**Mathematical settings** draft

(to be published in *Synthese*)

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.

**Homotopy model theory **pdf

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 simplicial sets and o-minimal structures.

**A Hidden Principle of Kant’s Transcendental Analytic **pdf

Kant assumes both universality and necessity to be *equivalent* criteria of *a priori* knowledge. This equivalence is never proved in the *Critique of Pure Reason*. Yet it is instrumental in ensuring essential steps of the Transcendental Analytic. (Two central examples are examined.) Hence a genuine hidden *principle* of transcendental philosophy, whose sole justification is provided by the Transcendental Analytic as a whole.

**Geometric Modal Logic **pdf

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. extended abstract

**Logical Contextuality in Frege **pdf

(forthcoming in* The Review of Symbolic Logic*)

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.

**Models As Universes**, forthcoming in *Notre Dame Journal of Formal Logic* 58/1 : 47-78 (2017). penultimate

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.

**Benacerraf’s Mathematical Antinomy**, in F. Pataut (ed), *New Perspectives on the Philosophy of Paul Benacerraf : Truth, Objects, Infinity*, Springer, 2016. pdf

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.

**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, pp. 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.

**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, pp. 283-300.

**Note:**There are a few typos in the published version. In particular, the sentence (s), on p. 291, should be read « (x) . x = x », and

*not*« (x) . x ≠ x ». All typos have been corrected in the online version. pdf

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.

**Une nouvelle sémantique de l’itération modale**, *Philosophia Scientiae* 18/1 (2014) : 185-203. pdf

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.

**Un principe caché de l’Analytique transcendantale : l’équivalence posée par Kant entre l’universalité et la nécessité**, *Philosophie* 121 (2014) : 29-49. pdf

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.

**Sur une application possible de la théorie de l’homotopie à la théorie des modèles**, *Annales de la Faculté des Sciences de Toulouse*, Série 6, XXII/5 (2013) : 1017-1043. pdf

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).

**Structured Variables**, *Philosophia Mathematica* 21 (2013) : 220-246. pdf

Using the framework of syntactic fibrations and drawing on Russell’s substitutional theory, I compare Russell’s and Tarski’s conceptions of variables.

**Le Nécessaire et l’universel. Analyse et critique de leur corrélation**, Vrin, Paris, 2013, 256 pp. summary

This book received the **Prix Jean Cavaillès 2015**: Prix Jean Cavaillès

Kant claimed any universal truth to be necessary, and *vice versa*. Since Kant onward, the equivalence of universality and necessity (as features of certain truths, in general called *a priori* truths) has always been maintained, without being properly proved. This book questions this equivalence and shows that both concepts ought to be reconsidered independently of each other.

**Reviews:**

In *History and Philosophy of Logic*, by F. Pataut: review HPL

In *Symposium*, by S. Gandon: review Symposium

In *Revue Internationale de Philosophie*, by R. Pouivet: review RIP

In *Klesis*, by J.-M. Salanskis: review Klesis

In *Critique*, n° 821 (octobre 2015), « Regards encore français sur la logique et les mathématiques », pp. 810-823.

**Diagrams as sketches**, *Synthese* 186/1 (2012) : 387-409. pdf

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.

**Generality of Logical Types**, *Russell : the Journal of Bertrand Russell Studies*, n.s. 31 (2011) : 85-107. pdf

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*32/3, (2011) : 241-264. pdf

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.

**Structures et généralité en théorie combinatoire : les mathématiques et les lettres**, *Les Etudes philosophiques*, 97/2 (2011) : 215-242. pdf

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. »

**Boa constructeur**, *Critique*, n° 666, 2002, p. 896-912. pdf

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.

**« Encadrés » in Alain Rey (dir.), Dictionnaire culturel en langue française, Paris, Le Robert, 2005 (4 tomes):**

Addition (*DCLF*, I, p. 96-97). pdf

Crise (*DCLF*, I, p. 1995-2003). pdf

Echec (*DCLF*, II, p. 254-256). pdf

Endroit (*DCLF*, II, p. 470-474). pdf

Fusée (DCLF, II, p. 1230-1234). pdf

Humour (*DCLF*, II, p. 1736-1742). pdf

Limite (DCLF, III, p. 66-69). pdf

Liquide (DCLF, III, p. 88-97). pdf

Logique (*DCLF*, III, p. 130-133). pdf

Ordre (*DCLF*, III, p. 1181-1190). pdf

Patience (*DCLF*, III, p. 1442-1445). pdf

Provisoire (*DCLF*, III, p. 2184-2186). pdf

Tennis (*DCLF*, IV, p. 1312-1314). pdf

Valeur (*DCLF*, IV, p. 1726-1734). pdf