Mathematics: Objectivity by representation

就物理世界而言，标准的实在论态度认为，对象独立于我们对它们的表征而存在，这种态度（初步看来）可能是合理的：如果事情进展顺利，我们以我们的方式表征物理对象，因为它们是某某。相反，正如我们想要论证的那样，在数学世界中，情况正好相反：如果事情进展顺利，数学对象是某某的，因为我们以我们的方式表示它们。这并不意味着数学不可能是客观的：数学表示可能会受到约束，这些约束将客观性强加于它们所构成的内容。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的名言来说："问题不在于数学对象的存在性，而在于数学陈述的客观性“（Dummett 1978，p. xxxviii）。我们要解决的问题涉及澄清数学推理和证明中表示的作用以及它们对数学本体论和理解的贡献的哲学问题。这是对数学哲学中一个经典问题的新的探索，它把理解与证明联系起来，并与数学本体论的构想联系起来。但我们的出发点既不是古典证明论，也不是古典形而上学。我们是通过打开科学实践转向的大门来看待这个问题的，在我们看来，这个问题既不是要找到数学推理的主题中立的形式化，也不是要为数学对象的存在提供一个新的论证。我们想知道数学（抽象）对象的适当域是如何通过诉诸不同种类的表示来构成的，以及对它们的适当推理是如何被许可的。 在这个意义上，在数学实践中，有关规定通过诉诸适当的表示来确定对象;（ii） 在什么意义上推理的严格性设想在内容（非正式）的角度可以依赖于这些规定;（iii） 在什么意义上，它是可能的，但（通过与科学调查的哲学研究相互联系）非正式证明的正式手段，允许使用逻辑和数学作为一种工具，认识论的特点。 我们还将我们的方法与数学和逻辑的经典基础方法进行了对比，如经典柏拉图主义和唯名论，它们都对数学对象持“存在主义态度”（它们都认为数学对象是否存在是至关重要的问题，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.