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February 21, 2007
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Re: Noncommutative Geometry Blog
casting me as ‘Maxim’s blogvoice’ will surely shut me up.
The last thing I’d want to do, sorry. There’s something I find attractive though in the idea of researchers having a group identity. Might it be possible that an excessive individualism stands in the way of progress? Of course, this is not to ignore the opposite problem of groups becoming rigidly insular and self-satisfied.
Re: Noncommutative Geometry Blog
David wrote:
There’s something I find attractive though in the idea of researchers having a group identity.
Perhaps mathematicians, more than philosophers, are suspicious of ‘schools of thought’. We may imagine ourselves more as individuals in search of truth, and bound to converge on the truth. If truth is one, who needs a multiplicity of ‘schools’?
I’m actually not sure what most mathematicians think about this, since they don’t talk about it much.
But, I can easily see why Lieven would bristle at being cast as the ‘blog voice of Kontsevich’s nongeometry’. Any self-respecting mathematician dreams of developing their own new insights, not being the spokesman for someone else, or some ‘school’.
I remember a vaguely related feeling of discomfort when you set up a distinction between the Khovanov school of categorification and some other school of which I’m supposedly an exponent. I can’t find the locus classicus, but here you mention ‘rival conceptions of categorification’. That made me feel queasy when I first read it, and it still does. At some point I remember pointing out that my student Aaron Lauda is working with Khovanov. The term ‘rival’ suggests there will be winners and losers. I expect that everything will fit into a big beautiful picture, with no real losers.
In philosophy there are certainly rival schools of thought, and agonistic combat between them. Maybe mathematicians try to avoid that. They may be missing out on something… but they may be getting something more important in return!
It’s an interesting issue.
(What about ‘Bourbaki’?)
Re: Noncommutative Geometry Blog
The locus classicus was the second post of the same name. I’ll grant you, as I did there, that using one person’s name to identify a current of thought is not a good idea. And yet I didn’t learn and repeated the mistake here, when there was a perfectly good alternative in the term ‘nongeometry’.
But the interesting question, as you point out, is whether there is anything to be gained or lost by mathematicians forming themselves into groups or schools, perhaps even if only for limited periods.
It must be a question of degree as there has to be a certain amount of group identity in place already, if only mediated through PhD supervisors. Its unrecognised presence can be dangerous.
One argument against a strong individualism is that it wants to assess people according to a fixed set of externally applied criteria. On the other hand, if a group has a certain amount of resources, it can spend them on different members according to their strengths. Where it might be hard for someone outside to see that a young researcher has achieved much, perhaps few journal publications, the leaders of the group may perceive a profound talent.
In a discipline such as philosophy where approaches to the same subject matter are very diverse, this notion of the possibility of equitable external assessment is very stultifying. The chance of fresh ideas from outside the dominant modes appearing in journals is limited by the small probability that an editor will happen to send a submission to sympathetic referees. It relies on the outmoded ‘encyclopaedic’ conception of enquiry, that good work can be seen to be good by any competent person.
Re: Noncommutative Geometry Blog
Imagine you have 300 researchers divided into 2 schools. School A has 100 members and school B has 200 members. Say each researcher submits one paper a year, and will be randomly assigned two referees. Also suppose that the paper will be accepted if and only if the referees belong to the same school as the author. Then school A can expect to have 11 papers published, while school B will have 89 papers published.
Now divide the researchers into schools of 50 members and 250 members with the same conditions applying. Now school A expects less than 1.5 papers and school B over 173 papers.
Could this phenomenon be behind Réné Thom’s comment?
Je voudrais dire mon étonnement que la collectivité scientifique ne possède pas en son sein des critiques, à la manière des critiques littéraires ou artistiques. Sans doute les barrières à la publication (la “peer review”) jouent-elles – partiellement – ce rôle. Mais dès qu’un groupe – - un “paradigme Kuhnien” - a conquis sociologiquement une position dominante, ces barrières tendent à perdre leur pouvoir de discrimination; bénignes pour les tenants du paradigmes, féroces pour les étrangers. Je suis convaincu qu’il y aurait, dans l’appréciation globale de la production scientifique, place pour un type d’esprit indépendant, non exempt de préjugés, mais soucieux en tout premier lieu de rigueur intellectuelle. Si la philosophie n’avait pas divorcé d’avec la science depuis longtemps, on aurait dû trouver chez les philosophes des sciences, les épistemologues, des individualités capables de tenir ce rôles. (‘Vertus et Dangers de l’Interdisciplinarité’: 636-643).
