Another awesome mathematics article that loses me about 10-15% of the way in do to my own technical limitations. Any tips from HN on how to improve my ability to get thru, say, 45-50% of these types of articles?? Generally speaking, not specific to the math in OP article

It's been way too many years since uni for me to do any rigorous math, so what tends to help me is to try and get an approximate intuition instead. As a specific example, this article talks a lot about manifolds. Since I didn't study math in English, I don't know what that is, so I go and look it up. A simplified, but intuitive model [0] might be:

> A manifold is a space that is locally Euclidean, but globally might be complicated, e.g. a torus or sphere, or etc.

Okay, so that's reasonably simple. As an intuition, if we imagine a 2D character living on the surface of the sphere, if they walk forward, from their perspective they just move forward in 2D, but from our outside perspective, they're moving in 3D on the curved surface of a sphere.

Once I have this, I try and read until I get lost again. I don't try to rigorously solve or follow through with each equation, but to rather approximately understand what the idea is.

[0] https://math.stackexchange.com/q/1211762/128941

You just don't know enough about the foundations the article is building off of to follow it. What I do in such a case, is forget about the article at hand, and make a list of things that I don't understand and then try and learn about them all individually. It's okay to just bounce off of a technical article and use it as motivation to learn more about the subject from other sources.

For example in this article, I got completely stuck on how the Exp and Log functions he's talking about relate to the usual definitions of those functions, so now I'm going down that rabbit hole.

Author of this post. I have an undergraduate in Applied Mathematics and my training in the "definition -> proposition -> proof" style of mathematics probably comes through in the article more than I wanted it to.

That being said, I began studying Differential Geometry and Lie Groups as part of my graduate degree in Electrical Engineering. Engineers think about problems very differently than mathematicians and I've benefited a lot from taking a more geometric-based and visual approach to learning in the years following my undergraduate.

So, my prescription would be to play around with math ideas when you see them. Create a script to draw what you are trying to visualize. This was my first time using the `manim` library and I gained a deeper appreciation and intuition for the ideas presented in the article even though I've studied them dozens of times!

Overall, learning math is a slow and deliberate exercise. Don't get down on yourself if you don't understand something at first glance. Feel free to pause, verify an idea (either visually or with a formal proof) and then continue on a more firm base of understanding.

Learning maths is all about having the proper prerequisites (and time and effort...). The concepts all build on simpler ones in a hierarchy leading down to our basic ideas of numbers and space, so if you're missing any of those simpler ones you simply won't be able to understand anything more advanced or exotic except in a very fragmentary or superficial way.

For differential geometry the prerequisites are linear algebra and multi-variable calculus, and the prerequisite for multi-variable calculus is single variable calculus, and the prerequisites for each of those is basic algebra, trigonometry, and elementary geometry.

You don't need to know everything about each of these to get things at the next level, but a thorough grounding in the basics is essential in my experience. There's a reason every STEM field begins with calculus and linear algebra - they're used everywhere in anything at a higher level. Once you get through those you will find things open up for you.

I don't know your level so it's difficult to make any concrete recommendations, but in general I find Lang's books to be clear and efficient sources. His Short Calculus covers all the basics of single variable calculus in less than 200 pages, instead of ~500 pages like many intro to calc books. Similarly his Calculus of Several Variables is ~300 pages instead of 500-700. Alternatively a mathematical methods for physicists book like the one by Riley, Hobson, and Bence might suit you. It's huge (~1300 pages), but you can pick out the chapters you want to learn from and it builds up from the basics to some quite sophisticated mathematics and has references if you want more depth on some topic, and great problems.

I find I don't really learn anything from watching videos. They can be complementary, but most of the learning with maths comes from doing problems after reading through an introduction to the concepts

What is it you’re trying to get out of reading these types of articles? In my experience, it’s hard to really retain info unless I’m actively working on a related problem, so reading something like this out of the blue is mostly just entertainment (and a way to check existing knowledge).

Study more math? It’s hard and takes a long time; there isn’t really a royal road.

spend 1-2 years learning consistently _any_ undergrad math. id recommend focusing on “mastering” multivariable calculus, dependencies included. but keep taking peaks of higher level stuff or just articles as these. doing this is important as the math can be contextualised in many ways and being exposed to those ways helps you internalise intuitions

To be clear I am not being sarcastic in saying this but the only method I’ve found to work with any consistency is: Try, try again.