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Workshops and conferences

3rd Paris-Nancy Philosophy of Mathematics Workshop	Jacques Dubucs, t.b.a. Ivanh Smadja, t.b.a. Luca Incurvati, "The graph conception of set" Gianluigi Oliveri, "On a New Argument for Realism in Mathematics" Michiel van Lambalgen, "The completeness of Kant's Table of Judgements and its  consequences for philosophy of mathematics." Jonathan Payne, "Abstraction with domain  expansion" Iris Loeb, "Questioning Constructive Reverse  Mathematics" Giovanni Sambin, "Real and ideal in mathematics: a dynamic view"	Nancy	September, 27th-28th, 2011
2nd Paris-Nancy Philosophy of Mathematics Workshop	Patricia Blanchette: Metatheory in Frege Francesca Boccuni: Plural logicism Jacques Bouveresse (TBC) John Burgess: Structure and rigor Gabrielle Crocco: Gödel and the problem of descriptive phrases Gilles Dowek: What is a theory? Mirja Hartimo: Husserl and contemporary trends in philosophy of mathematics Simon Hewitt: Faulting and Fixing Frege Ignasi Jané: An attempt at a faithful interpretation of set theory Osvaldo Ottaviani: "Forma dat esse rei". Transcendental approaches to philosophy of mathematics Francesca Poggiolesi: On the importance of being analytic. The paradigmatic case of the logic of proofs Davide Rizza: Applied Mathematics without mappings	Paris	November, 17th-19th, 2010
The Number Concept: Axiomatization, Cognition and Genesis	Andrew Arana: Purity in Arithmetic: Formal and Informal Issues Andrey Bovykin: The Next Generation of Ideas in Metamathematics of Arithmetic Véronique Izard: Two key premises of the concept exact numbers: Exact equality and successor function C.S. Jenkins: Arithmetic and Naturalism Matthew Katz: Writing and Reading Mental Magnitudes Felix Mühlhölzer: On the reference, interpretation and application of arithmetical terms Charles Parsons: The Kantian legacy in twentieth century foundations of mathematics Panu Raatikainen: Neo-Logicism and its Logic Daniel Sutherland: Kant on the Conditions of Arithmetical Cognition Sabetai Unguru: The Greek Concept of Number and Its Demise	Nancy	November 3rd-5th, 2010
8th IP Fellows' Seminar	Jean-Baptiste Joinet: Negation: from ideality to interaction Dirk Schlimm: Pasch's ideal of proof Mark van Atten: Intuitionism as phenomenology: a critique of Rota Irina Starikova: Intuition, Visualization, Geometric Shifts & Proofs	Paris	March 24th, 2010
"Mathesis metaphysica quadam": Leibniz, between Mathematics and Philosophy	Herbert Breger: The substructure of Leibniz's metaphysics Michel Serfati: Mathematics, metaphysics and symbolism in Leibniz: the principle of continuity Vincenzo De Risi: Leibniz's studies on the Parallel Postulate Philip Beeley: In deliberationibus ad vitam pertinentibus. Method and Certainty in Leibniz’s Mathematical Practice Richard Arthur: Leibniz’s Actual Infinite in Relation to his Analysis of Matter Samuel Levey: Leibniz's analysis of Galileo's paradox Emily Grosholz: The Representation of Time in Galileo, Newton and Leibniz Eberhard Knobloch: Analyticité, équipollence et la théorie des courbes chez Leibniz	Paris	March 8th-10th, 2010
7th IP Fellows Seminar	Andrew Arana: The complexity of pure and impure proof Jeremy Avigad: Understanding, formal verification, and the philosophy of mathematics Walter Dean: Models and recursivity Sean Walsh: Weierstrass, Frege, and Husserl on the Equality of Numbers	Paris	January 25th, 2010
6th IP Fellows Seminar	Davide Crippa: As simple as possible: Descartes´algebraic interpretation of Pappus´norm Henri Gallinon: The ideal of truth Richard Pettigrew: Prospects for a foundation for mathematics in category theory Andrei Rodin: Renewing foundations	Paris	November, 9th, 2009
Paris-Nancy Philmath Workshop	Paola Cantù: On Real and Ideal Elements in Mathematics Harvey Friedman: Conceptus Calculus Volker Halbach: Computational Structuralism Annika Kanckos: Hilbert’s second problem: a possible and necessary consistency proof proof Michael Potter: More on Replacement Davide Rizza: Discernibility by Symmetries Stewart Shapiro: Structures and Logics: a Case for Relativism Hourya Sinaceur: Objets mathématiques Claudio Ternullo: Why did Cantor Believe in the Truth of the Continuum Hypothesis? Sean Walsh: The Justification of Mathematical Induction	Nancy	October 21st-22nd, 2009
5th IP Fellows Seminar	Sean Walsh: The Role of Interpretability Results in the Justification of Axioms Andrew Arana: The geometric/algebraic distinction Walter Dean: Informal provability and Goedel’s modal interpretation of intuitionism	Paris	September 8th, 2009
The Imaginary, the Ideal and the Infinite in Mathematics	Andrew Arana: Purity and the identity of problems Michael Detlefsen: Freedom and Creativity in Mathematics Wilfrid Hodges: The literal meanings of statements in mathematical textbooks Paolo Mancosu: Measuring the size of infinite sets of integers: Was Cantor's conception of infinite number inevitable? Felix Mühlhölzer: Parsons on Mathematical Intuition and Natural Numbers Karl-Georg Niebergall: Assumptions of infinity in weak set theories and calculi of individuals Michael Potter: Classes as ideal elements and limitation of size Matthias Schirn: Frege's philosophy of geometry Ivahn Smadja: The Discrete and The Singular: Some Kroneckerian Ideas about Arithmetization Jamie Tappenden: Branching at Branch Points: Interpretations of "Riemann Surface" as a Fork in the Road to Contemporary Mathematics Albert Visser: Relativization and Thinking Modulo: How does it help us in Theory Reduction?	Pont-à-Mousson	June 25th and 26th, 2009
