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April 19, 2012

Mathematical Cultures

Posted by David Corfield

My friend and fellow philosopher Brendan Larvor mentioned an upcoming conference in September 2012 in London. This is the first of a series of three on Mathematical Cultures.

The first conference will gather research that explores and maps the variety of and connections among contemporary mathematical cultures. These can be research cultures, but may also include practitioner cultures (e.g. among engineers, economists, social scientists, etc.) and mathematical cultures among instructor and student groups (e.g. primary/secondary/tertiary teachers, school pupils, mathematics students at all levels). The project will not invite contributions on historically or culturally remote mathematical cultures except as these illuminate contemporary mathematical culture in developed societies.

Confirmed speakers include two professional mathematicians, Alexandre Borovik and Norbert Schappacher. There is a call for papers with a deadline of 1 June May 2012.

Perhaps it’s time to dust down the material I was collecting on the two cultures.

Posted at April 19, 2012 10:29 AM UTC

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Re: Mathematical Cultures

Dear David,

perhaps you will be also interested in the following blog: http://owl-sowa.blogspot.co.uk/

Posted by: Stewie Orf on April 19, 2012 11:22 AM | Permalink | Reply to this

Re: Mathematical Cultures

At nLab:two cultures you remind us of a quote taken from

  • Elkies, Introduction to analytic number theory (pdf)

which contains the following sentence

An ambitious theory-builder should regard the absence thus far of a Grand Unified Theory of analytic number theory not as an insult but as a challenge.

I suppose the use of the phrase “Grand Unified Theory” borrowed from physics it is not entirely a coincidence. It is precisely the same situation there:

An ambitious theory-builder should regard the absence thus far of a precise formulation of parts of physics not as an insult but as a challenge.

Then I have another comment which I might have made before:

I am sceptical that the dichotomy “theory building” and “problem solving” is accurate. Because to the extent that it is done well, theories are built in order to solve problems, not instead of solving problems. It’s a bit like claiming a dichotomy between the activity of building a tunnel boring machine and digging with a spade.

Posted by: Urs Schreiber on April 19, 2012 1:37 PM | Permalink | Reply to this

Re: Mathematical Cultures

I agree it wasn’t the greatest choice of a pair of names. Still there’s something intriguing in Tao’s remarks about the dichotomies: structured/pseudorandom and closed/open conditions.

I see we were once wondering about pseudorandomness cropping up closer to home, in the homotopy groups of the spheres.

Posted by: David Corfield on April 19, 2012 2:55 PM | Permalink | Reply to this

Re: Mathematical Cultures

I am sceptical that the dichotomy “theory building” and “problem solving” is accurate. Because to the extent that it is done well, theories are built in order to solve problems, not instead of solving problems.

I mostly agree with you, but not completely. I would say that to the extent that it is done well we build theories and solve problems in order to understand.

Also, theories and problems are hard to separate since problems are embedded in theories. What I mean is this: the problem of classification of groups makes no sense if you don’t know what a group is. The notion of a group is part of a theory.

Posted by: Eugene Lerman on April 19, 2012 4:06 PM | Permalink | Reply to this

Re: Mathematical Cultures

[ edit: only after posting the following I noticed that I overlapped with Eugene’s message above ]

Still there’s something intriguing in Tao’s remarks about the dichotomies: structured/pseudorandom and closed/open conditions.

I may have asked the following before, or mentioned it before in similar discussion, but I forget what your reply was, and I would be interested to hear what the reply of anyone following the structured/pseudorandom dichotomy is:

how do you see this dichotomy related to the dichotomy

theory / models

general abstract / exceptional

?

To name an example of what I mean here: the theory of groups is most structured / general abstract. Nevertheless, the collection of the models of the theory, say of all finite models or of all smooth models, etc. namely the finite groups, the Lie groups, etc. has the most intricate structure, with plenty of exceptional phenomena appearing: the exceptional groups.

The case of homotopy groups that you point to is similar: the theory of homotopy types of stable objects in an \infty-topos is most structured / general abstract. Nevertheless, when we look at specific models of the theory, they show the most intricate phenomena: already the initial object in the stabilization of the canonical \infty-topos, the sphere spectrum, shows plenty of exceptional structure

These examples may seem to be a a different style than those of PDEs, but I think PDEs fit into this in the same way. Today we know that in terms of D-geoemtry the theory of PDEs is almost at the same foundational depth as the previous examples, existing in any \infty-topos with just a tad more extra structure. Nevertheless, the models of this theory can show the most intricate and non-structured behaviour.

