A new way of thinking

I am sitting in a lecture hall, surrounded by people, most of whom I suspect have received university-level training in mathematics.

Full disclosure: I have not. With my background in Psychology and Cognitive Neuroscience, I have always been a “user” of mathematics. I received training in Statistics and learned which formula to apply to solve a specific problem. Throughout my life, I always felt a bit intimidated by maths. However, this changed in the last years of high school (thanks to my great maths teacher).

And here I am, listening to the mathematician Emily Riehl as she describes alternative ways of proof writing. She talks about category theory as a mathematical foundation system and using computer-based proof assistants to keep track of your steps in the proof. The whole process seems extremely meticulous. I understand maybe 40% of her talk, and even less as she walks us through a particular proof that relies on something called the Yoneda Lemma. To me sounds this sounds more like a character straight out of a science fiction story rather than anything related to math.

Side note: It actually is the fundamental theorem of category theory, named after the Japanese mathematician Nobuo Yoneda.

I do find it remarkable how fast and effortlessly she speaks about various mathematical objects and their relations, without ever stumbling over her words. The examples that she gives are drawn from her own research in category theory. In a nutshell, category theory provides a high-level framework to group mathematical objects with respect to their relations (for an in-depth / mathematically more rigorous explanation see: https://emilyriehl.github.io/files/context.pdf)

It feels like a new language to me. Like in the beginning when I started learning Italian and I couldn’t tell just from listening when one word ended and the next one began. Learning maths feels somehow similar. It just takes time, exposure, and patience to start making sense of it.

So, what are my takeaways from this talk?

1. It is cool to get an insight into how research in mathematics looks like
2. Turns out, no one actually teaches you proof writing - not even in many maths degrees
3. There are some great resources out there (‘An Infinite Descent into Pure Mathematics’: https://infinitedescent.xyz/)

Here is some work by Emily Riehl (targeted at a non-mathematician audience):

1. Explaining Infinity at 5 levels of difficulty: https://www.youtube.com/watch?v=Vp570S6Plt8
2. Blogpost on Category Theory: https://www.scientificamerican.com/article/infinity-category-theory-offers-a-birds-eye-view-of-mathematics1/