13th French Philosophy of Mathematics Workshop

ATTENTION : Change of rooms !

The sessions of the Interepisteme workshop and the FPMW13 conference will be held on the 6,7,8 on the Valrose Campus (as announced), but in the

Salle du Théâtre

Grand Château
Université Côte d'Azur, Parc Valrose, Nice.

https://univ-cotedazur.fr/portails/portail-sciences-de-la-vie/documents-utiles/plan-du-campus

The Saturday morning session (the 9th) will be held in the

Salle du Conseil

of the Carlone Campus.

https://univ-cotedazur.fr/portails/portail-sciences-de-lhomme-et-de-la-societe/campus-carlone-et-campus-saint-jean-dangely

Practical information (location, access, accommodation) is given at the bottom of this page.

The conference is in person, the presentations will not be broadcasted on the web.

Registration is free but mandatory, in particular to ensure the management of sanitary constraints and lunch breaks as well as the conference dinner. Please register here: https://fpmw13.sciencesconf.org/registration before September 24, 2021.

A sanitary pass (green pass) will be required to access the conference room.

Program :

Thursday, October 7
Chair: Jessica Carter
9:00 - 10:30 : Valeria Giardino (CNRS, Institut Jean-Nicod) : Experimenting with triangles
11:00 - 12:30 : Matt Hare (Kingston University, London) : The Effective as the Actual in Jean Cavaillès

Chair: Jean-Baptiste Joinet
14:00 - 15:30 : Dominique Pradelle (Sorbonne University, Archives Husserl) : Idéalités mathématiques et structure d'horizon
16:00 - 17:30 : Eduardo Giovannini and Georg Schiemer (Wien Universität, Institut für Philosophie) : Hilbert's Early Metatheory Revisited

Friday, October 8th
Chair: David Waszek
9:00 - 10:30 : Patrick Popescu-Pampu (University of Lille, Laboratoire Paul-Painlevé) : Absences textuelles
11:00 - 12:30 : Joan Bertran-San Millán (Universidade de Lisboa) : Peano's structuralism

Chair: Emmylou Haffner
14:00 - 15:30 : Silvia De Toffoli (Princeton University) : Diagrams and the Aprioricity of Mathematics
16:00 - 17:30 : Round table :  What are the Current Trends in Philosophy of Mathematics? 

Saturday, October 9th
Chair: Sébastien Poinat
9:00 - 10:30 : Hourya Sinaceur (CNRS, IHPST) : Le Rein analytischer Beweis de Bolzano
11:00 - 12:30 : Walter Dean (University of Warwick) : Machine intelligence and intrinsic mathematical difficulty

Duration of the talks: 60 minutes of presentation and 30 minutes of discussion.
The languages of the workshop will be French and English.

Scientific committee: Andrew Arana, Julien Bernard, Paola Cantù, Viviane Durand-Guerrier, Christophe Eckes, Sébastien Gandon, Emmylou Haffner, Brice Halimi, Thomas Hausberger, Jean-Baptiste Joinet, Jean-Pierre Marquis, Baptiste Mélès (dir.), Marco Panza, Frédéric Patras, Jean-Jacques Szczeciniarz

Organizing Committee: Paola Cantù (CNRS, CGGG UMR 7304, Aix-Marseille University), Jean-Luc Gautero (CRHI, Université Côte d'Azur), Frédéric Patras (CNRS, LJAD UMR 7351, Université Côte d'Azur), Sébastien Poinat (CRHI, Université Côte d'Azur)

The conference is organized by the Université Côte d'Azur with the support of the Université, the Laboratoire Jean-Alexandre Dieudonné and the 5th axis of the MSHS. It is also financially supported by the Groupement de Recherche (GDR) en Philosophie des Mathématiques, of which it is one of the annual activities

https://philmath.hypotheses.org/

For more information, please send an email to: fpmw13@sciencesconf.org.

On October 6, the INTEREPISTEME workshop in philosophy of mathematics will take place in Nice (same room and same building). If you also wish to participate in this meeting, you will find the program at the following address: https://episteme.hypotheses.org/

Practical information


For those of you who do not know the Université Côte d'Azur in Nice, a map of Nice is available here

https://cartes.nicecotedazur.org/portal/apps/webappviewer/index.html?id=acac745ce5e84798a43fa639b7535f49

You should search for "28 Avenue Valrose" and you will find the location of the scientific campus whose entrance is at the intersection of Joseph Vallot and Valrose avenues.

You can ask the reception of your hotel for the best way to get there. An easy solution is to take the TRAM, line 1, and stop at "Valrose Université". The location of the campus is also well indicated on the tram map.

https://www.lignesdazur.com/ftp/document/plan-autour-du-tram-ligne-1-08-2021-web.pdf


For those of you arriving at the airport, there is a tram line that leads directly from the airport to the city center.