And so on. Isn’t this what the dichotomy is all about? The general abstract versus the concrete particular?

To me it seems it is, but maybe I am not understanding well what others have in mind here (also I haven’t carefully read all that has been written in this context).

Accordingly, I am not so happy about talking of two “cultures”, as if it were two different ways of going about the same problem. Instead, it seems to me that there is one entity – mathematics – with two main facets to it (or maybe actually 3 of them, if we follow Lawvere’s classification above).

In the end, we want to understand the world we live in. Clearly, it’s a very exceptional instance of some general abstract pattern. The task of science and mathematics is to sort out both: what is exceptional about it, and what is general abstract.

For instance, that symmetry groups play a role is because groups are a foundational structure. That we see particular such groups may be exceptional random structure – or may be not. That’s still an important open question that people are trying to figure out. It’s a question of true or false, not of culture.

Of course every researcher may choose to focus on one aspect or the other. But such choices are usually called choices of subfield not choices of culture. There is the subfield of general abstract structures in mathematics, and the subfield of concrete particular structures. Maybe there is even a subfield of concrete general structures.

Posted by: Urs Schreiber on April 19, 2012 4:27 PM | Permalink | Reply to this

Re: Mathematical Cultures

Can we analyse it all in these terms? What about the Hall marriage theorem discussion? So we say that the theorem’s proof only works in very specific concrete general settings? It was debated there whether strength of logical system corresponds to category theoretic distinctions.

Does Tao’s closed/open distinction tally with anything category theoretic?

Posted by: David Corfield on April 19, 2012 6:07 PM | Permalink | Reply to this

Re: Mathematical Cultures

Can we analyse it all in these terms? What about the Hall marriage theorem discussion?

I am not sure if I understand what the issue is.

It seems that the discussion proceeds from observing the phenomenon that often after we write “The following are equivalent.” It continues with “The implication 1) \Rightarrow 2) is evident, so we now discuss 2) \Rightarrow 1).” Right?

I can’t quite see yet what this has to do with “two cultures”. (Except that I sense the hidden insinuation that one of the two cultures can’t prove anything non-evident…)

Posted by: Urs Schreiber on April 20, 2012 11:31 PM | Permalink | Reply to this

Re: Mathematical Cultures

Ah yes, the hidden insinuation. The harder direction is non-functorial, and must rely on situation specific reasoning. The easier functorial direction requires simple general reasoning.

If one notes that making the trivial trivially trivial may require plenty of hard work, does that forestall the insinuation?

But I wonder if that’s all there is to it. Is it just that those of a ‘Hungarian disposition’ are attracted to the kind of mathematics that doesn’t generalise easily? Could they enter into any specific category (concrete general) and find aspects specific to that setting?

Posted by: David Corfield on April 21, 2012 2:03 PM | Permalink | Reply to this

Re: Mathematical Cultures

Ah yes, the hidden insinuation. The harder direction is non-functorial, and must rely on situation specific reasoning. The easier functorial direction requires simple general reasoning.

But maybe only because there wasn’t enough theory-building beforehand to see that also the apparently harder direction follows some general reasoning.

Seriously, I feel unsure what the discussion is really about. Maybe instead of you asking me questions, let’s turn this around:

if you find the time to “dust down the material you were collecting”, as you envision, how would you today summarize the content of the (what is it?) issue (?) of the two cultures, the interesting aspects of it and what there is to learn from it? Maybe I’d need a clear crisp summary to see what motivates your interest here.

Posted by: Urs Schreiber on April 21, 2012 4:12 PM | Permalink | Reply to this

Re: Mathematical Cultures

Maybe I’d need a clear crisp summary to see what motivates your interest here.

Ok. So I’d like a clearer sense of answers to:

How are the interests of current mathematical researchers distributed within the space of mathematics? What is the connectivity of this distribution? How is the distribution evolving? How ought it to evolve? What are the internal goods of mathematics?

Gowers in his essay is pointing out that his area of research makes little connection with the monumental constructions of algebraic geometry, algebraic number theory, etc., but that people make the further mistake of imagining his ‘combinatorics’ to involve a sequence of unconnected problems, whose solution requires one-off triumphs of ingenuity. He gives Ramsey theory as an example:

Theorem. For every positive integer kk there is a positive integer NN, such that if the edges of the complete graph on NN vertices are all coloured either red or blue, then there must be kk vertices such that all edges joining them have the same colour. The least integer NN that works is known as R(k)R(k).

