 9:38 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

Minhyong, do you know what the situation is for function fields over finite fields? Is there an anabelian theorem, and if so, is there some kind of direct reconstruction of the additive structure of the field?

Posted by: James on April 20, 2007 2:07 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

Yes, everything works over function fields. A fairly readable survey of the ideas emphasizing the ‘reconstruction’ viewpoint is contained in the write-up of Florian Pop’s lectures at the Arizona winter school on arithmetic geometry, 2005.

Posted by: Minhyong Kim on April 21, 2007 5:27 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

I’ll have to qualify one remark from the previous post. The part where one recovers the multiplicative group F* from GF is also not very direct as it stands. It’s one of those things that seemed obvious, but when I actually thought it through following the post, struck me as not so. Perhaps I’ll make some more confident remarks on this point later on. In the meanwhile, it remains true that the multiplicative structure of F* is much closer to commutative Galois theory than the additive structure.

Posted by: Minhyong Kim on April 20, 2007 5:20 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

.. my friend Shin Mochizuki has the striking view that anabelian geometry is primarily concerned with the relation between the additive and multiplicative structures of rings.

Minhyong’s friend has also apparently claimed to prove the ABC-conjecture.

It seems there are a couple of notes (here and here) outlining his overall approach.

Posted by: David Corfield on September 5, 2012 3:53 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

Disclaimer: I’m claiming “pure” mathematics is any activity where one has decreed that the entities in the problem “work this way” and then proceeds, even if the problem is pursued partly for some other reason. To me, “applied” really means one is constantly rejecting/modifying parts of the mathematics “on physical grounds”.

Part of the justification for theory building is that it’s the equivalent of building an army tank division rather than using a popular uprising: by optimally structuring things you can accomplish exactly the same things but much, much easier. A good example of this is the theory of generating functions and precise asymptotic estimation of coefficients. People like Philippe Flajolet are doing incredibly intricate and – to my mind – deep work putting together theories that can pulverise some of the combinatorial problems that occur in algorithm analysis. This is theory to solve problems, rather than give pedantically correct names to everything. (Indeed, I remember being slightly miffed that TWF-185 seems to down-play Knuth-Patashnik-Graham and Wilf references compared to “the precise definitions” given by Joyal. I’ve not followed up any of those references because they aren’t accessible on-line, but it’d be interesting to know if these precise definitions add any extra “pulverising power” to the theory that’s used to deal with these sorts of combinatorial problems.)

In contrast, one area I’d like to work in seems to scream out for an organising theory, namely the mathematics/algorithmics of scene reconstruction given some subset of camera parameters and scene parameters/structures. Part of the reason it’s an area “I’d like to work in” rather than “an area I do work in” is that there seems to be little structuring theory with sudden appearance of different matrix decompositions and different classifications of the degenerate conditions, etc, at seemingly completely unrelated ways in the different algorithms, so it seems like currently to get to the cutting (maybe bleeding :-) ) edge you’ve got to study each algorithm from scratch. I’ve seen some work that claims to provide a theory for such things, but they all look like “getting the names right” theories rather than “army tank division theories”. (I haven’t had the time for comprehensive reading in this area so no links and an admission I might be missing good theories.)

I know this view of what makes a good theory is probably inappropriate for some areas like the foundations of mathematics, but it seems like a good one in other areas.

Posted by: dave tweed on April 19, 2007 12:10 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

Oops: should have been a much less absolute final sentence: “I know this view of what makes a good theory is probably inappropriate for some areas like the foundations of mathematics, but it seems like a good one in some other areas.”

Posted by: dave tweed on April 19, 2007 12:23 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

Did you ever read Rota’s endorsement of species in his preface to Combinatorial Species and Tree-like Structures by Bergeron, Labelle and Leroux?

Joyal’s definition of “labeled object” as a species discloses a vast horizon of new combinatorial constructions, which cannot be seen if one holds on to the reactionary view that “labeled objects” need no definition. The simplest, and the most remarkable application of the definition of species is the rigorous combinatorial rendering of functional composition, which was formerly dealt with by handwaving – always a bad sign. But it is just the beginning.

Posted by: David Corfield on April 19, 2007 4:10 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

A labeled graph – or any “labeled” combinatorial construct – is a functor from the groupoid of finite sets and bijections to itself.


Given that finite sets are a lot like finite vector spaces with a chosen basis, am I wrong in feeling that this suggests a close relationship between “labeled combinatorial constructs” and Schur functors?

Hm, we know that the latter do have a combinatorial description, in terms of Young diagrams.

Is this a coincidence?

Posted by: urs on April 19, 2007 4:51 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

Given that finite sets are a lot like finite vector spaces

Or at least finite pointed sets, as the finite vector spaces over the ‘field with one element’.

Posted by: David Corfield on April 19, 2007 5:21 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

But sets are the modules of the “field without elements” (0.4.13 (p. 21) and 3.4.12(h) (p. 145) of this).

