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Published online by Cambridge University Press: 28 May 2006
Husserl began to publish on problems in the philosophy of mathematics and logic soon after he received his Ph.D. in mathematics in 1881, and he continued to publish on them throughout his lifetime. Although much has happened in the foundations of mathematics since the turn of the century, many of Husserl's ideas are still relevant to recent issues in the philosophy of mathematics. In this essay, I argue that a number of the views on mathematics that are part of Husserl's transcendental phenomenology are more compelling than current alternative views in the philosophy of mathematics. In particular, I indicate how Husserl's views can be used to solve some basic problems in the philosophy of mathematics that arise for (naive) Platonism, nominalism, fictionalism, Hilbertian formalism, pragmatism and conventionalism.
A PRÉCIS OF PROBLEMS IN THE PHILOSOPHY OF MATHEMATICS
Many of the basic problems in the philosophy of mathematics center around the positions just mentioned. It will not be possible to discuss these problems in any detail here, but at least some general indications can be given.
A major difficulty for Platonism has been to explain how it is possible to have knowledge of immutable, acausal, abstract entities like numbers, sets, and functions. Once it is argued that these entities are abstract and mind-independent, there seems to be no way to establish an epistemic link with them that is not utterly mysterious. The apparent insurmountability of this problem might persuade one to abandon Platonism altogether in favor of some form of nominalism. The nominalist will at least not have the problem of explaining how knowledge of abstract entities or universals is possible, because on this view there simply are no abstract entities or universals. There are only concrete spatio-temporal particulars, and it is argued that however one ends up construing these, there will be no great mystery about how we could come to know about them. One could work quite naturally, for example, with a causal account of knowledge.
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