## Monday, August 30, 2021

### A tension in some of my recent work

Here is a tension in some recent work of mine. In Chapter 6 of Infinity, Causation, and Paradox, I argue that (a) the Axiom of Choice for countable sets of reals (ACCR) is true, and (b) this version of the Axiom of Choice plus causal infinitism implies a nasty paradox, so we should accept causal finitism instead. The argument for ACCR makes use of the premise that mathematical entities exist necessarily. But in “Might All Infinities Be The Same Size?”, I argue that for all we know, some mathematical entities exist contingently. Thus, the latter paper undercuts the argument of Chapter 6 of the book.

Fortunately, the argument of Chapter 6 of the book looks like it might be fixable. The argument for ACCR proceeded as follows:

1. For any set of non-empty countable sets of reals, it is metaphysically possible that there is a choice function.

2. If possibly there is a choice function, then necessarily there is a choice function.

The argument for (1) is elaborate, but it is (2) that the considerations in my article block.

But we can try to proceed as follows. The paradox in Chapter 6 of the book requires a choice function for a particular collection of non-empty countable sets of reals (reals generatable by a certain infinitary coin-tossing process). By (1), there is a possible world w′ where that particular collection of sets does have a choice function. So it seems all we need to do is to run the paradox in w′, and we should be done.

There are probably other areas in Chapter 6 where some tweaking (or more than that!) is needed to make things work with mathematical contingentism, and hence my cautious wording.

swaggerswaggmann said...

But then there is possibly a World w'' where there is no such function. So the argument don't work here and causal finitism is false.

Alexander R Pruss said...

The argument schema is:
1. If causal infinitism is true, then such-and-such a paradox is metaphysically possible.
2. Such-and-such a paradox is metaphysically impossible.
3. So causal infinitism is not true.

This only requires the paradox to work in one world. Indeed, NONE of the paradoxes I consider work in the actual world, since in the actual world there are no grim reapers, infinite sequences of die throws, etc., etc.

swaggerswaggmann said...
This comment has been removed by the author.
swaggerswaggmann said...

No. As there is a world where they don't work, so nothing to prevent infinitism, so you agree that in this world causal finitism can be false, and is. My point.

You fail to consider that your argument as a simple symmetry...

All that assuming that what you paint as paradox are not simply misunderstanding on your part, as many philosophers replying to you have hinted. A element being at two place at one, moving and immobile is paradoxical, but nature consults no philosophers.

ZsoltnagY said...

I know, that this is just a comment on a blogpost. But this is not correct here from you, Alexander Pruss:
The argument schema is:
1. If causal infinitism is true, then such-and-such a paradox is metaphysically possible.
2. Such-and-such a paradox is metaphysically impossible.
3. So causal infinitism is not true.

I guess, that you rather mean this by that:
The argument schema is:
1. If causal infinitism is false, then such-and-such a paradox is metaphysically possible.
2. Such-and-such a paradox is metaphysically impossible.
3. So causal infinitism is not false (or it is actually true).

I also guess, that this wouldn't have been approved by Aristotle, since that argument formulated in categorical logic is simply invalid from the gekko:
CL-1) Causal finitism is false.
Hence, all infinite causal chains are metaphysically possible.
CL-2) Some infinite causal chains are paradoxical.
Hence, Some infinite causal chains are not metaphysically possible.
CL-3) Therefore, all infinite causal chains are metaphysicall not possible.
So causal finitism is true.

If some infinite causal chains are not metaphysically possible, then not all infinite causal chains are metaphysically possible.
This is the correct logical equivalence and inference.
If you, Alexander Pruss, want to prove causal finitism to be true, then show and prove that properly. Prove that all infinite causal chains are not metaphysically possible and not just “some”.
Otherwise don't masquerade this kind of hasty generalisation as a "normal" modus tollens.
Thank you.

Alexander R Pruss said...

I really do mean "infinitism" in the argument as sketched.

Premise 1 is a material conditional. My preferred approach to premises like 1 is that there are nonparadoxical infinite causal chains that differ from the paradoxical one only in terms of some rearrangement or numerical parameters, and then I apply a rearrangement principle to conclude that the paradoxical one should be possible as well.

Zsolt Nagy said...
This comment has been removed by the author.
Zsolt Nagy said...

Yeah, but not all rearrangements are equal to each other especially not rearrangements of numerical parameters e.g. Riemann rearrangement theorem.
But from the Riemann rearrangement theorem doesn't follow, that all series behave like that: changing the limit of the series by rearrangement.
Some series doesn't change behaviour regarding the limit by any rearrangements. Namely any absolutely convergent series doesn't change its behaviour regarding limits for any rearrangements.
So how about "such-and-such" cases, where the infinite causal chain doesn't entail any contradictions or paradoxes for any rearrangements?
I guess, that neither you, Alexander Pruss, nor causal finitism can nor is willing to answer that question.