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March 7, 2011

Symposium: Sets Within Geometry

Posted by David Corfield

There is to be a Symposium – Sets Within Geometry – held in Nancy, France on 26-29 July, 2011. Confirmed speakers are: FW Lawvere (Buffalo), Yuri I. Manin (Bonn and IHES), Anders Kock (Aarhus), Christian Houzel (Paris), Colin McLarty (CWRU Cleveland), Martha Bunge (Montreal), Jean-Pierre Marquis (Montreal) and Alberto Peruzzi (Florence).

Statement of aims:

Those who have come together to organise this Symposium believe that the ultimate aim of foundational efforts is to provide clarifying guidance to teaching and research in mathematics, by concentrating the essential aspects of past such endeavors. By mathematics we mean the investigation of the Relations between Space and Quantity, of the reflected relations between quantity and quantity and between space and space, and the development of our knowledge of these in other words Geometry.

Using tools developed by Cantor and his contemporaries, much more explicit forms of the relation between space and quantity were developed in the 1930s in the field of functional analysis by Stone and Gelfand, partly through the notion of Spectrum (a space corresponding to a given system of quantities). In the 1950s Grothendieck applied those same tools, around the notion of Spectrum, to algebraic geometry by using and developing the further powerful tool of category theory . Further developments have strongly suggested that it is now possible to incorporate the whole set-theoretic “foundation” of Geometry, explicitly as part of that space-quantity dialectic, in other words as a chapter in an extended Algebraic Geometry.

Posted at March 7, 2011 10:22 AM UTC

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Re: Symposium: Sets Within Geometry

Thanks, David. I had meant to forward this announcement to the blog, too, but didn’t get around to it.

I’d be interested in seeing the titles and topics of the invited talks, to see how participants intend to try to fill with life what – if I am not mistaken – is Bill Lawvere’s formulation of his grand vision of topos theory and geometry, for instance when it says:

By mathematics we mean the investigation of the Relations between Space and Quantity, of the reflected relations between quantity and quantity and between space and space, and the development of our knowledge of these – in other words: Geometry.

Posted by: Urs Schreiber on March 7, 2011 6:45 PM | Permalink | Reply to this

Re: Symposium: Sets Within Geometry

I met up with Michael Wright, one of the organisors of the symposium, last week at this workshop. He tells me some form of recording will be made of the talks.

For those who don’t know, Michael has an extraordinary collection of material going back around 40 years, recordings of talks and interviews by those connected with category theory. He is planning to make these recording available at the Archive for Mathematical Sciences & Philosophy, which collects together material on history and philosophy of mathematics and science.

The Archive currently contains over 35,000 recordings, including approximately 2,500 video recordings.

It will take some time to make this material available. There will be some gems to discover there, especially for us in Recordings on Category Theory, Topos Theory and related topics.

Posted by: David Corfield on March 7, 2011 9:24 PM | Permalink | Reply to this

Re: Symposium: Sets Within Geometry

A small correction: Yuri Manin is not at the IHES

Posted by: Thomas on March 9, 2011 7:58 PM | Permalink | Reply to this

Re: Symposium: Sets Within Geometry

NOTE: change of date to 26-29 July.

Posted by: David Corfield on April 15, 2011 4:32 PM | Permalink | Reply to this

Re: Symposium: Sets Within Geometry

They now have talk titles and abstracts on the conference’s webpage.

This talk here may be related to some of the discussion about (,1)(\infty,1)-logic that we had here (maybe the corresponding article was mentioned here before, I forget. Is there an electronic copy available anywhere?):

Jean-Pierre Marqus, Abstract geometric sets and homotopy types –The metaphysics of abstraction

Abstract

In this paper, I explore the metaphysical aspects of abstract sets given via geometric means, namely in the usual terminology, as homotopy 0- types. I am not claiming that homotopy types should be taken as the ultimate foundations for mathematics, but rather that homotopy types exhibit what it means to be abstract for sets in such a foundational framework. I will therefore first rehearse some basic facts about homotopy types, what we know about them and how we know it. What I want to emphasize is not so much what homotopy types are, but rather the type of being they have in the mathematical realm. Once this is clarified, we can zero in on homotopy 0-types, namely sets. I will compare and contrast how these sets differ from the ones that are usually thought of as constituting the foundations of mathematics, e.g. in ZF. Furthermore, as homotopy types already indicates, one cannot apply one and the same criterion of identity for all mathematical entities. There are inherent dimensions arising naturally in the abstract geometric framework.

