December 2007
In December 2007 The Archive recorded the following events or has made arrangements to share recordings made by others.
29th NOVEMBER – 3rd DECEMBER 2007
Trends in The Mathematical Representation of Space : An International Symposium to Honour the 50th Anniversary of the Appearance of Alexandre Grothendieck’s Tohoku Paper
An International Symposium forming part of the 2007 – 2008 Program of The Boston Colloquium for The Philosophy of Science. Boston UNIVERSITY, Boston, Massachusetts, USA
29th NOVEMBER 2007 : Arrival and Welcome of Speakers.
Memorandum and Notes of informal discussions between speakers : Pierre CARTIER, F.W. LAWVERE and Colin McLARTY
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30th NOVEMBER 2007 : Boston University Department of Philosophy
3:00 PM : Invited Lecture by Pierre CARTIER (CNRS/IHES France)
On The Naturalness of Mathematics : Towards A Relativized Ontology for Mathematics (Followed by Questions and General Discussion)
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See following Abstract of Talk :
The question of the nature of mathematical concepts and ideas has often been hotly debated. Every science has its object or objects – a being or world of beings to be studied, classified and theorised. The objects of mathematics are clearly somewhat special, being in some sense more abstract than those of other disciplines. But mathematicians experience the need to give a reference to their concepts, in order to provide justification for their reasoning about them and to support their constructions and guide their intuitions. Over the centuries the favoured candidates for thus securing reference have changed drastically – from geometrical figures and the tools to design and construct them, then to an algebra based on numbers and symbols upon which various operations are to be defined and performed, later transitioning to ideal objects of various types, such as points at infinity, imaginary numbers, ideal divisors in a number theory whose subject matter has become more and more structural in character. In the late 19th and early 20th Century Cantorian set theory provided a new paradigm for the unification of this realm of referents, providing a representation in principle for all kinds of mathematical objects. But a naively realistic point of view about sets led to inconsistencies. These can be addressed by shifting to a formalistic point of view, but the naturalness of mathematical constructions is thereby lost and the artificiality of these constructions leads to a “Glass Perlen Spiel”
One way out of this impasse makes use of the reflexivity of mathematical thinking. Being a science of abstraction(s), mathematics can reflect on its own modus operandi and review its own activity in mathematical terms. This historical trend of symbolism and formalism has led to the creation of metamathematics. We view metamathematics as a level of constructive activity within mathematics itself, not as a separate science or autonomous justification of mathematics. We propose a view of mathematics as an elaborate and diverse family of projects of construction – involving the fabrication of tools with a conceptual purpose. 20th century developments in formal logic, programming languages, computer science and information theory have been one of the most impressiveoutcomes of this trend. Mathematics can benefit from taking itself as its own object of study. this offers a radically new perspective on and new kinds of proposals for, mathematical ontology.
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This Invited Lecture was a separate event, taking place before the Commencement of the Symposium.
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The Symposium : 1st – 3rd DECEMBER 2007 : Boston Colloquium For the Philosophy of Science : Photonics Centre, Boston University, Boston, Massachusetts
Day One : 1st DECEMBER 2007
Talk 1 : Pierre CARTIER (CNRS/IHES Paris)
Categories and Sets Living in A Contradictory World.
(Revised Title : Categories, Sets and Universes : Living on the Edge of Contradiction)
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Talk 2 : Colin McLARTY (CWRU Cleveland)
On Generality and Unity in Grothendieck’s Conception of Geometry
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LUNCH
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Talk 3 : F.W. LAWVERE (S.U.N.Y. at Buffalo)
Euler, Maxwell and Grothendieck : Three Phases in the Search for the Correct Mathematical Representation of the Cohesion and Variation involved in our Notion of Space
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Talk 4 : Jean-Pierre MARQUIS (University of Montreal)
From Klein to Kan : The Algebra of Space and the Space of Algebras
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Talk 5 : Michaela IFTIME (Boston University and MCPHS) :
Grothendieck’s Universe.
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Day Two : 2nd DECEMBER 2007
Talk 1 : Lou CRANE (KSU)
What Does Quantum Gravity Tell Us About The Nature of Space/Spacetime ?
