MathObRe (Mathematics: Objectivity by representation) is a French-German project (ANR-DFG funded) in Philosophy and History of Mathematics and Logic, coordinated by:
>> Colloque final du projet : 3-7 octobre 2017, Nancy.
>> Séminaire hebdomadaire le lundi, 10H-12H, salle 313, 3e étage 91, av. de la Libération, Nancy [see more]
>> Séminaire commun Nancy-Paris [see more]
>> Workshop Frege and Neo-Fregeanism - IHPST (Paris), June 13th 2014 [see more]
>> Colloque Pragmatism and the Practical Turn in Philosophy of Science - September 11-13 2014 [see more]
>> Group meeting, Munich Center for Mathematical Philosophy (MCMP), November 10-12, 2014 [http://www.mcmp.philosophie.uni-muenchen.de/events/workshops/container/math_obj_rep/index.html]
>> Colloque Representation and axiomatization: power and limits - IHPST (Paris), March 18-21, 2015 (http://www.ihpst.cnrs.fr/activites/colloques/representation-and-axiomatization-power-and-limits). A conference co-founded by the ANR-DFG project “HYPOTHESES: Hypothetical Reasoning - Its Proof-Theoretic Analysis” (http://ls.informatik.uni-tuebingen.de/hypotheses/), which focuses on the study of formal proofs with respect to the use of hypothetical reasoning>> Colloque 1935-2015, quatre-vingts ans de philosophie scientifique - Cerisy, July 13-19, 2015
>> Workshop on duality - Wuppertal, September 3-5, 2015
>> 3rd Congress of the Association for the Philosophy of Mathematical Practice (APMP) - Paris, Institut Henri Poincaré, November 2-4, 2015 (http://conference-apmp.sciencesconf.org/)
>> French Philosophy of Mathematics Workshop - Paris, November 5-7, 2015 (http://www.math.univ-toulouse.fr/FPMW/index.php)
>> Workshop Rhétorique en mathématique - Nancy, MSH Lorraine (salle internationale), November 27, 2015 [see more]>> Séminaire commun Paris-Nancy-Munich - Paris, January 21, 2016 & Nancy, January 22, 2016 [see more]
>> Colloque Foundations of Mathematical Structuralism - Munich Center for Mathematical Philosophy (MCMP), October 12-14, 2016 [http://www.mathematicalstructuralism2016.philosophie.uni-muenchen.de/program/index.html]
As far as the physical world is concerned, the standard realist attitude which conceives of objects as existing independently of our representations of them might be (prima facie) plausible: if things go well, we represent physical objects in the way we do because they are so-and-so. In contrast, as we want to argue, in the mathematical world the situation is reversed: if things go well, mathematical objects are so-and-so because we represent them as we do. This does not mean that mathematics could not be objective: mathematical representations might be subject to constraints that impose objectivity on what they constitute. If this is right, in order to understand the nature of mathematical objects we should first understand how mathematical representations work. In the words of Kreisel’s famous dictum: “the problem is not the existence of mathematical objects but the objectivity of mathematical statements” (Dummett 1978, p. xxxviii).
The problem we tackle concerns the philosophical question of clarifying the role of representations in mathematical reasoning and proofs and the way they contribute to mathematical ontology and understanding. This is a fresh inquiry concerning a classical problem in philosophy of mathematics connecting understanding to proofs and to the way the ontology of mathematic is conceived. But our starting point is neither classical proof theory nor classical metaphysics. We are rather looking at the problem by opening the door to the practical turn in science.
In our perspective the question is then neither to find a topic-neutral formalization of mathematical reasoning, nor to offer a new argument for the existence of mathematical objects. We rather wonder how appropriate domains of mathematical (abstract) objects are constituted, by appealing to different sorts of representations, and how appropriate reasoning on them are licensed.
Accordingly, we plan to analyse:
(i) in which sense in mathematical practice relevant stipulations determine objects by appealing to appropriate representations;
(ii) in what sense inferential rigor conceived in a contentual (informal) perspective can depend on these stipulations;
(iii) in what sense it is possible to characterize nevertheless (by interlinking philosophical studies with scientific investigations) informal provability by formal means, which allows using logic and mathematics as a tool for epistemology.
We also contrast our approach with classical foundational approaches of mathematics and logic, like classical Platonism and Nominalism, which both share an “existential attitude” facing mathematical objects (they both take as crucial the question whether they exist or do not exist, though giving opposite answers) and consider mathematical reasoning as topic-invariant.
See the complete project: MathObRe_full_project.pdf