Re: Noncommutative Geometry Blog
Perhaps this is to labour the point, but you see how in this artificial example recognition of schools would help. In the first case, access to journals could be split 1:2, and the school A referees could choose the best papers of their school, whereas under the original conditions, they’ll accept all 11 they see, however bad.
Re: Noncommutative Geometry Blog
Very interesting discussion,
I’ve been fascinated for a while by the question of why mathematics hasn’t ended up suffering badly from the problems afflicting theoretical physics in recent years. Without some sort of external source of discipline (traditionally experiment in the case of theoretical physics), how does a field avoid drifting off into the study of more and more esoteric and unfruitful problems? One sees this happening to various extents in some mathematical fields, but at the top levels of the math research community there somehow are mechanisms in place which limit this phenomenon.
As in any field, there is some degree of “groupthink” among mathematicians. People get trained in one technique, one way of looking at mathematics, and have trouble understanding and appreciating others. But there remains a certain level of awareness that other points of view are valid and important, and an openness to adopting ideas from them if there is good evidence that they will help solve problems that one’s own techniques can’t.
I do like the Rene Thom quote…
Re: Noncommutative Geometry Blog
Peter Woit asked in part:
I’ve been fascinated for a while by the question of why mathematics hasn’t ended up suffering badly from the problems afflicting theoretical physics in recent years. Without some sort of external source of discipline (traditionally experiment in the case of theoretical physics), how does a field avoid drifting off into the study of more and more esoteric and unfruitful problems?
Constructivist mathematicians have long criticised modern mathematics for being meaningless or decadent; few would go so far today as Kronecker did, but compare Errett Bishop’s Schizophrenia in contemporary mathematics. (As a mathematical pluralist, I would merely claim that constructivist mathematics is more meaningful than classical mathematics, as it uses fewer assumptions and so applies in more situations. Mathematics making classically false assumptions, as Brouwer did, may be more or less meaningful, depending on context.)
Re: Noncommutative Geometry Blog
Godel proved (I think) that any mathematical system (that is reasonably interesting) has propositions that can neither be proved nor disproved, and hence either the affirmative or the negative can be added as axioms. So the potential for meaningless elaboration is certainly there. Hence I claim that Mathematics must either (A) be about understanding the real world [Applied Mathematics]; or (B) be about understanding one or more parts of Mathematics, which must on all branches (no loops) lead back in a finite number of steps to (A).
Posted by:
Robert Smart on February 27, 2007 4:40 AM | Permalink
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Re: Noncommutative Geometry Blog
Robert Smart wrote at last:
I claim that Mathematics must either (A) be about understanding the real world [Applied Mathematics]; or (B) be about understanding one or more parts of Mathematics, which must on all branches (no loops) lead back in a finite number of steps to (A).
I don’t want to be dogmatic about what Mathematics ‘must’ be about, but I sympathise with this. When possible, I usually go ahead and do whatever Mathematics seems natural and interesting; if it ends up being useless, well, at least it’s still Art. But when I no longer know which ideas are right and which are wrong, then I go back and ask how the Mathematics is supposed to be applied. Even for Pure Mathematicians (several levels removed along (B), at best), this can be explained: When lost, I remind myself what are the motivating examples. Trace motivation back far enough, and you will always get to the real world, if for no other reason than that we mathematicians are real beings.
von Neumann quote
From the intro in Feb’07 “Computers and Mathematics with Applications”:
… physics has always been and will continue to be the well-spring of mathematics. As von Neumann aptly observes “As a mathematical discipline travels far from its empirical source, or still more, if it is a second or third generation only indirectly inspired by ideas coming from ‘reality’, it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely l’art pour l’art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical source, or after much ‘abstract’ inbreeding, a mathematical subject is in danger of degeneration”.
