Σ10 alethic Platonism

Here is an interesting metaphysical thesis about mathematics: Σ₁⁰ alethic Platonism. According to Σ₁⁰ alethic Platonism, every sentence about arithmetic with only one unbounded existential quantifier (i.e., an existential quantifier that ranges over all natural numbers, rather than all the natural numbers up to some bound), i.e., every Σ₁⁰ sentence, has an objective truth value. (And we automatically get Π₁⁰ alethic Platonism, as Π₁⁰ sentences are equivalent to negations of Σ₁⁰ sentences.)

Note that Σ₁⁰ alethic Platonism is sufficient to underwrite a weak logicism that says that mathematics is about what statements (narrowly) logically follow from what recursive axiomatizations. For Σ₁⁰ alethic Platonism is equivalent to the thesis that there is always a fact of the matter about what logically follows from what recursive axiomatization.

Of course, every alethic Platonist is a Σ₁⁰ alethic Platonist. But I think there is something particularly compelling about Σ₁⁰ alethic Platonism. Any Σ₁⁰ sentence, after all, can be rephrased into a sentence saying that a certain abstract Turing machine will halt. And it does seems like it should be possible to embody an abstract Turing machine as a physical Turing machine in some metaphysically possible world with an infinite future and infinite physical resources, and then there should be a fact of the matter whether that machine would in fact halt.

There is a hitch in this line of thought. We need to worry about worlds with “non-standard” embodiments of the Turing machine, embodiments where the “physical Turing machine” is performing an infinite task (a supertask, in fact an infinitely iterated supertask). To rule those worlds out in a non-arbitrary way requires an account of the finite and the infinite, and that account is apt to presuppose Platonism about the natural numbers (since the standard mathematical definition of the finite is that a finite set is one whose cardinality is a natural number). We causal finitists, however, do not need to worry, as we think that it is impossible for Turing machines to perform infinite tasks. This means that causal finitists—as well as anyone else who has a good account of the difference between the finite and the infinite—have good reason to accept Σ₁⁰ alethic Platonism.

I haven't done any surveys, but I suspect that most mathematicians would be correctly identified as at least being Σ₁⁰ alethic Platonists.