Tuesday, March 28, 2017

More remarks on Aristotelian set theory

If we have an Aristotelian picture of abstracta, we should expect that what mathematical objects exist differs between possible worlds.

For the Aristotelian, abstract objects are abstractions from concrete things. So we shouldn’t expect the same full panoply of sets regardless of what concrete things there are. For instance, suppose that the universe contains exactly three point particles, A, B and C. Then we can immediately abstract from these particle positions distance ratios like AB : BC, AC : AB and AC : BC. These ratios are then represented by real numbers. So we are going to have these real numbers. More sophisticated abstractive processes may well generate other real numbers: for instance, we will have a real number representing the ratio of the height of the triangle drawn from A to the base BC. And given a real number, we might be able to use purely abstract processes to generate further real numbers: given a and b, we may generate a + b and ab, say. But there is no reason to think that these abstract processes will generate the same collection of real numbers regardless of what the three particle positions we start with are.

So, what real numbers exist should vary between possible worlds. But every real number defines a subset of the natural numbers (just write the real number in binary, and let the nth bit decide if n is in the subset or not). If the real numbers vary between possible worlds, so do the subsets of the natural numbers. In particular, we should expect that in different possible worlds, a different set counts as ``the power set’’ of the natural numbers.

Furthermore, what bijections there are between sets will vary between possible worlds. Thus, if we see the question of whether two sets have the same count of members as having the same answer in every world where the two sets exist, we cannot take the standard Cantorian account of the size of a set. Instead, we may want to generate the concept of sameness of size from bijections in different worlds. Thus, we may try to say that two sets A and B are the same size at level 0 provided that there is a bijection between A and B. Then we say that A and B are the same size at level n provided that possibly there is a set C that is the same size as A at level p and the same size as B at level q and n ≥ 1 + p + q. Finally, we say that A and B are the same size simpliciter provided that they are the same size at some finite level. This is complicated, and I haven’t checked under what assumptions it generates a transitive relation (it’s plausibly reflexive and symmetric).

Anyway, the point is this: It is an interesting and not easy philosophical project to work out the set-theoretic consequences of Aristotelianism. This could make a good dissertation.


M. J. Glaeser said...

Then, if our world is fundamentally discrete (as some Quantum Gravity theories postulate), then is it the case that (assuming Aristotelianism) there aren't trascendental numbers in the actual world?

Alexander R Pruss said...

Could be, but it could also be that all the finitely describable reals one gets as abstractions for free, and that will include some transcendentals.

Anonymous said...

In particular, we should expect that in different possible worlds, a different set counts as "the power set" of the natural numbers.

That sounds funny to me. Mathematics tacitly assumes a platonic background, or at least talks that way, so surely in order to translate "a power-set is the set of all subsets" into Aristotelian language, we ought to say, "a power-set is the set of all possible subsets". In other words, we're not referring to sets that actually exist in the Aristotelian sense, but all the ones that could exist in the Platonic sense. Interestingly, this seems to show that the abstraction "the real numbers" could exist even though all the real numbers themselves do not (Aristotelianly). (Or maybe that's not so interesting — I'm not sure!)

Alexander R Pruss said...

One might try to ground some mathematical objects in causal powers. That would allow for real numbers that aren't exhibited in nature, as long as they could be so exhibited. And given my causal modal theory, this could do everything.