Metaphysical generality and mathematical genericity, submitted to Synthese.
Crampe mentale, flexion et réflexion. Commentaire d'un texte tiré du Cahier bleu de Wittgenstein, Les Etudes philosophiques N°151, 2024, p. 43-54.
Geometric Modal Logic, Notre Dame Journal of Formal Logic, 64/3, 2023, p. 377-406.
This paper raises the issue of higher-order possible worlds, i.e., possible worlds about the way the (lower-order) possible worlds might be. It combines Kripke modal semantics and Riemannian geometry so as to formalize full-fledged higher-order possible worlds while establishing correspondence results between modal axioms and metric properties.
Métaphysique des modalités, in D. Lefebvre & K. Trego (éds), Les usages du possible, Peeters, 2023, p. 281-306.
Y a-t-il un sens à dire d’une situation impossible qu’elle aurait pu être possible? Ou bien d’une situation contingente, qu’il est possible qu’il soit nécessaire qu’elle soit contingente? De manière générale, les modalités d’ordre supérieur sont-elles admissibles? L’article initie une reconsidération positive de cette question, en montrant qu’elle est à la fois fondamentale et historiquement plus ouverte qu’on pourrait le penser.
A categorical aspect of the analogy between quantifiers and modalities, in Logic in Question, Birkhäuser, coll. «Studies in Universal Logic», 2023, p. 675-690.
The formal analogy between quantifiers and modalities is well-known. Its obvious limitation is, too: it holds on the condition of confining quantification to a single variable. This paper first presents that analogy in the terms of dynamic logic, which allows one to overcome the limitation just mentioned and to translate fragments of first-order logic into the language of propositional modal logic. It then sets out how the modal semantics for first-order logic which is established by dynamic logic lends itself to a classical algebraic development based on "cylindric algebras." Finally, it is shown how this algebraic semantics can be presented in a categorical setting, and how the algebraic counterparts of first-order structures correspond to subfibrations of the "syntactic fibration" representing first-order logic.
Homotopy Model Theory, The Journal of Symbolic Logic, 86/4 (Dec. 2021), p. 1301-1323.
Any structure for a first-order language can be turned into a simplicial set. This prompts a bundle of connections between model theory and homotopy theory. An adjunction result is proved between a subcategory of simplicial sets and the category of o-minimal structures.
La logique, science recherchée, Revue de Métaphysique et de Morale 2020/2, N°106, p. 145-164.
Settings and misunderstandings in mathematics, Synthese, Vol. 196 N°11, 2019, p. 4623-4656.
This paper considers combinatorics as a source of good examples of what a misunderstanding in mathematics can be, which will cast some light on mathematical understanding in general. The second goal of the paper is to reconsider the "identity problem" faced by the structuralist interpretation of mathematics. The common thread is the notion of setting. The study of a mathematical object almost goes together with a device to set ideas, the understanding of which is an essential component of mathematical knowledge. It is claimed that the recognition of mathematical settings allows one to classify most of understandable misunderstandings in mathematics and also to solve the identity problem.
Accointance par procuration, Les Etudes philosophiques 2019/3, N° 193, p. 369-384.
Logical Contextuality in Frege, The Review of Symbolic Logic, 11(1), 2018, p. 1-20.
Discussing Frege's "logical universalism," I claim that the universality of logic (the fact that logical truths purport to be about everything) and the radicality of logic (the fact that logic precedes any other theory) ought to be distinguished. Drawing on a suggestion in Frege's "Foundations of geometry," I then argue, contra Wilfrid Hodges and William Demopoulos, that Frege can make sense of the notion of non-logical constant. The general point is that Tarski's semantics is but one implementation of Hilbert's concept of reinterpretation of a formal theory.
Choses en soi et vérités absolues, in E. During & E. Alloa (éds), Choses en soi, Paris, PUF, 2018, p. 519-534.
L'objet de cet article est de différencier deux questions concernant le domaine de ce qui est en soi : la première concerne l'existence de choses en soi ; la seconde, la validité de vérités en soi, ou encore absolues. La première est celle de savoir si ce qui existe pour moi n'existe que relativement à ma capacité d'y avoir accès ; la seconde, celle de savoir si ce qui est vrai pour moi ne l'est que relativement à ma capacité de l'établir. La thèse défendue est que ces deux questions sont complètement distinctes, en ce sens que la réponse donnée à l'une est indépendante de la réponse donnée à l'autre.
Mondes logiques, Les Etudes philosophiques 2018, N°2, p. 267-279.
Models As Universes, Notre Dame Journal of Formal Logic 58/1, 2017, p. 47-78.
Every model of ZFC can be shown to contain, in a sense to be precised, another model of ZFC. The paper explains that fact and draws conclusions based on it about logical consequence from ZFC. A modal logic of internal models of ZFC is set out at the end.
The Concept of "Essential" General Validity in Wittgenstein's Tractatus, in Sorin Costreie (ed), Early Analytic Philosophy. New Perspectives on the Tradition, Springer, The Western Ontario Series in Philosophy of Science 80, 2016, p. 283-300.
In the Tractatus (6.1231-6.1232), Wittgenstein describes the general validity of logical truths as being "essential," as opposed to merely "accidental" general truths. He does not say much more, and little have been said about it by commentators. How to make sense of the essential general validity by which Wittgenstein characterizes logic? This paper aims to elucidate this crucial concept.