Problem. (i) Does there exist a constant a>2a \gt \sqrt{2} such that R(k)a kR(k) \ge a^k for all sufficiently large kk. (ii) Does there exist a constant b<4b \lt 4 such that R(k)b kR(k) \le b^k for all sufficiently large kk?

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.

Odd isn’t it that a Fields’ medallist needs to defend himself like this? But he must have heard the charge. Google Books is cutting it off, but we get the gist from this statement by Saunders Mac Lane

There are also all manner of small problems, some natural, some concocted, problems are a vehicle of competition and display. In some schools (often in those influenced by Hungarian traditions) the business of mathematics seems to be just the formulation and the eventual solution of hard problems, with perhaps more attention to the difficulty than to the… (Mathematics: Form and Function, 430)

Gowers defends his interest by observing that any breakthrough in Ramsey theory is likely to involve a new general principle of potential broad significance to combinatorics, but that this new principle will most probably not be expressible as general theory. If so, this suggests that not all general principles may be expressed as what you call, following Lawvere, the general abstract.

So there are questions of how best to characterise the space of existing mathematical research, e.g., is it just to be expected that some parts won’t connect? I presume that you’re not going to be listening out intently to find out whether someone has shaved some epsilon off the bounds of the Ramsey number, whereas, you might want to hear of someone in pp-adic Langlands couch their work in a way that makes you see what pp-adic cohesiveness must be. But, as Elkies and Tao suggest, maybe there are useful bridges to be build between more combinatorial or analytic areas and the grand theories. If so, will we need to find a common characterisation? What if the best way to characterise the structure of the ‘combinatorics’ part of mathematics is not category-theoretic?

And then there are questions of value. Beyond something being valued because it relates to your interests, do you think pieces of research have value in terms of the good of mathematics itself? Do you think anyone should care about lengths of arithmetic progressions in random subsets of the integers? How do you feel about

Theorem. For every δ>0\delta \gt 0 there exists NN such that every subset of 1,2,...,N{1, 2,...,N} of size at least δN\delta N (that is, density at least δ\delta) contains an arithmetic progression of length three?

What could make such a result more valuable? If it tied in with a “noisy additive cohomology”?

Posted by: David Corfield on April 23, 2012 10:34 AM | Permalink | Reply to this

Re: Mathematical Cultures

I presume that you’re not going to be listening out intently to find out whether someone has shaved some epsilon off the bounds of the Ramsey number,

I might well be – if I knew, or as soon as I know, that it is part of the answer to a question that I care about.

Lest the impression is falsely gotten between the lines of our discussion that I spend my life just doing general abstract math, maybe let me remind you (you all) that I have a physics degree, have a good bit of my publications in mathematical physics, and am quite used to spending days with chasing signs, and integrands and the like. Sometimes I think that because I was brought up in an academic world bare of any general abstract, I came to appreciate it more than others who take it for granted. I had to fight for it, nobody around me appreciated it when I was a student. And I had several revelations where mountains of concrete particular tedious work collapsed and disappeared because finally the right general abstract perspective was found.

When I learned string theory, like any other student, I was exposed to the anecdote of Jacobi’s abstruse identity which is maybe a good illustration for our discussion.

Jacobi in 1892 figured out that the following is true:

12x( n=1 (1+x n1/21x n) 8 n=1 (1x n1/21x n) 8)=8 n=1 (1+x n1x n) 8. \frac{1}{2 \sqrt{x}} \left( \prod_{n=1}^{\infty} \left( \frac{1 + x^{n-1/2}}{1-x^n} \right)^8 - \prod_{n=1}^\infty \left( \frac{1- x^{n-1/2}}{1-x^n} \right)^8 \right) = 8 \prod_{n=1}^\infty \left( \frac{1+x^n}{1-x^n} \right)^8 \,.

This is abstruse, isn’t it? About as abstruse as Ramsey theory, maybe. Jacobi proved it – god knows how and why – it didn’t mean anything to anybody, and he moved on to prove other things… until a century later suddenly it turns out that it is the truth of this identity that makes superstring theory tick. And suddenly it is clear what it is this weird equation is actually saying: it counts the number of bosonic and of fermionic states of the superstring, and asserts that they are equal. This is exceptional.

