This important book by a major American philosopher brings together eleven essays treating problems in logic and the philosophy of mathematics.

Charles Parsons; X*—Mathematical Intuition, Proceedings of the Aristotelian Society, Volume 80, Issue 1, 1 June 1980, Pages 145–168, https://doi.org/10.1093/ari.

The subject of this paper is the philosophical problem of accounting for the relationship between mathematics and non-mathematical reality. The first section, devoted to the importance of the problem, suggests that many of the reasons for engaging in philosophy at all make an account of the relationship between mathematics and reality a priority, not only in philosophy of mathematics and philosophy of science, but also in general epistemology/metaphysics. This is followed by a (rather brief) survey of the major, traditional philosophies (...) of mathematics indicating how each is prepared to deal with the present problem. It is shown that (the standard formulations of) some views seem to deny outright that there is a relationship between mathematics and any non-mathematical reality; such philosophies are clearly unacceptable. Other views leave the relationship rather mysterious and, thus, are incomplete at best. The final, more speculative section provides the direction of a positive account. A structuralist philosophy of mathematics is outlined and it is proposed that mathematics applies to reality though the discovery of mathematical structures underlying the non-mathematical universe. (shrink)

The main goal of this paper is to urge that the normal platonistic account of mathematical truth is unsatisfactory even if pure abstract entities are assumed to exist (in a non-Question-Begging way). It is argued that the task of delineating an interpretation of a formal mathematical theory among pure abstract entities is not one that can be accomplished. An important effect of this conclusion on the question of the ontological commitments of informal mathematical theories is discussed. The paper concludes with (...) a sketch of a non-Platonistic theory of mathematical truth which utilizes an unanalyzed notion of logical possibility. (shrink)

This paper is concerned with the genesis of mathematical knowledge. While some philosophers might argue that mathematics has no real subject matter and thus is not a body of knowledge, I will not try to dissuade them directly. I shall not attempt such a refutation because it seems clear to me that mathematicians do know such things as the Mean Value Theorem, The Fundamental Theorem of Arithmetic, Godel's Theorems, etc. Moreover, this is much more evident to me than any philosophical (...) view of mathematics I know of — including my own. So I am going to take mathematics as my starting point. (shrink)

In Science without numbers Hartry Field attempted to formulate a nominalist version of Newtonian physics?one free of ontic commitment to numbers, functions or sets?sufficiently strong to have the standard platonist version as a conservative extension. However, when uses for abstract entities kept popping up like hydra heads, Field enriched his logic to avoid them. This paper reviews some of Field's attempts to deflate his ontology by inflating his logic.