A Young Math Enthusiast's Fear

Solution 1:

In going from high-school to, say, graduate${}^\color{Blue}\dagger$ level math, the higher math being "nowhere close to the math that appealed to" you is probably a very real threat. However, if your pleasure in learning analysis and algebra is any indication - instead, it will be math that you love even more.

A few things that I've noticed changing as I've learned more math are:

• Generality: Everyone knows about integers, rationals, reals, maybe complex numbers, but the next step up, conceptually, is to look at rings and fields and then modules and so forth. In higher math, the generality of our constructs increases a lot. Often we then narrow our focus again and end up looking at 'cousins' of the things we were originally studying. Other times we run into problems and it becomes the question of the decade precisely how to successfully go about a particular campaign of generalization and overcome the relevant obstacles.
• Branching: What a lot of people don't understand is that mathematics isn't linear, and doesn't progress in a rigid sequence. It branches out into many different areas, and in exploring these branches they can "feel" radically different. You can have a crush on one branch while hating another branch. In some cases, you have a love-hate relationship, or "it's complicated," etc.
• Reinterpretation: With a little bit of dabbling in different areas of math, it's possible that a single problem/idea can be attacked/framed from many different angles, using very different concepts. Sometimes this can seem "natural" and easily "motivated," while other times alien and bizarre. Frequently this sort of thing is a bit of a pastime for some mathematicians.
• Richness: In summary, mathematics becomes richer. In scaling conceptual mountains, we build concept on top of concept on top of seventeen more concepts until we're left studying situations that are saturated with structure, and when we hike back down the other side we come across the exotic or pathological; in branching we discover a high degree of diversity we hadn't previously imagined, each with comparable feel and texture to them; and then when we study wide and far we find that even our familiar notions have multiple sides to them.

$~~ {}^\color{Blue}\dagger$Yeah, disclaimer: I'm not actually there yet. :-)

Solution 2:

While this question may be slightly inappropriate for this sort of website, I have no problem with it and so I'll provide my own personal experiences. The highest level math I have made it to so far would be topics in Manifolds, Geometric Topology, Category Theory, and Model Theory, so keep this in mind, I am by no means a Mathematician!

From what I understand so far, Mathematics is a long-haul lifestyle which requires a sort of dedication and habit of living that I don't see in other disciplines. To be successful requires a mastery of the content you learn in classes and also additional self-study on topics that you are interested in and want to think about on your own time. If you are not even in university yet and are studying Real Analysis and Abstract Algebra then you are ahead of the game (making it through your undergraduate with a high GPA really is a game). My advice to you would be to balance the amount of time you spend on your courses and thinking about the content that is being taught to you, practicing the exercises, etc. and actually doing some real mathematics. By "real mathematics" (at least from my perspective) I mean, find a topic that is interesting to you, go pick out a graduate textbook on it from your library and read through some of it to get a general feel for the ideas and terminology (refer to earlier material if necessary), then attempt to develop your own theory! Given some basic definitions, try and derive some of the motivational ideas, perspectives, geometric intuitions, and maybe even some of the basic results in the field on your own. For budgeting your time, I would suggest working on courses and studying for what is directly pertinent to your GPA during the school week and then on weekends go all out reading whatever you want and trying to be as creative as possible!

If you are intending to get accepted into a "prestigious" (aka, that institution happens to have attracted really good mathematicians to work there) university, you need to balance the get-the-highest-GPA-possible act with being creative, staying interested, and doing additional research. Ideally, before you head to Graduate school you should have some publications and research done and a fairly high GPA. There are multiple avenues to head towards to get published research, but normally they come in the form of summer research projects facilitated by a professor who helps you research the topic, gives you guidance, and teaches you some of the basics to do research in the field.

Just as a little nudge to look up some topics I personally find very interesting:

• Geometric Topology: This is probably the field that fascinates me the most because of the mental imagery and "raw idea" I get from it. The basic foundations for the subject are off of the idea of a "manifold" which I could roughly describe as a type of surface, or a generalization thereof, in which you can embed other structures and define certain properties on, such as a metric (a notion of distance).

• Category Theory: If you have seen some Abstract Algebra you may have a bit of motivation to understand some introductory Category Theory, which to me provides very deep insight into the internal structure of mathematics and the possible relations which can occur. It is hard to describe, depending on what you have researched so far but I assume you know about functions $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$. The basic idea is that you consider a category of objects which are somehow associated to each other by "arrows" or "morphisms" (more general than the idea of a function $f$), then you can relate morphisms in categories to one another by "functors" and then proceed to develop other ideas such as a "natural transformation" that describe how functors can interact.

