Platonism is the philosophy that mathematical objects exist, that they are mind independent and that their existence is abstract, not spatio-temporal. In the last post in this series, I made an important distinction between this type of platonism which is a form of "object realism" and a more watered down type of "semantic realism". Which holds simply that mathematical statements about abstract objects are true or false independently of whether or not we believe in them.
A semantic realist might say that mathematics is "discovered" rather than "invented", in the same way that Michelangelo thought that he hadn't created the David, he just picked it out of the stone where it already pre-existed. I argued last time that Godel's incompleteness theorem entailed both that semantic realism is true and that mathematical objects, if they are real must be independent of human thought or language. Here we're going to consider the main argument for object realism from Frege and Quine.
It's important for the argument that we're clear on what the existential implications of standard, predicate logic are. The orthodox view which comes from Quine's criteria of ontological commitment, is that we're committed to the existence of entities which can be written as bound variables under a quantifer, and an affirmation of that theory requires that those proposition are true. To give an example, the variable x in $\exists x\left ( Px \right )$ or the variable y in $\forall y \left ( Sy \right )$ is an ontological presupposition of a theory, if these statements are true in that theory.
I think it's perfectly true that the bound variables of an existential quantifer have a reference in the real world but for universal quantifers, this is too strong a commitment. Under Quine's position, a statement such as "all werewolves are howlers" iff true carries a commitment to the existence of werewolves and this would've been fine in Aristotelian logic but it seems more natural to interpret it as "if a werewolf exists, it is a howler". Still, this weaker criteria is enough.
Frege's ontology is similar, except that Frege places significance on the use of singular terms. If I made a statement such as (a) there are at least three cities larger than Hong Kong, and if that statement were true, the truth conditions of that statement entail that I'm committed to the existence of Hong Kong, and at least three other cities. Now, consider the statement (b) there are at least three numbers larger than 14, the semantic structure of both statements is the same, "there are at least three Gs larger than F", so the truth conditions of both statements ought to be roughly the same.
So in fact we have two Frege-Quine type arguments, that can be summarised as
(1) there are propositions in the language of mathematics that refer to mathematical objects as the reference of singular terms or which treat such objects as the variables of an existential quantifer.
(2) there are true propositions in mathematics.
(3) we are committed to the existence of those things which are expressed in true propositions and that are either the reference of singular terms or the variables of existential quantifers.
C: We are committed to the existence of mathematical objects.
That would include all sorts of mathematical objects that we quantify over in first order logic, such as integers, functions, geometric shapes and so forth but it also includes sets. Since we quantify over sets in second order logic, for example $\exists x \exists S\left ( x\epsilon S \right )$ which reads for some x, for some S, x is an element of the set S. I've described how some of the basics of set theory work in another post.
Arguments against premise one used to be a common anti-platonist position, called "paraphrase nominalism" broadly that covers the views of Hilbert and von Neumann that mathematics can be reduced to pure logic, so that reference to mathematical entities can be paraphrased away. Unfortunately for nominalists, this position contradicts Godel's first incompleteness theorem, as almost everyone today now agrees.
Rejection of the second premise is called fictionalism and there are different types of fictionalists just as there are different types of platonists or nominalists, the simplest version would mirror J.L. Mackie's moral error theory. In that, simple fictionalism would accept that mathematical statements refer to abstract objects, but since mathematical objects don't exist according to fictionalism, all mathematical statements are false. This is the best way to attack the argument and my response would have to be something along the lines of the Putnam-Quine indispensability argument but this I'll have to consider in another post.
Premise three is a reaffirmation of the basic criteria of ontological commitment that we began with. I'm open minded to changing or refining whatever criteria we use but whatever the final theory is, it's not going to be radically different from what we already have, and it will roughly line up to assigning ontological commitments to an existential quantifer. Otherwise, you have a criteria which quantifies over non-existent objects which gets you into a number of difficult paradoxes but I don’t want to discuss those in this particular article.
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