This one here also sounds interesting, but I can’t quite guess from the abstract what precisely it is about:

Colin McLarty, Cohomology in the Category of Categories as Foundation

Abstract

The elimination of the axiom scheme of replacement from the construction of injective resolutions makes possible a rigorous naive treatment of cohomological number theory in a natural version of the Category of Categories axioms.

Posted by: Urs Schreiber on July 23, 2011 5:57 PM | Permalink | Reply to this

Re: Symposium: Sets Within Geometry

Marquis’s paper was available once. I see he’s redoing it. I know he favours Makkai’s FOLDS approach.

Posted by: David Corfield on July 23, 2011 6:12 PM | Permalink | Reply to this

Re: Symposium: Sets Within Geometry

I can’t guess from Marqus’ abstract what his talk is about either. I don’t even know what it means to talk about “what type of being something has in the mathematical realm.” Can anyone enlighten me?

Posted by: Mike Shulman on July 29, 2011 2:10 AM | Permalink | Reply to this

Re: Symposium: Sets Within Geometry

I followed the link that David posted here and then on to here. As far as I can tell Marquis is interested in homotopy types because for him they are examples of mathematical entities that are best not thought of as “sets with structure.”

Posted by: Eugene Lerman on July 30, 2011 8:32 PM | Permalink | Reply to this

Re: Symposium: Sets Within Geometry

Mike wrote:

I don’t even know what it means to talk about “what type of being something has in the mathematical realm.” Can anyone enlighten me?

I can’t tell in which way you’re claiming not to understand this phrase. You might simply have no idea what Marquis is talking about, because you’re not used to how philosophers talk. Or you could be asserting that his way of talking is too vague to meet your standards of clarity.

If it’s the latter, I can’t help you. But if it’s the former, I can. Lots of philosophers spend a lot of time talking about ontology: the different ways in which things can exist. Ways of existing are also called ‘types of being’. Marquis is saying he plans to talk about this issue for homotopy types. That would be my guess from this passage, anyway.

Posted by: John Baez on July 31, 2011 4:56 AM | Permalink | Reply to this

Re: Symposium: Sets Within Geometry

Thanks, John, that helps some. I knew that philosophers like to spend lots of time talking about what mathematical objects are. But are you saying that some philosophers think that some mathematical objects exist in a different way than other mathematical objects do? E.g. that homotopy types exist in a different way than ZF-sets do?

Posted by: Mike Shulman on July 31, 2011 6:22 PM | Permalink | Reply to this

Re: Symposium: Sets Within Geometry

As Eugene pointed out, the wish to present mathematics as dealing with more than sets with structure is part of the story here. The abstract of the paper by Marquis he mentions – Mathematical forms and forms of mathematics: leaving the shores of extensional mathematics – says

In this paper, I introduce the idea that some important parts of contemporary pure mathematics are moving away from what I call the extensional point of view. More specifically, these fields are based on criteria of identity that are not extensional. After presenting a few cases, I concentrate on homotopy theory where the situation is particularly clear. Moreover, homotopy types are arguably fundamental entities of geometry, thus of a large portion of mathematics, and potentially to all mathematics, at least according to some speculative research programs.

Posted by: David Corfield on August 5, 2011 11:47 AM | Permalink | Reply to this

Re: Symposium: Sets Within Geometry

Lots of philosophers spend a lot of time talking about ontology: the different ways in which things can exist.

One thing I like about category theory is that it tends to allow to formalize such questions and turn ideas into definitions and heuristics into theorems. I would hope this could be done here, too. It usually makes discussions become more fruitful than otherwise.

Posted by: Urs Schreiber on August 1, 2011 1:53 PM | Permalink | Reply to this

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