Platonism, fictionalism and all that

There was recently an interesting discussion of platonism and fictionalism as philosophies of mathematics.  This was at “The Electric Agora” blog.  I added a couple of comments myself.

Yesterday, I went back to take another look.  That was mostly to see if there were any additional comments.  And there were two, both by Robin Herbert.  But comments are now closed for that post.  So I’ll say something here.

First some links:

• First comment by Robin Herbert;
• Second comment by Robin Herbert.

Both comments add to the discussion and are worth reading.

Is fictionalism true?

In his first comment, Robin says:

So the argument that fictionalism must be true because the axioms are only conventions appears to make the same mistake as saying the truth or falsity of “if A then B” depends on the truth or falsity of A.

To me, this seems weird.  I have said that I am a fictionalist.  I have never said that fictionalism is true.  I’m not at all sure that I know what it would even mean to say that fictionalism is true.

Mathematical Fictionalism

Massimo Pigliucci has a post on mathematical platonism, so I thought it appropriate to discuss that in conjunction with my own version of mathematical fictionalism.

Pigliucci begins with three principle of platonism, which he takes from the SEP entry:

1. Existence: There are mathematical objects;
2. Abstractness: Mathematical objects are abstract;
3. Independence: Mathematical objects are independent of intelligent agents and their language, thought, and practices.

Here’s the parallel principles for my version of fictionalism:

1. Mathematical objects are useful fictions.  They have no actual existence, but it is useful to talk about them as if they existed.
2. Mathematical objects are abstract.  I take this as a consequence of their being fictions.
3. Mathematical objects are mental constructs, so are not strictly independent of the intelligent agents who talk about them.  However, if some alien intelligence exists — let’s call them Martians, to have a name — were to construct their own mathematics for reasons analogous to why we construct mathematics, then many of their mathematical fictions would have truth conditions analogous to those of our mathematics.

My fictionalist version of independence is weaker than the platonist version, though it seems adequate for mathematics.