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October 6, 2006

So Irigaray was Right?

Posted by David Corfield

A very long time ago, while teaching in a Cultural Studies department, I taught on an introductory course on postmodernism. One of the thinkers we treated was Luce Irigaray:

Irigaray examines the systematic suppression of feminine and maternal concerns from the history of Western philosophy in Ce sexe qui n’en est pas un (This sex which is not one) (1977), arguing that valorization of the masculine is destructive to the fluid multiplicity of feminine sexuality.

Now, Irigaray was one of the targets of Sokal and Bricmont:

Alan Sokal and Jean Bricmont’s Fashionable Nonsense criticizes Irigaray, as a general example of what they believe is the anti-scientific tendency of “postmodernism”. They cite her analyses of E=mc 2E=mc^{2} as a “sexed equation” (because it privileges the speed of light) and her argument that fluid mechanics has been neglected by “masculine” science that prefers to deal with “masculine” rigid objects rather than “feminine” fluids. (Wikipedia)

No surprise for Irigarayans then but that a woman, Penny Smith, should sort out the Navier-Stokes equations. All they need show now is the irrelevance of the fact that, as with any mathematician, Smith’s work emerges from a vast communal effort. E.g., according to Christina Sormani, Smith told her:

It was Marsden’s book [Applications of Global Analysis in Mathematical Physics] which talked about the geometric theory of both Einstein Cauchy and NS Cauchy that got me into both problems.

And then there’s the small worry that relativity and hydrodynamics may be quite closely related.

Without wishing to suppress the fluidity, what opportunities are there for categorified Navier-Stokes?

Posted at October 6, 2006 10:03 AM UTC

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Re: So Irigaray was Right?

I really enjoyed Penny Smith’s comments on sci.physics.research when I was a moderator there. So, I’m rooting for her success in proving global existence for the Navier-Stokes equations.

However, my informants say we should reserve our applause for a while. None of the Navier-Stokes experts have come forward to say they’re verified her argument yet.

So, it’s way premature to start pondering the inherently feminine nature of fluid dynamics, as opposed to the masculinity of “rigid bodies”.

More interesting to me is this:

There’s just one really interesting equation of physics that’s been proved to have solutions that last for all time without blowing up: the multiparticle Schrödinger equation for charged particles. The Navier-Stokes equation would make two!

Nobody has even proved that Newtonian point particles interacting via Newtonian gravity have globally well-defined time evolution for almost all initial conditions! It’s pathetic. The situation with Maxwell’s equations coupled to classical charged point particles is even worse, and interacting quantum fields are worse still, except in low-dimensional toy models. With general relativity we know there are singularities.

Of course, there are lots of idealized systems - like the harmonic oscillator, vacuum Maxwell equations, and free quantum fields - that are known to have solutions lasting for all time. But, these are all “noninteracting”.

Someday I want to write a paper about this scandal. It must say something about physics.

Posted by: John Baez on October 6, 2006 5:46 PM | Permalink | Reply to this

Re: So Irigaray was Right?

“There’s just one really interesting equation of physics that’s been proved to have solutions that last for all time without blowing up: the multiparticle Schrödinger equation for charged particles. The Navier-Stokes equation would make two!”

But this kind of results do exist for 2D Navier-Stokes, an equation with pattern forming behavior that is at least as interesting as in 3D, and it has pattern forming behaviour opposite to the 3D version (kolmogorov v. kraichnan cascades). It is fascinating to see that the behavior of the system changes completely going from 2D to 3D (like e.g. Poincare Bendixson th.). I am curious if you would classify this result as interesting for the topic of existence of solutions of interacting systems? Or is the impossibility to form knots in 2D a reason why these systems are noninteresting for this topic?? Intuition suggests me that the freedom or lack of freedom of objects to do `crazy stuff’ is important. Isn’t it true that in graph colouring and tilings, homology and obstruction theory are used (by Conway?) to make this rigorous?

Posted by: lauret on October 11, 2006 7:56 PM | Permalink | Reply to this

Re: So Irigaray was Right?

Before trying something wacky like categorifying the Navier-Stokes equations, it’d be good to categorify some famous exactly solvable differential equations. Usually the presence of an exact solution reveals the presence of some rock-hard algebra lurking under the mantle of smoothness: the iron fist under the velvet glove. Algebra is what we can categorify best, so far.

Jeff Morton learned a lot by categorifying the harmonic oscillator, introducing “sets with complex cardinality” to extend my work with Jim. What’s the next good one to try? Maybe a q-deformed version? - or maybe the Korteweg -de Vries equation, which has the algebra of loop groups lurking beneath it. It may be a bit shallow compared to full-fledged fluid flow… but it displays a nice mix of masculine and feminine qualities.

