I want to develop some related thoughts to my last post, in a different (and hopefully more optimistic) context. For a long time I thought I wasn't interested in studying mathematics at the graduate level. I used two basic arguments to support this. The first is that I didn't even understand what graduate level math was talking about. Upper level math features things like groups and rings and differential geometry and I don't know what else, but even the basic vocabulary is totally foreign to me. The second is that, in my understanding, upper level math isn't even about numbers anymore, and I like numbers. I have since concluded that I was wrong, wrong, wrong.
First of all, it's not even true that all upper level math is unfamiliar to me. Looking at graduate math department course descriptions, I can find lots of courses on subjects like statistics and Fourier analysis and things that I actually do have an understanding of and interest in. In some cases, like Fourier, my understanding is a result of the engineering courses I took after I basically decided I should get a degree in engineering rather than math. I may not have known I was interested in the subject before I took the engineering courses, but with the benefit of hindsight it's clear that I should have skipped the engineering and gone straight for the math.
Second of all, I don't know where I got the idea that if I was unfamiliar with the subject, I wouldn't be interested in it. (This may be related to the being smart vs. working hard distinction from last time. The existence of topics I'm not familiar with may imply that I'm not smart, so if I was being motivated by seeming smart rather than by working hard, that was reason to stay away.) Regardless, this idea is clearly ridiculous. I studied some abstract algebra last fall. I went in not knowing what groups or rings were, and now I have at least the basics down. Getting the basics is less important to me than the fact that the whole time I was studying, I was thinking, "This is great. Why didn't I take this course a long time ago?" Lack of knowledge is not an excuse. There was always plenty of evidence that I was interested in all the math I did know, so it's not surprising that this interest extends to subjects I don't know.
Abstract algebra also dents the objection about not being about numbers. Abstract algebra is about sets, which are made up of things, which are not necessarily numbers. So a lot of abstract algebra isn't really about numbers. But one of the central ideas of abstract algebra is that different sets behave the same, so when you're talking about one set you are really talking about all similar sets. Which means it's often convenient, when talking about any arbitrary set, to find a similar set of numbers and talk about that one instead.
One interpretation is that abstract algebra isn't really about numbers. Another interpretation is that abstract algebra allows you to talk about all kinds of things which are not numbers as if they in fact are numbers. The numbers don't go away. Instead abstract algebra almost makes the numbers more powerful by making them more universal.
Last fall, speed was more important than depth in my crash course in abstract algebra. I ended up with a pretty cursory understanding of the subject. But now I'm looking forward to going back and studying it in more depth. It's clear I was making excusing for why I shouldn't be doing this before. But the evidence is pretty overwhelmingly that those excuses were wrong. Now I'm looking for opportunities to study more math instead, and I'm a lot happier as a result.