Problems of general epistemology with special pertinence to mathematics

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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.