Sciences, Philosophie, Histoire – UMR 7219, laboratoire SPHERE

I. Developing research at the interface between philosophy of mathematics and philosophy of logic

• The formalization of mathematical theories in an appropriate logical and linguistic context; Which of course implies a discussion about the choice of contexts. What is the framework adapted to the formalization of mathematics?

• The logical analysis of the properties of mathematical proofs. Is it possible to logically characterize certain properties that mathematicians attribute informally to their demonstrations?

• The research will also cover questions related to constructive or bounded finite versions of arithmetic, relationships between logicist versions and concepts more directly related to number theory and model theory. Different but connected to the above, we will also favor research on the relations between evidence and program (Curry-Howard correspondence) and computability.

II. To develop research on the question of the foundations of mathematics

• Reflections on the theory of sets and its semantics.

• The questions raised by the powerful and developing alternative of category theory

• The "philosophical" approaches to the question of foundations. We mean by this the different programs born recently, the purpose of which is to justify or explain the presuppositions specific to certain mathematical theories or, at least to clarify the epistemic modality.

• The history of the programs of foundation of mathematics is obviously within the thematic perimeter of the project.

III. Promote any research concerning questions usually asked in metaphysics and in the philosophy of language about mathematics

• The opposition Platonism / nominalism (as well as that which is close, but which does not cover the first, realism and anti-realism) -or perhaps it would be better to say, the oppositions between the different kinds of Platonism and the different kinds of nominalism - and the question of the ontological status of mathematical objects.

• The role of the notion of truth in mathematics and / or the possibility of fixing a conception of truth such that one can say, in an appropriate way, that the theorems of mathematics are true (we think, for example, Between truth and guaranteed assertability, super-assertability, probability, etc.).

• The nature of mathematical knowledge, admitting that there is one, in the proper sense of the term ’knowledge’: is there a specificity of mathematical knowledge in relation to other forms of knowledge? And if so, how to define it?

• The question of what can mean a pragmatic conception of mathematics.

• The question of the historicity of mathematics and the evolution of mathematical knowledge, or, more generally, of progress and / or conceptual change and / Or the transformation and / or substitution of theories in mathematics: is the evolution of mathematics of the same nature as the evolution of other sciences? Are there scientific revolutions in mathematics or not?

IV. Promote the development of the philosophy of mathematical practice in all its aspects - notably historical and didactic

• Appreciation of the virtues of a mathematical argument: by comparing different cases of studies, we try to specify criteria for evaluating arguments (in particular evidence) or forms of evidence (for example, examples or counterexamples) that mathematicians commonly accept but which escape (or partially escape, or seem to escape) attempts at formalization.

• Studies of new forms of application of mathematics.

• Role of computers in mathematical evidence and feasibility of evidence.

• New approaches to mathematical practices. In particular, mathematical practices are being questioned today by the cognitive sciences and sociology. In this respect: the question of the relevance of this research to philosophy and, on the other hand, the new avenues of research aroused by these approaches for the philosophy of mathematics.

• The relationship with the history and didactics of mathematics touches on the project as closely as the disciplines have been most attentive to the fine texture of mathematics.

Note on the necessarily non exhaustive nature of these tracks of research, and return on the motivations and the interest of this grouping of research:

• All these tracks are currently being explored by project members. Most can be at once by philosophers and mathematicians. Their diversity shows that no dogmatic limitation is imposed a priori on our project. On the contrary, it is this wealth, this very abundance which gives value to our project, as an attempt to constitute a unifying framework and a place of confrontation. Let us add that the many interactions between these different directions can only be local, because they arise from the growth, the development and the crossing of the tracks of research, and can not be summarized, let alone prescribed, overhanging. The interest of this GDR, once again, is to allow this community which is strengthening in France to continue to develop and find its own tracks of development.

We shall conclude with two remarks and two difficulties which, in our opinion, are at the heart of the philosophy of mathematics today, but which are also at the center of contemporary mathematics and which call for and justify a real interdisciplinarity:

• a) Mathematics is developed by hyper-specialization and an extreme increase in the technicality of their objects, to the point of making communication between different disciplinary communities impossible (posing a remarkable philosophical and sociological problem); At the same time, at different levels of stratification, mathematics weaves relations of connections and unity between these various specialties. How can we understand these two opposing movements?

• b) How can the philosophy of mathematics, given the extreme conceptual technicality recognized by even the greatest active mathematicians, succeed in taking into account this powerful and difficult level of mathematical reflection without being absorbed, carried away by this conceptual flow and maintain a level of autonomy that does not come to be added artificially or even abstractly to the field of real practice of mathematics?