The Trick Is The Thing, Part II

I’ve enjoyed learning about quadratic equations and trigonometry for my Maths degree, and am struck by how many incremental steps along the way contributed to the total edifice of understanding.

Alex Strick van Linschoten


March 25, 2023

I’m approaching the end of the MU123 ‘Discovering Mathematics’ module and have really been enjoying some of the recent areas we’ve studied. In particular, quadratic equations really captured my attention and imagination. I’m starting to work on the trigonometry unit and it looks set to be equally amazing. What’s even better is that I think I’m starting to see (the beginnings of) some of the connections between concepts and units that I previous wrote about which makes everything that much more compelling.

To broaden those connections even further, in the FastAI course we often are reminded of how lots of parts of deep learning are (just) clever tricks. Those tricks might take the form of how you process the data, or how one thing is combined with another, but together these things combine together to form a more meaningful whole. So it is, I am discovering, with mathematics.

In a degree format like the one I’m working my way through, you often don’t encounter the insights and ‘tricks’ in a similar context to when and how they were first discovered. Instead of encountering problems that require solutions, you first cover the solutions and then apply those to some problems. I was struck today, though, at the cumulative weight of these incremental improvements.

To take one example, we have trigonometry which is — as I currently understand it, one day of study into the unit (please don’t email me) — the study of how angles and lengths relate to one another. We learn how we can use sin and cos and tan to calculate lengths and angles when we don’t have all the information about a particular geometric shape. When we start off, we’re only talking about right-angled triangles, but then later on we start thinking about all kinds of triangles and the trick is to drop a perpendicular so that we are still actually talking about right-angled triangles. So we’ve made this mental leap which allows us to solve more interesting problems.

There are many other examples of this from how we work with quadratic equations, to how something like the VAE (variational autoencoder) helps us train faster in the world of deep learning and Stable Diffusion.

As someone watching the breathtaking pace of developments in generative AI, I’m struck by how we can observe the same thing there, but at an accelerated pace compared to the world where many of these mathematical techniques were discovered. The internet combined with a number of smart people thinking about this problem space are proving fertile ground.

From trigonometry to quadratic equations to deep learning, the insights and tricks developed along the way build upon each other to form a more meaningful whole. This idea of clever tricks is not unique to mathematics or deep learning, but it is a universal concept that applies to many areas of human knowledge and understanding. These tricks may take the form of new methods or approaches that allow us to solve problems more efficiently or effectively.