Aristotelian abstraction

On the Aristotelian account of abstraction, we abstract the forms of things via perception. But here is a problem. First of all, we sometimes have better knowledge of things not perceived than of things perceived. Let’s say you read a detailed scientific book about pangolins, though you’ve never encountered one, while I once saw a koala in a zoo, but don’t much know anything more about them than that they are cute marsupials. It seems quite problematic to say I have the form of the koala in my mind, but you do not have the form of the pangolin. For that would make the mental possession of the form of the thing an unimportant flourish to what really epistemically matters, since clearly you are better off epistemically for reading the book about pangolins than I am for seeing the koala.

Of course, presumably, the author of the pangolin book presumably saw the pangolin. But that doesn’t seem to matter much, either. There are many dinosaur species where the paleontologist knows more about the species than I know about koalas, and yet no human ever perceptually interacted with a dinosaur, but typically only with minerals that have displaced their carcasses.

What should someone committed to the Aristotelian story about abstraction say? Two options come to mind, a modest and an expansive one.

Modest option: It really doesn’t really matter much epistemically whether one has the form of a pangolin or a koala in one’s mind. What matters is the propositional knowledge. However, it is essential to our possession of the propositional knowledge that our thoughts have intentionality, that they refer to the world. And it is crucial to securing intentionality that we have Aristotelian abstraction at the interface between the world and the mind. The triceratops expert’s intentionality with respect to triceratopses is derivative from the causal chain from triceratopses to their corpses to their fossils to the light reflecting from the fossils into the expert’s eyes to (eventually somehow) the expert’s thoughts. Where the causal chain crosses from the world to the mind, we need a transmission of form to ensure intentionality for the whole chain. Maybe the transmission happens when the minerals that have replaced the bones transmit their mineral forms to the expert’s mind, in some mysterious way mediated by the light.

Expansive option: Everything in the world carries the forms of everything nondivine that is causally upstream from itself. The triceratops corpse somehow has the form of the triceratops in it, and so do the bones of the triceratops, and so do the minerals that replace the bones, and so does the light modulated by reflection from the minerals, and so do the electrical impulses in the nerves from the retinal receptors to the brain. It is easier to extract the form when it comes from seeing a live koala than when it comes from seeing fossils, so that a layman can do the former while the scientist is needed to do the latter. Similarly, when you read a book about pangolins, the ink on the page, being causally modulated by the author’s knowledge, carries the form of the pangolin (in addition to any forms naturally contained in the ink, say those of the particles constituting the ink).

Medievals already said something like this about the modulated light, so it’s just a matter of extending the story. And while the problem I am concerned about is one that does not require any science or technology beyond what the medievals had—the medievals knew about the gaining of expertise from books (if anything, they might have overestimated it!)—in a modern setting we it feels like an Aristotelian has to do this. After all, it is hard to deny that one can gain the same kind of knowledge of koalas by looking at them through high-end augmented reality goggles as from directly seeing them with our eyes. Thus, on a form-transmission story, the electrical potentials of the capacitors in the computer memory in the goggles have to somehow carry the form of the koala. But given the many ways that computer memory can be realized (think of magnetized rings versus capacitors versus markings on a CD), it seems plausible that any effect will have to have to be admitted to carry the form of its cause. It is tempting to say the form is only found in substances where there is sufficient data to reconstruct significant information about the causally-upstream object the form is of, but I think is not tenable. Presumably each bacterium is a substance, and we could have a type of biological memory in augmented reality glasses where each bit of information is stored in a different bacterium.

This story coheres very nicely with the essentiality of origins. It is a kind of a reversal of the principle of proportionality of causation on which the reality of the effect is actually or eminently found in the cause: the formal reality of a cause is found in the effect. I think it’s a defensible story. But I also find it hard to believe.

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.