Posted by: David Corfield on April 20, 2007 3:50 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

Urs writes:

Given that finite sets are a lot like finite vector spaces with a chosen basis, am I wrong in feeling that this suggests a close relationship between “labeled combinatorial constructs” and Schur functors?

No, you’re exactly right! This will be a big part of the Tale of Groupoidification. Joyal’s species (aka ‘structure types’) are functors

F:FinSet 0SetF: FinSet_0 \to Set

Schur functors can be seen as functors

F:FinSet 0VectF: FinSet_0 \to Vect

But, they can also be seen as functors

F^:VectVect\widehat{F}: Vect \to Vect

(note - no subscript 00 indicating ‘groupoid’ here).

This sets up a fascinating web of relationships. It’s also very interesting to study all the other variants of these ideas, like qq-deformed species:

F:FinVect(q) 0SetF: FinVect(q)_0 \to Set

where FinVect(q)FinVect(q) is the category of finite-dimensional vector spaces over the field with qq elements, and FinVect(q) 0FinVect(q)_0 is the corresponding groupoid. Just as ordinary species are a categorification of the Hilbert space for the harmonic oscillator, qq-deformed Hilbert spaces are a categorification of the Hilbert space for the qq-deformed oscillator!

Posted by: John Baez on April 20, 2007 10:06 PM | Permalink | Reply to this

Purposiveness and Hemispheres; Re: The Two Cultures of Mathematics

One puzzle to me in Rota’s endorsement is:

“Advances in mathematics occur in one of two ways. The first … always comes as an unexpected application of theories that were previously developed without a specific purpose.”

Developing a thoery cannot strictly happen “without a specific purpose” if developed by a human being. Automated Theorem Proving is less clear.

Clarification requires some distinction between conscious and unconscious purpose. A mathematician “just playing around” in some domain of mathematics is operating, in the view of a Kuhnian philospher, within a paradigm. That paradigm carries intentions and assumptions which may be outside the conscious awareness of the mathematician, yet still constitute a purpose, even if only to maintain the paradigm.

When I previously (upthread? supra?) praised Terrence Tao’s use of motivation (i.e. number wants to be a prime) I was trying to deal with this notion of purpose in a parallel sense. Our brains are dual-processors at the hemispheic level of resolution, as reflected by the intent of Mauchley and Eckert in building the first dual processor computer, BINAC, in a secret project for the U.S. Air Force circa 1949, as Mauchly explained to me at some length (when we worked with Ted Nelson in first implementing Hyptertext, circa 1976) , and is now declassified.

If we are right-handed, it is our left hemisphere that does logical, analytic, sequential tasks (I’m oversimplifying) and our right hemisphere that does aesthetic, global, parallel, gestalt tasks.

In my opinion, the Tao approach to theory development and problem solving alike is to use both hemispheres of the brain, unifying conscious and unconscious “purpose.”

I have things to say about Joyal and species, but that is less important to me here and now than the issue of motivation OF a mathematician, and imputed BY a mathematician to the objects of inquiry.

Posted by: Jonathan Vos Post on April 19, 2007 4:59 PM | Permalink | Reply to this

Re: Purposiveness and Hemispheres; Re: The Two Cultures of Mathematics

The comment about using both hemispheres of the brain reminded me of a recent book on Teichmueller theory which has a forward by W Thurston. In it, he talks about the pleasure of using one’s “full brain” in mathematics and how many mathematicians don’t.

When I read it a few months ago, it seemed like the kind of thing the philosophically minded people around here would like. I think this is the book:

Posted by: James on April 22, 2007 1:56 AM | Permalink | Reply to this

Re: Purposiveness and Hemispheres; Re: The Two Cultures of Mathematics

Now I see the forward is actually available online:

Also, he uses the phrases “full mind”, “whole mind”, and “whole brain”, but never “full brain”.

Posted by: James on April 22, 2007 2:01 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

I hadn’t, and it makes a lot of sense. What I was trying to get at by hyperbole in the original post was that you can have theories that “explain” things and “get the names right” and they’re good things to have developed. There seemed to me a sort-of subtext in your post that most theories were of this sort (which is probably a mis-reading based partly on the “aesthetic” point), so I was pointing out that there are other theories which are more about systematizing powerful tools. (Indeed, it’s really a spectrum rather than a binary choice.) I’m not putting down explanatory theories but rather pointing out there are also some really rather splendid “tank division” theories around. (Indeed, arguably differential/integral calculus and linear algebra are more commonly used to “get answers” rather than “explain”.)

When I talk about “getting the names right” theories I just mean the cases where pains are taken in definitions yet it’s not clear that this buys you anything more than precise definitions. This is in contrast to, say, the more rigorous development of calculus in the nineteenth century which actually clarifies the errors in some “paradoxes/contradictions” that were “constructible” before.