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Talk 2 : Gonzalo REYES (Univ of Montreal)
An Axiomatic Approach to Einstein’s Field Equations in The Framework of Synthetic Differential Geometry
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LUNCH
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Talk 3 : Shawn WESTMORELAND (KSU)
The Twin Paradox Without Acceleration
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Final Session of Symposium : Round Table Discussion of Trends in the Mathematical Representation of Space on the 50th Anniversdary of Grothendieck’s Tohoku paper
Panel members : Pierre CARTIER, F.W. LAWVERE, Colin McLARTY, Jean-Pierre MARQUIS, John STACHEL, Lou CRANE
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Closure of Symposium
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Symposium Dinner
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29th November – 3rd December 2011
Notes and Memoranda of Discussions between Symposiasts outside the Formal Sessions of The Symposium. Includes Memorandum of Discussions with F.W. Lawvere, P. Cartier, J-P. Marquis, Colin McLarty and others on such topics as Stone Representation vis-a-vis position of Hilbert Space in Functional Analysis, reasons for bad functoriality of Quantisation; Grothendieck’s implicit proposals for “Bypassing Logic” and his 1973 aide-memoire to Duskin; the Adjoint Functor Theorem and the background to Kan’s discovery of it; Category-theoretic perspectives on Differential Geometry; and many other topics to be further detailed here.
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Also link here for Notes of a December 2007 discussion with John Stachel during Boston Colloquium Meeting of various topics in Differential geometry and The Theory of General Realtivity and particularly of points re General Covariance raised by Simon Saunders in his Essay in Festschrift for John Stachel & other topics
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1st DECEMBER 2007
Seminaire maMuPhi ENS : Francis Borceux : Des jets aux infiniment petits : quand l’intuition se mue en rigueur [1][5]
Video : www.diffusion.ens.fr/index.php?res=conf&idconf=1881
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François Nicolas : Quelques raisonances musicales de l’exposé de Francis Borceux
[1][5] Intuitivement, une fonction f:R→R est continue en un point a lorsqu’une variation infinitésimale de x au voisinage de a provoque une variation infinitésimale de f(x) au voisinage de f(a).
L’approche que F.W. Lawvere et A. Kock ont donnée de la notion d’infiniment petit est la suivante.
Si x est petit, x2 est encore plus petit. Si x est très, très petit, x2 devient vraiment minuscule. Appelons donc “infiniment petit” un nombre x tel que x2=0. L’idée provient de la “théorie des jets” due à Ehresmann.
Considérons toutes les fonctions passant par un point du plan, que rien ne nous empêche de prendre comme origine: donc f(0)=0.
Avoir la même tangente à l’origine est une relation d’équivalence: une classe d’équivalence s’appelle un “jet”. Le propre d’un tel jet est que si on l’élève au carré, on trouve le jet nul (la classe d’une fonction à tangente horizontale).
Divers auteurs ont prouvé qu’en travaillant dans des topos ad hoc, on peut construire des anneaux R admettant des éléments de carré nul, que l’on peut penser comme étant les infiniment petits et grâce auxquels on peut développer la géométrie différentielle.
The links to the recordings of these MaMuPhi seminars are by kind permission of the Seminar organisers and speakers, who retain the copyright to these recordings.
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1st DECEMBER 2007
Universite de Paris 7 : Halle des Farines
Dr Franck DEDRZEJEWSKI : Diagrammes et Categories
Extracts from Transcript of Thesis Defence, Text of Thesis and subsequent comments and discussions with Dr Dedrzejewski and others to be provided via links here subject to permission of Dr Dedrzejewski and Members of the Jury (R. Guitart, D. Lecourt, E.Klein, A.Papadopoulos. J-M. Salankis, J Szczeciniarz)
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3rd DECEMBER 2007
Renaud CHORLAY : The Distinction Between Local and Global Notions in Mathematics and in particular in Geometry:
The recognition, Crystallization, and fuller articulation of the distinction within different branches of mathematics from the work of Riemann to the development of Sheaf Theory c. 1850 – 1953
A Ph.D thesis defended at REHSEIS on 3rd December 2007.