IP workshop	Juliet Floyd: Proof: Mathematical Knowledge, Sense, and Context Akihiro Kanamori: Mathematical Knowledge and Complexities in Proof	Paris	May 26th, 2009
Joint IP-REHSEIS workshop	Colin McLarty: Philosophy of Mathematics and Current Mathematics Steve Awodey: Ideals of Proofs	Paris	May 13th, 2009
4th IP Fellows Seminar	Paola Cantù: A Theory-relative Notion of Ideality Sebastien Maronne: Two Criteria for the Real/Ideal Distinction in early modern Geometry Mattia Petrolo: Classical logic under focus: ideality, proofs and computation Renaud Chorlay: Preference for the axiomatic: a case study	Nancy	April 30th, 2009
The Fundamental Idea of Proof Theory	Per Martin-Löf: Proof theory as conceived by Hilbert and logic in the traditional sense Albert Visser: Look again. Syntax is no syntax Göran Sundholm: Proofs as chains of mental operations; could Brouwer be right after all? Peter Schroeder-Heister: Bidirectional reasoning Dag Prawitz: Conflicting intuitions about deductive reasoning Rafael Nunez: Towards the cognitive foundations of proof Wilfried Sieg: Uncovering aspects of the mathematical mind	Paris	April 15th - 16th, 2009
Joint IP-REHSEIS workshop	David Rabouin: Leibniz on infinitesimals Bruno Belhoste & Karine Chemla: Poncelet’s ideal elements in geometry: between Carnot and Chasles Ivahn Smadja: Kronecker on ideal numbers and mathematical substance Brice Halimi: Ideal elements and extensions: the case of nonstandard set theory	Paris	March 5th, 2009
3rd IP Fellows Seminar	Sebastien Maronne: Ideal elements and projective geometry in early modern mathematics Oliver Schlaudt: Abstraction and Ideation. A constructivist approach to the nature of mathematical concepts Paul McCallion: Ideal numbers vs ideal numerical properties John Mumma: Contentful reasoning and rigor in elementary geometry	Paris	February 27th, 2009
2nd IP Fellows Seminar	Mattia Petrolo (IP Fellow): Ideal proofs and logical constructivity: From intuitionistic to classical logic Renaud Chorlay (IP Fellow): Ways out of the Grey Paola Cantù (IP Fellow): Ideal numbers and magnitudes: a matter of degree? Andrei Rodin (IP Fellow): How Mathematical Concepts Get Their Bodies: The example of Forcing Agustin Rayo (IP visitor): Towards a Trivialist Account of Mathematics * *You can view a draft of Pr. Rayo's paper at: http://web.mit.edu/arayo/www/km.pdf	Paris	January 22nd, 2009
Geometrical Thinking (workshop)	Jeremy Gray: Geometrical Thinking: the case of minimal surfaces Michael Hallett: Geometry and Number Douglas Jesseph: 'The Very Soul of Mathematics':  Barrow's Mathematical Lectures and the theory of Ratios in the Seventeenth Century Mary Domski: Kant on the Imagination and Geometrical Certainty Victor Pambuccian: Elementary geometries, groups, and fields: Similarities and differences Henk Bos: The early modern tradition of geometrical problem solving as context for new ideas about the nature of geometry	Nancy	December 15th - 16th, 2008
Fall IP Fellows Seminar	Dr Paul McCallion: Ideality and the integers Dr. Fabien Schang: An abstract object in logic: logical value, and its philosophical import Dr. John Mumma: The real and the ideal in complex projective space	Nancy	October 29th, 2008
Mathematical Understanding (workshop)	Andrew Arana Jeremy Avigad Karine Chemla Leo Corry Mic Detlefsen José Feirreros Jeremy Gray Gerhard Heinzmann Jesper Lützen Paolo Mancosu Ken Manders Philippe Nabonnand Marco Panza Thomas Ryckman Stewart Shapiro Erhard Scholz Ivahn Smadja Mark Steiner Jan van Plato Jean-Jacques Szczeciniarz Titles: t.b.a.	Paris	June 9th - 13th, 2008
Proof, Justification and Learning (workshop)	Pr. Sergei  Artemov: Justification logic Denis Bonnay: How formal is a formal proof? Kevin Kelly: Why Relations of Ideas are Matters of Fact: A Unified Theory of Theoretical Unification in Formal and Empirical Reasoning Pr. Rohit Parikh: Epistemology, pure and applied Manuel Rebuschi & Tero Tulenheimo: IF logic and the epistemology of mathematical objects	Nancy	May 26th - 27th, 2008
Visual Reasoning in Mathematics and the A Priori (workshop)	Pr. Marcus Giaquinto: Synthetic a priori knowledge in geometry: recovery of a Kantian Insight Dr. Valeria Giardino: Diagrammatic reasoning in mathematics: cognitive issues Dr. John Mumma: Does rigor require that everything be laid down in advance?  Diagrammatic vs. axiomatic proof in elementary geometry Pr. Lisa Shabel: Representation and Reasoning: Kant on Symbols, Diagrams and Mathematical Demonstration	Nancy	May 5th - 6th, 2008
Proof and Inference: IP inaugural symposium	Pr. Michael Detlefsen: Varieties of Completeness Pr. Dag Prawitz: The Validity of inference	Paris	Feb. 8th, 2008

