The Mental Hurdles of Thinking Abstractly
I promised to write about category theory, not that anyone would care, but I want to follow through.
I’m still trying to learn it, so bear with me.
Category theory has been a challenge for me to wrap my head around, but I think the core concepts are gaining traction in my mind.
Much of mathematics involves sets and functions. A set is a collection of things, where each thing is unique. A function is a relationship between two sets, a domain and a codomain, where every object in the domain set corresponds to exactly one object in the codomain set.
These limitations of sets and functions, uniqueness and each object in the domain having exactly one relation into the codomain, mean that sets and functions aren’t as abstract or general as they could be.
Enter category theory. The language it uses sounds silly, almost unscientific. “Here we have some objects, and look, there’s arrows between them.” “Objects”, and “Arrows”, hardly sound like mathematical terms.
When I was first learning abstract algebra, what struck me is that the concepts weren’t very advanced, they were quite basic in fact. We were rotating squares and polygons, we were simply describing natural symmetries that any child grows familiar with when they first learn shapes.
With category theory, it’s like de ja vu. I thought I had learned how to think abstractly before, but once again my mind is being stretched and contorted trying to think about very basic stuff.
Alice In Wonderland
I have been fortunate to find a great series of youtube lectures, by Dr. Martin J. M. Codrington, that provide a gentle introduction to category theory. Gentle is good, ‘cause I have to hear something more than once to learn it.
I have years of formal music education and training, thanks to high school band classes, so Dr. Codrington’s practical example of choice, music theory, made a lot of sense to me. Defining the relations between pitch classes, pitch class names, and the modes used for different scales is a subtle exercise. It teases out a lot of the motivation for the mathematical tools of category theory.
While watching the lectures, Dr. Codrington’s gentle voice, and the dizzying mathematical abstractions he was presenting, lured me into the realm of dreams.
There, the relations between pitch classes, pitch class names, and musical modes morphed into relations between race, politics, and identity.
Perhaps it was left-over thoughts from reading Brian Andersen’s discussions of racial issues, Jazz music, and American Politics. Perhaps it was the fact I was surprised to see Dr. Codrington was black when I saw his picture.
What I was learning in category theory is that relationships between collections of things are often more subtle and complex than we recognize. From a half-dream state, my mind morphed mathetical musings into a visceral and critical social issue, involving #BlackLivesMatter protesters, Bernie Sanders groupies, and our American legacy of social problems along racial boundaries.
Since then, my mind has been trying to find some category theory way to describe relationships between racial identity, cultural affinity, social affiliation, genotypic heritage, phenotypic expression, and political alignment. It’s a messy subject for sure. One that’s hard to broach without making some ignorant mistake. I’m not purporting to be knowledgable on any of this, just striving for some basic awareness.
Why did Bernie’s message resonate more with young white people? What is the significance of these parallel political racial movements like #blacklivesmatter? How can Trump politics not be about ignorance and racism?
Category theory is exciting and powerful. I hope to learn more about it. Meanwhile, my mind is distracted by a number of these more important, open-ended, unanswered questions.
