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)的领域进行理性探究的一个例子，这个领域不依赖于感官的证据，但确实产生了知识。这引出了关于知识和理性信念的三个一般性问题。既然抽象事物是不可感知的，没有可感知的痕迹，没有可感知的物体，在时空中也没有位置，怎么可能有任何类型的抽象事物的知识呢？这种认识的本质是什么？2.不以感官证据为基础，怎么可能有对客观真理的认识呢？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，这样的知识如何解释？3.我们能说明理性信念的获得既包括在最终前提(公理和原始数据)中理性信念的可能性，也包括通过推理获得的理性信念吗？理性信念获得的本质是什么：理性的信念获取方式一定会导致真正的信念吗？或者，在某些情况下，理性的信念获取方式很大程度上会导致错误的信念？在对过去的尝试进行批判性审查之后，我的研究旨在找到这些问题的适当解决方案。这些问题的难度使一些哲学家抛弃了传统的数学观，转而认为数学是有用的谬误主体，或者是有用的虚构，或者被重新解释为关于具体事物的真理主体。我的工作将以三篇文章的形式发表在专业期刊上，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.