Ask HN: Best resources to gain math intuition?

It doesn't shock me that there are holes. I noticed that some math topics are very important to engineering and physics coursework, but given short shrift in the math department. Examples are the way that complex numbers are used, and specific kinds of differential equations such as the general harmonic oscillator.

My college physics coursework actually had its own "math methods" class, intended to fill some of those gaps, and to get us prepared for the higher level physics courses.

Mathematics is a very general tool. As with any very general tool, a lot of the devil is in the details of how to use it in any particular domain.

For this reason, in-sourcing mathematics service courses is best for everyone. The very best math-adjacent departments in every field tend to do this either directly or indirectly. E.g., in the direct model, many CS departments internalize the Discrete Mathematics course and some combinatorics. And an example of the indirect model is Mathematics departments that hire Math Finance professors to cover the service load for econ/fin/bus depts.

I think this in-sourcing (either directly or indirectly) is best for everyone -- mathematics depts don't do a good job at teaching those service courses and often don't do a great job of it in any case. Unfortunately, most departments don't have the headcount (in students or faculty) for a specialized mathematics curriculum, so they have to share the math faculty with N other majors to predictable effect.

For me it was trying to frame as much as possible in a geometric lens. Draw pictures of anything and everything as much as possible. Graphs, number lines, 3d animations, whatever it takes.

Also, understand the "bold print" from the "fine print". I mean to say, every theorem will make a general statement, and then have a bunch of conditions where it holds. So worry first about the bold print, then try to understand the fine print.

Much of mathematics is about making the strongest and most general statements in bold print with as little fine print as possible. Basically the less fine print you have, the more important probably your result is.

Try to be around problems that requires a high degree of math skills to solve. let the problems drive you.

intuition is usually built by spending hundreds of hours thinking, eating, and drinking the problems you're truly interested in to arrive to a solution.

If you spend enough time doing this, you'll know exactly what areas you need to inquire about as your understanding to the problem becomes clearer and gaps start to narrow down.

If you don't know what you need to learn you probably haven't spent enough time on the problem or just read the solution which made you feel this way, not realizing that the solution was done by a person that could have spent 100x the time you spent on it and built the intuition you're asking about.

For me, math intuition takes practice and a lot of visuals, graphing and diagrams. Some people get it through proofs and equations, but I feel like the way math is taught in school doesn’t usually do a great job of developing intuitions. Coding it up forces me to learn it, and playing with it and tweaking a lot along the way helps develop the intuition.