In this deft and vigorous book, Mark Balaguer demonstrates that there are no good arguments for or against mathematical platonism (ie., the view that abstract, or non-spatio-temporal, mathematical objects exist, and that mathematical theories are descriptions of such objects). Balaguer does this by establishing that both platonism and anti-platonism are defensible positions. In Part I, he shows that the former is defensible by introducing a novel version of platonism, which he calls full-blooded platonism, or FBP. He argues that if platonists endorse FBP, they can then solve all of the problems traditionally associated with their view, most notably the two Benacerrafian problems (that is, the epistemological problem and the non-uniqueness problem).
In Part II, Balaguer defends anti-platonism (in particular, mathematical fictionalism) against various attacks, chief among them the Quine-Putnam indispensability argument. Balaguer's version of fictionalism bears similarities to Hartry Field's, but the arguments Balaguer uses to defend this view are very different. Parts I and II of this book taken together clearly establish that we do not have any good argument for or against platonism.
In Part III, Balaguer extends his conclusions, arguing that it is not simply that we do not currently have any good argument for or against platonism, but that we could never have such an argument, and indeed, that there is no fact of the matter as to whether platonism is correct (ie., whether there exist any abstract objects).
This lucid and accessibly written book breaks new ground in its area of engagement and makes vital reading for both specialists and anyone else interested in the philosophy of mathematics or metaphysics in general.
Industry Reviews
"Platonism and anti-platonism in mathematics is an impressive work. Balaguer presents forceful arguments for the viability of both FBP and fictionalism, and against the feasibility of any substantially different Platonist or anti-Platonist position. ... an admirable achievement."--The Bulletin of Symbolic Logic
"Brings several interesting new ideas to the philosophy of mathematics. The most important is a new position concerning the existence of mathematical objects--his Full-Blooded Platonism. This is the novel and radical claim that any mathematical object that can consistently exist does exist. Balaguer has done an excellent job of expounding and defending this idea. I find his philosophical style unique in its honesty and boldness; for he manages to make his
initially provocative claims quite plausible without having to weaken or qualify them. This is an important work from which not only philosophers of mathematics but also epistemologists and metaphysicians
will profit."--Michael D. Resnik, University of North Carolina, Chapel Hill
"In the end the author, quite as much as Goodman and Quine, though in a different way, rejects scientific standards and classical mathematics, and remains true to the spirit, if not the letter, of nominalism...In addition to his ultimate and intermediary conclusions, the author reaches conclusions on many lesser controversial questions, and offers a number of insights on the way."--John P. Burgess, Princeton University, The Philosophical Review
"Platonism and anti-platonism in mathematics is an impressive work. Balaguer presents forceful arguments for the viability of both FBP and fictionalism, and against the feasibility of any substantially different Platonist or anti-Platonist position. ... an admirable achievement."--The Bulletin of Symbolic Logic
