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March 19, 2012

Circles Disturbed

Posted by David Corfield

Apostolos Doxiadis and Barry Mazur have edited a book with Princeton University Press called Circles Disturbed: The Interplay of Mathematics and Narrative. Its chapters are largely provided by participants of a workshop held in Delphi in 2007, whose number included John Baez and myself.

John posted about his talk Why Mathematics Is Boring. However, he didn’t develop it into a chapter for the book. I posted about the meeting here, and you can read a draft of my chapter.

The introduction to the book is available. As it mentions, the accompanying interviews, which were conducted by workshop participants on each other, are to be made available. Two are already uploaded, including my interview of Barry Mazur.

Posted at March 19, 2012 11:17 AM UTC

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16 Comments & 1 Trackback

Re: Circles Disturbed

The available link doesn’t work and the uploaded link produces the Description of the book, not the 2 interviews - at least for me.

Posted by: jim stasheff on March 19, 2012 1:56 PM | Permalink | Reply to this

Re: Circles Disturbed

Whoops, thanks. I missed some ‘http’s. The link for the interviews should take you to the Thales and Friends main page. Interviews are on the right. They would make better sense if you could read the corresponding article as well, since they were carried out just after the interviewee had given their talk.

Posted by: David Corfield on March 19, 2012 2:52 PM | Permalink | Reply to this

Re: Circles Disturbed

Now I see there’s a special page for the interviews.

Posted by: David Corfield on March 19, 2012 4:57 PM | Permalink | Reply to this

Re: Circles Disturbed

Now the interview Colin McLarty had with me is available.

Posted by: David Corfield on April 18, 2012 4:46 PM | Permalink | Reply to this

Re: Circles Disturbed

Very interesting! I like the way that it’s visibly a conversation between two people who respect each other and are interested in listening as well as talking. In that respect it’s not much like a traditional interview.

I also like Colin’s example of Bollywood movie characters as datatypes.

Posted by: Tom Leinster on April 18, 2012 11:14 PM | Permalink | Reply to this

Re: Circles Disturbed

Thanks, Tim. Colin and I go back a number of years. We jointly ran an NEH Summer Seminar – Proofs and refutations in mathematics today – in Cleveland for 6 weeks in 2001.

I thoroughly recommend his chapter in the book. All my preconceptions of what Gordan meant when he said “This is not mathematics, it is Theology!” about Hilbert’s proof of the Basis Theorem were overthrown.

Posted by: David Corfield on April 19, 2012 8:55 AM | Permalink | Reply to this

Re: Circles Disturbed

Thanks for this quote by Rota, which you have on page 18:

“What can you prove with exterior algebra that you cannot prove without it?” Whenever you hear this question raised about some new piece of mathematics, be assured that you are likely to be in the presence of something important. In my time, I have heard it repeated for random variables, Laurent Schwartz’ theory of distributions, ideles and Grothendieck’s schemes, to mention only a few. A proper retort might be: “You are right. There is nothing in yesterday’s mathematics that could not also be proved without it. Exterior algebra is not meant to prove old facts, it is meant to disclose a new world. Disclosing new worlds is as worthwhile a mathematical enterprise as proving old conjectures.”

I can relate to that. “What has exterior algebra ever done for us?”. Similar to: “What have the Romans ever done for us?” #, isn’t it?

In both cases, the people who ask feel the need to fight occupation, which makes them lose sight of their own gain.

Posted by: Urs Schreiber on March 19, 2012 9:16 PM | Permalink | Reply to this

Re: Circles Disturbed

I think I’ve mentioned this before on an older Cafe post, but I can never read that Rota quote without thinking of another passage later in Indiscrete Thoughts where he is less favourable towards distribution theory, quoting Calderon to bolster his remark.

(Let me make it clear that I *like* Rota’s retort; I just think that quoting it is a double-edged sword.)

And I don’t think the comparison to the Life of Brian quote really works, even though it is always nice to see the quote deployed. Perhaps “what has exterior algebra ever done for us?” is more like “what has [name of famous work of literature] done for us?”

Posted by: Yemon Choi on March 20, 2012 12:33 AM | Permalink | Reply to this

Re: Circles Disturbed

You mean this?

The last time the theory of summability of divergent series was declared obsolete was after the publication of Laurent Schwartz’s theory of distributions. This theory was supposed to do away with a lot of things: summability Banach spaces, functions. Instead, the theory of distributions, after acquiring some degree of acceptance among specialists in the more absurd reaches of partial differential equations, has dropped into oblivion (while Banach spaces are flourishing more than ever). Sometime in the fifties Alberto Calderón remarked: “As soon as people realize that they cannot make changes of variables, the theory of distributions will be in trouble.”

As ever, one has to say to oneself “Rota was exaggerating for effect”. But “dropped into oblivion”, really?

Posted by: Tom Leinster on March 20, 2012 3:31 AM | Permalink | Reply to this

Re: Circles Disturbed

Thanks, Tom - that’s the one. I couldn’t find my copy of IT and didn’t want to risk misquoting.

To get away from this particular example, for a moment - if there is, as Spinal Tap have taught us, a fine line between stupid and clever, perhaps there is also a fine line between “exaggerating for effect” and “talking out of one’s ****” ?