And in more than one sense. In fact there is a long list of similar (if maybe not quite as impressive) very non-general-abstract exceptional identities in all kinds of fields of math whose truth turns out to be the reason for the supersymmetry of some QFT in some dimension. Supersymmetry is a major source of exceptional structures in math. I am in principle interested in each and every one of these – number of hours in the day being the limiting factor.

Gowers defends his interest by observing that any breakthrough in Ramsey theory is likely to involve a new general principle of potential broad significance to combinatorics, but that this new principle will most probably not be expressible as general theory. If so, this suggests that not all general principles may be expressed as what you call, following Lawvere, the general abstract.

This paragraph I do not quite understand. I should probably read Gowers’ orginal. I find already the distinction between “a general principle of broad significance” and a “general theory” subtle, and much more so if both of these together with this distinction is now conjectured. This makes me think that I can’t really know what this discussion is about.

you might want to hear of someone in p-adic Langlands couch their work in a way that makes you see what pp-adic cohesiveness must be.

You know, I am not that interested in cohesiveness in itself. I will probably not spend a whole lot of energy in understanding whatever random model of cohesion somebody suggests. I am interested in cohesion as a means to an end. It so happens that it turns out to be a major machine that helps me whith what I am really interested in: understanding physics. And it helps me distinguish which parts of physics are general abstract, which are exceptional. For instance we learned from cohesive \infty-topos theory that the structure of Chern-Simons theory and its higher dimensional siblings is hard-wired at the very, very foundation of mathematical structure, much more fundamental even than the fundamental role it plays in physics really suggests. This changed my perspective on these action functionals. At the same time, we are busy tabulating the exceptional structures found here: while the structure of \infty-Chern-Simons theory is fundamental general abstract, every single one of them comes from some cocycle, and these oranize, as usual, as various infinite series with plenty of exceptional instances. All of them most interesting to study, no general nonsense explaining their properties.

maybe there are useful bridges to be build between more combinatorial or analytic areas and the grand theories.

Beyond something being valued because it relates to your interests, do you think pieces of research have value in terms of the good of mathematics itself?

I am not sure I see how anyone can value mathematics beyond it relating to his or her interest. Would I say: here is a completely uninteresting piece of mathematics, but I value it because it meets the following abstract criteria? These criteria will be imposed because they are interesting to somebody for some reason.

Mathematics is all about being intersted in something, isn’t it?

Do you think anyone should care about lengths of arithmetic progressions in random subsets of the integers?

I am glad that somebody cares about it, so that the moment I discover why I need it, there is somebody who already knows about it!

I am every second day discovering some remote piece of math that I never knew what it was good for and suddenly I see how it springs out of, say, cohesive topos theory. Then I am glad that somebody already worked it all out. Parts of my writing of the “cohesive document” was work along these lines: “Look, this fits here, and this fits here, and this fits here.”

Posted by: Urs Schreiber on April 24, 2012 12:22 AM | Permalink | Reply to this

Re: Mathematical Cultures

Does the story of Jacobi’s abstruse identity tell us anything more than that an apparently obscure result can become important? The lesson is that we need to be careful in our assessment of value because connections may be forged in the future to other pieces of theory. But it doesn’t tell us about what is a worthy thing to forge to (other than that being a central result for string theory is good) or how this forging takes place, and these are the topics of Gowers’ essay.

Let’s leave aside the contentious value topic altogether, since it’s the topic of Brendan’s second conference, and think about differences of method. What is the glue that holds what Gowers calls ‘combinatorics’ together? You say you should probably read Gowers’ original essay, which is certainly true, but if you only have a couple of minutes and want to know what interests me in it, here is an extract.

He writes,

In general, as one gains experience at solving problems in an area such as combinatorics, one finds that certain difficulties recur. It may not be possible to express these difficulties in the form of a precisely stated conjecture, so instead one often focuses on a particular problem which involves those difficulties. The problem then takes on an importance which goes beyond merely finding out whether the answer to it is yes or no. This explains why it was possible for so many of Erdos’s problems to have hidden depths.