Posted by: John Baez on October 6, 2006 11:34 PM | Permalink | Reply to this

Re: So Irigaray was Right?

Anyone feeling in a partial differential equation kind of mood should take a look at Sergiu Klainerman’s PDE as a Unified Subject, an end of the millennium paper with something for everyone (mathematician, physicist, or philosopher).

Posted by: David Corfield on October 7, 2006 12:04 AM | Permalink | Reply to this

Re: So Irigaray was Right?

In your lecture you’re treating the ascent up the ladder of p-brane dynamics. Can this analogy be completed:

classical particle dynamics:classical string dynamics::fluid dynamics:*** ?

Something like stringy fluid dynamics, perhaps for an incompressible polymer fluid.

Posted by: David Corfield on October 7, 2006 8:40 AM | Permalink | Reply to this

Re: So Irigaray was Right?

Can this analogy be completed:

classical particle dynamics:classical string dynamics::fluid dynamics:*** ?

Something like stringy fluid dynamics, perhaps for an incompressible polymer fluid.

Hm, interesting question. I have never thought about what the 2-particle analog of continuum dynamics could be.

Part of the reason is that from the physics point of view one tends not to be interested too much in the classical 2-particle - and much less in its continuum theory.

The only exception is the one you mention: polymer dynamics.

But, for instance, the back-of-the-envelope-order-of-magnitude computation of entropy of black holes from strings is done (using the string/black hole correspondence principle) by regarding a classical string - much like an ideal polymer - randomly stretched out and of such a length, that the mean diameter of its collection of points is precisely the Schwarzschild radius for the given mass of a string of that length.

The entropy of the string - regarded as the entropy of a random walk - can then be regarded as the entropy of the corresponding black hole.

So that’s one use of classical or semi-classical strings. Another major use has been as a first order check in AdS/CFT computations.

But in both cases the physics does not make one want to study a continuum dynamics of such classical strings.

On the other hand, there is another analogy which deserves to be completed, namely

1-particle dynamics : ordinary field theory :: 2-particle dynamics :: ???

This has received a lot of attention. The “???” here is called string field theory. It is so far only understood to some degree at the classical level.

I bet there is a nice systematic way to complete this second analogy in full detail. Essentially, it is all about categorifying the concept of second quantization.

On the other hand, whether the first analogy you ask about has an “interesting” completion, I find hard to say.

Posted by: urs on October 7, 2006 4:02 PM | Permalink | Reply to this

Re: So Irigaray was Right?

Perhaps when we have time on our hands in the depths of winter you can explain to us what Roman Jackiw is trying to achieve.

Posted by: David Corfield on October 8, 2006 12:18 PM | Permalink | Reply to this

fluid dynamics

what Roman Jackiw is trying to achieve.

I wasn’t aware of that text before.

Now I have browsed through it a little.

It seems that the main point is that the standard action (the Nambu-Goto action) for the relativistic (p+1)(p+1)-dimensional “pp-brane” (in either of its incarnations, plain vanilla, or supersymmetric, nonabelian, noncommutative) can, for certain low dimensions of target space, be seen to lead to dynamics equivalent to that of certain fluid models - in particular the Chaplygin gas (in the non-relativistic limit).

This is not all that surprising - not after the authors have pointed out that this gas is known to be just the non-relativistic limit of the Born-Infeld model. The latter is used extensively in the study of brane dynamics.

At this point it looks a little coincidental to me, that these Nambu-Goto action functionals can be related to the Chaplygin gas. But maybe I am missing something.

I have only skimmed the text, but the authors don’t seem to discuss the meaning (if any) of this coincidence, it seems.

What I see is, in the introduction of section 5, a discussion that fluid dynamics of nonabelian particles should be necessary for describing quark-gluon plasmas.

I assume, then, that the well known nonabelian generalization of the Nambu-Goto action (usually considered as describing “noncommutative dd-branes”) together with the general relation of the NG-action to the Chaplygin fluid model, here serves as a warm-up example for a nonabelian fluid model that happens to be relatively accessible and/or well understood.

But, as I said, I am not qualified to really judge this approach. Maybe somebody reading this here can tell us more about it.

Posted by: urs on October 9, 2006 8:05 PM | Permalink | Reply to this

Re: So Irigaray was Right?

How strange! I had been reading some of Sokal’s stuff just yesterday, and on hearing of the story, I naturally made the same association as well.

Don’t encourage the PoMos! ;)

But on thinking about it, it does raise a _real_ sociological issue. Consider Irigaray’s underlying assumptions:
1) Physicists are sexist
2) Solids are ‘male’, fluids ‘female’
3) Solids and fluids are equally easy to model

Now, most level-headed people would refute the statement on #1 or #2. But physicists seem to always go with #3. Admittedly, most folks wouldn’t be able to determine #3.

