Mathematical Fictionalism

by Neil Rickert

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.

Mathematical truth

The SEP article on fictionalism has this to say about truth:

Thus, the idea is that sentences like ‘3 is prime’ are false, or untrue, for the same reason that, say, ‘The tooth fairy is generous’ is false or untrue—because just as there is no such person as the tooth fairy, so too there is no such thing as the number 3.

I see that as an unnecessarily restrictive view of truth.  I am more inclined to say that “3 is prime” is true in the same sense that it is true that Sherlock Holmes lived at 221B Baker Street.  That is to say, mathematical truths are statements that are to be understood within the context of mathematics, and not to be considered statements about the physical world.

Mathematics and behaviorism

I see my fictionalist view of mathematics as connected with my behaviorism.  Counting is one of our behaviors.  I see arithmetic as arising from an analysis of counting.  For the purpose of that analysis, we idealize our behavior so that in ideal counting we never make mistakes and there is never ambiguity as to whether an item is something to be counted.  We can then see mathematical objects as fictional entities that are stand-ins for the real world object that we might be counting in an practical counting scenario.  Our study of arithmetic allows us to draw general conclusions that should be applicable to practical counting situations.  Whether or not mathematical objects exist has no importance for this purpose.  What does matter, is that the relationships between counts that we can prove mathematically, should apply to practical applications of counting.

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