I have a pair of pants that have a pocket that has a pocket. Not the kind of small change, house key specific pocket you sometimes get sitting up near the belt. This one is deep down within the softer fabric of the lining. It is so well hidden, so well integrated, that I sometimes forget it’s there. It’s only those occasions when I want to warm up my hands a little, by sliding into my empty pockets, that I get to wondering whether they really are all that empty.
The inner pocket has nothing in it. But the outer pocket does, namely, the inner pocket itself! The inner pocket is also exactly the same shape as the outer pocket, just a scaled down version. If I put my index finger into the lowest corner of the inner pocket, the very fabrication of the peculiar nesting means my index finger is at the same time also resting at the lowest corner of the outer pocket.
It is strange the way that putting nothing inside itself can create something. Take an empty pocket, copy it, and shrink it, then put that miniature copy inside the original pocket. With this hierarchy we can distinguish two distinct objects; the inner pocket that contains no things, and the outer pocket that contains one thing. It’s not too difficult now to imagine putting this two-level hierarchy of pockets into another, bigger pocket. Or even going the other way and sewing a smaller pocket within the inner pocket, to make an inner inner pocket. Then an inner inner inner pocket, and so on...
This notion of recursively nesting identical copies of a thing within itself is at the heart of the formal construction of the whole numbers, from which every other object in modern mathematics can be built. A theory of pockets could get us there too, but mathematicians have already developed a far more sophisticated theory of things they call sets.
We can think of a set as an abstraction of a pocket. It’s a thing that you put stuff in. In the same way that the pockets on your pants come for free, we generally take for granted that we already have access to a set with nothing in it. That is, we make the claim that there exists an empty set. After that we just have to outline the rules of how two sets can be combined, then we’re free to define our way through the rest of the mathematical landscape.
While this origin story has a tidy self-consistency, it doesn’t really reflect the way many mathematical concepts have naturally developed. This can lead to a confusing experience when searching along the genealogy of a particular mathematical object.
For example, a good understanding of quadratic equations would involve, at least, an understanding of the Cartesian plane. But the Cartesian plane is built from the number line, so we might first want to get a handle on that. Then again, the number line is built from the whole numbers, which we know are built from sets. And sets are built from ... nothing? What then of quadratic equations? We seemed to have stepped back through the family tree to find that there never was a tree to begin with! Having emptied our pockets we find only a pile of empty pockets! What would Descartes say about his eponymous plane?!
Of course, looking to sets for the meaning of quadratic equations is like putting your face right up to a painting and wondering where the art went. We all know that fruit is made of tiny cells, but we don’t expect to see anything that is uniquely characteristic of an apple when we look at it under a microscope. Water can be cool or warm, slippery and sloshy, but a single molecule of water can be none of these things. An ecologist doesn’t need to know about quarks to understand trophic cascade. Even though large, complex objects have smaller building blocks, the properties of those larger things belong to that scale, and cannot be sought at the foundations.
Indeed, Descartes and many others made substantial contributions to coordinate geometry in the centuries before Cantor created the theory of sets. The Cartesian plane was a response to the desire to understand the space we live in, not purely a derivation from axiomatic set theory. We see quite often a mathematical object or concept will be initially inspired by our intuition about how we experience the world. Then an evolution similar to that of handheld tools takes place. The more a concept gets used, the more precise we want to make it, and the more powerful and useful it becomes.
In this way, we can picture a particular mathematical idea as a tool, and the whole body of mathematical knowledge as a toolbox. We fashion a tool to meet a need from our environment. We might then find that this new tool looks like a combination of things we already have, or that we are able to use it to build an even more specific tool. The process of identifying and creating dependencies between formalized mathematical concepts makes them more robust, and helps us to imagine beyond what we might be capable of with our senses alone.
I think that it is curious, but not too surprising, that there is a bottom that we eventually reach when rummaging through the mathematical toolbox. The unpacking must end somewhere, and we have seen that the end is a fully unpacked box. What more would we expect, the meta-emptiness of “not even an empty box”?
But to turn around and look back up at the sprawling, unfathomably complex menagerie of mathematical variety, all bootstrapped from thin air, it seems there is no end in this direction.
We are limited only by our imagination.