It seems to come down to this: people who know what they’re doing (Rota? Zeilberger? Gelfand?) exaggerate for effect; people who don’t know what they’re doing, talk out of their ****s. ;) Either that, or this is one of those irregularly declined verbs …

Posted by: Yemon Choi on March 21, 2012 9:43 AM | Permalink | Reply to this

Re: Circles Disturbed

Now I’m curious. Is it really true that you can’t make changes of variables with distributions? What does this even mean?

Let’s see. If we’re talking about integrating functions on measure spaces, I take “change of variables” to mean the following. Let α:XY\alpha\colon X \to Y be a map of measurable spaces. Then any measure μ\mu on XX gives rise to a (“pushforward”) measure α *μ\alpha_* \mu on YY, such that

X(gα)dμ= Ygd(α *μ) \int_X (g \circ \alpha) d\mu = \int_Y g\, d(\alpha_* \mu)

for any g:YRg\colon Y \to \mathbf{R} such that gαg\circ\alpha is μ\mu-integrable.

So, what would change of variables for distributions mean? I suppose it would say that if XX and YY were suitable “spaces” (open subsets of R n\mathbf{R}^n?) and α:XY\alpha\colon X \to Y a map of a suitable sort (C C^\infty and proper to be on the safe side?) then any distribution uu on XX somehow gives rise to a “pushforward” distribution α *u\alpha_* u on YY, such that

gα,u=g,α *u \langle g\circ\alpha, u \rangle = \langle g, \alpha_* u \rangle

for all suitable functions gg on YY.

Is that plausible? Ignoring all analytical details, I’d imagine that α:XY\alpha\colon X \to Y induces a linear map Test(Y)Test(X)Test(Y) \to Test(X) of spaces of test functions, by composition. The dual of this linear map would be something like Dist(X)Dist(Y)Dist(X) \to Dist(Y), since the space Dist(X)Dist(X) of distributions on XX is a subspace of the dual of Test(X)Test(X). And that’s what we want. (If you interpret “Test(X)Test(X)” as meaning “C(X)C(X)” then pushforward measures can be described like this, I guess.)

So I suppose something must go wrong with the seminorm estimates. Can an expert enlighten me?

Posted by: Tom Leinster on March 24, 2012 5:21 PM | Permalink | Reply to this

Re: Circles Disturbed

I must admit that I’m not completely sure I get what the quotes are talking about either, but I think that the point is that you can’t do what you describe with the function α\alpha as a distribution: it has to be smooth. That is, if you try to define a function α:XY\alpha \colon X \to Y with the property that α *g\alpha_* g is defined and is a distribution for all distributions gg on YY then α\alpha has to be a smooth function. You can’t do it with α\alpha being a distribution. So it’s not change-of-variables of distributions that can’t be done, it’s change-of-variables by distributions.

(I could be wrong on both the interpretation and the conclusion.)

Posted by: Andrew Stacey on March 26, 2012 1:15 PM | Permalink | Reply to this

Re: Circles Disturbed

And I don’t think the comparison to the Life of Brian quote really works, [..] Perhaps [it] is more like “what has [name of famous work of literature] done for us?”

Not if you follow my reading of the analogy, which was:

In both cases, the people who ask feel the need to fight occupation, which makes them lose sight of their own gain.

This is, in my experience, the reason for why strong concepts are being rejected, in the sense of the Rota quote: people feel their work is being “occupied” by, say, exterior algebra, category theory, etc., if they start admitting that some statements follow from the general theory. So they go into resistence.

Posted by: Urs Schreiber on March 21, 2012 10:55 PM | Permalink | Reply to this

Re: Circles Disturbed

Something very striking for me is the increase in mathematical exposition in the 8 years since I started writing the piece. Imagine a time before blogs, MathOverflow, the nLab,…

Posted by: David Corfield on March 20, 2012 9:55 AM | Permalink | Reply to this

Re: Circles Disturbed

Returning to Thales and Friends… and the aim of thinking about mathematics as/in culture, it just so happens I have a network grant to work on this:

http://sites.google.com/site/mathematicalcultures/home

I’ve got speakers lined up to talk about more established themes and historical cases, but it strikes me that there are developments afoot that students of mathematical practice should think about. One is the rise of webstuff as David says, his examples plus polymath and Math 2.0. The other, related, is the Elsevier boycott, which has led to mathematicians talking about the socially distributed aspect of their practice more publicly than usual. If anyone should want to fly a kite or two on these topics, don’t be shy…

Posted by: Brendan Larvor on April 17, 2012 1:21 PM | Permalink | Reply to this
Read the post Mathematical Cultures
Weblog: The n-Category Café
Excerpt: Conference on mathematical cultures
Tracked: April 19, 2012 10:33 AM

Re: Circles Disturbed

By the way, one reason I never developed my piece into a chapter for the book is that I never got an email warning me that it was due. I was sort of glad about that, since I was really overburdened at the time, trying to finish up all my work on higher categories and higher gauge theory. But it left a pang of guilt which I am trying to assuage now by coming clean!

Posted by: John Baez on April 20, 2012 5:05 AM | Permalink | Reply to this

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