Then

It is perhaps helpful to consider various different ways that one branch of mathematics can be of benet to another. Here is a list, in order of directness (but not exhaustive), of how area A can help area B.
(i) A theorem of A has an immediate and useful consequence in B.
(ii) A theorem of A has a consequence in B, but it takes some work to prove it.
(iii) A theorem of A resembles a question in B sufficiently closely to enable one to imitate or adapt the A-proof and answer the B-question.
(iv) In order to solve a problem in A, one is led to develop tools in B which are of independent interest.
(v) Area B contains definitions which resemble those of area A. (To give just one example, sometimes one wishes to define a notion of independence which behaves in some but not all ways like linear independence in vector spaces.) Area A then suggests fruitful ways of organizing and tackling the results and problems in B.
(vi) If one gains an expertise in area A, then one picks up certain habits of thought which enable one to make a significant contribution to area B.
(vii) 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.

So, I find that an interesting thought that whatever allows for that transfer of skill from A to B, especially towards the end of the list, is difficult to formulate explicitly. When I first read the essay, I felt myself doubt the possibility that there could be no general formulation explaining such connections, that it was a matter of time before a general theory emerged. But who is to say? Then what kinds of general formulations are possible? Are there non-category-theoretic frameworks for such general formulations?

To make any progress on these, we would have to look at concrete examples, perhaps taking Gowers’ own lead:

As for (vi) and (vii), I can speak from personal experience, having worked recently on several problems outside Banach space theory. Although I have not applied Banach-space results, my experience in that area has enabled me to think about problems in ways that would not otherwise have occurred to me. And I am far from alone in this, as many mathematicians who have worked on Banach spaces have worked successfully in other areas as well, such as harmonic analysis, partial differential equations, C*-algebras, probability and combinatorics.

Incidently, when calling for cross-cultural work towards the end of the essay, Gowers writes

collaboration of this kind would require greater efforts on the part of problem-solvers to learn a bit of theory, and greater sympathy on the part of theoreticians towards mathematicians who do not know what cohomology is.

Interesting how cohomology is taken to represent the other side.

Posted by: David Corfield on April 25, 2012 9:50 AM | Permalink | Reply to this

Re: Mathematical Cultures

I see that Gowers says explicitly that his list is not exhaustive, but all the same I’m quite surprised not to see anything along the lines of “the definitions of area A provide a useful organizing principle for definitions and theorems in area B”. That seems to be very often the role played by A=category theory.

I felt myself doubt the possibility that there could be no general formulation explaining such connections, that it was a matter of time before a general theory emerged. But who is to say?

An excellent point. I would guess that I probably speak for many when I say that one of the things that first attracted me to category theory was that it does provide a general formulation of such connections. If I think back (with an effort) to a time before I had heard the word “category”, I think it was not at all obvious to me that relationships of this form could ever be formalized precisely. That’s why it was such a revelation to me to realize that there was a formal definition which included both groups and group homomorphisms, rings and ring homomorphisms, modules and module homomorphisms, etc.

One might say that what mathematics does is to see connections between different things and make them precise in a general theory by rising to a higher level of abstraction, and so it’s only natural that this activity could act upon itself as well. But there’s even less reason to expect that all relationships outside mathematics can be formalized mathematically, so why should it be the case inside mathematics? Maybe this is a case of “unreasonable effectiveness” leading us to mistakenly expect universal effectiveness.

Posted by: Mike Shulman on April 25, 2012 3:37 PM | Permalink | Reply to this

Re: Mathematical Cultures

Not, of course, that it should put people off trying for an explicit general theory.

What’s so fascinating about reading Albert Lautman’s work from the 1930s is that he doesn’t know that there’s a general framework just a few decades ahead which will make sense of what he knows are similarities between theories, e.g., covering spaces and field extensions. So he takes the common idea being expressed to be ‘above’ mathematics.

Posted by: David Corfield on April 25, 2012 4:46 PM | Permalink | Reply to this

Re: Mathematical Cultures

Does the story of Jacobi’s abstruse identity tell us anything more than that an apparently obscure result can become important?

Yes, it does, I think. Whether it’s “important” that the superstring is consistent is something easily debateable and debated. The point here is rather: the abstruse identity turns out to be a concrete particular of a (more) general abstract: namely of supersymmetry representation theory.

The lesson is that we need to be careful in our assessment of value because connections may be forged in the future to other pieces of theory.

I thought of a different lesson. Getting back to the point I raised at the beginning: I am still wondering if the “two cultures” are not simply the two fields of study that every mathematician must necessarily be both interested in, that of the general abstract and of its concrete models.