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Third Paris-Nancy PhilMath Workshop

The third Paris-Nancy PhilMath Workshop (P-NPMW 3) will take place in Nancy, September 27-28, 2011, at the Maison des Sciences de l'Homme Lorraine (91 avenue de la Libération, salle Internationale).

Steering Committee: Andrew Arana, Mark van Atten, Michael Detlefsen, Paolo Mancosu , Marco Panza, David Rabouin, Jean-Jacques Szczeciniarz.

Program Committee: Denis Bonnay, Gabriella Crocco, Jacques Dubucs, Sebastien Gandon, Gerhard Heinzmann, Brice Halimi, Philippe Nabonnand, Andrei Rodin, Stewart Shapiro, Sean Walsh.

The schedule is as follows:

Tuesday, September 27th

10:30-10:45  welcome of participants 10:45-12:15  Jacques Dubucs, t.b.a. 12:15-14:00  lunch break

14:00-15.30  Ivanh Smadja, t.b.a. 15.30-17:00  Luca Incurvati, The graph conception of set 17:00-17.15  coffee break 17.15-18.45  Gianluigi Oliveri, On a New Argument for Realism in Mathematics 19:30 workshop dinner

Wednesday, September 28th

9:00-10:30   Michiel van Lambalgen, The completeness of Kant's Table of Judgements and its consequences for philosophy of mathematics 10:45-12:15  Jonathan Payne, Abstraction with domain expansion 12:15-14:00  lunch break

14:00-15.30  Iris Loeb, Questioning Constructive Reverse Mathematics 15.30-17:00  Giovanni Sambin, Real and ideal in mathematics: a dynamic view 17:30-18:30  organization of the P-NPMW 2012

For further information, please contact Andrei Rodin (rodin@ens.fr).

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Second Paris-Nancy PhilMath Workshop

The second Paris-Nancy PhilMath Workshop (P-NPMW 2) will take place in Paris, November 17-19, 2010, at the Ecole Normale supérieure (45 rue d'Ulm, salle Dussanne) and the University Paris Diderot (Paris 7 rive gauche, salle Klimt). This is the second in an annual series of workshops on the philosophy of mathematics organized by a team of scholars from Paris, Nancy and elsewhere in France.

The schedule is as follows:

Wednesday, November 17th (Ecole Normale supérieure, salle Dussanne)

Session I (10h00--11h15) Gilles Dowek What is a theory?

Session II (11h30--12h45) Francesca Poggiolesi On the importance of being analytic. The paradigmatic case of the logic of proofs

Session III (14h15--15h30) John Burgess Structure and rigor

Session IV (15h45--17h00) Francesca Boccuni Plural logicism

Thursday, November 18th (Ecole Normale supérieure, salle Dussanne)

Session V (9h00-10h15) Gabrielle Crocco Gödel and the problem of descriptive phrases

Session VI (10h30--11h45) Mirja Hartimo Husserl and contemporary trends in philosophy of mathematics

Session VII (12h00--13h15) Osvaldo Ottaviani "Forma dat esse rei". Transcendental approaches to philosophy of mathematics

Session VIII (14h15--15h45) Patricia Blanchette Metatheory in Frege

Session IX (16h00--17h00) Simon Hewitt Faulting and Fixing Frege

Friday, November 19th (Paris 7 rive gauche, salle Klimt)

Session X (9h00--10h15) Jacques Bouveresse TBC

Session XI (10h30--11h45) Davide Rizza Applied Mathematics without mappings

Session XII (12h00--13h15) Ignasi Jané An attempt at a faithful interpretation of set theory

For further information, please contact Marco Panza (Marco.Panza@univ-paris-diderot.fr).

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The Number Concept: Axiomatization, Cognition and Genesis

Nancy, November 3-5. A conference organized by Mchael Detlefsen, Gerhard Heinzmann and Sean Walsh, funded and supported by the IP project, the Poincaré Archives and the University Nancy 2.

Schedule and abstracts are available at: http://poincare.univ-nancy2.fr/Activites/?contentId=7073

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8th IP Fellows Seminar

The IP Fellows' Seminar will meet March 24th, 2010 from 09h30 to 17h25, at the Ecole Normale Supérieure, salle Favard in the new building of the ENS (46 rue d'Ulm, 75005, Paris), on the ground floor (enter the building and find a passage on your right).

The program is as follows:

Session I: 9h30--10h55 Dr. Jean-Baptiste Joinet (Philosophy, U of Paris 1; Preuves, Programmes et Systèmes), "Negation: from ideality to interaction" Abstract: Starting from the formalist/intuitionist debate about the ideality of negation in the first third of the XXth century, I will investigate the functional approaches of negation as a logical connective and the contemporary interactional approaches of negation as a protological   relation.

Session II: 11h00--12h25 Pr. Dirk Schlimm (Philosophy, McGill U), "Pasch's ideal of proof" Abstract: In his "Lectures on Newer Geometry" (1882) Moritz Pasch (1843-1930) clearly formulated the demand that deductions must be independent from the meanings of the non-logical terms involved, and he continued to elaborate on this view throughout the rest of his life. His growing concern for the justification of mathematical arguments led him to investigate the notion of consistency, to distinguish non-logical from logical components of expressions, and to the view  that "is part of the essence of pure deduction that every proof can be 'atomized', i.e., resolved into steps of certain kinds, or that it consists of a single such step" (1917). In this talk I will present some of Pasch's reflections on the ideals of mathematical rigor, which he hoped would lead to nothing less than a "renewal of logic" (1918).

Session III: 14h30--15h55 Dr. Mark van Atten (CNRS; U of Paris 1; IHPST), "Intuitionism as phenomenology: a critique of Rota" Abstract: In his paper `Husserl and the reform of logic', Gian-Carlo Rota contents, with Husserl, that the way of genetic phenomenology is inevitable to help us out of foundational predicaments in mathematics and the sciences. To my mind, the most successful application of (in effect) genetic phenomenological reflection to mathematics so far has been Brouwer's development of intuitionistic mathematics. In this talk, I will first explain and defend the claim that mathematical intuitionism should be considered a form of genetic phenomenology. This will then lead me to criticize another claim of Rota's, namely, that `The existence of mathematical items is a chapter in the philosophy of mathematics that is devoid of consequence' (in `The Barber of Seville, or the useless precaution'): not on the ground that the intuitionistic view on existence leads to a different ontology than that of classical mathematics, although it does, but on the ground that considerations on the mode of existence of mathematical objects have enabled intuitionism to introduce certain new principles of mathematical reasoning.

Session IV: 16h00--17h25 Ms. Irina Starikova (Philosophy, U of Bristol), "Intuition, Visualization, Geometric Shifts & Proofs'' Abstract: It is widely accepted that visualisations, either actual or in imagination, are useful illustrations in mathematical proofs, but there is no universal agreement that they may be more than just illustrations. In this paper, I make two claims: one is that visualisations play an important heuristic role in mathematics and in mathematical proofs in particular, and in addition, they may reliably play a justificatory role as well. The second claim is that another important part the visual plays in mathematics is facilitating new intuitions which may lead to significant conceptual reconstructions in the area and even be considered as foundational in respect to a particular subject. In my talk, I use several examples. One of them is from geometric group theory, which is now considered as one of the powerful and rapidly developing subjects in modern mathematics  and investigates the relations between geometric and algebraic properties of groups. The methodology of linking geometry with other areas of mathematics raises the question of distinction and interconnection between geometry and visual intuition, which I also bring to the discussion.  Finally, I consider the role of the use of visual representations of mathematical objects in the development of the concept of mathematical proof.

