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April 29, 2008

Returning to Lautman

Posted by David Corfield

I mentioned in an earlier post that Albert Lautman had a considerable influence on my decision to turn to philosophy. I recently found out that his writings have been gathered together and republished as Les mathématiques, les idées et le réel physique, Vrin, 2006, a copy of which arrived through the post the other day. It’s remarkable how much contemporary mathematics Lautman covers – class field theory, algebraic topology, analytic number theory, etc.

At the same time as I was reading Lautman I became excited by category theory, via Colin McLarty’s Uses and Abuses of the History of Topos Theory, British Journal for the Philosophy of Science 1990 41(3):351-375, and Saunders Mac Lane’s Mathematics: Function and Form, and noticed an affinity with Lautman’s thinking, supported by a remark made by Jean Dieudonné in his 1977 Preface:

La “montée vers l’absolu” qu’il y discerne, et où il voit une tendance générale, a pris en effet, grâce au langage des catégories, une forme applicable à toutes les parties des mathématiques: c’est la notion de ‘foncteur représentable’ qui joue aujourd’hui un rôle considérable, tant dans la découverte que dans la structuration d’une théorie. (p. 36)

That Lautman worked with Claude Chevalley and Charles Ehresmann may not be unconnected.

The 2006 edition includes an interesting introduction – Lautman et la dialectique créatrice des mathématiques – by Fernando Zalamea, who devotes a section to interpreting Lautman through category theoretic spectacles. In successive posts I’ll take a look at how amenable are his mathematical examples to this treatment.

I’ll end now with a quotation Zalamea has selected from Lautman, which nicely expresses what might be done instead of Anglophone philosophy of mathematics:

La philosophie mathématique, telle que nous la concevons, ne consiste donc pas tant à retrouver un problème logigue de la métaphysique classique au sein d’une théorie mathématique, qu’à appréhender globalement la structure de cette théorie pour dégager le problème logique qui se trouve à la fois défini et résolu par l’existence même de cette théorie.

Yes, to what problems or questions is mathematics a response?

Posted at April 29, 2008 4:18 PM UTC

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Re: Returning to Lautman

For Isaac Newton, mathematics was the response to the question: “How to understand the mind of God, and the divine laws which regulate the universe?”
(For the devout believer Newton, “divine laws” was meant literally, not metaphorically.)

A century later, mathematics at Oxford and Cambridge was a response to the question: “How to appreciate and approach the divine by reasoning in a manner supported by the divine?” Georg Cantor may have had similar motivations, as perhaps did Russian mathematicians in the first half of the 20th century, influenced as they were by a Russian Orthodox spiritual movement.

Posted by: Peter on April 30, 2008 10:28 AM | Permalink | Reply to this

Re: Returning to Lautman

To add to the Early Modern philosophers mentioned here (an excerpt of the book is now available), we find that Lautman commonly quoted Malebranche

L’étude des mathématiques est la plus pure application de l’esprit à Dieu.

In our ‘disenchanted’ world, we might ask what replaces God.

But the interesting thing about Lautman was that he didn’t remain at this ultimate question of what all of mathematics is responding to, but delved into specific theories as exemplifying certain themes, such as global-local relations.

Posted by: David Corfield on April 30, 2008 11:56 AM | Permalink | Reply to this
Read the post Ambiguity Theory
Weblog: The n-Category Café
Excerpt: Galois theory as the study of ambiguity
Tracked: May 19, 2008 2:14 PM
Read the post Melliès on Functorial Boxes
Weblog: The n-Category Café
Excerpt: Functorial boxes
Tracked: August 22, 2008 2:40 PM
Read the post Zalamea on Sheaf Logic
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Excerpt: Zalamea seminar announcement
Tracked: October 3, 2011 2:50 PM

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