Studying representation theory of super Lie algebras is something very much towards the general abstract end of the pattern. But every single instance it considers may be very peculiar. So what at Jacobi’s times may have seemed to be a quasi-random result is now understood to be of interest as a model for a nice structured theory. So if I am interested in the general abstract of susy representation theory, I will necessarily also be interest in various abstruse identities that may seem to be from a “different culture”. But they are not.

Posted by: Urs Schreiber on April 27, 2012 12:08 PM | Permalink | Reply to this

Re: Mathematical Cultures

I am still wondering if the “two cultures” are not simply the two fields of study that every mathematician must necessarily be both interested in, that of the general abstract and of its concrete models.

But what if there are areas of maths where the general abstract component isn’t as neat and expressible as in our area? Gowers isn’t suggesting that the algebraic geometer looks just at the general abstract and the combinatorist just the concrete pseudorandom stuff. His thought is that in his branch of combinatorics what is common between areas is not easily expressed.

(vii) 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.

Perhaps we’ve got so used to the nut soaking approach that we don’t consider this possibility.

I believe the reason the Tricki was set up was to codify some of the commonality. Maybe that’s a good way to think of the culture divide, one leads to nLab, the other to Tricki. Flicking through some principles there, we find:

But maybe there are even less precisely expressible ideas, like maybe it’s hard for you to convey how learning one language helps you learn another, or learning to play one sport helps you with another.

Posted by: David Corfield on April 27, 2012 2:35 PM | Permalink | Reply to this

Re: Mathematical Cultures

@ Urs: The lesson is that we need to be careful in our assessment of value because connections may be forged in the future to other pieces of theory.

Jim:Amen! viz my theses originally dismissed as entirely self-contained and of no interest even within math


@Urs:I thought of a different lesson. Getting back to the point I raised at the beginning: I am still wondering if the “two cultures” are not simply the two fields of study that every mathematician must necessarily be both interested in, that of the general abstract and of its concrete models.

Jim:must necessarily?? may should be but certainly not in reality

Posted by: jim stasheff on April 27, 2012 3:02 PM | Permalink | Reply to this

Re: Mathematical Cultures

To be honest, I can’t remember the last time I did anything with a concrete particular, except to look for possible counterexamples to a general abstract statement I was considering. Peter May always used to tell me when I was a graduate student “someday we’ll get you to calculate something”; occasionally I’ve thought I was on the verge of doing so, but it never panned out.

Is it enough if a mathematician interested in the abstract general has an appreciation for the concrete particular and the concerns of the people who study it? Civilization is all about division of labor…

Posted by: Mike Shulman on April 27, 2012 6:11 PM | Permalink | Reply to this

Re: Mathematical Cultures

On the other hand, I do think there are also less-easily-codified principles and techniques even in “our area” of math — and possibly the prevalence of techniques which can be codified sometimes leads us to discount or ignore those that can’t, to our detriment.

For instance, what is “homotopy theory”? Nowadays the fashion is to do everything with a model category. But several people have said to me recently, and I agree with this, that homotopy theory is a collection of tools and ideas that goes beyond model categories, and too much focus on model categories is blinding. Not all useful notions of “fibration” and “cofibration” fit into a model-categorical framework, or even a category-of-fibrant-objects framework. Quasifibrations are an important tool in classical homotopy theory. I recently had occasion to use another kind of fibration that wasn’t part of a model category.

I don’t mean to accuse anyone in particular of too much focus on model categories; I’m just saying that when you’re the sort of person who likes abstract generals (like me), it’s easy to get seduced by a particular abstract general and ignore the commonalities that it doesn’t capture.

Of course, one of the goals of a category theorist is to find new abstract generals which do codify commonalities that haven’t been precisely codified before. Perhaps the wide applicability of general abstract ideas in “our area” is at least partly due to the fact that “we” are the sort of people who do this sort of thing, and so we’ve spent lots of time applying it to our own subject.

Posted by: Mike Shulman on April 27, 2012 6:42 PM | Permalink | Reply to this

Re: Mathematical Cultures

Mike S wrote:
I don’t mean to accuse anyone in particular of too much focus on model categories; I’m just saying that when you’re the sort of person who likes abstract generals (like me), it’s easy to get seduced by a particular abstract general and ignore the commonalities that it doesn’t capture.