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"Mathesis metaphysica quadam": Leibniz, between Mathematics and Philosophy

The Ideals of Proof (IP) Project (ANR) and REHSEIS (UMR 7219, SPHERE) are pleased to announce a workshop on interrelationships between mathematics and philosophy in the thought of Leibniz. The workshop will take place Monday, March 8 through Wednesday, March 10, 2010. All meetings the first two days will be in the salle Dusanne of the ENS (45 rue d'Ulm, Paris). On the third day, the meetings will be in the Klee room, which is room 454A of the Condorcet building, on the Grands Moulins campus of the U of Paris-Diderot. The talks and discussions are free and open to the public. All interested persons are warmly invited to attend.

The schedule for the workshop is as follows.

Monday, March 8th

Session I (9h00-10h50) Herbert Breger (Leibniz-Archiv, Hannover) "The substructure of Leibniz's metaphysics" Abstract: Leibniz believed that his philosophy was nearly mathematical or could be transformed into mathematical certainty. We know that he thereby alluded to his project of a characteristica universalis.  Although we are not convinced of Leibniz's claim, we can recognize a substructure of Leibniz's metaphysics which is mathematical or is built on notions of philosophy of mathematics.

Session II (11h00--12h50) Michel Serfati (IREM, Université Paris VII) "Mathematics, metaphysics and symbolism in Leibniz: the principle of continuity" Abstract: On the basis of the epistemological analysis of several Leibnizian mathematical examples I will first attempt to show how Leibniz’s “principle of continuity” (which is, in fact, a meta-principle) belongs to the conceptual framework of what he calls “symbolic thought”, at least insofar as its mathematical implementations are concerned. I will then show how the ambiguity of the mathematical and metaphysical status of the principle engendered controversies between Leibniz and some of his correspondents. In light of these several examples we will also appreciate Leibniz’s stand vis-à-vis a central question, underlying this study, namely the architectonic character in mathematical thought of the use of signs, and especially this crucial point, namely the capacity of the mathematician to remain at the symbolic level, to continue to work at that level, retarding as much as possible the appeal to meanings ; this is a primordial prescription for Leibniz – deriving from what he calls the “autarchic” character of the sign. Finally, it will be shown how this seventeenth century “principle” remains to these days fully operational in research and teaching. Let us repeat, emphasizing Leibniz’s glory: these epistemological procedures, this “continuity schema” seen as interiorized methodological guide, this specific conception of continuity as a regulative principle – all these concepts that are so familiar to present day mathematicians that one can hardly imagine that they were not so before, were in fact ignored before him, especially by Descartes. Nowadays, therefore, 300 years after Leibniz, the “principle of continuity” belongs to an interiorized set of methodological rules. Like the principle of homogeneity (considered as normative) or that of generalization-extension (considered as a standard procedure of construction of algebraic or topological objects), they constitute part of the daily mathematical practice.

Session III (14h30--16h20) Vincenzo De Risi (Humboldt Fellow, Technische Universität, Berlin) "Leibniz's studies on the Parallel Postulate" Abstract: The development of non-Euclidean geometries is often regarded as an exemplar topic to consider the relations between philosophy and mathematics. Although the philosophical import of the foundational studies on the Parallel Postulate was clearly recognized only in the second half of the 19th century, there is no doubt that a number of mathematicians and philosophers were well aware of the metaphysical and epistemological issues raised by the theory of parallelism already in the 17th and 18th centuries. I would like to show the most important contributions of Leibniz to the geometrical foundations of a theory of parallels and their philosophical consequences for a full-fledged monadology, framing his considerations in the more general conceptual development which eventually gave birth to the modern concept of space.

Tuesday, March 9th

Session IV (9h00-10h50) Philip Beeley (Linacre College, Oxford) "In deliberationibus ad vitam pertinentibus. Method and Certainty in Leibniz’s Mathematical Practice" Abstract: Already in his earliest philosophical writings, Leibniz was concerned to account for the success in the application of the mathematical sciences to our understanding of nature. In this context he developed a sophisticated concept of negligible error which would later stand him in good stead in his mathematical writings, particularly those which are now seen to have played a decisive role in the emergence of modern analysis. The paper examines the historical and methodological context of Leibniz’s error concept and shows how it serves to exemplify the profound ways in which Leibniz’s philosophical deliberations on the application of mathematics informed his mathematical practice.

Session V (11h00--12h50) Richard Arthur (Department of Philosophy, McMaster University) “Leibniz’s Actual Infinite in Relation to his Analysis of Matter” Abstract: In this paper I examine some aspects of the relationship between Leibniz's thinking on the infinite and his analysis of matter. I begin by examining Leibniz's quadrature of the hyperbola, which demonstrates the connection between his work on infinite series, his upholding of the part-whole axiom, and his consequent fictionalist understanding of the infinite and infinitely small. On that conception, to say that there are actually infinitely many primes, for example, is to say that there are more than can be assigned any finite number N. There is no such thing as the collection of all primes, nor a number of them that is greater than all N, although one can calculate with such a number as a fiction under certain well-defined conditions. I then show how this conception fits with Leibniz's conception of the infinite division of matter, relating it to his principle of continuity, and  contrasting his position on the infinite and the composition of matter with those of Georg Cantor.

