Jean-Pierre Marquis

Category theory and the foundations of mathematics: Philosophical excavations

The aim of this paper is to clarify the role of category theory in the foundations of mathematics. There is a good deal of confusion surrounding this issue. A standard philosophical strategy in the face of a situation of this kind is to draw various distinctions and in this way show that the confusion rests on divergent conceptions of what the foundations of mathematics ought to be. This is the strategy adopted in the present paper. It is divided into 5 sections. We first show that already in the set theoretical framework, there are different dimensions to the expression ‘foundations of’. We then explore these dimensions more thoroughly. After a very short discussion of the links between these dimensions, we move to some of the arguments presented for and against category theory in the foundational landscape. We end up on a more speculative note by examining the relationships between category theory and set theory.

Mathematical forms and forms of mathematics: leaving the shores of extensional mathematics

In this paper, I introduce the idea that some important parts of contemporary pure mathematics are moving away from what I call the extensional point of view. More specifically, these fields are based on criteria of identity that are not extensional. After presenting a few cases, I concentrate on homotopy theory where the situation is particularly clear. Moreover, homotopy types are arguably fundamental entities of geometry, thus of a large portion of mathematics, and potentially to all mathematics, at least according to some speculative research programs.

A View from Space: The Foundations of Mathematics

Suppose we were to meet with extraterrestrials and that we were able to have a discussion about our respective cultures. At some point, they start asking questions about that something which we call “mathematics”. “What is it?”, they ask. Tough question. How should we answer them?

Menger and Nöbeling on Pointless Topology

This paper looks at how the idea of pointless topology itself evolved during its pre-localic phase by analyzing the definitions of the concept of topological space of Menger and Nobeling. Menger put forward a topology of lumps in order to generalize the definition of the real line. As to Nobeling, he developed an abstract theory of posets so that a topological space becomes a particular case of topological poset. The analysis emphasizes two points. First, Mengers geometrical perspective was superseded by an algebraic one, a lattice-theoretical one to be precise. Second, Mengers bottom–up approach was replaced by a top–down one.

Mathematical engineering and mathematical change 1

Abstract In this paper, I introduce and examine the notion of “mathematical engineering” and its impact on mathematical change. Mathematical engineering is an important part of contemporary mathematics and it roughly consists of the “construction” and development of various machines, probes and instruments used in numerous mathematical fields. As an example of such constructions, I briefly present the basic steps and properties of homology theory. I then try to show that this aspect of contemporary mathematics has important consequences on our conception of mathematical knowledge, in particular mathematical growth.

Category Theory and Structuralism in Mathematics: Syntactical Considerations

Thus, to be is to be related and the “essence” of an “entity” is given by its relations to its “environment”. This claim is striking: it seems to describe perfectly well the way objects of a category are characterized and studied. Consider, for instance, the fundamental notion of product in a category C: a product for two objects A and B of C is an object C with two morphisms p 1: C → A and p 2: C → B such that for any other pair of morphisms f: D → A and g:D→B, there is a unique morphism h:D→C such that f = p 1 h and g = p 2 h. What is crucial in this specification is the pair of morphisms and the universal property expressed by the condition, for it is those which are used in proofs involving products. Thus to be a product is, in an informal sense, to be a position in a category. It is to be related in a certain manner to the other objects or positions in the category. Moreover, a product for two objects is defined up to isomorphism and it does not make sense to ask what is the product of two objects. It simply does not matter as far as mathematical properties are concerned. Now, if mathematics can be developed within category theory and if we can show that all the crucial concepts are given by universal properties, or, equivalently, come from adjoint situations, then we would have substantiated the above claim considerably.

If Not-True and Not Being True are not Identical, Which One is False?

In classical logic, truth and falsity are highly symmetric and this symmetry is captured by the negation operator which “transforms” truth into falsehood and vice-versa. Moreover, the negation operation represented in the semantics by a unary operator on the two-element Boolean algebra is lifted to the higher-order operation of complementation on sets. However, if we abandon bivalence, then symmetry is problematic. For one thing, we are forced to distinguish between not-true, which is a truth-value, from not being true, the complement of the singleton set consisting of the truth, which is not the false and not even a truth-value. In classical logic, these two notions collapse into one. Once they are distinguished, their relationships have to be settled and in particular the links between complementation and negation have to be clarified. The purpose of this paper is to explore some of these relationships in a specific context, namely topos theory.