Of course, one of the goals of a category theorist is to find new abstract generals which do codify commonalities that haven’t been precisely codified before. Perhaps the wide applicability of general abstract ideas in “our area” is at least partly due to the fact that “we” are the sort of people who do this sort of thing, and so we’ve spent lots of time applying it to our own subject.

***
ah, that explains why I’ve been struggling - I’m a homotopy theorist who uses cat ONLY if they help me pursue what I’m after. Can’t think when I’ve used a model cat structure explicitly. The `we’ above obviousl does not include me.

Posted by: jim stasheff on April 28, 2012 6:11 PM | Permalink | Reply to this

Re: Mathematical Cultures

The ‘we’ above obviously does not include me.

I hope that this didn’t make you feel offended or excluded. (It’s always difficult to judge feelings on the Internet.) The intent was quite the opposite — it’s easy for ‘us’ to get unreasonably attached to our abstract generals, and so it’s important for ‘us’ to have people like you around. And the purpose of putting ‘we’ in scare quotes was precisely to avoid implying that it ought to include everyone in the discussion. Personally, I’m extremely glad that you frequent this blog.

Posted by: Mike Shulman on April 30, 2012 3:40 AM | Permalink | Reply to this

Re: Mathematical Cultures

Not offended at all. Happy to know that you welcome a traditionalist who speaks cat `in sich dicht’ so non-fluently.

Posted by: jim stasheff on April 30, 2012 2:21 PM | Permalink | Reply to this

Re: Mathematical Cultures

To be honest, I can’t remember the last time I did anything with a concrete particular, […] Peter May always used to tell me when I was a graduate student “someday we’ll get you to calculate something”

You computed π 1(S 1)\pi_1(S^1).

Posted by: Urs Schreiber on April 27, 2012 7:27 PM | Permalink | Reply to this

Re: Mathematical Cultures

Flicking through some principles there, we find:

  • If a parameter is generating undesirable boundary terms, try averaging over many choices of that parameter

  • Use conservation laws and monotonicity formulae to obtain long-time control on solutions

Or

  • If you need a factorization, check if the small object arugument applies.

  • If you want to know that it’s an isomorphism, try to see if it is one under Yoneda embedding.

Posted by: Urs Schreiber on April 27, 2012 7:42 PM | Permalink | Reply to this

Re: Mathematical Cultures

You computed π 1(S 1)\pi_1(S^1).

No, I mimicked someone else’s computation of π 1(S 1)\pi_1(S^1) inside a different foundational system. But let’s not argue semantics. (-:

If you need a factorization, check if the small object argument applies.

If you want to know that it’s an isomorphism, try to see if it is one under Yoneda embedding.

My first thought was things like that too, but those are actually just of the form

If you need X, check whether you can apply theorem Y whose conclusion is X.

Not particularly deep; it just involves knowing what theorems exist in the subject. Whereas I get the impression that the principles David quoted are not expressed by a single theorem.

Posted by: Mike Shulman on April 27, 2012 8:39 PM | Permalink | Reply to this

Re: Mathematical Cultures

My first thought was things like that too, but those are actually just of the form

If you need X, check whether you can apply theorem Y whose conclusion is X.

Every trick is a theorem once you specify all the necessary conditions that make it work. Conversely, the small object argument certainly has the feel of a trick. That’s why it’s not called the “small object theorem”, I guess!

It bugs me that much of this discussion seems to rest on a bad cliché of what work in fields is like that are counted to the “second culture”. As if the fact that a field has a nice conceptual structure doesn’t mean that the majority of the published proofs are still fiddly technical proofs involving lots of tricks and clever ideas. (I just spent a whole semester seminar on a huge fiddly technical proof involving lots of tricks and clever ideas that, nevertheless – once done – establishes a nice fact in higher category theory.)

Maybe I am reading something into it that isn’t there. Above we already saw David get upset about me getting upset about somebody suggesting that if its trivial, then it’s part of the “second culture”.

But I think I’ll leave it at that now. I have to look after what turns out to be a much more fiddly and more technical proof than I originally envisioned ;-)

Posted by: Urs Schreiber on April 27, 2012 10:01 PM | Permalink | Reply to this

Re: Mathematical Cultures

There’s another difficult to formulate skill. Reading people’s intentions. Why would you suppose I was upset ? I was rather sympathising with your thought, it having occurred to me as well that that insinuation was there.

