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:
- in which sense in mathematical practice relevant stipulations determine objects by appealing to appropriate representations;
- in what sense inferential rigor conceived in a contentual (informal) perspective can depend on these stipulations;
- 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.