Philmathmed 2026, Università di Roma La Sapienza, INDAM, April 27-29

[CANTU TESTA Paola]

PHILMATHMED 9 (2026). The narrative of the infinitesimal
April 27-29, 2026
Sala conferenze INdAM piazzale Aldo Moro 5 Roma

PROGRAM

Monday April 27
14h00 Welcome
14h30-16h00 Frédéric Patras
16h30-18h00 Fabien Carbo-Gil
18h00-19h30 Round table chaired by Thierry Paul

Tuesday April 28
9h30-11h00 Paola Cantù
11h30-13h00 Matteo Broggio
15h00-16h30 Claudio Bartocci
17h00-18h30 Tommaso Peripoli

Wednesday April 29
9h30-11h30 Round table chaired by Thomas Hausberger
with Viviane Durand-Guerrier, Laura Branchetti, Annalisa Cusi and Francesco Contel
11h30-12h00 General discussion

REGISTRATION: https://indico.math.cnrs.fr/event/15767/

WEBSITE: https://philmath.hypotheses.org/philmathmed-9-2026

ABSTRACTS

Claudio Bartocci, Models for infinitesimals
Origins and development of the main ideas of synthetic differential geometry will be discussed. Put in a nutshell, this theory aims at reproducing basic intuitive notions of calculus and differential geometry taking advantage of the intuitionistic internal logic of a suitable topos. It will be stressed that synthetic differential geometry, as originally envisaged by F.W. Lawvere in 1967 and further developed by E.J. Dubuc, A. Kock, I. Moerdijk, G.E. Reyes and others, has its roots in the pioneering work of A. Weil and C. Ehresmann. 

Matteo Broggio, From Infinitesimal Analysis to Intensive Reality: Hermann Cohen’s Neo-Kantian History of Calculus
The neo-Kantian philosopher Hermann Cohen was undoubtedly a maverick within the late nineteenth-century mathematical debate. Beginning in the first half of the 1880s, he diverged from the new orthodoxy of the “great triumvirate”, composed of Cantor, Dedekind and Weierstrass, and attempted to develop an alternative account of calculus’ foundations, based on the notion of intensive, or infinitesimally small, magnitude. At first, Cohen’s approach received only lukewarm support. However, in recent years, because of its non-conformity with the epsilontic doctrine, Cohen’s interpretation of calculus has regained scholarly attention in the context of non-standard analysis. Building on the current state of research, the talk aims to examine Cohen’s philosophy of mathematics, with particular attention to three aspects: (1) Cohen’s neo-Kantian methodology; (2) his interpretation of calculus as presented in Das Prinzip der Infinitesimal-Methode und seine Geschichte (1883); (3) the early reception of this work among German mathematicians. The question of the compatibility between Cohen’s conception of infinitesimals and non-standard analysis will ultimately be left open for discussion.

Paola Cantù, Veronese's non-Archimedean Continuity 
The paper focuses on the history of the “Archimedes axiom”, a name that was first given by Stolz in 1882 to a mathematical sentence occurring in Euclid’s Book V to express an intuitive content concerning quantity measurement in proportion theory. The name was later associated to the analysis of continuity in Hilbert’s Grundlagen, where it first expressed continuity tout court, and subsequently just a component of continuity, as in the second edition of the text, where Hilbert included it between the axioms of the fifth group together with a maximal axiom. It will be shown that beyond Euclid and Hilbert, a plethora of lesser-known authors essentially contributed to the axiomatic formulation of continuity (Stolz, but also Veronese and Baldus, for example). It will be claimed that alternative axiomatic formulations of the Archimedean axiom not only offered more rigorous formulations of some intuitive notion of continuity but allowed the latter to be specified and decomposed into simpler components. Different axiomatic presentations, which have followed one another historically and often emerged in a distant dialogue between various authors, have gradually clarified and sometimes broken down into distinct elements the intuitive content initially associated with the idea of measuring magnitudes, and then progressively refined with reference to certain minimal properties of continuity.

Fabien Carbo-Gil, Les infinitésimaux depuis la perspective ensembliste
De nos jours, c’est la théorie des ensembles qui sert usuellement de fondement à l’analyse et à la théorie de l’intégration, via notamment les notions topologiques de limite et de mesure. A la fin du XIXe siècle, la théorie Cantorienne des ensembles émerge dans un contexte de redéfinition du continu mathématique marqué notamment par un rejet des infinitésimaux. Nous souhaitons défendre l’idée que malgré cette opposition historique, la théorie des ensembles fournit des éléments qui éclairent la réflexion philosophique sur la nature et la fonction des infinitésimaux en mathématiques à plusieurs niveaux. 
Dans un premier temps, nous comparerons l’axiomatique ensembliste standard ZFC avec des axiomatiques non-standard qui autorisent l’existence d’infinitésimaux. Cette comparaison nous permettra de justifier le caractère à la fois théoriquement dispensable mais fécond en pratique de l’usage des infinitésimaux.
Dans un second temps, nous montrerons que la question de la légitimité de l’usage de l’infini pour comprendre le continu, qui a joué un rôle central dans les discussions autour des infinitésimaux à partir du XVIIe siècle, demeure une question centrale de la théorie des ensembles actuelle. Nous aborderons en particuliers quelques développements de la théorie des ensembles liés au problème du continu et aux grands cardinaux.

Frédéric Patras, Finite integration, from Euclid to Hilbert and beyond 
A very interesting problem, dual in a certain sense to the approach of integration through differential calculus, is the possibility to develop a finite theory of integration. "Infinitesimals" appear in that context although in a very specific form. The talk will discuss some key steps and ideas of the theory, featuring their meaning for the philosophy of mathematics and surprising connections with other domains of mathematics.