Posted by: David Corfield on April 27, 2012 11:10 PM | Permalink | Reply to this

Re: Mathematical Cultures

I think I glimpsed what might be at stake in all this. Let’s accept for the moment that at ends of a spectrum there are areas of maths which approximate Gowers’ description and then areas where elaborate theory underscores commonality, think higher algebra-higher geometry. Now people being what they are, there is a great sensitivity to criticism and a readiness to defend. So what form can criticism take?

Combinatorics: “This Hungarian maths is just a bunch of unconnected problems. Each requires ingenuity to solve, but there’s no genuine subject here.” Reply: “Actually there are commonalities, but they’re hard to see. People who know how to work in one area will be able to work well in another, even if its hard to say how.”

Theory-intense: “There’s a lot of theory for its own sake, and often when you ‘succeed’ it’s mere to show that a part of maths is really trivial.” Reply: “The theory is not for its own sake (Lurie), anyway showing the trivial to be trivially trivial is important work (Freyd).”

Look at the reaction to Urs’ comment

One reason why I am after having an adjunction is because this makes Galois theory become almost a tautology.

Mike replies

I think the word “tautologous” is kind of misleading and belittling here.

Then they agree that showing such a thing involves plenty of work.

Posted by: David Corfield on April 28, 2012 4:46 PM | Permalink | Reply to this

Re: Mathematical Cultures

Every trick is a theorem once you specify all the necessary conditions that make it work.

That depends what you mean by a “trick”. A lot of what Gowers is talking about are general strategies like the probabilistic method. I.e., to construct an object XX with some extremal or near-extremal property, try building XX randomly. This is rather too vague to make into a theorem in a useful way.

Posted by: Mark Meckes on April 27, 2012 11:13 PM | Permalink | Reply to this

Re: Mathematical Cultures

David asks

Why would you suppose I was upset ?

Okay, so I mis-interpreted your

Ah yes, the hidden insinuation.

I thought this was short for “It is telling that you would see a hidden insinuaton there, where there is none.”

Good to know that this isn’t what you meant.

Mike replies (and Mark similarly):

Every trick is a theorem once you specify all the necessary conditions that make it work.

I think the point being made was exactly that that isn’t necessarily the case. It isn’t necessarily always possible to specify precise conditions under which any given “trick” works.

Hm. if there is no way to specify at least one set of conditions under which a trick works, then it is a trick that does not work. :-)

Conversely, whenever you apply any reasoning to anything, you can always go back and check which axioms you used in the reasoning. Throw away the axioms that you didn’t use, and you have formulated a theorem a bit more general than your original trick.

Of course at this point we would really need concrete examples to make progress.

Posted by: Urs Schreiber on April 29, 2012 10:16 PM | Permalink | Reply to this

Re: Mathematical Cultures

if there is no way to specify at least one set of conditions under which a trick works, then it is a trick that does not work.

Yes, of course. I meant that there is no precise most general set of conditions under which a trick works. E.g. the “same trick” works in situations A, B, and C, but the minimal assumptions in the three cases have formally very little in common.

Posted by: Mike Shulman on April 30, 2012 3:46 AM | Permalink | Reply to this

Re: Mathematical Cultures

Every trick is a theorem once you specify all the necessary conditions that make it work.

I think the point being made was exactly that that isn’t necessarily the case. It isn’t necessarily always possible to specify precise conditions under which any given “trick” works. Certainly there’s no a priori reason to expect that to be true.

Posted by: Mike Shulman on April 28, 2012 2:48 AM | Permalink | Reply to this

Re: Mathematical Cultures

I think you miswrote: the sphere spectrum is not the initial object in the category of spectra, though it is the unit object for the monoidal structure. But that doesn’t really change the point you were making.

Posted by: Mike Shulman on April 22, 2012 2:36 PM | Permalink | Reply to this

Re: Mathematical Cultures

Thanks, yes. My fingers must have been thinking of ring spectra when they typed this.

Posted by: Urs Schreiber on April 22, 2012 5:53 PM | Permalink | Reply to this

Re: Mathematical Cultures

Please note I got the call for papers deadline wrong. It’s 1 May 2012.

Posted by: David Corfield on April 23, 2012 10:50 AM | Permalink | Reply to this

Re: Mathematical Cultures

The idea of mathematical cultures is evidently all the rage.

Posted by: David Corfield on April 27, 2012 9:28 AM | Permalink | Reply to this

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