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.

Truth and axioms in mathematics

There’s been some discussion of truth in mathematics in the comments to my previous post.  Here, I want to expand a little on my view and express puzzlement at the idea that axioms are themselves true or false.

In response to a question, said “Actually, I take axioms to be neither true nor false, and I take the truth of mathematical theorems to be relative to the assumed axioms.”  Let me restate that in terms of the Peano axioms for ordinary arithmetic.

1. The Peano axioms are neither true nor false.  Rather, they are definitional statements.  They define that part of mathematics known as Peano Arithmetic (or PA, or simply arithmetic).
2. Theorems proved in PA are true in a relative sense.  Their truth is relative to the PA axioms.  They are true as used within PA, but perhaps not even meaningful outside of PA.

Why I don’t like philosophy of mathematics

I recently posted a link to an explanation of the philosophy of mathematics.  While I thought that Balaguer’s explanation was very good, I also remarked that I don’t find the philosophy of mathematics to be useful.  In this post, I’ll say why I don’t find it useful.

Toward the end of his explanation, Balaguer presents the following argument for platonism:

1. Semantic platonism is true–i.e., ordinary mathematical sentences like ‘2 + 2 = 4’ and ‘3 is prime’ are straightforward claims about abstract objects (or at any rate, they purport to be about abstract objects). Therefore,
2. Mathematical sentences like ‘2 + 2 = 4’ and ‘3 is prime’ could be true only if platonism were true–i.e., only if abstract objects existed. But
3. Mathematical sentences like ‘2 + 2 = 4’ and ‘3 is prime’ are true. Therefore,
4. Platonism is true.

Balaguer, who says he is a fictionalist and not a platonist, questions step 3 in that argument.  However, it seems to me that step 2 is already mistaken.  People simply do not use “true” in the way that step 2 supposes.

Nagel’s “Mind and Cosmos” – not quite a review

I have been reading Nagel’s book, “Mind and Cosmos:Why the Materialist Neo-Darwinian Conception of Nature Is Almost Certainly False“, so naturally I want to say something about it.  However, this won’t be the usual kind of review.  There’s no need for that.  There are already plenty of reviews available for this book, some of them scathing critiques and some of them offering high praise.

For myself, I disagree with much of what Nagel writes.  But I find it interesting nonetheless.  Readers of this blog will have noticed that I disagree with a lot of traditional philosophy.  And Nagel particularly emphasizes some of those parts where I disagree.  So, in a way, this highlights my disagreement.  If I were to suggest an alternative title for Nagel’s book, it might be:

• “What’s wrong with philosophy” on steroids

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.

On metaphysics

In an earlier post, I hinted that I would discuss the two essays by Massimo Pigliucci on naturalized metaphysics.  So that will be the goal of this post.  For convenience, I shall refer to those two essays as NM1 and NM2.

• NM1: Surprise! Naturalistic metaphysics undermines naive determinism, part I
• NM2: Surprise! Naturalistic metaphysics undermines naive determinism, part II

The goal

It is not my aim here to argue that Pigliucci is wrong.  Rather, the aim is to present how I look at the questions he is discussing.  Partly, this is because I have rather non-typical views, and am sometimes asked to explain them.  Partly, it is because I have indicated my dislike for metaphysics, and some have suggested that we cannot actually do without metaphysics.  So perhaps the discussion here will help my readers better understand my viewpoint.