Posted by: dave tweed on April 21, 2007 11:54 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

Dave Tweed wrote:

This is theory to solve problems, rather than give pedantically correct names to everything. (Indeed, I remember being slightly miffed that TWF-185 seems to down-play Knuth-Patashnik-Graham and Wilf references compared to “the precise definitions” given by Joyal…)

I love Concrete Mathematics and Generatingfunctionology! I focused on Joyal’s species because they fit into my overall game plan, which is categorifying known mathematics and using this to discover new connections between seemingly different ideas.

Combinatorists like to count structures on finite sets. But what is ‘a structure on finite sets’, exactly? Joyal realized that it’s a map assigning to each finite set some collection of ‘structures’ on that set — but in a way that’s functorial with respect to bijections. So, it’s a functor

F:FinSet 0SetF: FinSet_0 \to Set

where FinSet 0FinSet_0 is the groupoid of finite sets and bijections. Joyal called such a functor a species, and showed that many generating function techniques work directly at the level of species, without the need to ‘decategorify’ and use generating functions.

So far, this is just a matter of more deeply understanding what people were already perfectly good at doing. The fun starts when we realize that the category of species

hom(FinSet 0,Set)hom(FinSet_0, Set)

is very much like the Hilbert space for the quantum harmonic oscillator. There is, in fact, a very precise relation! So, we can understand things like the canonical commutation relations and Feynman diagrams in a purely combinatorial way. The ultimate payoffs of this are still unclear, but it’s certainly cool.

Even better, when we replace the groupoid of finite sets by the groupoid of finite-dimensional vector spaces over the field with qq elements, we get a categorified version of the qq-deformed Fock space!

This is the beginning of a larger story, including a theory of categorified quantum groups — and much more as well. Quite generally, it seems large wads of fancy linear algebra (like quantum groups, and Hecke algebras, and Hall algebras) is secretly combinatorics and incidence geometry in disguise! I’m starting to tell this now in This Week’s Finds — I call it The Tale of Groupoidification.

So, I beg your forgiveness and reemphasize the remark on my This Week’s Finds homepage, namely:

Mostly I try to write about subjects I actually understand, which limits the selection tremendously.

I could never be good at counting finite sets, like a true combinatorist. In fact my passion runs in the opposite direction: taking known math and replacing natural numbers by finite sets — that is, categorification!

Dave Tweed wrote:

… it’d be interesting to know if these precise definitions add any extra “pulverising power” to the theory that’s used to deal with these sorts of combinatorial problems.)

The main thing Joyal’s work does is open up new realms of inquiry. It’s not about “extra pulverising power”.

Posted by: John Baez on April 20, 2007 9:51 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

I wasn’t trying to be accusatory so no apology remotely necessary. I was primarily being a bit hyperbolic in the post to highlight that not all mathematical theories are primarily used for explanation. I was only a bit “miffed” at the semi-implied suggestion in the TWF that techniques for actually extracting answers to problems via generating functions were just “fun” when I think they have some deep mathematical work in them as well. My reply above to David C expands on this.

Posted by: dave tweed on April 22, 2007 12:13 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics

I am certainly of the Theory camp, but I think that’s just because of how my brain is wired. I imagine, those who would disparage others in the “combinitorics” camp, do so for the same reasons that rednecks disparage Mexicans … they don’t understand what they’re saying.

The fact of the matter is that Combinitorial problems are often VERY hard. And if you tend to “miss the trees for the forest” as many theoriticians do, then those problems will likely cause you more grief than you are willing to endure.

That is certainly true of me. I feel confortable imagining how a manifold can have a differentiable structure, but even a simple sudoku puzzle can give me the heebee-jeebee’s :)

Posted by: Saij on April 19, 2007 7:57 AM | Permalink | Reply to this

The Narrative Self; Re: The Two Cultures of Mathematics

Part of my self-identity is as a Mathematician, even though I am one of the least expert of anyone on this blog. I have several times pontificated on narrative structure in mathematical Proof. Stepping back, here is a short article on the Philosophy of Narrative Identity, complete with intriguing references.

Don Quixote and The Narrative Self

Stefán Snaevarr asks, are our identities created by narratives?

Once upon a time a philosopher wrote an article called ‘Don Quixote and The Narrative Self’. He commenced by saying: In this essay, I will discuss the question of whether our selves are constituted by narratives, ie stories. Are we like Don Quixote, whose self was created by his reading of medieval romances: are we Homo quixotienses, the narrative self? Or are we rather like the protagonist of Sartre’s novel Nausea, Antonin Roquentin, whose life did not form any narrative unity? Are we in other words rather Homo roquentinenses?

The idea that our life is a story is by no means new. Thus the great bard Shakespeare said that life “…is a tale told by an idiot, full of sound and fury, signifying nothing.” (Macbeth) However, it took philosophers some time to discover the philosophical import of this view of life. It was actually a German chap called William Schapp who first gave this age-old idea a philosophical twist….

Posted by: Jonathan Vos Post on April 21, 2007 3:18 AM | Permalink | Reply to this
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