### 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.

## Re: Searching for a New Epistemology in Berlin

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