Posted by:
Robert Smart on May 11, 2007 2:44 AM | Permalink
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Re: von Neumann quote
But how many “men with an exceptionally well-developed taste” are there? How can we recognise them (taking “men” to mean “people”)? Does von Neumann include himself?
Re: Noncommutative Geometry Blog
N. Blogbaki !?
Posted by:
Florifulgurator on February 23, 2007 1:27 PM | Permalink
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Re: Noncommutative Geometry Blog
Re Mathematicians are wary of “schools”: Very true. The man himself said it well: mathematicians are fermions, physicists are bosons. It’s one reason I left physics for mathematics: in mathematics, if a town gets too crowded, you can pick up your rucksack and go see what you can find in the wilderness. There’s always more room. In physics, (at least the high energy theory world I know most), ignoring the fashions is likely career suicide.
Certainly the physics model is self-perpetuating, but how did it get started? My own theory (probably not original) is that, in the old days (pre 1974), physicists were, rightly, suspicious of fancy calculatin’. There is a real world we urgently want to understand, and spending time on unrealistic toy models, or going back over a known calculation to understand it better, was a manifest waste of time. Not unlike mathematics before the nineteenth century: there is one true geometry, one true theory of numbers, and we just need to attack it more vigorously to yield truths.
Beginning in the nineteenth century, mathematicians started to appreciate the value of the crazy made up ideas, not just for their intrinsic merit, but for the light they shed on classical questions. For example, complex analysis made mysterious aspects of real analysis clear, and it addressed problems in number theory that were inaccessible to just thinking harder within number theory. Old questions of Euclidean geometry made more sense in the context of Riemannian and non-continuum geometries. We’ve had a century or two to get used to this way of working now.
Up until recently, physicists have been able to stick to the old model. If they had been interested in, say, the 4-dimensional Poincaré conjecture for some reason, I suspect they would have slogged away at in 4 dimensions until they got somewhere. Three dimensions might have been considered, but thinking about 5 and higher would have just been crazy talk…
Now physicists have been forced to think about, say, quantizing d-dimensional objects in n-dimensional spaces: the post 19th century mathematics model, where you look for some insight by looking at a large space of theories instead of just the one you’re interested in. But their sociology hasn’t caught up: it’s still geared to old days, when a new experimental result would appear, and everyone would rush in to the be the first to calculate the new number. So they’re not spreading out to cover the territory the way mathematicians would, nor are they trying to consolidate and reconceptualize the foundations, as mathematicians have learned to do when they head out onto uncertain ground.
That’s my theory, but now that I write it down, I don’t find it very convincing. Does anyone here have other ideas, about how this particular math-physics cultural difference came to be?
Posted by:
yagwara on February 24, 2007 11:08 PM | Permalink
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Re: Noncommutative Geometry Blog
It is easy to recognise the “collective cry of agony on the part of physicists trying and - so far - failing to find a theory that goes beyond the Standard Model and general relativity”, as John puts it. But what is the mood in the rest of physics like? Are people working in condensed matter physics suffering this angst?
As to mathematics, it’s interesting to compare Hermann Weyl
Mathematics is not the rigid and petrifying schema, as the layman so much likes to view it; with it, we rather stand precisely at the point of intersection of restraint and freedom that makes up the essence of man itself.
(p. 136, ‘The Current Epistemogical Situation in Mathematics’ in Paolo Mancosu (ed.) From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s, Oxford University Press, 1998, pp. 123-142).
with von Neumann’s concern, expressed in “The Mathematician’, that mathematics may degenerate by following the path of least resistance, if it doesn’t continually return to empirical sources. People often forget the proviso he then adds, ‘unless it is developed by men of exquisite taste’.
Perhaps both Weyl and von Neumann are partially right. I said somewhere in my book that mathematics has an extra degree of freedom compared to physics. Perhaps it swings between being underconstrained and being just right, while physics swings between being overconstrained and just right.
