Mathematics: Objectivity by representation

就物质世界而言，标准的现实主义态度认为对象是独立于我们对它们的表示而存在的，这可能(表面上)是可信的：如果事情进展顺利，我们以我们所做的方式表示物理对象，因为它们是某某。相比之下，正如我们想要争论的那样，在数学世界中，情况是相反的：如果事情进展顺利，数学对象就是某某，因为我们用我们所做的来表示它们。这并不意味着数学不能是客观的：数学表示可能会受到约束，对它们构成的内容施加客观性。如果这是正确的，为了理解数学对象的本质，我们应该首先理解数学表示是如何工作的。用Kreisel的名言：“问题不是数学对象的存在，而是数学陈述的客观性”(Dummett 1978，P.XXXVIII)。我们解决的问题涉及澄清表示在数学推理和证明中的作用以及它们对数学本体论和理解的贡献的方式的哲学问题。这是对数学哲学中的一个经典问题的新探索，这个问题将理解与证明联系起来，并与数学本体论的构思方式有关。但我们的出发点既不是经典证明论，也不是经典形而上学。我们通过打开科学实践转向的大门来看待这个问题。在我们看来，问题既不是寻找数学推理的主题中立的形式化，也不是为数学对象的存在提供新的论点。因此，我们计划分析：(I)在数学实践中，相关的规定在何种意义上通过诉诸适当的表述来确定对象；(Ii)在内容(非正式)视角下构思的推理的严密性在何种意义上可以依赖于这些规定；(Iii)在何种意义上(通过将哲学研究与科学调查相联系)通过形式手段来描述非正式证明能力，从而允许使用逻辑和数学作为认识论的工具。我们还将我们的方法与经典的数学和逻辑的基础方法进行了比较，如经典的柏拉图主义和唯名论，这两种方法都对数学对象持一种‘存在主义态度’(它们都将存在或不存在的问题视为关键，尽管给出了相反的答案)，并认为数学推理是主题不变的。

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 mathematics 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.