I’ve been thinking of rebooting my Aristotelian metaphysics series, and I thought I might put this as the preamble or something, so you might also consider this a draft of that.
This stuff is mostly just observations of analogies; I’m not sure if I would consider any of it strictly “proven.”
EDIT: Realized I forgot to add tags.
Aristotelian thought distinguishes two ways in which a given attribute can exist in multiple things: formally and analogically. An attribute exists in two things formally if it is in both of them in exactly the same way; so two red objects both have redness formally, since there are no two ways that something can be red (if we’re specific about what hue we mean by “red”). On the other hand, an attribute exists in two things analogically if it exists in both of them in different ways. For example, both humans and octopuses can be considered to have “hands” in a sense, but obviously a human’s hands are very different from an octopus’s tentacles.
Aristotelian thought also gives us an archetypal form of definition. This form works by considering a genus of things that are assumed to be known to the listener, and delimiting a species from within that genus by means of a specific difference that is common to everything in that species. So for example, a mammal can be defined as a species of animal whose females bear live young and feed their newborns with milk. (Of course, this isn’t technically accurate because platypuses and echidnas lay eggs.) This definition takes a category, animals, that is assumed to be known to the listener, and then delimits the category of mammals by means of their common characteristic of bearing live young and nursing their newborns. Here, the genus is animals, the species is mammals, and the specific difference is bearing live young and nursing their newborns.
Notice that “genus” and “species” are relative terms, since a category can be a genus relative to another, narrower category and a species relative to another, broader category. Further, genera are recursive, since a species within a genus is itself (potentially) a genus; the genus is made up of genera.
Now, math makes a lot of use of things called “sets,” which are rather vaguely defined collections of objects. As you work with them, you gain an intuitive grasp of them, but they’re never really rigorously defined. All you can really say about a set is, as the Wikipedia page says, that it’s a collection of well-defined objects (ironic, considering that the set itself is vaguely defined). Thus a set can contain anything. You can define a set consisting of the numbers 7, 12, 13, 19, and 20, just because you like those numbers. Or you can define a set consisting of red, green, and blue. Or you can refer to the set of all even numbers, or the set of all rational numbers, etc. Or the set of all people who wear their hair in a topknot.
A set can also be broken down into subsets, where every member of the subset is also in the original set (referred to as a superset). So the set containing 2, 4, and 6 is a subset of the even numbers, while the even numbers are a superset of the set containing 2, 4, and 6.
Incidentally, it’s also perfectly acceptable to have a set of sets. In fact, the set of all subsets of a given set is called the power set of that set.
This brings up an interesting question: Is it possible to form a set of all sets? As it turns out, the answer is no, because it results in Russell’s paradox. Every set is either a member of itself or not; for convenience, we can refer to these as self-inclusive sets and self-exclusive sets. The set of all self-exclusive sets would then have to be a subset of the set of all sets. But is the set of all self-exclusive sets a self-exclusive set, or a self-inclusive set? If it’s self-exclusive, then it would have to be a member of itself—which then implies that it must be self-inclusive. By the same token, if it’s self-inclusive, then that means that it’s not a member of itself, which means that it must be self-exclusive. Either way, we get a contradiction. Therefore, the set of all self-exclusive sets can’t possibly exist, and therefore the set of all sets, which must be a superset of the former, also can’t exist.
This leads us to the concept of classes, which is even more vaguely defined than sets. Basically, a class is a group of objects that all have something in common somehow, but that we can’t necessarily represent as a set. “All sets” would then be a class, but not a set.
Now, one interesting point that’s often glossed over in math textbooks is that there’s a very obvious difference between sets like “7, 12, 13, 19, and 20” and sets like “the even numbers.” Formal math doesn’t have a term for distinguishing these two types of sets as far as I know, but for convenience, let’s call the former type of set a scoop (from the action of arbitrarily scooping random things out of a jar) and the latter a proper set. We can then say that a scoop only exists because somebody decided it does, while a proper set actually has a kind of inner coherence. Why is this?
Well, thinking back to Aristotle gives us a clue. The members of a scoop don’t necessarily have anything in common. But the members of a proper set have some common characteristic that they all share formally. And we can take this as a kind of “definition” of proper sets: A proper set is a grouping of objects that all share some common characteristic formally (but see below—I don’t think it’s actually possible to give a rigorous definition of proper sets).
And recalling the Aristotelian contrast of formal vs. analogical and the mathematical contrast of set vs. class immediately brings another connection to mind: A class would just be a grouping of objects that all share some common characteristic analogically.
And now that we’ve gotten ourselves into a math-and-Aristotle-y sort of mood, we might as well go a bit further. Recall how genera and species behave recursively—any species within a genus can potentially be a genus itself, and any genus can potentially be a species of another genus. Well, notice that the relation of supersets and subsets behaves in exactly the same way—any subset can potentially be a superset of another set, and any superset can potentially be a subset of another set. And further notice that a species is delimited by some characteristic that all its members share formally. In other words, a species is a proper set. So a definition is nothing other than a delimitation of one proper set from within another.
And this shows why it’s not possible to define proper sets—we would have to delimit the proper set of all proper sets from some other proper set, which is impossible because, as shown above, there is no proper set of proper sets. But the collection of all sets is a class, which tells us that proper sets are an analogical concept.