Tuesday, February 24, 2015

Mathematics and the actual infinite

  1. If mathematical realism is true, there is an actual infinite.
  2. The best alternatives to mathematical realism require the possibility of an actual infinite.
  3. So, probably, an actual infinite is possible.
What needs justifying is (2). Here, I say that the best alternative to mathematical realism is either fictionalism or some version of structuralism. Structuralism says that mathematics describes possible structures. But if there cannot be an actual infinite, then there is no possible structure that is described by arithmetic. On the other hand, fictionalism is very problematic when it is impossible for anything like the fictional story to be true.

4 comments:

Peter said...

This is Greek to me:

«But if there cannot be an actual infinite, then there is no possible structure that is described by arithmetic.»

Thank you, if you'd like to describe this a bit more.

Michael Gonzalez said...

What does "anything like" mean? Infinite sets are like finite ones, and the rules can be generalized. Very high numbers may not actually be instantiated, but smaller numbers are, and the rules are the same. I'm just not sure exactly what problem with Fictionalism you are picking out.

John West said...

Peter,

But if there cannot be an actual infinite, then there is no possible structure that is described by arithmetic.

I just took this as saying that what it means for a structure to be possible, is that the structure could have been actual.

Though, if I'm completely missing the mark, I would also appreciate further description.

Alexander R Pruss said...

Michael:

What I meant by "anything like" was something like this: Maybe there couldn't be actually infinitely many numbers, but there could be infinitely many numerals, or pigs, or some other stand-in isomorphic to what we think of as numbers.

(I was hedging with the "anything like" because of a Kripkean worry that all fictional characters are impossible for the reason that Kripke thinks unicorns are impossible. The worry disappears if instead of possibility we talk of conceivability in the two-dimensionalist semantics sense.)

The rules for dealing with infinities are not the same as for finite ones. The cardinality of an infinite set is often (maybe always--depends on whether the Axiom of Countable Choice is true) unchanged by addition or subtraction of a finite amount.