Then a good idea for physicists when feeling overconstrained and mathematicians when feeling underconstrained is to turn to each other. But as Yagwara suggests you’d have to have the social resources to do this in the best way.
Re: Noncommutative Geometry Blog
I didn’t quite have von Neumann right:
I think that it is a relatively good approximation to truth - which is much too complicated to allow anything but approximations-that mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure. But, once they are so conceived, the subject begins to live a peculiar life of its own and is better compared to a creative one, governed by almost entirely aesthetical motivations, than to anything else and, in particular, to an empirical science. There is, however, a further point which, I believe, needs stressing. As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from “reality” it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely l’art pour l’art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical source, or after much “abstract” inbreeding, a mathematical subject is in danger of degeneration. At the inception the style is usually classical; when it shows signs of becoming baroque, then the danger signal is up. It would be easy to give examples, to trace specific evolutions into the baroque and the very high baroque, but this, again, would be too technical.
So it was ‘exceptionally well-developed taste’.
Re: Noncommutative Geometry Blog
I like your characterization of mathematics and physics as prone to under- and over- constraint respectively. I know in that in mathematics, when you’re doing more invention than discovery, then you’re on the wrong track.
Maybe this is one way of looking at the difficulties facing fundamental physics: suddenly the biggest danger is underconstraint, while the physics culture is still geared to work in conditions of overconstraint.
And yeah, as you pointed out, we’re just talking about fundamental physics here. As far as I am aware, condensed matter physicists, atomic physicists, classical mechanics, and general relativists are working away on tough problems quite happily, without hand wringing or rending of garments.
Posted by:
yagwara on February 25, 2007 6:25 PM | Permalink
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Re: Noncommutative Geometry Blog
Now that you mention it, the notion of constraint serves to explain many of the differences in style between mathematics and physics. Not having sensible objects to refer to, mathematicians have learned (the hard way) to make careful definitions so we can all agree what we are talking about. Without experiment to keep us honest, we’ve learned to reason closely.
I would argue that abstraction itself is a response to underconstraint, and not only in an organizing and simplifying role. To take a notorious example, Grothendieck, contrary to common perception, had no interest in generalizing for its own sake. His apparently inexhaustible appetite for abstraction has always been driven by a desire — as is clear from his writings — to really understand the underlying mechanism of a proof or construction, the real thing underneath clumsy appearances.
But, does any of this explain mathematicians’ and physicists’ differing affinities for clustering? Not that mathematicians don’t form groups, but their groups tend to be loosely knit associations of autonomous individuals, wed by some common purpose. And not that physicists are the borg, but they (not just the high-energy variety) really do form much tighter groups that talk more and move together more.
I will defend the mathematicians’ way of working in one on one combat with all challengers — but in leaving physics I do admit I miss the scientific gossip.
Posted by:
yagwara on February 25, 2007 7:00 PM | Permalink
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Re: Noncommutative Geometry Blog
Of course the tendency of physicists to be bosons comes from experimental results. What is not so much recognized is that the interpretation of experiments depends on theory, and is largely a matter of taste. The overall effect is that physics is excessively self pruning.
Re: Noncommutative Geometry Blog
Thanks for the notes on this.
Posted by:
Martisa Vignali on February 15, 2012 11:36 PM | Permalink
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Re: Noncommutative Geometry Blog
A Question
Let X be a locally compact hausdorff space. is there a compactification Y of X such that Y is homotopic equivalent to X.
motivation: the closed unit bass as compactifivation of open unit ball
The noncommutative analogy:
Let A be a non unital C* algebra. Is there a unitization of A with the same homotopy type as A
Re: Noncommutative Geometry Blog
David,
thanks for the link to this ‘noncommutative geometry blog’ and yes it seems to be aimed at Connes-verts…
about interaction : on numerous occasions i’ve asked people to contribute to a group-blog on NOG/NAG and even keep a domain name noncommutative.org handy to host it. I’m willing to drop my own blog in favour for such a group-effort. so far, nobody showed any interest…
finally, i hope to follow my own approach to noncommutative geometry. casting me as ‘Maxim’s blogvoice’ will surely shut me up.