数学真理的问题(英文)

The Problem of Mathematical Truth

在当前数学实践中,数学知识(如果有这样的知识的话)是通过在定义和公理的基础上证明定理来获得的。问题在于该怎样理解证明中所得到的东西是如何构成知识的,具体而言,即是要给出一个关于数学真理和数学知识的统一的解释,该解释能够揭示两者的内在联系。此处的困难是,根据贝纳塞拉夫的为人熟知的论证,由于塔斯基语义学认为真与对象的联系(通过单称词项或通过量词)是不可消去的,因此在数学中无法将塔斯基语义学与完整的认识论相结合:数学知识要么是通过证明得到的,这种情况下数学知识与数学对象是无关的,因此我们就无法解释数学真理;要么数学对象是数学真理的构件,从而数学知识不是通过证明得到的,这种情况下我们就无从理解数学知识。接着,本文通过一系列阶段,将这些困难一直追溯到最基本的逻辑观念,即将之看作形式的和纯粹解释性的:如果数学是从概念出发仅仅使用逻辑的推理实践,依照康德,那么数学应该是分析的,也即,仅仅是解释性的,根本就不是通常意义上的知识。我认为,这对数学真理是真正困难的问题。本文概括了四种回应,其中仅有一个有希望解决我们的困难,也即皮尔斯和弗雷格的回应。根据他们的方案,逻辑是科学,因此是实验性的和可错的;符号语言是有内容的,尽管并不涉及与任何对象的关联;证明是构成性的,因此是富于产出的过程。通过充分发展这些观点,我们将有可能最终解决数学真理的问题。

In current mathematical practice, mathematical knowledge （if it is achieved at all） is achieved by proving theorems on the basis of definitions and axioms. The problem is to understand how what is achieved thereby constitutes knowledge; more specifically, it is to develop a unified account of mathematical truth and mathematical knowledge, one that reveals their inner connection. What stands in our way, according to a very familiar argument of Benacerraf＇s, is that in mathematics there seems no way to combine a Tarskian semantics, according to which truth involves ineliminable reference to objects （either by way of singular terms or by way of quantifiers）, with an adequate epistemology： either mathematical knowledge is by way of proof, in which case mathematical objects are irrelevant to mathematical knowledge and then we have no account of mathematical truth, or mathematical knowledge is not by way of proof because mathematical objects are constitutive of mathematical truth, but then we have no resources for understanding mathematical knowledge. I then trace the difficulties, in a series of stages, all the way down to our most basic conception of logic as formal and merely explicative： if mathemat- ics is a practice of reasoning from concepts by logic alone then it ought, according to Kant, to be analytic, that is, merely explicative, not knowledge properly speaking at all. This, I submit, is the really hard problem of mathematical truth. Four responses are outlined, but only one holds out promise of resolving our difficulties, namely, that of Peirce and Frege. According to them, logic is a science, and hence experimental and fallible, symbolic language is contentful despite involving no reference to any objects, and proof is a constructive and hence fruitful process. Adequately developed, these ideas will enable us finally to resolve the problem of mathematical truth.