Too Old

I'm 24 and I've studied a bit of math in university. I took a linear algebra and intro calc class and did alright in those. I've always been good in math because I've always found it deeply interesting.

My question is, does anyone think it is too late in my life to try to get into math seriously and major in it? Right now, I've just been taking general science courses trying to get a feel for what interests me. Physics/mathematics are what I find most appealing.

Does anyone have any advice on this?

Well, even though I'm a pure math major, that hardly qualifies me to advise you. Still, I'll just blather on about something until it takes, roughly, the form of advice. And please note that all of what I'm about to say might only pertain to pure mathematics, I don't know enough about applied mathematics to know if it still applies. In other words, this might be useless to you =P Anyway, as a math student I've noticed a rather large distinction in HOW students are interested in math. Now, what I'm going to describe are perfect models of these types of students, and there are plenty of students who fall short of these descriptions. Nevertheless, I think that they can still fit into one of the categories based on their priorities for understanding math.

The predominant type of interest in math, which consists of most of the math majors I've talked to, is at a technical level. These are the students who are concerned with applying mathematical "machinery" (theorems/heuristics) to prove specific problems. That is, they enjoy exploring the details of the various areas of math and tend to be very creative and clever with how they apply what they know. In fact, some of them might have participated in math competitions, and perhaps a subset of those that did might possibly have participated in the USAMO or IMO (google them). Those who attend problem solving seminars at their university and participate in the Putnam competition (again, google) likely exhibit this point of view (well, perhaps that's a hasty generalization...). If I had to summarize this viewpoint in one word I would say it's "local."

On the other hand, you have those who might be called "theorists." They enjoy math at a conceptual level rather than a technical one. They like learning of the larger framework which situates whatever it is they work on, and absolutely love deep results (sorry, I know that sounds really fucking tacky). This kind of math might not be as "flashy" as what the clever problem solvers concern themselves with, but I believe it leads to more significant mathematics. I can't stress to you how few of these people I've met, but they exist. In fact, I'd call myself one, but that would be a bit pompous, yeah? The entire reason I bring this up is because it's easy to get discouraged with all the "brilliant technicians" (type 1) that will surround you. Just know that there is more than one way to think about math. In fact, there's two. Haha... ugh.

Since you took linear algebra I'm hoping that you can follow this paragraph, but don't worry if you can't. I'm only trying to give you an example of how the latter of these viewpoints can be rather enlightening.I remember when I was taking linear algebra one of the last things we covered was the spectral theorem for normal operators. The theorem was framed as one about matrices, and the rest of the students (by way of their questions) didn't seem to understand the significance of it, they were concerned only with all the possible ways you could prove it and other technical details. The way our teacher presented it was: a matrix being normal is equivalent to it being unitarily diagonalizable. However, a conceptual rephrasing of this gives the spectral theorem as a classification theorem. It classifies the linear operators for which there exists an orthonormal basis of eigenvectors (perhaps the most pleasant you can have). I find that much nicer to think about personally!

Continuing on with linear algebra, here's some specific advice. I know you said you took linear algebra and I think that's great, but it's often the case that one's first exposure to linear algebra is how engineers think about it (this is what happened to me). The way mathematicians think about it tends to center more on abstract vector spaces and linear transformations between them. In my opinion, the only worth of matrices are as representations of linear transformations, and you should only argue with matrices if you must (i.e. when you have an unbelievable basis as described above). Arguing by way of linear operators provides a basis-independent (i.e. coordinate-independent) approach to linear algebra. This is nice for two reasons. First, you don't repeat yourself by proving a bunch of special cases (dependent on certain coordinates) of a more general result. Second, it's often a much cleaner (but more abstract) way of presenting linear algebra and gives you access to the big picture, which is always nice. Honestly, I think there are only six or seven major topics in linear algebra, it just doesn't seem that way when taking a typical linear algebra course because they shove a lot of arbitrary shit into it. Check out the books "Finite-Dimensional Vector Spaces" by Paul Halmos and/or "Linear Algebra Done Right" by Sheldon Axler. Both are fantastic in their presentation (and quite concise), but if they're too terse at times then try supplementing with Wikipedia. Hope all of that helped. I didn't even mean to type as much as I did!