Platonism, fictionalism and all that

by Neil Rickert

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:

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.

When I say that I am a fictionalist, what I mean is simply that when I am doing mathematics, I am treating mathematical entities as if they are useful fictions.  It is useful to be able to give an existence proof for the solution of an equation.  And, in talk about that, it is convenient to be able to say “a solution exists.”  But it has always seemed to me that this is not the ordinary real-life meaning of “exist”.  It is, instead, a special meaning of “exist” that is used in talk about mathematics.  So I can see that as saying that it exists in the fictional world of mathematical entities.  I don’t have to require any actual existence beyond that.

If I meet a mathematical platonist, I don’t say to him “your platonism is false”.  I’m skeptical of platonism, but I’m not going to criticize a mathematician who holds to a platonist philosophy.  It doesn’t really matter to me whether platonism is true or false, or whether fictionalism is true or false.  And I doubt that it even makes sense to talk about whether fictionalism or platonism is true or false.

Fictionalism or formalism?

In his second comment, Robin writes:

I started as a Fictionalist. If you asked me to define mathematics I would have said that is easy, mathematics consists of choosing a set of symbols and then making up some rules to manipulate them by and then proceeding to manipulate them by those rules.

Well, that’s weird, too.  What Robin is defining as “fictionalism” is what I would refer to as “formalism”.  And I see formalism as quite distinct from fictionalism.

If I say

  • 3 + 4 = 7

I am not thinking of “3”, “4” and “7” as symbols to which I am applying rules.  Rather, I am thinking of “3”, “4” and “7” as names of counts, and in thinking about that arithmetic expression I am thinking about counts rather than about symbols.  But nothing was actually counted.  So they are names of fictional counts.  And that’s why I think of myself as a fictionalist.  When I say “3 + 4 = 7”, I am really saying something about what should ideally be the consequences of counting behavior.  So my fictionalism is related to my behaviorism.

My understanding is that a platonist would say that “3”, “4” and “7” actually exist in a platonic world of mathematical entities, and that counting selects an entity from that platonic world.

Is Robin a platonist?

According to Robin’s comments, he was once a fictionalist.  But that did not work out for him and he is now a platonist.  But Robin explains how he thinks of mathematics.  And, on reading that explanation, it seems to me that Robin actually is a fictionalist and not a platonist.

When he says that he was once a fictionalist, but changed to a platonist, I am reading that as “he was once a formalist, but changed to being a fictionalist”.

In his first comment, Robin expresses an “if … then” view.  Everything we claim to be true about mathematics is an “if … then” kind of truth.  That is, if we assume the axioms to be true, then it follows that the theorems are true.  My view is similar.  But platonists don’t talk that way.  The platonists with whom I have discussed mathematics, seem to believe that there is an actual truth to be found.  And if we cannot prove it from our axioms, then maybe we have the wrong axioms.  They tend to see mathematics as a science of the platonist world, with axioms as analogues of the scientific laws.

On meaning

Clearly, Robin and I disagree on the meaning of “fictionalism” and “platonism”.  Just as clearly this is not a fighting disagreement.  We can disagree without being disagreeable about it.

I see this as just one illustration of why I take meaning to be subjective.  That seems to disagree with the view of most philosophers.  Maybe I’ll come up with a separate post on this issue.

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