Analogy at the foundations of mathematics

Consider the Benacerraf identification problem in philosophy of maths: there are multiple different ways of “defining” natural numbers in terms of sets, so there is no way of determining which definition is the “correct” one. This is not just a problem about natural numbers but they’re a useful notion to introduce the problem with. In fact, it’s not even just a problem with sets, since there are even multiple foundations from which to choose to define the natural numbers (I’m thinking about type theory and category theory here, considered as foundations).

Now, I’m broadly Aristotelian about mathematical objects: we abstract quantity and structure from reality, isolate certain aspects of these (which we call axioms), and extend these abstracted notions beyond our experience. One of these abstraction-and-extensions is the natural numbers. Presumably something similar could be said for things like sets, groups, rings, topological spaces, metric spaces, the real numbers, the rational numbers, and so on. So, as far as I’m concerned (as are some other non-Aristotelians, I’m sure), the natural numbers are intelligible wholly apart from any set-theoretic “definition” of them.

So what do we do when we “define” natural numbers (or anything) in terms of sets? For starters, I don’t think we define them at all. Rather we characterize numbers in terms of sets (although we nonetheless talk about this in terms of defining or constructing). But what does that mean?

Consider a concrete example: characterize 0 as ∅ (the empty set), 1 as {∅} = {0}, 2 as {∅, {∅}} = {0, 1}, 3 as {0, 1, 2}, and more generally n = {0, 1, 2, …, n-1} for any natural number n. For the sake of brevity, call these sets which we characterize the natural numbers with the natural sets. Now the natural sets have certain features or aspects, partly due to being sets and partly due to their relations with one another (relations they might not share with non-natural sets). Similarly, the natural numbers have their own features or aspects. There are aspects of the natural numbers that aren’t aspects of the natural sets, and there are aspects of the natural sets which aren’t aspects of the natural numbers. The important thing is that there is a correspondence between some of the natural number aspects and some of the natural set aspects. Call this the correspondence of aspect. The extent and nature of the correspondence will depend entirely on which sets we choose and how we choose to relate them to the numbers. Our considered characterization, for example, quite naturally captures the notion of order between natural numbers. That is, n < k (considered as numbers) if and only if n ∈ k (considered as sets).

So far so good, but we can push this further. What does correspondence of aspect consist in? There’s a mathematical way to make this precise: isomorphism. But we’re talking about the very interpretation of mathematics itself, so that won’t work.

Perhaps we could look to metaphysics, particularly Scholastic metaphysics. Here we find the notion of analogy, and it turns out to be quite useful. Much could be said about this, but roughly analogical predicates fall between strictly univocal and strictly equivocal predicates. They are not used in exactly the same sense when predicated, but they are nonetheless related. An example is the notion “seeing” when we see a table (say) and when we see how a conclusion follows from the premises of an argument. Clearly we don’t “see” in exactly the same way in both cases (like the “ball” in “rugby ball” and “soccer ball“), but yet the two seeings aren’t completely unrelated either (like the “ball” in “matric ball” and “rugby ball“). We say that my seeing the table is analogous to my seeing the conclusion. Much can be said about analogy, but for our purposes it will suffice to note that this particular kind of analogy is called analogy of proper proportionality. In these cases we have a relation between relations: our eyes are to the table (relation 1) as our intellect is to the conclusion (relation 2). We write this as “eyes : table :: intellect : conclusion”.

Ok, now come back to our correspondence of aspect. It seems to me that this consists in analogy of proper proportionality (and perhaps sometimes in other kinds of analogy, but I don’t see where they’d apply at this point so I’m sticking with proper proportionality for now). For instance, the characterization we’re considering captures that each natural set is to ∈ as each natural number is to <. If you feel like conveying this in a particularly obfuscated way for just 0, then you could write “∅ : ∈ :: 0 : <“.

So, in summary, we don’t define numbers in terms of sets (strictly speaking). We establish a correspondence of aspect using analogy between sets and numbers, and study those aspects from the perspective of sets. More generally, when we try use sets as a foundation what we’re doing is finding correspondence of aspect using analogy between various structures or quantities and sets and studying them from that perspective. And depending on the correspondence and sets in view, we might end up studying slightly different collections of aspects of these quantities and structures. Although I suspect for many purposes the differences will be irrelevant.

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