6 Octobre 2021 (October 6, 2021)

Salle du Théâtre, Grand Château.Université Côte d’Azur, Parc Valrose, Nice.

10.30 Welcome

11.00-12:15 Carlo Cellucci (Università Roma 1)* Introducing Heuristic Philosophy of Mathematics*

14:00 – 15:15 Jessica Carter (Aarhus University) *Pragmatic realism and social ontology*

15:30 – 16:45 Georg Schiemer (Wien Universität) *Symmetries and implicit structure*

17:00 – 18:15 Andrea Sereni (IUSS, Pavia) *Chaos in Heaven. Fictionalism, Definitions and Mathematical Creation*

Organisateurs: Paola Cantù (CNRS / Université Aix-Marseille), Frédéric Patras (CNRS/ Université Côte d’Azur

The workshop will not be broadcasted on the web.

Registration is free but mandatory. Please get registered by sending an email to paola dot cantu at univ-amu dot fr

*Green pass required to access the conference room.*

*Résumés*

*Jessica Carter (Center for Science Studies at Aarhus University: Pragmatic realism and Social Ontology*. According to some interpretations of the Philosophy of Mathematical Practice is a focus on human agents and their activities when formulating positions on the ontology and epistemology of mathematics. Recently J. Cole has formulated a position on the ontology of mathematics. Cole draws on tools from Social Ontology to say that mathematical domains are introduced by collective declarations and that they serve a representational function. He further argues that mathematical objects so introduced are objective, atemporal and exists by necessity. J. Ferreirós, on the other hand, formulates an epistemology of mathematics that is based on the view that mathematics is the outcome of various activities of human agents. In both accounts is the idea that mathematics consists of different “levels” or strata and that different levels interact in various ways. Mathematics dealing with ‘hypothetical states of things’ (Carter 2014) proposes a similar idea. In general, Mathematicians formulate hypotheses. Case studies from contemporary mathematics reveal that when new hypotheses (and the objects they refer to) are introduced they relate in various ways to previously studied domains. Inspired by readings of C.S. Peirce, I identify three types of “introduction processes” referred to as abstraction, generalization and abduction and argue that they conform to this description. In addition, a pragmatic view of reality is explored in the case of mathematics, that “the reality of a substance depends on the truth of statements concerning a more primary substance”. The main part of the talk will consist in an elaboration and critical assessment of the above formulated views: that it is possible to characterize the mathematical universe as consisting of different levels that are interconnected in various ways and the idea that there is a “bottom” level that is somehow related to the world of sensory appearances. In addition, I will express certain concerns about the historical arguments offered by Cole in support of his claim that mathematical domains exist by necessity and that they are atemporal.

*Carlo Cellucci (Università Roma 1) : Introducing Heuristic Philosophy of Mathematics*. The purpose of the talk is to outline the main aspects of heuristic philosophy of mathematics, which is supposed to be an alternative to mainstream philosophy of mathematics, the philosophy of mathematics that has been dominant in the past century. I will argue that heuristic philosophy of mathematics must not be confused with the philosophy of mathematical practice, because the latter is essentially continuous with mainstream philosophy of mathematics. What is more important, I will argue that heuristic philosophy of philosophy is not subject to the shortcomings of mainstream philosophy of mathematics. As an example, I will compare the approaches of heuristic philosophy of mathematics and mainstream philosophy of mathematics to some aspects of mathematics.

*Georg Schiemer Symmetries and implicit structure*. According to a dominant view in modern philosophy of mathematics, mathematics can be understood as the study of the abstract structure of objects such as groups, number systems, graphs, or topological spaces. But what precisely is the relevant intrinsic struc- ture of such mathematical entities? How can we think about their structural content? In the present present talk, I will compare two general ways to think about the implicit structure of entities of pure mathematics. According to the first approach, the struc- tural properties of such objects are specified with reference to formal languages, usually based on some notion of definability. According to the second approach, structures are determined in terms of invariance under symmetries. For instance, the structural prop- erties of a given mathematical system are often said to be those properties invariant under certain structure-preserving mappings between similar systems (see, e.g., [3]). In the talk, we will investigate these two approaches by drawing to a particular mathe- matical example, namely the study of incidence structures in finite affine and projective geometry. Given this case study, I give a philosophical analysis of the conceptual differences be- tween the two methods to express implicit structure. The talk will focus on two issues. The first concerns the conceptual motivation for treating mathematical structures in terms of the notions of definability and invariance under symmetries. As will be argued, both methods capture some form of “topic neutrality” underlying the structuralist ac- count of mathematics. In the case of invariance, this is due to the fact that mathematics is indifferent to the intrinsic nature of mathematical objects and thus also indifferent to arbitrary switchings of such objects in a given system. In the case of definability-based approaches, the relevant topic neutrality is related to the the “formality” of logic and the fact that adequate logical definitions should be reducible to statements about the primitive mathematical structure (see [1]). Secondly, we discuss the relevance of the two ways to think about implicit structure for our understanding of mathematical structuralism. Here, in particular, the focus will be on the notion of the equivalence of mathematical structures. Building on the existing literature on the topic, I will discuss two notions of structural equivalence that take into account not only the (axiomatically defined) primitive structure, but also its implicit structural content. The first notion is motivated by the idea of definable implicit structure and based on the notion of interpretability (compare, e.g., [2]). The second notion, in turn, is motivated by the invariant approach and based on the concept of “transfer principles” between structures. *References*

[1] D. Bonnay. Logicality and invariance. The Bulletin of Symbolic Logic, 14(1):29–68, 2008.

[2] T. Button and S. Walsh. Philosophy and Model Theory. Oxford: Oxford University Press, 2018.

[3] J. Korbmacher and G. Schiemer. What are structural properties? Philosophia Mathematica, https://doi.org/10.1093/philmat/nkx011, 2017.

*Andrea Sereni. Chaos in Heaven. Fictionalism, Definitions and Mathematical Creation*Mathematical fictionalism is a welcome view for those who want the benefits of a literal reading of mathematical discourse while dispensing with its apparent problematic ontological commitments. Fictionalism can and has been argued for in a number of different ways, and remains a controversial position. We argue that among possible motivations – epistemic, attitudinal, ontological – the most plausible route should be considered to be semantical. Fiction-introducing principles are much alike (and sometimes are) mathematical definitions. While fiction-introducing principles may involve acts of creation, they have rarely been compared to creative definitions. The paper investigates what the debate on creative definitions, in particular as regards Frege’s concerns and Dedekind’s remarks on this score, can contribute to the assessment of contemporary mathematical fictionalism.