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.

Two models of mathematics

I've been thinking lately about high-level parallels between three activities I engage in on a regular basis: philosophy, mathematics and computer programming. One of the obvious things that all of these have in common is that abstraction has a role, though what role it has differs between the three and within each, depending on what one is doing.

One model of mathematics is the abstractive model. "Aristotelian" is a label that comes to mind. Natural numbers abstract from pluralities, ratios abstract from pairs of natural numbers, fields abstract from arithmetic, morphisms abstract from homomorphisms, categories abstract from just about everything. There is an old category theory joke that mathematicians are so absent-minded because mathematics is all about the application of forgetful functors. Taking the joke's thesis literally probably isn't going to give an adequate picture of, say, the number theorist or harmonic analyst are doing, but the basic picture is clear: mathematicians abstract away, or forget, structure. Pluralities have all sorts of structure: these four cows have legs, spots, give milk, and have one dominant cow, but we forget about everything but the "four" to get numbers. When we go from pairs to ratio, we forget everything but the multiplicative relationship. And so on up.

The other model of mathematics is the constructive model (using the term very loosely, and without a commitment to constructivism). Here, we build our way up to more complex structures out of simpler ones. The most impressive example is just how much of mathematics can be seen as about sets and membership.

The abstractive model makes one think of very high level programming languages, of paradigms like functional programming or object-oriented programming. The constructive model makes one think of assembly language programming (some will lump C in here, and I guess that's not unfair). The models can be thought of as models of practice. If so, then they are complementary. Both with computers and mathematics, we need both the abstractive and the constructive practices. We need Java and assembly programmers; we need category theorists and real analysts. Different solutions are appropriate to different problems. You are unlikely to do a lot of abstraction when producing hand-optimized code for an 8-bit microcontroller, but you won't want to produce a banking system in this way. The abstractive approach ("abstract nonsense", as it is fondly called) is just the right thing for many (but not all) problems in algebra, but a constructive approach is likely to be the better solution for many (but not all) problems in harmonic analysis. And of course really good work can involve both, and a well-rounded programmer and mathematician can work in both modalities.

But besides thinking of the two models as models of practice, one can think of them as models of ontology. There is an abstractive ontology of mathematics. Mathematical facts are simply abstractions from facts about concrete things. And there is a constructive ontology, the most prominent starting with sets and showing how the mathematics we care about can be made to be about sets.

With computers, it's clear that the analogue to the constructive ontology is simply the truth of the matter. Computers are built up of transistors, the transistors run microcode, the microcode runs higher level machine code, and all the way up to Java, Haskell and the like. That elegant Haskell one-liner will eventually need to be implemented with microcode. And we have tools to bridge all the levels of abstraction. The proponent of the set-theoretic ontology in mathematics says that the analogy here is perfect. But that is far from clear. We need to take seriously the idea that mathematical abstractions are not built up out of simpler mathematical entities, but are either fundamental or are grounded in non-mathematical entities and pluralities (and their powers, I will say).

Nothing new here. Just laying out the territory for myself.

Adding and multiplying

Two oranges plus three oranges equals five oranges. Two oranges times three oranges equals...? That just sounds malformed. One can add objects but one can't multiply them, it seems.

I suppose one could do a Cartesian product of sets, though, and say that two oranges times three oranges equals six pairs of oranges. If you're a mereological universalist, the "pairs of oranges" might be genuine though unnatural objects; otherwise, you might take them to be abstracta. So addition is either more concrete or more natural than multiplication.

Is there a point to these observations? Not really. They just struck me.