I’ve recently been having some cool email exchanges with a professionally-trained mathematician, Newcomb Greenleaf. He now teaches in the Individualized BA program at Goddard College, and is on the board of the Yoga Science Foundation.
Just for the fun of it, I’m including a few excerpts here.
Here are some thoughts on category theory but, first, a big disclaimer: I am a total dilettante amateur in mathematics. No formal training whatsoever--or, rather, my formal training ended at Venice High School, where I flunked beginning algebra three times in a row (much to the chagrin of my parents!). I only know what I've picked up on my own through books and the Internet. I'm also acutely aware of how easy it is to "see the Virgin Mary in your danish", i.e., see what you want to see in science and math results. Newton thought the attractive power of universal gravity was a reflection of God's love (so by that logic, is dark energy proof of God's hate?). Maupertuis was convinced that the Principle of Least Action, proved God's existence. (Voltaire wrote a parody of him called Doctor Akakia.) Even Leibniz, who by all accounts was probably one of the most versatile western intellects of all time, believed that mod-2 arithmetic demonstrated how God could make something (1) out of nothing (0). These guys represent the cream of professionals, and I'm just a dilettante amateur! So having stated all of this by way of caveat, here are the parts of category theory that seem to resonate with Buddhism and my personal meditation experience.
Connection is a huge theme in Buddhism. It's elaborated philosophically in the doctrine of pratītyasamutpāda. The slogan is "This being that is." Indeed in the Mahayana formulation, there aren't even entities--just connections. As you know, category theory is all about arrows--different flavors of arrows connected in various ways. Paralleling Mahayana, it's even theoretically possible to do away with the objects themselves. So, in a sense, both Buddhist philosophy and category theory seem to say "it's arrows all the way down."
Moving on to a different theme, in the way I like to formulate impermanence, binary contrast is very important--different ways in which self-cancelling polarities can mold experience. In my system, this is formulated in terms of expansion-contraction (check out more about this here, here, and p. 56 on here...). One important class of categories are "pointed categories" or "categories with a zero object". The arrows in these categories all start from zero and return to zero--which sounds an awful lot like how I experience consciousness working). I talk a bit about the history of zero in my article Algorithm and Emptiness--here's the link. I also talk about related issues on pp. 151-159 and pp. 173-183 in my Five Ways manual.
So to the extent that category theory generalizes group theory, the theme of mutually cancelling polarities is a major theme, and that maps on to my meditative experience rather nicely. Category theory then goes on to generalize invertibility itself with the notion of adjoint functors. Although the language is just coincidental, I love phrases like "forgetful functor" and "free functor." In contemplative practice, we first forget our specific identity, which then frees us up to assume arbitrary identities. Once again, I'm very aware that the language used here is merely coincidental, but I do find it amusing.
Moving on, at the most universal level, arrows can always be reversed. There's this incredibly beautiful duality principle that pervades every facet of category theory. Pythagoras, Lao-Tzu, Hegel, and Marx would all be pleased.
I felt privileged to allow the elegance of your mathematics to enhance the punch of the dharma. Tangibly, they have inspired me to re-examine my understanding of the foundations of math in categorical terms. I've had to recall my rather checkered history with categories. In graduate school I was good friends with Peter Freyd, whose thesis became the first book about category theory, which up 'til then was treated in the context of algebra or geometry or topology. It's called Abelian Categories, and I find that it's still in print. I learned to think categorically I recall Peter's excitement and particularly his emphasis that, "We used to think that category theory just did away with the elements. Now we see that it also does away with the objects." and it's nice to see that insight again.
To which I responded:
I'm totally awestruck/starstruck that you were good friends with Peter Freyd.