Empiricism and Mathematical Knowledge

Citation

Leeper, Gideon Jack (2016) Empiricism and Mathematical Knowledge. Other, California Institute of Technology. doi:10.7907/7B38-2T90. https://resolver.caltech.edu/CaltechTHESIS:11042016-141839141

Abstract

Empiricism asserts that sensory perceptions and experiences are the basis of all knowledge, i.e. essentially everything we know has either been directly perceived or deduced from perceived knowledge. This would seem to conflict with the wealth of mathematical knowledge that we have accrued as a civilization. Mathematics consists primarily of facts concerning ideal objects, such as topological spaces, groups, and categories, which we have never encountered in physical reality, and cannot reasonably expect to. Then from an empiricist perspective, it seems impossible for us to know so much about these objects, when there is no sensory experience on which we can base our knowledge. I first assert that mathematical knowledge is not knowledge in the sense used by empiricists, and thus its existence does not conflict with their views. However, I argue that first-hand mathematical knowledge, specifically one's belief in the validity of a proof, in some sense has an empirical component.

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