Tommaso Peripoli, Descartes as a Challenge to Naïve Platonism
My talk aims to show that naive platonism in the ontology of mathematics faces a serious challenge from the history of mathematics. In the development of mathematics, a deep and enduring interplay between mathematics and philosophy can be observed. Rigorous philosophical inquiry plays a crucial role in justifying the acceptance of certain mathematical entities or, conversely, in challenging their ontological legitimacy. This provides a form of justification for ontological beliefs that cannot be adequately grounded in the mere observation of mathematical practice, nor derived from mathematics alone. Briefly, according to mathematical platonism, mathematical entities exist independently of our minds. In its most naive version, platonism asserts that we can interpret mathematical assertions at face value: all mathematical entities present in our mathematical theories exist without further specifications regarding their existence (Linnebo, 2023). However, if we look closely at the history of mathematics, things are not so simple. In particular, considering early modern mathematics, naive platonism does not seem to be the best philosophical account for a correct understanding of mathematical entities. René Descartes’ work illustrates this tension well. In La Géométrie, he denied the possibility of accepting indivisibles, yet in other works he invoked infinitesimals, even though he considered them unknowable (e.g., he used indivisibles to find the area of the cycloid). Descartes believes that mathematics cannot use entities that it does not fully understand, such as indivisibles. However, in the Meditations (and in calculating the area of the cycloid), he also admits the possibility of using indivisibles. The ontological status of indivisibles is problematic. On the one hand, certain mathematical procedures seem to presuppose their existence; on the other hand, they are neither clearly defined nor rigorously characterized. Descartes himself maintained that they are in principle undefinable, since in his view infinity, properly speaking, is accessible only to God (Cfr., Letters to Marsenne (15 feb.1635, 13 nov.162). To sum up, my aim is to challenge naive platonism by showing that there are cases in which some mathematicians use entities in their theories and then deny their existence in the platonic world of mathematics.
Short Bibliography:

• Linnebo, Øystein (2023), "Platonism in the Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Summer 2024 Edition), Edward N. Zalta & Uri Nodelman (eds.), URL = <https://plato.stanford.edu/archives/sum2024/entries/platonism-mathematics/>.

• Panza, M. & Sereni, A. (2013). ‘Plato's Problem: An Introduction to Mathematical Platonism’. New York: Palgrave-Macmillan.

• Descartes, René (1644). Principia philosophiae. First edition. Amstelodami: Apud Ludovicum Elzevirium.

• — (1975). The Geometry of René Descartes. Trans. by David Eugene Smith and Marcia L. Latham. Original work published 1637. Chicago and London: Open Court.

• Jullien, Vincent (2006). “Les frontiéres des math ́ematiques cart ́esiennes”. French. In: Philosophie naturelle et géométrie au XVIIe siécle. Ed. by Vincent Jullien. Paris: Honoré Champion, pp. 311–353.

• — (2015a). “Descartes and the Use of Indivisibles”. In: Seventeenth-Century In- divisibles Revisited. Vol. 49. Science Networks. Historical Studies. Birkhäuser, pp. 165–175.

• — (2015b).“Explaining the Sudden Rise of Methods of Indivisibles”.In:Seventeenth- Century Indivisibles Revisited. Vol. 49. Science Networks. Historical Studies. Birkhäuser, pp. 1–18.

Round table. Didactical issues in proof and proving: students, infinitesimals and ChatGPT
Chair : Thomas Hausberger, University of Montpellier. Contributors: Viviane Durand-Guerrier, University of Montpellier, Laura Branchetti, University of Milan, Annalisa Cusi and Francesco Contel, Sapienza University of Rome
Are there gaps in the decimal line? In this panel, Viviane Durand-Guerrier will first address this question through the lens of the triad discreteness–density–continuity. She will then highlight its connections with didactical issues related to a fixed-point conjecture.
In a second contribution, Laura Branchetti will examine the transition from calculus to modern analysis, which raised a central ontological question concerning the status of infinitesimals. Initially used in Leibnizian calculus as effective yet ontologically ambiguous entities, infinitesimals were later replaced by the formalism of limits. She will analyze interactions with ChatGPT in the context of an AI-assisted mathematical activity, focusing in particular on issues of argumentation and proof involving the use and interpretation of infinitesimals.
Finally, Annalisa Cusi and Francesco Contel will further explore this direction by discussing GenAI-mediated interactions from the perspective of a reconceptualization of the instrumental genesis framework. Their contribution will focus on the issue of biased representations of mathematical knowledge and its didactical implications for activities involving the exploration of conjectures and the development of proofs.

Comité scientifique : Claudio Bartocci (DIMA, Università degli Studi di Genova) ; Francesca Biagioli (Università degli Studi di Torino) ; Paola Cantù (CNRS/Univ Aix-Marseille) ; Thomas Hausberger (IMAG Montpellier) ; Sébastien Maronne (IMT Toulouse) ; Frédéric Patras (CNRS /LJAD Nice) ; Silvia de Toffoli (IUSS Pavia), Thierry Paul (LYSM, CNRS Rome)

Comité d’organisation : Thierry Paul (LYSM, CNRS Rome), Thomas Hausberger (IMAG Montpellier) ; Frédéric Patras (CNRS /LJAD Nice)

Partenaires : RT CNRS SHS PHILMATH; ANR AXDEF; LYSM Laboratoire Ypatia des Sciences Mathématiques (CNRS Mathématiques).