But I think it’s telling of the scientist mindset to refute the least subjective argument. And also of the mindset of (at least certain) postmodernists to show a complete disregard for any such idea.

(Of course, maybe I’m just being self-righteous white male physicist defending my hegemony :)

Posted by: Sven on October 7, 2006 7:39 PM | Permalink | Reply to this

Re: So Irigaray was Right?

Perhaps it’s more plausible for primatology:

1) Primatologists of the 1950s were sexist.

2) It’s equally easy to see male and female involvement in dominance relations between chimpanzees.

At least these PoMo writers made us more aware of how our thoughts and actions may be structured by language. But by putting out some dubious, extreme claims, they may have helped us to avoid noticing more insidious everyday cases. As regards mathematics, we should be concerned about the issues Alexandre Borovik discusses, relating to the environment within the mathematics community.

Posted by: David Corfield on October 8, 2006 11:53 AM | Permalink | Reply to this

Re: So Irigaray was Right?

Agreed. And I agree with Borovik too. Having studied both Chemistry (with a 50-50 distribution) and Physics (70-30) the attitude differences are immediately obvious - with both the men and women. And it’s not beneficial to either. I was definitely happier with a more even balance.

But it’s not as simple as that either, because in the higher years, that 50-50 distribution in Chemistry turned into 70% men in Phys-Chem and 70% women in Biochem.

There’s definitely something about the math. But I don’t think it’s neurological or pure competitiveness (Chemists generally have a ‘we suck at math’ self-image).

I think it’s 99% self-confidence. Not in general, or in knowledge, but in reasoning skills. Some of the best female students I’ve had would still ask for help more often than the worst male ones. Most of the time they didn’t need help. They needed confirmation that they were on the right track. Hardly strange then they’d avoid Math, with its long reasoning chains and binary answers.

As usual it’s simple enough to state the problem and a lot harder to come up with what to do about it. ..If only the teaching of science had had the same progress of science itself..

Posted by: Sven on October 9, 2006 7:56 AM | Permalink | Reply to this

Re: So Irigaray was Right?

Hardly strange then they’d avoid Math, with its long reasoning chains and binary answers.

Something strikes me as odd in this. I think I know what it is. This is a great characterization of mathematics at the lower and middle levels.

I’ve been at the doctoral/postdoctoral level for years. I see plenty of mileposts used to break up the long chains, and I see plenty of more philosophical grey areas than simple binary answers. Of course a given track is either logical or not, but there are many paths to the best theorems. Which one is best in a given situation – particularly which one generalizes to which wider context – is far from binary.

As much as I hate to sound like I agree with the Theory crowd, maybe the problem really is in the sociology of mathematics education. Giant impersonal classes teach the main path and discourage alternatives (I’ve already had to slap my grader around three times this semester for marking correct answers different than the book’s as wrong). Through the upper undergraduate levels of many (if not most) schools you’re exactly right about long chains of reasoning and binary answers. If you’re correct that those aspects dissuade women from mathematics, then they do so in the early stages before what I’d consider to be the true character of the field shines through.

Posted by: John Armstrong on October 9, 2006 1:20 PM | Permalink | Reply to this

Re: So Irigaray was Right?

Essential then to let the “true character of the field shine through” as much and as early as possible. Even at age 7.

Posted by: David Corfield on October 10, 2006 8:30 AM | Permalink | Reply to this

Re: So Irigaray was Right?

Right indeed. I try to do what I can: since I completely lack the temperment for working with 7-year-olds I try to improve mathematics teachers. Very little possible around here, but if I finally get a job at a school with a bigger math-ed program I’ll be involved.

Posted by: John Armstrong on October 10, 2006 2:21 PM | Permalink | Reply to this

Re: So Irigaray was Right?

Sven writes:

Some of the best female students I’ve had would still ask for help more often than the worst male ones.

I notice that too. This is one reason more male students fail my undergraduate math classes: they can be completely clueless yet still not ask for help.

I think part of it is that male adolescents feel a kind of blow to the ego when they have to ask questions of an elder male, in a way that females don’t.

It’s vaguely related to this joke:

Q: Why does it take millions of sperm to fertilize one egg?

A: They don’t want to stop and ask for directions.

Posted by: John Baez on October 9, 2006 6:12 PM | Permalink | Reply to this

Re: So Irigaray was Right?

Penny Smith has withdrawn her paper claiming to prove global-in-time existence of solutions for the Navier-Stokes equation, citing a “serious flaw”. Try the remarks by “Euler” on Peter Woit’s blog for more details.

Posted by: John Baez on October 8, 2006 10:01 PM | Permalink | Reply to this

Re: So Irigaray was Right?

Posted by: David Corfield on October 8, 2006 10:36 PM | Permalink | Reply to this

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