A new argument for causal finitism

I will give an argument for causal finitism from a premise I don’t accept:

1. Necessary Arithmetical Alethic Incompleteness (NAAI): Necessarily, there is an arithmetical sentence that is neither true nor false.

While I don’t accept NAAI, some thinkers (e.g., likely all intuitionists) accept it.

Here’s the argument:

2. If infinite causal histories are possible, supertasks are possible.

3. If supertasks are possible, for every arithmetical sentence, there is a possible world where someone knows whether the sentence is true or false by means of a supertask.

4. If for every arithmetical sentence there is a possible world where someone knows whether the sentence is true or false by means of a supertask, there is a possible world where for every arithmetical sentence someone knows whether it is true or false.

5. Necessarily, if someone knows whether p is true or false, then p is true or false.

6. So, if infinite causal histories are possibly, possibly all arithmetical sentences are true or false. (2-5)

7. So, infinite causal histories are impossible. (1, 6)

The thought behind (3) is that if for every n it is possible to check the truth value of ϕ(n) by a finite task or supertask, then by an iterated supertask it is possible to check the truth values of ∀xϕ(x) (and equivalently ∃xϕ(x)). Since every arithmetical sequence can be written in the form Q₁x₁...Q_kx_kϕ(x₁,...,x_k), where the truth value of ϕ(n₁,...,n_k) is finitely checkable, it follows that every arithmetical sequence can have its truth value checked by a supertask.

The thought behind (4) is that one can imagine an infinite world (say, a multiverse) where for every arithmetical sentence ϕ the relevant supertask is run and hence the truth value of the sentence is known.

Provability and truth

The most common argument that mathematical truth is not provability uses Tarski’s indefinability of truth theorem or Goedel’s first incompleteness theorem. But while this is a powerful argument, it won’t convince an intuitionist who rejects the law of excluded middle. Plus it’s interesting to see if a different argument can be constructed.

Here is one. It’s much less conclusive than the Tarski-Goedel approach. But it does seem to have at least a little bit of force. Sometimes we have experimental evidence (at least of the computer-based kind) for a mathematical claim. For instance, perhaps, you have defined some probabilistic setup, and you wonder what the expected value of some quantity Q is. You now set up an apparatus that implements the probabilistic setup, and you calculate the average value of your observations of Q. After a billion runs, the average value is 3.141597. It’s very reasonable to conclude that the last digit is a random deviation, and that the mathematically expected value of Q is actually π.

But is it reasonable to conclude that it’s likely provable that the expected value of Q is π? I don’t see why it would be. Or, at least, we should be much less confident that it’s provable than that the expected value is π. Hence, provability is not truth.

The problem of the stone

I like to use the stone argument as a warm-up in a philosophy of religion class. But it's actually kind of tricky to use. Here's a natural way to put it:

1. Either God can or cannot make a stone he can't lift.
2. If God cannot make such a stone, then there is something God can't do.
3. If God can make such a stone, then there is something God can't do, namely lift the stone.
4. So, there is something God can't do.

But in this formulation, (3) can be easily rejected. It does not follow follow from God's merely being able to make such a stone that there is something God can't do, just as it doesn't follow from God's being able to make a unicorn that there is a unicorn. The correct conditional is:

5. If God does make such a stone, then there is something God can't do, namely lift the stone.

But if we replace (3) by (5) in the argument, the argument ceases to be valid.

This means that the stone argument isn't actually an argument against omnipotence. If all that was in view was omnipotence, one could say: "Sure, God can create such a stone. Were he to create it, he wouldn't be omnipotent. But he hasn't created such a stone and he is omnipotent." Rather, we should take the stone argument as an argument against essential omnipotence. And that makes the argument a little less suited for warm-up classroom use, because one has to introduce the notion of an essential property.

What I actually did in class today is I gave the argument in the invalid form. Alas, nobody caught the invalidity. Though, interestingly, one student was unsure of disjunction-elimination in general.

I also emphasized that the stone wasn't really a problem for omnipotence, but for particular attempts to define omnipotence. I think it's important to to distinguish those atheological arguments that are problems for theism from those that are problems for particular ways of defining theism. The inductive problem of evil is an argument against theism; the stone argument is only an argument against particular formulations.