Session VI (14h30--16h20) Samuel Levey (Department of Philosophy, Dartmouth College) "Leibniz's analysis of Galileo's paradox" Abstract: In "Two New Sciences" Galileo shows that the natural numbers can be mapped into the square numbers, yielding the seemingly paradoxical result that the naturals are both greater than and equal to the squares. Galileo concludes that in the infinite the terms 'greater', 'less', and 'equal' do not apply. Leibniz rejects this on the ground that it would violate the axiom, from Euclid, that the whole is greater than the part, and argues instead that the paradox shows the idea of an infinite whole to be contradictory. Russell and others have rejected Leibniz's analysis as resting on a crucial ambiguity in Euclid's Axiom, and Galileo's paradox is typically resolved by abandoning a key premise. Leibniz has some defenders who point out that Russell's criticisms proceed from a distinctly post-Cantorian standpoint. In this paper I argue that Leibniz's analysis involves a more subtle error, and even granting Galileo's premises as well as Euclid's Axiom it does not follow that the idea of an infinite whole is contradictory. This follows only on a "strong" definition of 'infinite', whereas Leibniz's own (now standard) definition allows infinite wholes consistently with Galileo's paradox. The details involved cast light well into the recesses of Leibniz's philosophy of mathematics.

Wednesday, March 10th

Session VII (9h00--10h50) Emily Grosholz (Department of Philosophy, Pennsylvania State University) "The Representation of Time in Galileo, Newton and Leibniz" Abstract: I revisit the representation of time in the writings of Galileo, Newton and Leibniz, to see whether they treat time as concrete, choosing representations that allow us to refer with reliable precision, or as abstract, choosing representations that allow us to analyze successfully, locating appropriate conditions of intelligibility. I interpret the conflict between Leibniz and Newton as due to differences in their choice of representation, and argue that the conflict has not been resolved, but remains with us.

Session VIII (11h00--12h50) Eberhard Knobloch (Institut für Philosophie, Wissenschaftstheorie, Wissenschafts- und Technikgteschichte, Technische Universität Berlin) "Analyticité, équipollence et la théorie des courbes chez Leibniz" Abstract: Avant et après l'invention de son calcul différentiel, Leibniz s'appuyait sur les notions d'analyticité et d'équipollence de lignes et de figures. Ces deux notions jouent un rôle essentiel dans les mathématiques leibniziennes. Les deux parties de la géométrie ont besoin de deux différents types d'analyse tandis que l'existence de l'analyse entraîne la géométricité des objets. Plus tard, Leibniz a donné une autre justification pour la géométricité d'une courbe. Qu'est-ce que veut dire 'analytique' ou 'équipollent'? La conférence essaiera de clarifier cette question. En plus,, elle va expliquer la classification leibnizienne des courbes, en particulier celle de la 'Quadrature arithmétique du cercle etc.', leur géométricité et va démontrer quelques théorèmes généraux sur les courbes analytiques simples.

For further information, please contact either of the workshop organizers, Michael Detlefsen (mdetlef1@nd.edu) or David Rabouin (David.Rabouin@ens.fr).

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7th IP Fellows Seminar

The IP Fellows' Seminar will meet January 25th, 2010 from 09h00 to 16h20, at the Ecole Normale Supérieure, salle Dussane, 45 rue d'Ulm, 75005, Paris.

The program is as follows:

Session I (9h00--10h20) Andrew Arana, IP Fellow, Assistant Professor of Philosophy, Kansas State University "The complexity of pure and impure proof " Abstract: It has often been claimed that pure proofs are less valuable than impure proofs because of the relative "difficulty" or "complexity" of pure versus impure proofs. Whatever advantages pure proof may have over impure proof would be countered by disadvantages if impure proof is systematically "easier'' or "simpler" than pure proof. In order to evaluate this claim, it is helpful to identify precise measures of proof complexity for which the claim may be tested. In this talk we single out a "topological" measure of proof complexity studied in recent work by Alessandra Carbone, and evaluate the thesis for that measure.

Session II (10h30--11h50) Jeremy Avigad, Professor of Philosophy and Mathematical Sciences, Carnegie Mellon University "Understanding, formal verification, and the philosophy of mathematics" Abstract: The philosophy of mathematics has long been focused on determining the methods that are appropriate for justifying claims of mathematical knowledge, and the metaphysical considerations that make them so. But, as of late, a number of philosophers have noted that a much broader range of normative judgments arise in ordinary mathematical practice; for example, questions can be natural, theorems important, proofs explanatory, concepts powerful, and so on. Such judgments are often viewed as providing assessments of mathematical understanding, something more complicated and mysterious than mathematical knowledge. Meanwhile, in a branch of computer science known as "formal verification," interactive proof systems have been developed to support the construction of complex formal axiomatic proofs. Such efforts require one to develop models of mathematical language, inference, and proof that are more elaborate than the simple foundational models of the last century. In this talk, I will explain how these models illuminate various aspects of mathematical understanding, and discuss ways that such work can inform, and be informed by, a more robust philosophy of mathematics."

Session III: 13h30--14h50 Walter Dean, Assistant Professor of Philosophy, U of Warwick "Models and recursivity "

Session IV: 15h00--16h20 Sean Walsh, IP Fellow, Joint PhD Program in Logic and Foundational Studies, U of Notre Dame "Weierstrass, Frege, and Husserl on the Equality of Numbers" Abstract: From the early 1860s until the mid 1880s, Weierstrass lectured on introductory complex analysis once every two years. One of the persistent features of these lectures was a long introductory section on the development of extensions of the natural number concept and the treatment of infinite sequences. This introductory section of Weierstrass' lectures was discussed by both Frege and Husserl in their published and unpublished writings on the philosophy of arithmetic. In this talk, I focus on contextualizing and evaluating Weierstrass, Frege, and Husserl's views on Hume's Principle and the equality of numbers."

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6th IP Fellows Seminar

The IP Fellows' Seminar will meet November 9, 2009 from 09h00 to 17h15, at the Ecole Normale Supérieure, salle Jules Ferry, 29 rue d'Ulm, 75005, Paris).

The program is as follows:

Session I (9h00--10h30) Andrei Rodin "Renewing foundations"

Session II (10h45--12h15) Henri Gallinon "The ideal of truth"

Session III (14h00--15h30) Davide Crippa "As simple as possible: Descartes´algebraic interpretation of Pappus´norm"

Session IV (15h45--17h15) Richard Pettigrew "Prospects for a foundation for mathematics in category theory"

Paris-Nancy Philmath Workshop

The first Paris- Nancy Workshop in the Philosophy of Mathematics will take place in Nancy, October 21-22, in the building of the Maison des sciences de l'Homme Lorraine, 91, avenue de la Libération,  BP 454, F-54 001 Nancy cedex, 3e étage, salle Internationale (see the map). This workshop is co-organized by Henri Poincaré Archives, IHPST and REHSEIS, and supported by the IP project.

