August 14, 2023

On Mathematics

Tags: contemplation maths
2 minute read

When I was in university I strolled into a maths class on topology. It was extremely abstract, with definition after definition, and theorem after theorem presented without motivation.

Sometimes in maths one doesn’t need a motivation; one is just playing around. And that’s wonderful. And some folks seem totally fine doing that without talking about it. But I personally prefer being clear about what it is that we’re doing. If that’s just playing around without context, that’s fine. But if there’s context, such as a history of thought in the area, or potential applications, I find it really helpful to present these as well. Then one can dive in to very abstract concepts, but with a sense of orientation.

The worst habit in mathematics, in my opinion, is to present a final solution as a theorem, and to prove it without any context on how this theorem was arrived at in the first place. Often the process through which something was discovered can be extremely enlightening. Prof. Brad Osgood at Stanford is remarkly good at pointing this out, for instance in his class on the Fourier Transform and its applications.

I’ve found that pairing highly abstract mathematics with readings on history and context, or explicit awareness of the lack thereof, brings the whole subject to life. Topics that before seemed too abstract acquire meaning. And meaning drives understanding.

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