Newsletter 50: Space, the Final Frontier

If you have been reading this newsletter for a while, you know that space - the final frontier version - is one of my favorite topics. As a mathematician, I worked on spatial mathematics and as a cognitive scientist I studied how language and visual representations of space talk to each other.

Note: I don’t believe in representations anymore; there’s inner space and there’s outer space, but there isn’t any such thing as a spatial representation.

Physicists have thought about space a lot, especially in the twentieth century. Space plays a huge role in science fiction, fantasy and gaming which means that much of popular entertainment is spatialized. As visual creatures, space constitutes much of our experience. Despite it’s overweening importance, space has been poorly theorized, at least in my not so humble assessment.

For example, there’s no such thing as philosophy of space. There’s philosophy of physics in which space-time plays a big role, but there’s no such thing as the philosophy of space as such. In comparison, logic plays an outsize role in philosophy, with the concepts and terminology of logic infiltrating almost every philosophical discipline at least in the analytic avatar of western philosophy. You can’t call yourself a philosopher if you haven’t heard of first-order logic and know the difference between free and bound variables, between existential and universal quantifiers, between possibility and necessity.

Contrast that awareness of logic with the awareness of geometry or topology and you will notice a start difference. It’s a rare philosopher who would have heard of, let alone understood, Brouwer’s Fixed Point theorem. Not that I am particularly upset about the lack of knowledge of Brouwer’s work in topology in particular - though it’s surely true that more philosophers have heard of his work in intuitionism than on topology.

Anyway, the reason I am bemoaning lack of spatial insight in philosophy is because I was reminded recently of Bertrand Russell’s claim that those who read Plato don’t know mathematics and those who do mathematics don’t read Plato. I am guessing he believed he was one of the exceptions, but his approach to mathematics was as an extension of logic, which is severely limiting.

The foundations of mathematics were my first serious intellectual interest; I was interested in Godel’s proof of incompleteness before I was interested in Brouwer’s fixed point theorem. The reason is simple: Godel’s proof was a major cultural watershed and along with Heisenberg’s uncertainty principle, it has become a staple of popular writing about science. Douglas Hofstadter’s Godel, Escher, Bach did much to popularize those ideas and show their influence in areas as far as afield as music and the study of the mind. No day goes by without us hearing that the mind/brain is a computer of some sort or the other.

While the foundations of mathematics have had a close relationship with logic, our understanding of space hasn’t played such an influential role. Paradoxically, spatial ideas have proven far more influential in the practice of modern mathematics than logical ideas. The list of Fields medallists has several topologists and geometers but exactly one set theorist (who wasn’t one by training). This gap between the great technical advances in spatial modeling and the rather puny footprint in spatial metaphysics remains a problem, for it’s impossible today to proclaim without sounding mystical or incoherent that The Mind is Space.

We need an account of space grounded in mathematics but independent of the specificities of physics or perception or architecture or any other discipline that depends on spatial insight. What would it take us to get there?