Problems of general epistemology with special pertinence to mathematics

数学似乎是对抽象实体（如数字、函数和结构）领域进行理性探究的一个例子，它不依赖于感官的证据，但确实产生了知识。这就提出了关于知识和理性信念的三个一般性问题。既然抽象的事物是不可感知的，不留下可感知的痕迹，不影响可感知的物体，在时空中没有位置，那么，我们怎么可能有关于它们的知识呢？这种知识的本质是什么？2.不以感官的证据为基础的客观真理的知识怎么可能存在呢？人们试图用语言意义的知识来解释这种知识。这种尝试能成功吗？如果不是，那么如何解释这些知识呢？3.我们能否给出一个理性信念获得的解释，它既包含对最终前提（公理和原始数据）的理性信念的可能性，也包含通过推理获得的理性信念？理性信念获得的本质是什么：获得信念的理性方式必须倾向于导致真正的信念吗？或者，在某些情况下，获取信念的理性方式在很大程度上会导致错误的信念？我的研究旨在对过去的尝试进行批判性审查后，开发这些问题的适当解决方案。这些问题的困难性使得一些哲学家放弃了传统的数学观，转而认为数学是一个有用的谬误体，或者是有用的虚构，或者是被重新解释为关于具体事物的真理体。我的工作将在专业期刊上发表三篇文章，这将有助于说明为什么我们不需要接受这样的结论。

Mathematics appears to be an example of rational inquiry about a realm of abstract entities (such as numbers, functions and structures) that does not depend on the evidence of the senses but does yield knowledge. This raises three general problems about knowledge and rational belief.1. How is it possible to have knowledge of abstract things of any kind, given that they are not perceptible, leave no perceptible traces, influence no perceptible bodies and have no location in space-time? What is the nature of such knowledge?2. How can there be knowledge of objective truths that is not based on the evidence of the senses? Attempts have been made to explain such knowledge in terms of knowledge of linguistic meanings. Can attempts of this kind succeed? If not, how is such knowledge to be explained?3. Can we give an account of rational belief acquisition that accommodates both the possibility of rational belief in ultimate premises (axioms and raw data) and rational belief acquired by inference? What is the nature of rational belief acquisition: Must a rational way of acquiring a belief tend to result in true beliefs? Or could there be circumstances in which rational ways of acquiring beliefs lead largely to false beliefs?My research aims to develop proper solutions of these problems, following critical examination of past attempts. The difficulty of these problems has led some philosophers to abandon the traditional view of mathematics and instead hold that mathematics is a useful body of falsehoods or is usefulfiction, or is to be re-interpreted as a body of truths about concrete things. My work, to be published as three articles in professional journals, will help to show why we do not need to accept such conclusions.