Friday, February 23, 2018

A slightly different causal finitist approach to finitude

The existence of non-standard models of arithmetic makes defining finitude problematic. A finite set is normally defined as one that can be numbered by a natural number, but what is a natural number? The Peano axioms sadly underdetermine the answer: there are non-standard models.

Now, causal finitism is the metaphysical doctrine that nothing can have an infinite causal history. Causal finitism allows for a very neat and pretty intuitive metaphysical account of what a natural number is:

  • A natural number is a number one can causally count to starting with zero.

Causal counting is counting where each step is causally dependent on the preceding one. Thus, you say “one” because you remember saying “zero”, and so on. The causal part of causal counting excludes a case where monkeys are typing at random and by chance type up 0, 1, 2, 3, 4. If causal finitism is false, the above account is apt to fail: it may be possible to count to infinite numbers, given infinite causal sequences.

While we can then plug this into the standard definition of a finite set, we can also define finitude directly:

  • A finite set or plurality is one whose elements can be causally counted.

One of the reasons we want an account of the finite is so we get an account of proof. Imagine that every day of a past eternity I said: “And thus I am the Queen of England.” Each day my statement followed from what I said before, by reiteration. And trivially all premises were true, since there were no premises. Yet the conclusion is false. How can that be? Well, because what I gave wasn’t a proof, as proofs need to be finite. (I expect we often don’t bother to mention this point explicitly in logic classes.)

The above account of finitude gives an account of the finitude of proof. But interestingly, given causal finitism, we can give an account of proof that doesn’t make use of finitude:

  • To causally prove a conclusion from some assumptions is to utter a sequence of steps, where each step’s being uttered is causally dependent on its being in accordance with the rules of the logical system.

  • A proof is a sequence of steps that could be uttered in causally proving.

My infinite “proof” that I am the Queen of England cannot be causally given if causal finitism is true, because then each day’s utterance will be causally dependent on the previous day’s utterance, in violation of causal finitism. However, interestingly, the above account of proof does not guarantee that a proof is finite. A proof could contain an infinite number of steps. For instance, uttering an axiom or stating a premise does not need to causally depend on previous steps, but only on one’s knowledge of what the axioms and premises are, and so causal finitism does not preclude having written down an infinite number of axioms or premises. However, what causal finitism does guarantee is that the conclusion will only depend on a finite number of the steps—and that’s all we need to make the proof be a good one.

What is particularly nice about this approach is that the restriction of proofs to being finite can sound ad hoc. But it is very natural to think of the process of proving as a causal process, and of proofs as abstractions from the process of proving. And given causal finitism, that’s all we need.


Wesley C. said...

Does causal finitism depend on the PSR, or is the PSR independent of causal finitism such that even if causal finitism were false, PSR could still be true?

Also, your argument about how causal finitism is related to argumentative causation, if we admit that PSR entails CF, may also make for a good argument in favour of PSR.

Namely, if PSR is false, then not only could there be a proof with infinitely many dependent steps, but there could also be conclusions that are true for no reason, which would wreak havoc all over the place.

Alexander R Pruss said...

A small subset of my arguments for CF depends on the PSR, but that's all.

IanS said...

Even in non-standard models, can’t the usual natural numbers be identified as the successors of zero? (This is not my area, so I may be way off beam.)

I’m not following your “proof”. “And thus X” follows only if the previous statements, taken together, imply X. I don’t see that they do. “And trivially all premises were true, since there were no premises.” Isn’t the point of a proof to show that the conclusion follows from true or accepted premises? On this view a “proof” without premises cannot be a proof, regardless of causal finitism.

On possible infinite proofs: if the conclusion depends only on a finite number of the steps, aren’t the others superfluous?

Alexander R Pruss said...

Proofs without premises are quite common. For instance, in classical natural deduction systems one can prove P or ~P without any premises.

Defining the natural numbers as successors of zero is problematic. The standard way to say it is this: "The natural numbers are the smallest set S such that (a) 0 is in S, and (b) the successor of every element of S is in S." However, in a non-standard model, that smallest set might actually be larger than our familiar set of standard natural numbers. In such a non-standard model, our familiar set of standard natural numbers won't exist as a set.

IanS said...

Thank you. I now see. I'm still a bit worried by a metaphysical, as opposed to a purely formal approach, but I don't know enough to take it further.