• Logic: It may seem sort of obvious, but I think it is very important to know formal logic. Not only does it help with developing your argumentative skills in proofs, but allows you to see some of the underlying structure necessary to define mathematics in general. Learning about axiomatic systems, such as Peano arithmetic, and seeing topics on the borderlines of mathematical logic and Set Theory (such as forcing) give a new perspective that shows you how weak some of the foundations of mathematics are. Expose yourself to the various seeming "paradoxes", such as the Banach-Tarski paradox, Russell's Paradox, etc. and look for ways that YOU might resolve them before looking up the answer.

I need to get back to learning about cohomology though, it is the weekend after all!

By the way, I highly recommend you watch the lecture series given by Satyan Devadoss called "The Shape of Nature", it gets into some ideas relating to convex geometry, topology, knot theory, etc. and is very accessible!

Solution 3:

Excellent question. Even though you already have plenty of excellent answers, and already accepted one, here's another one.

I'm not a mathematician. I'm just old. :) And from a geezer's perspective, I think you getting very close to the bullseye here:

But this being said, I do have a small concern. Because Higher Math is so very different from grade-school math, I fear that as I dive in deeper and deeper into mathematics, I might realize that the Math I do then has changed so much that it was nowhere close to the Math that appealed me. How substantial is this fear? Is it legitimate?

This sounds to me like a special case of the more general question: I love doing $X$ on my time, but will I love making a living out of doing $X$?

For example: "I love reading novels, but will I love making a living out of reading novels?"

Well, first, can you make a living out of doing $X$? Some people can, but not all. When $X$ = "reading novels", there are a few talented literary critics, for example, who can make a living out of reading novels, and writing down their reactions about them. Another way of making a living out of this $X$ may be to become an editor for a publishing house, but in this case one may be forced to read novels that one doesn't particularly enjoy. Some can make a living out reading novels by becoming literature professors, although, here again, the fraction of their time they spend reading novels they actually enjoy may be small. Some manage to express their enjoyment of novels by writing novels themselves, and getting others to pay them to read them... Many ways.

When $X$ = "doing math", the situation is roughly similar: many ways of making a living doing math, some easier to "land" than others, and some that would be more enjoyable to you than others. I think it would be a good idea for you to invest a small fraction of your efforts finding out the various ways that one can make a living out of doing math, and finding people who are actually doing so. Ask them what they like and not like about their jobs. Especially, ask them how much they enjoy the math that they actually do for work. Ask them whether the math they do is the one they'd like to do, and if not, how the desired and the actual differ.

One great aspect of enjoying math is that, as passions go, it takes so little to pursue it. A good book, pencil, and paper, and you're off! (In fact, often one doesn't even need any of these.) Plus, math is, for all practical purposes, endless. (It would take me at least many lifetimes to visit all the planets in the math universe...) So you need not worry about running out of math to enjoy.

One downside of enjoying math, though, is that many around you (probably most, actually) will not share your interest in the least, and this can be a bit of a bummer. But I'm happy to report that, thanks to the internet, this downside becomes more and more negligible every day. And of course, with a bit of luck you'll be able to land a career that not only let's you do the math you enjoy, but also surrounds you with colleagues who share your appreciation for mathematics.

Solution 4:

Imho, anyone who wants to study mathematics at university should try doing so.

If you do well, great, a mathematics degree is extremely marketable, even without any further graduate education. If you do poorly, all that additional mathematical experience will serve you well when you switch into another less abstract subject, like statistics, machine learning, finance, software development, etc.

You cannot of course plan on going to graduate school until you've seen how undergraduate mathematics studies go, but attempting the undergrad mathematics degree makes sense regardless. There is simply no downside unless you're considering between math and physics, which offers similar advantages to mathematics.

American undergrads increase their chances for graduate school admission by taking graduate courses as an undergrad, doing an REU, which occasionally yield a journal article, and competing in the Putnam exams.

In particular, the Math Subject GREs assume you know the basic first year graduate mathematics material, meaning doing well requires knowing a lot more than the basic mathematics degree requires.

Ergo, there is a potentially significant disadvantage to studying undergraduate mathematics at most liberal arts collages as opposed to large research universities because most small collages simply don't offer those graduate courses.