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.
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.
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.
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.
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."