On Branching Complexity

 The thing I've most learned about learning mathematics is the branching complexity. This is how at each level you think you've gotten the hang of it, but you soon find you were only in the shallow end of a pool that apparently extends forever. Every answer opens three questions. At first this feels like failure, like you're moving backwards. Eventually you realize it's just what mathematics is. The pool has no bottom because it was never a pool. It's an ocean, and the people who seem to have mastered it are treading water in a different spot, looking back at you with the same mixture of confidence and quiet vertigo.

I don't like thinking about a preprint I posted in 2022, so I won't talk about it in detail. Suffice to say it breaks my heart with embarrassment to think about it out there uncorrected. I posted it without feedback from anyone. I thought I had it all.

I soon realized how wrong I was.

What followed was years of actually learning mathematics while working full time. This is a specific texture of life that's hard to describe to people who haven't done it: the commute spent with a textbook, the Sunday mornings with handwritten notes before the week starts again, the slow accumulation of understanding in hours stolen from everything else. It is not the same as learning mathematics in a PhD program with time and infrastructure and people around you doing the same thing. It is slower, lonelier, and in some ways more clarifying, because every hour spent on it is chosen.

It led me somewhere I didn't plan. I started with spin mappings in 2022 and am now reading Serre on representations of finite groups. This is a thread I followed without knowing where it would end up led to each topic pulling on the next, each gap in understanding pointing toward something I had to go and learn before I could continue. The embarrassment of 2022 left a specific residue: I became very cautious about saying I've understood something. Careful to a fault, maybe. I won't proceed past a page I haven't actually absorbed. I won't make a claim I can't reconstruct from first principles. This slows things down considerably. I think it's the right way.

It's not really the case anymore that I don't know where I will end up. My map is more detailed now and I have specific things I want to understand, specific texts I'm working toward, a sense of the shape of the territory even if large parts of it remain unvisited. That's different from 2022, when I was moving mostly by feel. Having a map doesn't mean you know what you'll find. It means you know which direction to walk when you get lost.

The pool still extends forever. I've just stopped expecting to find the bottom.