Platonism and Anti-Platonism in Mathematics

Platonism and Anti-Platonism in Mathematics

Platonism and Anti-Platonism in Mathematics

Platonism and Anti-Platonism in Mathematics

Synopsis

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.

Excerpt

This book is a work in the philosophy of mathematics. But what does that mean? Well, it might mean either of two different things, for there seem to be two strands in this area of philosophy, two projects that people have taken to be the central project of the philosophy of mathematics. One is the hermeneutical project of providing an adequate interpretation and account of mathematical theory and practice; the other is the metaphysical project of answering the question of whether or not there exist abstract objects.

First, some terminology. An abstract object is an object that exists outside of spacetime or, being more careful, a non-spatiotemporal object, that is, an object that exists but not in spacetime. In any event, such objects are non-physical, nonmental, and acausal. The belief in such objects is called platonism, and the disbelief, anti-platonism. A mathematical object is just an abstract object that would ordinarily be thought of as falling in the domain of mathematics, for example, a number, function, or set. Finally, mathematical platonism is the view that there exist mathematical objects, and mathematical anti-platonism is the view that there do not exist such objects. For the sake of rhetorical elegance, I will often use 'platonism' and 'anti-platonism' to refer to these two views.

The hermeneutical and metaphysical projects are, of course, not entirely separate. Philosophers primarily interested in the former almost invariably advance views regarding the latter, that is, they adopt either platonism or anti-platonism. And philosophers primarily concerned with the latter almost invariably (indeed, in this case, we can probably drop the 'almost') advance views regarding the former, that is, they adopt some interpretation of mathematical theory and practice. The difference between the two is a difference in attitude about what is important. For instance, upon noticing that mathematicians are somewhat cavalier about ontology, philosophers of the hermeneutical bent might themselves be inclined to become cavalier about ontology, whereas philosophers of the metaphysical bent . . .

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