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Mathematics: Objectivity by representation

数学：通过表征实现客观性

基本信息

批准号：
246591146
负责人：
Professor Dr. Hannes Leitgeb
金额：
--
依托单位：
Laboratoire de Philosophie et dHistoire des Sciences
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2014
资助国家：
德国
起止时间：
2013-12-31 至 2017-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/246591146?language=en
关键词：
Mathematics Objectivity representation

项目摘要

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.

就物理世界而言，标准现实主义的态度认为物体是独立于我们对它们的表征而存在的这种态度（表面上看）可能是可信的：如果事情进展顺利，我们之所以这样表征物理物体，是因为它们是某某。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，数学对象就是某某，因为我们按照自己的方式来表示它们。这并不意味着数学不可能是客观的：数学表征可能受制于将客观性强加于它们所构成的东西之上的约束。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。用克瑞塞尔的名言来说：“问题不在于数学对象的存在，而在于数学陈述的客观性”（Dummett 1978, p. xxxviii）。我们处理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方式的哲学问题。这是对数学哲学中的一个经典问题的一个新的探索，它将理解与证明以及数学本体论的概念联系起来。但我们的出发点既不是经典证明论，也不是经典形而上学。我们更倾向于通过打开通往科学实践的大门来看待这个问题。在我们看来，问题既不是找到一个主题中立的数学推理的形式化，也不是为数学对象的存在提供一个新的论据。我们很想知道，数学（抽象）对象的适当领域是如何通过求助于不同种类的表象而构成的，以及如何在这些表象上进行适当的推理。因此，我们计划分析：(i)在数学实践中，相关规定在何种意义上通过诉诸适当的表示来确定对象；（ii）在何种意义上，从内容（非正式）角度构思的推理严谨性可以依赖于这些规定；（iii）在何种意义上，通过形式手段（通过将哲学研究与科学调查联系起来）来表征非正式可证明性是可能的，这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”（它们都把它们是否存在的问题视为关键，尽管给出了相反的答案），并将数学推理视为主题不变的。