Steering Committee: Michael Detlefsen, Jacques Dubucs, Sébastien Gandon, Gerhard Heinzmann, Jean-Jacques Szczeciniarz.

Program Committee: Michael Detlefsen, Marco Panza, Gabriel Sandu, Ivahn Smadja, Mark van Atten.

The program is as follows:

Session I (9h00-10h30) Hourya Sinaceur, IHPST, CNRS/Paris I "Objets mathématiques" (10h30-10h45) Break (10h45-12h15) Sean Walsh, IP, Nancy 2/Paris VII "The Justification of Mathematical Induction" (12h15-14h15) Lunch & Break

Session II (14h15-15h45) Stewart Shapiro, Dept. of Philosophy, The Ohio State University & University of St Andrews "Structures and Logics: a Case for Relativism" (15h45-17h15) Claudio Ternullo, University of Liverpool "Why did Cantor Believe in the Truth of the Continuum Hypothesis?" (17h15-17h30) Break (17h30-19h00) Paola Cantu, IP, Nancy 2/Paris VII "On Real and Ideal Elements in Mathematics"

Session III (8h30-10h00) Volker Halbach, Faculty of Philosophy, University of Oxford "Computational Structuralism" (10h05-11h35) Annika Kanckos, Dept. of Philosophy, University of Helsinki "Hilbert’s second problem: a possible and necessary consistency proof" (11h35-11h45) Break (11h45-13h15) Michael Potter, Faculty of Philosophy & Fitzwilliam College, University of Cambridge "More on Replacement" (13h15-14h30) Lunch & Break

Session IV (14h30-16h00) Harvey Friedman, Dept. Of Mathematics, The Ohio State University "Conceptus Calculus" (16h05-17h35) Davide Rizza, University of East Anglia "Discernibility by Symmetries"

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5th IP Fellows Seminar

The IP Fellows' Seminar will meet again September 8, 2009 from 10h30 to 17h15, at the Ecole Normale Supérieure (45, rue d’Ulm, 75005, Paris), Immeuble Rataud, salle INFO1

The program is as follows:

Session I (10h30--12h00) Sean Walsh "The Role of Interpretability Results in the Justification of Axioms"

Session II (14h00--15h30) Andrew Arana "The geometric/algebraic distinction"

Session III (15h45--17h15) Walter Dean "Informal provability and Goedel’s modal interpretation of intuitionism"

How to get there:

The Immeuble Rataud is a new recently constructed building of ENS; it is located behind the old historical building of the school. To get to Immeuble Rataud you should enter the old building by the main entry, cross the yard, and finally cross the part of the old building behind the yard. La salle (room) INFO 1 is located underground, so after entering the Immeuble Rataud take the elevator and go down to the level - 2.

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The Imaginary, the Ideal and the Infinite in Mathematics

On  June 25th and 26th the IP project will jointly sponsor a workshop with the TransCoop Project of the Alexander von Humboldt Stiftung. The name of the conference is "The Imaginary, the Ideal and the Infinite in Mathematics." Speakers include Andrew Arana, Michael Detlefsen, Wilfrid Hodges, Paolo Mancosu, Felix Mühlhölzer, Karl-Georg Niebergall, Michael Potter, Matthias Schirn, Ivahn Smadja, Jamie Tappenden and Albert Visser. 

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IP Workshop

An IP workshop will take place on May 26, 2009. Place: Salle Campanille (room 897C), Grands Moulins, University of Paris-Diderot. Time: 14h00--18h00

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Joint IP-REHSEIS workshop

A joint IP-REHSEIS workshop will take place in Paris, Thursday, May 13th, at 9h30 in the REHSEIS building, salle des séminaires [map here].

The schedule is as follows:

Session I (9h00--10h55) Steve Awodey (Carnegie Mellon University) "Ideals of Proofs" Under the "Propositions as Types" or "Brouwer-Heyting-Kolmogorov" interpretation, systems of type theory represent the proof theory of a logic, rather than just the property of provability, as modeled in conventional semantics. Thus for instance typed lambda calculus presents the proof theory of propositional logic, and dependent type theory that of (intuitionistic) first-order logic. New methods in algebraic logic show how to extend such systems of type theory by "ideal elements", in the traditional sense, and in much the same way that e.g. the complex numbers extend the reals. As in that case, the extended domain of "ideals" admits solutions to equations not solvable in the original system. One fundamental such equation is P(X) = X, where P is the powerset operation. An ideal solution to this equation is then a model of set theory extending the original system of type theory conservatively.

Session II (11h05--13h00) Colin McLarty (Case Western Reserve University "Philosophy of Mathematics and Current Mathematics" It has been asked “Why should philosophy of mathematics have to use the very latest math? Couldn't it all be done from Euclid's Elements?” The answer is almost yes. The great issues already appear in Euclid. But we need two caveats. First, mathematics is not just a body of theorems open to conceptual critique, but also a drive in which people make progress. That drive is hard to see when you focus only on already-solved problems. The second follows from the first: the issues may all appear in Euclid but very great answers have been given since then and are still being given by mathematicians. This talk will look at what the drive for purity of method meant to leading 20th century mathematicians, how it affected today's working ideas of “structure”, and what that says about the ontology and the foundations of mathematics.

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4th IP Fellows Seminar

The 4th IP Fellows Seminar will take place in Nancy on April 30th. All sessions will be held in la salle B12 of the IRT (Institut Régional du Travail), 138 avenue de la Libération, Nancy. Map: http://www.univ-nancy2.fr/presentation/plans/irt.html?depuis_id=557

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The Fundamental Idea of Proof Theory

The Ideals of Proof  chaire d'excellence research group is sponsoring a two day workshop April 15th and April 16th in Paris on The Fundamental Idea of Proof Theory.

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Joint IP-REHSEIS workshop

A joint IP-REHSEIS workshop will take place in Paris, Thursday, March 5th, at 9h30 in the REHSEIS building [map here].

