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Two Streams in Hatfield *July 6, 2009*

*Posted by dcorfield in Uncategorized.*

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Brendan Larvor and I ran a conference – Two Streams in the Philosophy of Mathematics from 1-3 July. I thought I’d put up a post here for post-conference discussion.

In accordance with one stream’s policy of encouraging dialogue with *real* mathematicians, we invited Yehuda Rav (Paris-Sud), Michael Harris (Jussieu) and this blog’s very own Alexandre Borovik. For me two of the most interesting issues to emerge during the conference was Borovik’s ‘phantoms’ and Harris’s ‘avatars’. The first of these may occur when there is a question as to whether a certain entity exists. Even if it does not, it may transpire that some counterpart of this nonexistent entity exists elsewhere. The setting of finite simple groups is a rich environment for this phenomenon.

In the case of avatars, on the other hand, they all exist, but they indicate the existence of a not yet expressible universal object. Grothendieck’s theory of motives is the classic example, and indeed it was here that he coined the term ‘avatar’ to describe an instantiation of a motive in a particular cohomological setting.

What I’d like to know is what can be said about these phenomena. What is the right language to formulate them? Do we have earlier cases of avatars or phantoms which we now know how to express? Might it be possible to understand both phenomena in the same framwork? I.e., perhaps there may be avatars which happen not to exist, but for which existing fellow avatars act as phantoms.

So that’s a small taste of two of the talks. There were fourteen others. Personally, I was very pleased to hear Ivor Grattan-Guinness speak about ‘notions’, such as symmetry, convexity, and linearity, continually reappearing in mathematics. My own talk focused on duality, but I gave it a Cassirerian gloss as a ‘principle’.

If anyone would like to share their thoughts on the conference, please feel free.

Dear David,

You ask: Do we have earlier cases of avatars or phantoms which we now know how to express? I wonder if Kronecker’s Jugendtraum, which is after all often pictured as a precursor to Langland’s programme, could be seen as an earlier case of avatar. I am not knowledgeable enough about these matters, but it seems to me that the relations it establishes between algebraic extensions and special values of purely analytical functions would be of no interest if one had constructed the extensions with non-constructive methods or if the analytical functions were shown to have an algebraic origin. These results could presumably be interpreted both ways, i.e., as generating number fields ‘analytically’ or as defining continuous quantities ‘algebraically’. (Hence the stupidity of reducing Kronecker to an evil constructivist wishing to stick to natural numbers.). This is not exactly what an avatar is meant to be as far as I understood Michael’s paper) but something close.

At all events, it is merely a suggestion for you and other mathematicians to ponder, I do not have the competence to say more.

And thanks for your part in organizing this rather refreshing conference.

As ever,

Mathieu

Hello Mathieu,

Good to meet you again in Hatfield.

Your choice of Kronecker’s Jugendtraum is a very apt one. Barry Mazur has written some wonderful notes here. It seems that it runs eventually into what Michael Harris was telling us, i.e., about motives.

Many thanks for the link to this fascinating paper, I withdraw my sceptical note, the Jugendtraum is indeed a forerunner. M.

I want to make two disjoint remarks after this great refreshing meeting in Hartfield,for which I am grateful to both organizors and to all participants for a stimulating atmosphere .

First one should not forget that only a part (the main one ? ) of mathematics is made of architectural (sometimes Gaudiesque ) constructions,plans, global views … but a crucial role is also often given a posteriori to small local experiments, discoveries , which start new fields. Both attitudes can mix eventually in the same period or the same actor.This corresponds to the classical ”Birds and frogs ” division,rather ” Foxes and hedgehoxes ” used by Freeman Dyson (See his Einstein’s lecture ,and readers’s remarks,Notices of the AMS,Feb.2009 and seq.) .Think for example in the start of non-periodical tilings ,or fractal studies or further in time Eulerian Koenisberg path. .. Myriad of examples ,and philosophical considerations should also apply to THEM !

It is fascinating to see how mathematics give meaning to non-mathematical words :this is the well informed story of infinity,but recent examples can be found in human sciences as well, in finance.. Think of decision,volatility,design…

There should be a philosophical perspective on that too:does it tell something on language ? on mathematics ? on both ?

Jean-Michel Kantor

[…] is from an early version of Chapter 7 and from my 2009 presentation on avatars at the conference Two Streams in the Philosophy of Mathematics organized by David Corfield and Brendan Larvor. Some of the first half has been preserved on p. […]