PhilMath Intersem est une coopération entre Notre Dame University et le laboratoire SPHERE (UMR7219), initiée avec le regretté professeur Michael Detlefsen.
PROGRAMME 2023
Les séances auront lieu l’aprèsmidi de 14h à 18h et suivie d’un apéritif. En dehors du 15 juin, chacun des deux exposés de l’aprèsmidi durera environ 1h, suivie d’une demiheure de questions.
Lundi, 5 juin  salle 454A du bât. Condorcet, Univ Paris Cité, 3 rue Elsa Morante Paris 75013

Juliette Kennedy, (University of Helsinki)
The supervenience of syntax on semantics in the foundational context
If a model class is a class of structures of the same similarity type closed under isomorphism, under what conditions can the class be said to have a natural syntax, or a natural logic? How to think about model classes that have no syntax, no notion of formula? More generally, does syntax always supervene on semantics? In this talk we present some old and new results dealing with these questions.

Gabriel Catren, (Université Paris Cité, SPHERE)
Abstraction, Equality, and Univalence
We shall propose a conceptualoriented discussion of the socalled Univalent Foundations Program, that is, of MartinLöf type theory enriched with a homotopic interpretation, together with the univalence axiom proposed by Voevodsky. In particular, we shall analyze whether Leibniz’s principle of the identity of indiscernibles holds or not in Univalent Foundations. We shall finally argue that univalence can be understood as a particular implementation of a constructive notion of abstraction that resolves so to speak Fregean abstraction.
Mercredi, 7 juin  salle 454A du bât. Condorcet, Univ Paris Cité, 3 rue Elsa Morante Paris 75013

Lavinia Picollo, (National University of Singapore) & Dan Waxman, National University of Singapore
On Arithmetical Pluralism
Arithmetical pluralism is the view that there is no one true arithmetic but many competing arithmetical theories, each true in its own language, all equally good from an objective standpoint. Pluralist views have recently attracted much interest but have also been the subject of significant criticism, most saliently from Putnam (1979) and Koellner (2009). These critics argue that, due to the possibility of arithmetizing the syntax of arithmetical languages, one cannot coherently say that arithmetic is a matter of `taste’ whilst consistency is a matter of fact. In response, some (e.g. Warren (2015)) have forcefully argued that Putnam’s and Koellner’s argument relies on a misunderstanding. In this paper we put forward a new argument on the side of the critics: appealing to internal categoricity results for arithmetic, we argue that arithmetical pluralism cannot coherently be maintained while supposing that the consistency of mathematical theories is a matter of fact after all.

Tim Button, (University College London)
Higherorder logic and internal categoricity
In Philosophy & Model Theory, Sean Walsh and I suggested that higherorder internal categoricity results could help to explain the precision of our mathematical concepts. In this talk, I will explain why higherorder results seem especially helpfully. I will discuss some weaknesses regarding firstorder results. I will comment on the reversemathematics of these higherorder results. And I will also consider some general reasons to embrace higher types within our homelanguage.
Vendredi, 9 juin  salle 628 du bât. Olympe de Gouges, Univ Paris Cité, 8 Rue Albert Einstein Paris 75013

Joel David Hamkins, (University of Notre Dame)
A deflationary account of Fregean abstraction in set theory, with Basic Law V as a ZFC theorem
The settheoretic distinction between sets and classes instantiates in important respects the Fregean distinction between objects and concepts, for in set theory we commonly take the universe of sets as a realm of objects to be considered under the guise of diverse concepts, the definable classes, each serving as a predicate on that domain of individuals. Although it is commonly held that in a very general manner, there can be no association of classes with objects in a way that fulfills Frege’s Basic Law V, nevertheless, in the ZF framework, it turns out that we can provide a completely deflationary account of this and other Fregean abstraction principles. Namely, there is a mapping of classes to objects, definable in set theory in senses I shall explain (hence deflationary), associating every firstorder parametrically definable class F with a set object εF, in such a way that Basic Law V is fulfilled:
εF=εG ⇔ ∀x (Fx ⇔ Gx)
Russell’s elementary refutation of the general comprehension axiom, therefore, is improperly described as a refutation of Basic Law V itself, but rather refutes Basic Law V only when augmented with powerful class comprehension principles going strictly beyond ZF, one amounting, I argue, to a truth predicate in Frege’s system. The main result therefore leads to a proof of Tarski’s theorem on the nondefinability of truth as a corollary to Russell’s argument, independently of Gödel. A central goal of the project is to highlight the issue of definability and deflationism for the extension assignment problem at the core of Fregean abstraction.

