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September 18, 2006

Searching for a New Epistemology in Berlin

Posted by David Corfield

I’m back from a 48 hour workshop - Towards a New Epistemology of Mathematics. This provided the opportunity to meet up with some old friends, and form some new acquaintances. I met fellow blogger Kenny Easwaran for the first time, and heard him talk about The Role of Axioms in Mathematics. Dirk Schlimm made us think about why Jordan only thought to show that all composition series of a finite group, as feature in the Jordan-Hölder Theorem, have the same collection of numerical values of the orders of quotients of successive terms, while Hölder later gave the stronger result that it was true of the isomorphism classes of such quotients. While you might think that Hölder was enabled to do so having moved on to consider abstract groups instead of the substitution groups treated by Jordan, Dirk showed that Jordan did in fact have the technical resources to prove the stronger result. Something higher-level inspires you to pose the richer question.

Alan Baker told us his thoughts on the use of experiment in mathematics. Brendan Larvor, author of a very good book on Lakatos, opened his talk with “Here we are nearly 45 years after Lakatos’s Proofs and Refutations, still looking for the new epistemology…”. This struck a chord as Chapter 7 of my book opens “Nearly forty years have passed since Imre Lakatos published his paper Proofs and Refutations…”, itself updated from its earlier appearance as a journal article in 1997. We imagined ourselves 15 years hence, “Nearly 60 years have passed…”.

One thing I think we desperately need is an annotated bibliography of existing work in these new directions. A few reinventings of wheels might thus be prevented. But even more desperately, and I only became more convinced of this through the workshop, we need to recognise that arguments for changes in the philosophy of mathematics must impact profoundly on the rest of philosophy. My allegiance to MacIntyre’s outlook demonstrates just how radical I take these changes to be. I’ll put my notes up here soon when I’ve straightened them out.

Meetings of this ilk often reach out to disciplines other than philosophy, such as didactics, psychology, sociology, and history. I generally enjoy listening to the historians most of all. Leo Corry always has interesting things to tell us about mathematics. His talk discussed what we could learn about mathematicians’ images of mathematics (in the sense that he defines it - see p.8 of this) from charting changes in the classification schemes of reviewing journals. He dwelt on such changes in the classification of articles in or around algebra.

Taking our conference dinner with the Carnap workshop gave me the chance to meet up with Steve Awodey again. I heard from him about a topological semantics for first-order modal logic, which is being discussed here. I learned from another participant that Carnap was in analysis both in Vienna and Chicago, and that Gödel was in analysis jointly with his wife in New York City. I knew that many English intellectuals of the 1920s were interested in psychoanalysis, and that Ramsey had travelled to Vienna for analysis, but I was surprised to find out about Carnap and Gödel. I wonder if Popper’s rivalry with the Circle had anything to do with his negative comments on psychoanalysis.

On 13 November, 1940, Carnap and Gödel met in Princeton to discuss the possibility of a theology consistent with scientific rationality, and Carnap took stenographic summarizing notes, reproduced here. The Leibnizian Gödel argues in favour of the thesis, while Carnap argues against it even invoking psychoanalysis:

CARNAP: But I maintain that the change of direction you propose would certainly be unproductive. This my assumption, which is of course not a proof, is supported by our knowledge of psychoanalysis and other fields of inquiry as to how the idea of God and all of theology and so on can be traced back to childhood experiences and beliefs.

This isn’t much of a supporting argument. Many scientific ideas can be traced back to childhood experiences and beliefs, yet this has no bearing on their validity. See section 2 of this for the childhood experience of the ecologist Arne Naess.

Unfortunately, I had very little time to look about Berlin, which I had never visited before. By waking up at 6 a.m. on the last morning, however, I did manage a stroll down Unter den Linden and felt something of the weight of history bearing down upon it.

Posted at September 18, 2006 11:27 AM UTC

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Re: Searching for a New Epistemology in Berlin

Alan Baker told us his thoughts on the use of experiment in mathematics.

Was this about using computers, or about using physically motivated conjectures #?

Posted by: urs on September 19, 2006 1:57 PM | Permalink | Reply to this

Re: Searching for a New Epistemology in Berlin

He was thinking more about computer experimentation, like this. Both cases are very interesting. I treat them in chapter 6 of my book. One question you can raise is whether these types are really different. They’re both about the behaviour of physical systems. But then the set of physical systems instantiating a given calculation don’t seem to form a natural kind.

Posted by: David Corfield on September 19, 2006 2:17 PM | Permalink | Reply to this

Re: Searching for a New Epistemology in Berlin

And then there are people who do experimental math in the sense of conjectural physics, and then use computer experiments on top of that to test the conjectures. Like this, for instance.

Hopefully everybody still knows what is actually being computed here…

While not directly related, this reminds me of an event that took place a few years ago. Experimentalists at the Brookhaven accelerator had found experimental evidence for a significant deviation of some physics from the standard model prediction. A couple of people quickly explained the observed deviation with “new physics”.

But somebody went through the pain of checking the long series of computations that were necessary to derive the required prediction from the standard model. Somewhere in the chain of computations computer code was involved, which was used to compute certain matrix elements.

It involved a Hodge duality operation and hence lots of signs. It turned out that one of these signs were coded incorrectly in the computer program.

That guy corrected the program and checked what the correction did to the final result. It turned out that with the corrected sign the final result changed by a tiny bit, just so that the deviation from experiment was no longer significant.

(see this and scroll down to Muon (g2g-2))

Moral: somebody needs to check the computer checks of the checks of our conjectures.

Posted by: urs on September 19, 2006 2:42 PM | Permalink | Reply to this

Re: Searching for a New Epistemology in Berlin

Polya was very aware of all the subtle influences on your degrees of belief in mathematical propositions, and this before computers had contributed a host of new forms of evidence.

Just think of what is at stake in the credence you give to the sketch of a proof. Some researchers at Warwick want to do an experiment on this. To participate go here.

Posted by: David Corfield on September 19, 2006 3:54 PM | Permalink | Reply to this

Re: Searching for a New Epistemology in Berlin

By the way, Tommaso Dorigo today has a blog entry on this g2g-2 experiment and its implications - or lack thereof - for physics beyond the standard model.

Posted by: urs on September 20, 2006 9:41 PM | Permalink | Reply to this
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