Mathematics: Objectivity by representation – MathObRe

The methodology of our research will combine abstract philosophical considerations with logical analysis. Still, historical inquiries (based on the analysis of texts and mutual influences), and attention to mathematical practice as such will also be exploited. This is why we include in the project mathematicians and historians of mathematics. We also asked for the collaboration of some internationally recognized specialists from France and Germany who do not belong to the three institutional partners, and appeal to other colleagues we are used to collaborate with, coming from abroad.
The sustained thesis seems, so to say, the limit to which converge the classical philosophical positions with regard to foundations of mathematics: The three main initiators of this project, initially engaged in pragmatic (Heinzmann), formalist (Leitgeb) and Platonistic (Panza) traditions, came to elaborate and share a common position facing to the difficulties of their original approaches. This is the main reason for the French-German collaboration. Its synergy should allow a better elaboration of this common position on the basis of the different starting points.

On the French side, the project is directly connected with M. Detlefsen’s chaire d’excellence “Ideal of Proofs” (ANR, 2007-2011; www.univ-nancy2.fr/poincare/idealsofproof/) directed by J.J. Szczeciniarz, G. Heinzmann and M. Panza, the latter two being involved in the present project. Our aim is to develop the research pursued in this project in a different direction: instead of focusing on “ideal elements” involved in mathematical theories and proofs, we focus on the role of representation, especially if provided by an affinity to empirical artefacts.
On the German side, the project is triggered by two streams of research that have emerged at the MCMP within the last two years: i) The study of ante rem structuralism in the philosophy of mathematics (the view that mathematics concerns structures, which are only determined up to isomorphism); ii) the questions whether logical concepts are constituted by logical rules, and theoretical concepts are determined by scientific theories.
In the last decades, many studies have aimed to overcome classical positions in mathematical ontology and epistemology as Platonism, nominalism, and classical pragmatism (as presented in Kitcher 1984). These efforts suggest looking at mathematical practice as a source for finding a solution to the problem these positions were willing to answer. This is today a crucial task for philosophy of mathematics, and for example witnessed by the activities and publications of the Association for the Philosophy of Mathematical Practice (http://institucional.us.es/apmp/index_members.htm), which counts as its members many participants to our project.
Even taken together, all these studies and events give no answer to our focal concern: clarifying the role of representations in mathematical reasoning and proofs and the way they contribute to mathematical ontology and understanding. We believe therefore, that our project could add important insights to the understanding of mathematical practice.

The outstanding features of our aims in the four scientific tasks (task (2) to (5)) are the following:
(a) Representational Identification
Parsons (2008) mentions, as examples of mathematical quasi-empirical objects, geometrical figures of Euclid’s plane geometry, natural numbers conceived as strings of strokes, and finite small sets. However, he only considers the second example. This suggests of developing other examples of representational identification This is the main feature of task (2).
(b) Varieties of Representational Identification
This task (3) has a comparative and specifically philosophical nature. Its basic feature is that of reflecting on the general notion of representational identification in its relations with the notion of intuition, and of comparing different varieties of representational identification of mathematical objects.
(c) What’s an informal proof?
Task (4) aims to investigate the relation between pragmatism and the practical turn by focusing on the concept of informal proof and aesthetic elements (in Goodman’s sense) in mathematical proof. Our main feature is to give some criteria, improved by mathematical practice, in order to decide which of different proofs of a theorem is more explanatory then an other.
(d) Theoretical terms and entities vs mathematical terms and entities
The main feature of task (5) is to investigated the thesis that mathematical objects would be determined by mathematical terms and theories in much the same way in which theoretical objects are determined by theoretical terms and theories according to Carnap’s analysis. Mathematical representation would thus end up prior to the ontology of mathematics.

Heinzmann, G. (1999), “Poincaré on Understanding Mathematics”, Philosophia Scientiae 3 (2), 43-60.
Heinzmann, G. (2013), L’intuition épistémique. Une approche pragmatique du contexte de justification en mathématiques et en philosophie, Paris : Vrin (Mathesis).
Heinzmann, G. (2015), Pragmatism and the Practical Turn in Philosophy of Mathematics: Explanatory Proofs, in : E. Agazzi, G. Heinzmann (eds), Pragmatism and the Practical Turn in Philosophy of Sciences. Milan : Franco Angeli Editore (in press).
Jullien, C. (2008), Mathématiques et esthétique – Une Exploration goodmanienne, Presses universitaires de Rennes, coll. Aesthetica, 2008.
Leitgeb, H. and J. Jadyman (2008), “Criteria of Identity and Structuralist Ontology”, Philosophia Mathematica 16/3, 2008, 388-396.
Leitgeb, H. (2009), “On Formal and Informal Provability”, in: O. Linnebo and O. Bueno (eds.), New Waves in Philosophy of Mathematics, Palgrave Macmillan, New York, 2009, pp. 263-299.
Leitgeb, H. (2011), “New Life for Carnap's Aufbau?”, Synthese 180/2, 2011, 265-299.
Panza, M. (2011a) “Rethinking Geometrical Exactness”, Historia Mathematica, 38, 1, 2011, pp. 42-95.
Panza, M. (2011b) “The Twofold Role of Diagrams in Euclid’s Plane Geometry”, Synthese, 186, n° 1, 2012, pp. 55-102.
Panza, M.& Sereni, A. (2013), Plato’s Problem, Palgrave, Basingstoke, 2013.
Rebuschi, M. (2006). “IF & Epistemic Action Logic”. In J. van Benthem et al. (eds.): The Age of Alternative Logics. Assessing Philosophy of Logic and Mathematics Today, Dordrecht, Springer, 261-281.
Sinaceur, H.B. (2008), Richard Dedekind: La création des nombres, Vrin, Paris, 2008.
Steinberger, F. (2013), «On the Equivalence Conjecture for Proof-Theoretic Harmony«, Notre Dame Journal for Formal Logic 54 (1), 79-86.
Wagner P. (2012), ed. Carnap’s Ideal of Explication and Naturalism, Basingstoke, Palgrave Macmillan.

As far 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.