The schedule is as follows:

Session I (9h30--11h00) David Rabouin "Leibniz on infinitesimals"

Session II (11h15--12h45) Bruno Belhoste & Karine Chemla "Poncelet’s ideal elements in geometry: between Carnot and Chasles"

Session III (14h00--15h30) Ivahn Smadja "Kronecker on ideal numbers and mathematical substance"

Session IV (15h45--17h15) Brice Halimi "Ideal elements and extensions: the case of nonstandard set theory"

General discussion (17h15--18h00)

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3rd IP Fellows Seminar

The third IP Fellows Seminar will take place in Paris, Friday, February 27th, at 9h30 in salle 417B of the Halle aux Farines building on the Grands Moulins campus of University Paris-Diderot.

The schedule is as follows:

Session I (9h30--11h00): Sebastien Maronne "Ideal elements and projective geometry in early modern mathematics"

Session II (11h15--12h45): Oliver Schlaudt "Abstraction and Ideation. A constructivist approach to the nature of mathematical concepts"

Session III (Focus Session) (14h00--15h30): Paul McCallion "Ideal numbers vs ideal numerical properties"

Session IV (Focus Session) (15h45--17h15): John Mumma "Contentful reasoning and rigor in elementary geometry"

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2nd IP Fellows Seminar

The next IP Fellows Seminar will take place Thursday, January 22nd. We are pleased to have as a special guest Pr. Agustin Rayo (Philosophy, MIT) who will give the final talk of the Seminar.

The schedule is as follows:

Session I (9:00--10:25): Mattia Petrolo (IP Fellow) "Ideal proofs and logical constructivity: From intuitionistic to classical logic"

Session II (10:30--11:55): Renaud Chorlay (IP Fellow) "Ways out of the Grey"

Session III (13:15--14:40): Paola Cantù (IP Fellow) "Ideal numbers and magnitudes: a matter of degree?"

Session IV (14:45--16:10): Andrei Rodin (IP Fellow) "How Mathematical Concepts Get Their Bodies: The example of Forcing"

Session V (16:15--17:45): Agustin Rayo (IP visitor) "Towards a Trivialist Account of Mathematics"

All sessions will take place in the salle Dussane in the old historical building of the ENS, 45, rue d'Ulm, 75005, Paris.

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Geometrical Thinking

Workshop to be held at the Université Nancy 2, Amphithéâtre Maison des Sciences de l'Homme Lorraine, December 15th - 16th, 2008. 

The schedule is as follows:

Monday 15

Session I (10h00-12h00): Jeremy Gray (Open University, Department of Mathematics and Statistics) "Geometrical Thinking: the case of minimal surfaces" Modern mathematics presents two obstacles to the  view that thinking geometrically is involved in any problem to do with size and shape, however formulated. The first is the view that modern mathematics is abstract and conceptual; it is no longer about naïve ideas of size and shape and no longer corresponds to any useful definition of geometric thinking. This objection is answered by the second obstacle, which asserts that there are fundamental ways of thinking mathematically, but that thinking geometrically is not one of them; it has been consumed by thinking in topological, analytic or algebraic terms. This paper takes as an example of modern mathematical practice the solutions by Jesse Douglas and Tibor Rado of the Plateau Problem in the 1930s, and argues that in any given setting forms of mathematical thinking can be identified by identifying the mathematical idea(s) that govern the argument. That involves reading mathematical arguments in a structured way that can distinguish between mathematical objects or concepts and how they are treated. Thus a geometric concept may be handled in an analytic way, which would not be per se an example of thinking geometrically, and another argument, although heavily analytic in parts, may overall form an example of geometric thinking. The claim will be made that many of the most interesting examples of thinking geometrically are of a mixed nature, but are no less geometric for that.

Session II (14h30-16h00): Michael Hallett (McGill University, Department of Philosophy) "Geometry and Number" What is the relationship between geometry and number? There were various attempts in the nineteenth century to effect a reduction of geometry to forms of analysis, one being the \emph{Erlanger Programm}. Against these, there  were the  various attempts to develop geometry synthetically. Hilbert's great transformation of geometry in 1899 was an attempt both to   pursue geometry synthetically as far as possible, and yet to show how number can be (in a sense) built in to geometry with synthetic geometrical principles. The result is a much more abstract, algebraic approach, which is at the same time an explanation of why analytic geometry works.

Session III (16h15--18h15): Douglas Jesseph (South Florida University, Department of Philosophy) "'The Very Soul of Mathematics':  Barrow's Mathematical Lectures and the theory of Ratios in the Seventeenth Century" This talk is concerned with Isaac Barrow's defense of the classical Euclidean theory of ratios against "innovators" who proposed alternative accounts of the nature of ratios. On Barrow's view, the theory of ratios is "the very soul of mathematics" and a defense of the classical doctrine is an essential part of his program to see geometry established as the one true foundation for all of mathematics.  I consider Barrow's arguments for the superiority of the classical theory of ratios and its role in geometry, as well as his reply to the objections of John Wallis, Thomas Hobbes, and Giovanni Borelli. In the end, I conclude that Barrow's defense of the classical approach to ratios is due in large part to his conception of geometric demonstration as founded in the consideration of true causes that are understood by attending to the motions by which geometric magnitudes are brought into being.

Tuesday 16

Session IV (10h00-12h00): Mary Domski (University of New Mexico, Department of Philosophy) "Kant on the Imagination and Geometrical Certainty" While it has long been recognized that the imagination contributes  to the universality of geometrical knowledge in the First Critique,  I suggest that that the peculiar interpretation of the imagination  that Kant forwards in the Critical period also helps secure the  certainty he attributes to geometrical knowledge.  To make my case,  I consider two accounts of geometrical certainty forwarded prior to  1787.  One account is from Kant himself, as presented in his 1764  Prize Essay, and the other is from Locke, as presented in his 1689  An Essay Concerning Human Understanding.  While neither Locke nor  the pre-critical Kant appeal to the imagination per se, each does  appeal to the power that the mind (or understanding) has to  perceive the geometrical objects presented before it, and each, as  we will see, links this power of mind to geometrical certainty.  I  want to highlight the problems that emerge from these accounts of  geometrical certainty in order to make better sense of why Kant, in  1787, dissolves any connection between the imagination’s power of  perception – our “degree of sensibility” – and the certainty of  geometrical knowledge.