Extracts from Transcript of Thesis Defence, Text of Thesis and subsequent comments and discussions with Renaud CHORLAY and others to be provided via links here subject to permission of Dr CHORLAY and Members of the Jury (E.Scholz, C. Houzel, JJ. Szczeciniarz et al.)
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3rd-4th DECEMBER 2007
Paris REHSEIS Histoire et Philosophie des mathematiques
2 Day SYMPOSIUM on “Algorithmes reciproques”
In Seminar Series/Research Axis “Algorithmes, Instruments, Operations, Algebre”
DAY ONE : Monday 3rd DECEMBER 2007
09:30-10:30 : Talk 1 : Christine PROUST (REHSEIS) :
Faire et define une “equation” : l’exemple du texte cuneiforme de Susa
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10:30-10:45 : Kethrine CHEMLA (REHSEIS) :
Revenir au point du depart dans les mathematiques de la Chine ancienne
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10:45-11:00 : General Discussion
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Coffee Break
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11:30-12:30 : Francois PATTE (Univ Paris 5 – Rene Descartes)
Le calcul de carre et de la racine carre des Karani
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12:30-12:45 : Agatha KELLER (REHSEIS)
Quelques reflexions generales sur les algorithmes d’inversion dans la litterature sanskrite avant le XIIeme Siecle
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13:45-13:00 : General Discussion
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Lunch
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14:30-15:30 : Marco PANZA (REHSIS) :
Les limites d’algebre : Newton et le probleme des quadratures et l’algorithme inverse des vitesses.
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15:30-15:45 : David RABOUIN (REHSEIS)
L’integration et la differentiation comme operations reciproques chez Leibniz
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Coffee Break
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16:30-17:30 : Marie-Jose DURAND-RICHARD (Univ Paris 8 and REHSEIS)
Recherche sur la reciprocite operatoire des fonctions : de Herschel a De Morgan
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17:30-17:45 : Dominique TOURNES (IUFM de La Reunion et IHES)
A propos du “retour des suites” dans l’analyse algebrique continentale
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17:45-18:00 : General Discussion
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DAY TWO : Tuesday 4th DECEMBER 2007
09:30-10:30 : Maarten BULLYNCK (University of Wuppertal and REHSEIS)
The theoryof repeated fractions as a partial inspiration for Section V of C.F. GAUSS’s Disquisitiones Arithmeticae
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10:30-10:45 : Catherine GOLDSTEIN (IMJ) 15:45-16:00 :
Inversion des procedure de calculs : les interpretations de la section V des Disquisitiones Arithmeticae entre algorithmes et structures
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Coffee Break
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11:30-12:30 : Ralf KROMER (LPHSAHP Nancy) : The Concept of an Adjoint Functor : Its origins and history from its discovery by Kan, Its role in mathematical practice and its significance for the philosophy and foundations of mathematics in particular its thematisation and foregrounding in the work of F.W. Lawvere and in his conception of the foundations of mathematics
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12:30-13:00 : General Discussion
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Plus Notes and memoranda of further discussions with R. KROMER, D. RABOUIN and R. CHORLAY
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6th DECEMBER 2007 :
Institute of Mathematics, University of Antwerp, Belgium :
Talk by Isaar STUBBE on recent work on 2-categories subsuming and generalising the Construction of Sheaves on A Quantaloid : Locally principaly generated elements. split idempotents, sublattice-based -vs- ordered sets viewpoints, Left and Right Adjoint module morphisms
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7th-8th DECEMBER 2007
Brussels Belgium : Belgian Royal Academy Meeting
STRUCTURE AND IDENTITY : Logical and Metaphysical Explorations
Day ONE : Friday 7th DECEMBER 2007
Opening Ceremony and Introductory Remarks by Dr Karin VERELST (VUB Brussels)
Talk 1 : Dr Thomas FORSTER (Cambridge)
Interpretation, Reduction and Logic
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Talk 2 : Jean-Yves BEZIAU (University of Neuchatel Swizerland)
Identity Logic and Structure
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Talk 3 : Graham PRIEST (University of Melbourne)
Identity and Difference : Their Structure in Unity : Contradiction in Identity or
(Alternative Title) : “Gluons” : Identityand Unities in Thought and in Reality : Contradiction in Identity
