Mathematical Objectivity by Representation (MathObRe)

Our project Mathematical Objectivity by Representation, started up from a dilemma formulated by Poincaré (1902) “How mathematics, which is both rigorous and non-tautological, can be either 1) rigorous, if non-deductive, or 2) non-tautological, if deductive?”.  We tried to answer this by studying the role and function of representational tools (such as diagrams, notations, formalisms) in mathematics, by a particular attention to the role they play within different sorts of structuralist accounts of mathematics, and their function in fixing mathematical objects. Structuralism is the philosophical thesis expressing that mathematics is not concerned with individual objects, but with “systems” or properties of objects that share a common structure. According to a widespread kind of structuralism, what the constants in mathematical propositions denote are not individuals, but just “positions” or “places” in this structure; they do not have an identity outside of the structure. Thus, the objectuel character of Mathematics is just a way of speaking. This made us address Poincaré’s question by investigating the ontological and epistemic status of mathematical objects qua abstract objects, and so wondering how it is possible to account for knowledge of such objects and their properties. Our idea was and is indeed that such a knowledge can only be acquired by investigating different representations of these objects.

A Road to Mathematical Practice

In the last decades, many studies have aimed to overcome classical positions in mathematical ontology and epistemology as Platonism, nominalism, formalism, but also strong anti-realism. The failure of these efforts suggests looking at mathematical practice as a source for finding a solution to the problem these positions were willing to answer. Rather than reasoning in general, from a metaphysical or, traditionally foundational perspective, we inscribed our inquiry within this orientation called today `Philosophy of Mathematical Practice’, with a strict and constant relation with many colleagues working within APMP (Association for the Philosophy of Mathematical Practice), whose one of the coordinators of the project (Marco Panza) has been one of the founder and whose he is presently member of the steering committee. The general idea of this orientation is tackling philosophical questions about mathematics (such as that of the role of formalism and representations in it; and their relation with the content of mathematical statements, and, then, with mathematical ontology), by looking at specific case-studies from mathematical activity and history. This is just what we did.

Some results

Among many other results, this approach has allowed us:

• to test on a number of significant examples of the general theory of mathematical intuition advanced by Heinzmann: are a fruitful tool for explaining the explicative characters of certain proofs and to relay structures to objects;
• to identify a form of platonism non subjected to the usual metaphysical objections related to the difficulty of arguing in favour of the existence of abstract (mathematical) objects and to propose a new understanding of  Frege’s logicism and platonism (Panza);
• ­to develop a new account of rational belief based on three assumptions: the logical closure of rational belief; the axioms of probability for rational degrees of belief; and the so-called Lockean thesis, in which the concepts of rational belief and rational degree of belief figure simultaneously (Leitgeb).

Our Scientific Production

Though many of our analyses and results have been presented in some yet unpublished talks, and require refinement before publication, the essential means of diffusion of the results of our projects has been provided by scientific publications in highly ranked scientific journals and collective books. A complete list is offered in the final list of publications. The works mentioned above offer a significant sample of our work.

Essential facts about the project

Our Franco-German project has been devoted to a philosophical topic concerning mathematics, and has been developed with constant contact with the international scientific community of philosophers (and historians) of mathematics (and logic). It has involved the Archives Poincaré in Nancy, the IHPST in Paris and the MCMP in Munich. It has been coordinated, for the French side, by Gerhard Heinzmann, full professor at the Univ. de Lorraine, Nancy (AP) and Marco Panza, research director at the CNRS, and Presidential Fellow at Chapman University, CA (IHPST), and, for the German side, by  Hannes Leitgeb, full professor at the Univ. of Munich (MCMP). It started in March 2014 and ended in October 2017. It has been financed by the ANR for 217.256 € (108.628€ LHSP and 108.628€ IHPST).