Sets and Descent, in A. Sereni & F. Boccuni (eds), Objectivity, Realism and Proof, Springer, Boston Studies in the History and Philosophy of Science 318, 2016, p. 123-142.
Algebraic Set Theory (AST) is an influential reconsideration of Zermelo-Frankel set theory (ZFC) in category-theoretic terms. This paper gets back to the original formulation by Joyal & Moerdijk. It explains in detail how the first axioms of AST (in this formulation) set up a framework linked to modern algebraic geometry. As a result, AST is shown to accomplish, not only an original and fruitful combination of set theory with category theory, but the genuine graft of a deeply geometric idea onto the usual setting of ZFC.
Benacerraf’s Mathematical Antinomy, in F. Pataut (ed), New Perspectives on the Philosophy of Paul Benacerraf : Truth, Objects, Infinity, Springer, 2016, p. 45-62.
This paper argues that Benacerraf’s dilemma can be compared to Kant’s mathematical antinomies of pure reason. Such a comparison is called for by strong analogies; It turns out to be precise and robust. The aim of the paper is to harness the analogy so as to transpose Kant’s solution for antinomies into Benacerraf’s setting and solve Benacerraf's dilemma.
Un principe caché de l’Analytique transcendantale : l’équivalence posée par Kant entre l’universalité et la nécessité, Philosophie 121 (2014), p. 29-49.
Kant never proves his thesis that universality and necessity are « inseparable » features (that no judgment can be universally true without being necessarily true, and vice versa). I show that this thesis is put to use instead, at several essential steps of Kant’s Analytic.
Une nouvelle sémantique de l’itération modale, Philosophia Scientiae 18/1, 2014, p. 185-203.
Saying that a proposition is necessarily necessarily true amounts to saying that the proposition is necessarily true whatever the range of all the possible worlds may be. This range then becomes a possible datum among others, which triggers the reference to higher-order possible worlds. This article aims at formalizing such a notion of high-order possible world, in sharp contrast to the Leibnizian heritage of a fixed closed totality of possible worlds, by using tools coming from Riemannian geometry.
Structured Variables, Philosophia Mathematica 21, 2013, p. 220-246.
Using the framework of syntactic fibrations and drawing on Russell’s substitutional theory, I compare Russell’s and Tarski’s conceptions of variables.
Sur une application possible du concept d’homotopie à la théorie des modèles, Annales de la Faculté des Sciences de Toulouse, Série 6, XXII/5, 2013, p. 1017-1043.
This paper endeavors to show the possible application to model theory of concepts coming from modern homotopy theory. In particular, the concept of simplicial set can be brought into play to describe the formulas of a first-order language L, the definable subsets of an L-structure, as well as the type spaces of a theory expressed in L. A comparison is sketched between categories of models (in the model-theoretic sense) and model categories (in the homotopy-theoretic sense).
Diagrams as sketches, Synthese 186/1, 2012, p. 387-409.
The notion of evolving diagram is introduced and detailed as an important case of mathematical diagram. An evolving diagram combines, through a dynamical graphical enrichment, the representation of an object and the representation of a piece of reasoning based on the representation of that object. Evolving diagrams can be illustrated in particular with category-theoretic diagrams, in the context of "sketch theory," a branch of modern category theory. It is argued that sketch theory provides a diagrammatic theory of diagrams, that it helps to overcome the rivalry between set theory and category theory as a general semantical framework, and that it suggests a more flexible understanding of the opposition between formal proofs and diagrammatic reasoning.
Structures et généralité en théorie combinatoire : les mathématiques et les lettres, Les Etudes philosophiques, 97/2, 2011, p. 215-242.
In the context of permutations on "the" n-element set, to swap 1 and 2 is to put 2 at the first place (place no. 1) and 1 at the second (place no. 2). Substitution theory is thus led to combine numerals and numbers. This can induce some (relatively) legitimate confusion. This paper shows that many cases of misunderstanding in mathematics can be compared to a confusion between numerals and numbers. On a positive side, it shows how important are all the numberings or parametrizations that turn out to be pervasive throughout mathematics, as devices to "set ideas."
Generality of Logical Types, Russell : the Journal of Bertrand Russell Studies, n.s. 31, 2011, p. 85-107.
Two kinds of generality can be attributed to logical types in Principia Mathematica, and ought to be clearly distinguished. The first one, external generality, pertains to the formality of types as introduced in the Introduction to the first edition. The variety of possible epistemic counterparts of each type is what substantiates and explains its formality. The second kind of generality, internal generality, bears on typical ambiguity and is shown to be formalizable within specific systems of modern typed lambda calculus.
The Versatility of Universality in Principia Mathematica, History and Philosophy of Logic, Vol. 32, n°3, 2011, p. 241-264.
In the Introduction of the first edition of Principia Mathematica, Russell says that one can account for all propositional functions using predicative variables only, that is, dismissing non-predicative variables. That claim is not self-evident at all, hence a « no loss of generality » problem, that this paper is devoted to solve. Two main points are put forward: Firstly, ramified types should be conceived of as highly fine-grained propositional forms; Secondly, the formal hierarchy of types lends itself to realizations in different epistemic universes.
Boa constructeur, Critique, n° 666, 2002, p. 896-912.
This is an assessment of Carnap's notion of construction in the Aufbau, on the occasion of the review of S. Laugier (ed), Carnap et la construction logique du monde, Paris, Vrin, 2001.