Nice has an impressive amount of hotels, the one reserved for the speakers is now full and the period is too busy for us to give uniform advice. A simple solution is to look for a hotel between the train station and the university, on tram line 1. Both the campus and the city center and seaside will be easily accessible on foot (or by tram).

Please note: on Saturday, October 9, the conference will be held in the Salle du Conseil on the Carlone Campus (formerly the "Faculté des Lettres, Arts et Sciences Humaines"), 98 boulevard Edouard Herriot (Nice). To access the Salle du Conseil, simply go up the main stairs, then turn right and go up to the 1st floor of building A (see map). The access for people with reduced mobility is 100 meters after the stairs, on the west side of the campus.  
 
To view the map, go to
https://univ-cotedazur.fr/portails/portail-sciences-de-lhomme-et-de-la-societe/documents-utiles

and click on "Campus Map", you will get it in pdf format.
 
To get to the Carlone Campus, you have different possibilities:

    By bus: the bus stop is called "Carlone" and is served by lines 6, 60 and 87.
    By streetcar: take the T2 line, Magnan stop, then take the bus (lines 6, 60 or 87) or walk to the Carlone Campus following the direction of "Faculté des Lettres" (10-15 minutes with a steep climb).
    By car: leave the highway at exit 50 Nice Centre, then continue on the Promenade des Anglais, turn left on Avenue Fabron, continue on Boulevard de Cambrai, at the traffic circle turn right on Boulevard Édouard Herriot.

See also

https://univ-cotedazur.fr/portails/portail-sciences-de-lhomme-et-de-la-societe/campus-carlone-et-campus-saint-jean-dangely

Joan Bertran San-Millan, Peano's structuralism
Recent historical studies have located in late nineteenth-century mathematics the first proponents of methodological structuralism. In this talk, I shall attempt to answer the question of whether Giuseppe Peano can be counted amongst the early structuralists. I shall focus on Peano’s understanding of the primitive notions and the axioms of arithmetic and geometry, and distinguish two phases in his axiomatisation of these theories. First, I shall argue that the undefinability of the primitive notions of arithmetic and geometry led Peano to the study of the relational features of the systems of objects that compose these theories. Second, I shall defend that Peano developed a schematic understanding of the axioms of arithmetic which, despite diverging in some respects from Dedekind’s construction of arithmetic, should be considered structuralist

Walter Dean, Machine intelligence and intrinsic mathematical difficulty. 
This paper provides a philosophical appraisal of recent work on mathematical proof discovery using methods from artificial intelligence.   We will begin by surveying techniques from traditional automated theorem proving (as applied, e.g., by McCune in his solution to the Robbins problem about Boolean algebras), SAT-solves (as applied, e.g., by Heule et al. in their recent solution to the Boolean Pythagorean Triple problem), as well as ongoing work which attempts to combine machine learning with logic-based techniques.   These studies will be used to inform a general argument about the limits of what might be accomplished with such techniques in light of computability and complexity-theoretic considerations.   We will finally suggest that these developments highlight several under-explored issues about attributions of difficulty to mathematical problems in relation to the open questions which we choose to investigate in practice.   This is joint work with Alberto Naibo (Université Paris 1 Panthéon-Sorbonne, IHPST).

Silvia De Toffoli, Diagrams and the Aprioricity of Mathematics
A widely held view among philosophers of mathematics is that neither diagrams nor visualizations can play a justificative role in proofs – call it the NO DIAGRAMS VIEW. To be sure, according to this view, diagrams and visualizations can be important and even indispensable for a subject to understand a proof. Still, their role remains enabling rather than justificative. One of the worries that immediately arise when challenging this view concerns the allegedly a priori status of proofs. In my talk, I will suggest that such worry is unwarranted. I will sketch a conception of the a priori according to which diagrams and visualizations can provide a priori justification.  I will conclude with a revindication of Kant’s claim that geometry is synthetic a priori. In so doing, I will contrast my interpretation of the Kantian claim with Giaquinto’s (2007) interpretation, which derives from his endorsement of a version of the NO DIAGRAMS VIEW.

Valeria Giardino, Experimenting with Triangles
There has been a long and extensive debate in philosophy about the functioning of thought-experiments in science (see Stuart et al. 2018). However, less work has been devoted to the specific case of thought-experiments in mathematics. Are there genuine thought-experiments in mathematics? According to Lakatos for example, proofs can be seen as thought-experiments suggesting a decomposition of the original conjecture into lemmas (Lakatos 1976). Would that mean that all reasoning in mathematics can be presented as a thought-experiment? In my talk, I will analyze the notion of thought-experiment in mathematics by considering three examples of reasoning with triangles (Klein, 1903; Giaquinto 2007; Bråting and Pejlare, 2008). This will bring me to the definition of a framework where mathematics is considered as threefold, in a continuous interaction between theory, experiment and technology (Rav, 2005); only some (thought) experiments can be considered as proofs, thanks to the constraints built-in or notified on the representations involved.