Gabriel Scherer, (INRIA)
Proof search and program identity
Programming language research produced a beautiful result called the CurryHoward Isomorphism, which establishes a strong link betweeen certain representations of formal proof and certain representations of formal computer programs. This isomorphism creates a bridge between proof theory and programming language theory that can transfer results and intuitions. In this talk, we hope to discuss notions of equivalence, identity and representation of proofs (in propositional intuitionistic logic) and programs (in the simplytyped lambdacalculus):
Equivalence. Programming languages have a natural notion of equivalence, two program fragments are equivalent if they behave in the same way when placed inside a largeer program. The notion of equivalence of proofs, on the other hand, has no obvious, clear definition.
Representation: proof theory has seen many different suggestions for representations of formal proofs; in particular, researchers try to capture the "identity" of proofs through representations where equivalent proofs have the same representation.
Some proof representations in particular, namely those based on "focusing", were suggested by studying the problem of proof search. We will discuss how these ideas can be transferred to formal programming languages, to suggest representations of certain programs that capture
their identity.
Mardi, 13 juin  salle 628 du bât. Olympe de Gouges, Univ Paris Cité, 8 Rue Albert Einstein Paris 75013

Matteo Bianchetti & Giorgio Venturi, (University of Pisa)
Formal Ontology and Mathematics. A Case Study on the Identity of Proofs
We propose a novel, ontological approach to studying mathematical propositions and proofs. We do so by developing what we call a “formal ontology” of proofs using semantic modeling tools (like RDF and OWL) developed by the computer science community. In this talk, (i) we describe this new approach and (ii), to provide an example, we apply it to the problem of the identity of proofs. In a nutshell, we will investigate the idea that the identity of proofs is connected to the ontology needed for its formalization. This will allow us to discuss another related problem: that of purity of methods and its topic conception illustrated by Detlefsen and Arana, in "Purity of proofs". After providing a few examples of proofs, analyzed through the lenses of formal ontology, we will end the talk by suggesting how this study can shed light on the concept of creativity in mathematics.

Paul Tran Hoang ,(South Puget Sound Community College)
Measuring Theory Through Structure
There is widespread sentiment among practicing logicians that the notion of biinterpretability deserves a privileged status as a criterion for when two (singlesorted or manysorted) firstorder theories are "theoretically equivalent." Despite this, the notion of biintepretability has largely been ignored by the philosophical literature.
The aim of this talk is to evaluate biinterpretability as a measure of theoretical equivalence. To do this, I first put forth an instrumentalist conception of theories according to which theories are instruments for pursuing a myriad of possibly incompatible scientific aims. I then explore one such aim, namely: to capture a class of mathematical structures of antecedent interest. With this aim in mind, I argue that there is good reason to think that biinterpretable theories are theoretically equivalent since they have corresponding structurally equivalent (and thus, share the "same" class of) models. This talk also addresses objections given or inspired by Tim Button and Sean Walsh, Hillary Putnam, and Kameryn J. Williams.
Jeudi, 15 juin  salle 628 du bât. Olympe de Gouges, Univ Paris Cité,
8 Rue Albert Einstein Paris 75013

Caroline Ehrhardt, (Université Paris 8)
What is a group? Historical circulation and identities of a mathematical object (18301900)
It is often said that the concept of group is due to Evariste Galois. In this paper, I would like to put this statement in perspective by asking what exactly is meant by it. More precisely, I will examine this mathematical object in the works of mathematicians who used it between the 1810s and the end of the 19th century. Why did they use it? How did they write it? To what point their works were link to Galois’? I hope to emphasize the fact the meanings given to a mathematical objects, as well as the practice it is associated to, are linked to historical contexts : what we call today “the group concept” is the result a historical process of readings and transmission through time of mathematical works.

Ivahn Smadja, (Université de Nantes)
"No mere play of wit”: Kummer’s chemical analogy revisited
In a letter to his former student Leopold Kronecker dated 14 June 1846, then in print a few months later, in a paper completed in September, Ernst Eduard Kummer developed a wellknown analogy between his newfound theory of ideal complex numbers and chemistry. Claiming that the “whole conceptual sphere of chemistry” evinced “striking agreement” with that of his extended number theory, he insisted that this analogy should not be deemed a “mere play of wit”. In his view, the reason for it would lay in the fact that both the “chemistry of natural substances” and the socalled “chemistry of complex numbers” should be considered as “realizations of one and the same fundamental concept of composition, although within different spheres of being”. The present paper aims at providing an interpretive framework for these puzzling statements. In so doing, it will shed light on shared concerns, common to both chemistry, then in flux, and mathematics, pertaining to individuation of either chemical substances or mathematical objects.
Elements of context will be adduced to put Kummer’s chemical analogy in perspective, starting with his early texts, the critical reviews he contributed to the Jahrbücher für wissenschaftliche Kritik, a journal founded in Berlin under Hegel’s aegis.

Ivan Marin, (Université de PicardieJules Verne & IMJ). Répondant : David Rabouin (Université Paris Cité, SPHERE)
Identification problems in and from Group Theory
Identification in mathematics can take various forms. One is related to the question of classification, very much in line with what happens in Natural Sciences (in the sense in which one can, for example, identify a given species by this or that feature in a classification). Another meaning is related to the way in which one may or may not recognize the identity between various objects (or various expressions of “the same” object). In this talk, I will explore how problems of identification appear in the framework of Group Theory. It will turn out that, in this setting, these two meanings are often intertwined, and that they are closely related to other identification problem in the seemingly remote question of identification of shapes and topological forms.