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Talk 4 : Catarina Dutilh NOVAES (University of Amsterdam)
A Lesson from Medieval Logic : How I learned to stop worrying and love the paradoxes
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Talk 5 : Elia ZARDINI (University of St Andrews) :
Identity and Transitivity
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Talk 6 : Peter CAWS (George Washington University)
Identity and Intentionality : Structures in The Transcendental Subject
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PM Talks in Parallel Sessions :
Anna-Sofia MAURIN (Lund University Sweden)
Tropes to the Rescue : Saving the Bundle Theory from Bradleyian regress
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Gertrude Van de VIJVER and Filip KOLEN (University of Ghent)
The Disentanglement of Structure and Substance : What Kant and Husserl Allow Us to Say about the Overcoming of Metaphysics
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DAY TWO : Saturday 8th DECEMBER 2007
Talk 1 : Michel SERFATI (Univ of Paris 7 Jussieu, IREM)
Mathematics and Individuation
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Talk 2 : Bob COECKE (University of Oxford)
One out of Many : Pictographic and Categorial Foundations for the Representation of Quantum Systems
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Talk 3 : Andrei RODIN (ENS Paris)
The Notion of Isomorphism Does Not Replace that of Identity in Category Theory (or anywhere else)
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Talk 4 : Christian DE RONDE (VUB Brussels)
Individuality and Identity in Modal Interpretations of Quantum Theory
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Talk 5 : Maarten VAN DIJK (University of Ghent)
A New Causal Structure for Nature : The Role of Conservation principles in the Scientific revolution of the 17th Century
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Talk 6 : Jean-Louis HUDRY (Edinburgh)
Paraconsistency, Bivalence and Identity
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Coffee break
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Talk 7 : Abraham STONE (California)
The Disunity of Beings and the Disunity of Being : Husserl versus Heidegger and Carnap
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Talk 8 : Richard ARTHUR (McMaster)
x+dx = x : Leibniz’s Archimedean Infinitesimals
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Closing Ceremony and Concluding Remarks by the Organisers of the Meeting, Karin VERELST and Wim CRISTIAENS
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Note/memorandum of further conversation with Abraham Stone and Eric Schleisser
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12th DECEMBER 2007
Talk in Paris (at IMJ Chevalaret ?) by Yves ANDRE
Details to be added.
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15th DECEMBER 2007 :
Seminaire MaMuPhi ENS
Ralf Kromer : La théorie des catégories : un outil d’analyse musicale aux yeux de la critique philosophique
Video : www.diffusion.ens.fr/index.php?res=conf&idconf=1882
The links to the recordings of these MaMuPhi seminars are by kind permission of the Seminar organisers and speakers, who retain the copyright to these recordings.
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15th DECEMBER 2007
10:00 AM : ENS Paris 45 Rue D’Ulm
Seminar in Physics Department in Series on Magnetohydrodynamics :
Sebastien GAITIER (Univ Paris XI) : The Solar Wind, turbulence and the Magnetohydrodynamic Hall Effect
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16th-20th DECEMBER 2007
Notes and Memoranda of Conversations and Discussions with Dr John MAYBERRY (Unversity of BRISTOL) about, inter alia : The Book of J.L. BELL on The Continuous and The Discrete in The History of Mathematics. The origins and development of the Concept of A Sheaf and the significance of Sheaf Theory for the Dialectics of Continuous/Discrete and Local/Global in Mathematics. The Yoneda Lemma and other topics. Hom Sets and criticisms of claims as to the foundational status of Category Theory. John Mayberry’s own views on the Foundations of Mathematics
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17th DECEMBER 2007
REHSEIS Seminar in Series “Recherches sur La Generalite”
Talk 1 : Evelyn FOX-KELLER
Practices of Generalisation in Mathematics, Physics, Biology and Evolutionary Theory compared and contrasted.
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Talk 2 : Jacqueline BONIFACE
Permanent and Accidental Properties in Kummer’s Theory of Ideal Numbers
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19th DECEMBER 2007
Seminar in History of Astronomy at Observatoire de Paris.
Details to be added
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Other entries for December 2007 to be added.