Eduardo Giovannini and Georg Schiemer, Hilbert's Early Metatheory Revisited
In this talk, we give a historically sensitive discussion of Hilbert's early metatheory of formal axiomatics. His work from the turn of the last century is often regarded as one of his most original contributions to the development of the "model-theoretic" viewpoint in modern logic and mathematics. We will re-assess Hilbert's role in the development of a model-theoretic conception of theories by focusing on two aspects of his early contributions to the axiomatic foundations of geometry and analysis. First, we examine Hilbert's understanding of mathematical languages and their interpretations; in particular, we argue that his early semantic views were informed by a particular notion of isomorphism. Second, we analyze Hilbert's reflections on the central concept of categoricity. For this purpose, we study a categoricity proof of the axiom system for real analysis sketched by Hilbert in the lecture course Logische Prinzipien des mathematischen Denkens from 1905.

Matt Hare, The Effective as the Actual in Jean Cavaillès
This paper will outline a reading of Jean Cavaillès' philosophical and epistemological works as developing a theory of logical time, orientated around two central concepts: concatenation [enchaînement] and the effective [l'effectif]. I will focus here on a double sense of the latter concept: 1) The effective as the actual; 2) The effective as the computable. I will trace the relation between these senses at two crucial points in the mathematical history of the concept, first during the reception of set theory in France among the French Empiricists (in particular with respect to methodological constraints advanced by Borel and Lebesgue), and second in the formation of the modern notion of effective calculability in the work of Gödel, Church and Kleene. Cavaillès treats the latter phase of this history at length in the posthumously published Transfini et continu (written 1940-1941), wherein Cavaillès claims that developments in the theory of recursive functions necessitate a fundamental revision of Kantian notions of intuition and of the transcendental, which are in turn described as different modalities of an effective process [procès effectif]. I will argue that Transfini et continu marks a shift in Cavaillès' thought concerning the effective, when contrasted with his earlier comments on Borel and Lebesgue, and one that points towards the role played by the concept in Cavaillès' final work, Sur la logique et la théorie de la science (1946). I will thus examine how Cavaillès' position changed in response to formal developments, and in doing so explore the relations between some important stages in the mathematical history of the effective and Cavaillès' often extremely condensed philosophical formulas concerning the concept.

Patrick Popescu-Pampu, Absences textuelles
En partant de l'article "La stabilité topologique des applications polynomiales" de René Thom, je présenterai quelques aspects des processus de recherche mathématique qui sont en général absents des textes présentant les résultats de la recherche. J'indiquerai aussi pourquoi je pense que leur inclusion enrichirait ces textes.

Dominique Pradelle, Mathematical idealities and horizon structure (in French)
The main purpose of this lecture will be to question the relevance of the Husserlian concept of horizon structure to characterize mathematical idealities, their ontological status and their mode of constitution: does ideal object consciousness imply a horizon structure? In other words, is the mathematical object reducible to a finite core of meaning, or does it envelop an indefinite horizon of determinacies and implicit relations, which commands an indefinite analysis? We will start from the opposition between the position of the young Derrida, for whom the mathematical object has the status of an ideal object, because it is transparent and exhausted in its phenomenality, and therefore almost immanent, and that of Desanti, for whom any elucidation of a mathematical ideality implies the unfolding of a thick space and of implicit dimensions of meaning. Next, we will elucidate the origin of the Derridean thesis from the classical thesis according to which thought finds in the mathematical object only what it has put there itself. We find this thesis in Leibniz (finite character of the analysis of mathematical notions), in Kant (character of arbitrarily composed notions of mathematical concepts) and in the Logic of Port-Royal and the "law of Port-Royal", which posits the equivalence between character of a concept and property of the objects it subsumes, as well as the inverse proportion between comprehension and extension of a concept. Finally, we will see how Bolzano, by clearly explaining the distinction between the character of a concept and the property of the corresponding objects, challenges the law of Port-Royal and the Derridean thesis, and restores all its relevance to the idea of horizon structure for mathematical objects.

Hourya Benis Sinaceur, L'analyse conceptuelle et le "Rein analytischer Beweis" de Bolzano
Que veut dire "analyse conceptuelle" lorsque cette expression est appliquée à la philosophie mathématique de Bernard Bolzano ? Il est principalement question de déterminer de manière précise la signification des concepts et propositions utilisés, d'établir des définitions pour les concepts primitifs, de rendre claires les connexions logiques entre propositions, de rapporter les vérités mathématiques à leurs fondements ultimes, de promouvoir l'exigence aristotélicienne de pureté des méthodes. Le présent exposé n'a pas pour but une réflexion généralisante sur ces thèmes bien connus. Je m'y applique simplement à un examen attentif de la facture du "Rein analytischer Beweis...", où l'on retrouve incarnée la perspective sémantique et strictement accomplie l'exigence de rigueur déductive. Je puise dans les premiers écrits de Bolzano des éléments directeurs pour cette lecture, qui redresse, selon moi, une interprétation récente.