Session V (14h30-16h00): Victor Pambuccian (Arizona State University, New School of Interdisciplinary Arts & Sciences) "Elementary geometries, groups, and fields: Similarities and differences" A first look at similarities between classical first-order geometries and algebraic structures reveals a strong affinity to group theory, as well as an affinity to certain first-order theories of fields. A second look, informed by what Pappus called the method of analysis, reveals the genuinely geometrical thinking, which turns out to not be reducible to algebraic reasoning, by being removed from any form of computing. From a formal logic perspective, geometrical thinking turns out to be characterized by its non-equational nature, geometrical results requiring  relation symbols to be expressed, and  proofs thereof cannot be reduced to systems of equalities and inequalities.

Session VI (16h15--18h15): Henk Bos (Universiteit Utrecht, Mathematisch Instituut) "The early modern tradition of geometrical problem solving as context for new ideas about the nature of geometry" The early modern period saw the rise of analytic geometry, which showed that algebraic and geometrical procedures could be very effectively combined. Such a combination challenged the classical perception of the nature of pure geometry and of the boundary lines separating it from the neighboring areas arithmetic, algebra and practical geometry. Analytic geometry, and later the calculus, did indeed shift most traditional boundaries between mathematical disciplines. There were, also before Descartes, various attempts to merge algebra and geometry. I note three aspects of these attempts. (1)   The primary use of algebra in pure geometry was heuristic; it was to serve as analysis, in the sense explained by Pappus, of which early modern mathematicians had become aware through the publication of Pappus’ Collectio in 1588. (2)   Pappus’ text had also made mathematicians aware that an analysis for geometry was already on hand in the classical mathematical corpus, namely the practice of reasoning in terms of  “given” elements in geometrical figures, for which Euclid’s Data, available in print, was a kind of elementary handbook. Around 1600 it became common to refer to this type of analysis as the “ancient analysis” (“analysis veterum”) reserving  “new analysis” for the heuristic use of algebra. (3)   Both the old and the new analysis appear to have functioned near-exclusively for finding solutions of geometrical problems, not in finding proofs of theorems. I use the name “Early modern tradition of geometrical problem solving” to denote an extensive part of 16th- and 17th-century pure geometry which focused on the solution of geometrical problems. This was done in the classical style of Euclid, Apollonius and Archimedes. In classical Geek geometry there were two kinds of propositions, theorems and problems. A theorem stated a geometrical truth and required a synthetic proof (“QED”). A problem put forward a geometrical task and required a construction, together with a proof that the constructed object indeed satisfied the conditions of the task (“QEF”). Each had its own formal structure. Early modern pure geometry did not add many new theorems to the classical corpus. But there was an intense interest in solving geometrical problems. I find it significant, though perhaps not surprising, that the great advances in early modern mathematics, originated and, for a large part, took place in the context of a formalized, primarily task-oriented part of geometry. In my presentation I would like to explore relations between the special nature of early modern problem solving and the tendencies, accompanying the search for heuristic methods, to shift or break through the traditional boundary lines of geometry.

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Fall IP Fellows Seminar

Seminar to be held at the University of Nancy 2, Campus Lettres et Sciences Humaines, room G-04, October 29th. 

The schedule is as follows:

Session I (10h30-12h00): Paul McCallion (IP Fellow) "Ideality and the integers"

Session II (13h30-15h00): Fabien Schang (IP Fellow) "An abstract object in logic: logical value, and its philosophical import"

Session III (15h15--16h45): John Mumma (IP Fellow) "The real and the ideal in complex projective space"

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Proof, Justification and Learning

Workshop to be held at the University of Nancy 2, May 26th--27th. 

The schedule is as follows:

Monday, May 26, 10am--12pm, Session 1: Sergei Artemov (Computer Science, Mathematics and Philosophy, Graduate Center, CUNY; Mathematics, Moscow State University) "Justification Logic"

Monday, May 26, 2pm--4pm, Session 2: Denis Bonnay (Département d'Etudes Cognitives, Ecole Normale Supérieure) "How formal is a formal proof?"

Monday, May 26, 4:15pm--6:15pm, Session 3: Kevin Kelly (Philosophy, Carnegie Mellon University) "Why Relations of Ideas are Matters of Fact: A Unified Theory of Theoretical Unification in Formal and Empirical Reasoning"

Tuesday, May 27, 10am--12pm, Session 4: Rohit Parikh (Computer and Information Science, Brooklyn College and Mathematics, Philosophy and Computer Science, CUNY Graduate Center) "Epistemology, pure and applied"

Tuesday, May 27, 2pm--4pm, Session 5: Manuel Rebuschi (Lab. d'Hist. des Sciences et de Phil., Archives Henri Poincaré, Université Nancy 2) and Tero Tulenheimo (Academy of Finland) "IF logic and the epistemology of mathematical objects"

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Visual Reasoning and the A Priori

Workshop to be held at the University of Nancy 2 (Campus Lettres et Sciences Humaines, salle G-04), May 5th--6th. The workshop is sponsored by the IP project and is the first in a series of workshops and conferences to be held over the next four years.

The schedule is as follows:

Monday, May 5, 10am Session 1: Marcus Giaquinto, Department of Philosophy, University College London "Synthetic a priori knowledge in geometry: recovery of a Kantian Insight"

Monday, May 5, 2pm Session 2: Valeria Giardino, Institut Jean Nicod "Diagrammatic reasoning in mathematics: cognitive issues"

Monday, May 5, 4pm Session 3: John Mumma, Department of Philosophy, Carnegie Mellon University "Does rigor require that everything be laid down in advance? Diagrammatic vs. axiomatic proof in elementary  geometry"

Tuesday, May 6, 10am Session 4: Lisa Shabel, Department of Philosophy, The Ohio State University "Representation and Reasoning: Kant on Symbols, Diagrams and Mathematical Demonstration"

Tuesday, May 6, 2pm Session 5: Round table

If you would like to attend these workshops and would like to take a meal with the group, please let Fabien Schang (schang.fabien@voila.fr) know so that we can plan accordingly.

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