<p>Well OK, there is something important which needs to be mentioned in a thread like this, and made very clear. It's somewhat come out when people contrast applied and pure math, but need be stated very clearly, I think.</p>
<p>Which is that people major in math for many highly different reasons. A major in nothing but abstract mathematics suggests aims for a research career, and the fact is very few people can make it in such a career. I hope to be one of them, because with some substantial reflection I assured myself that I vastly prefer abstract math. No matter what, math is not that easy a major, though the burden is considerably on those trying to make a career of pure math. </p>
<p>For someone who's not sure, the best way is to try out some classes, and usually you'll get a sense only after actually doing the work that it's not for you. Feel free to throw such classes out of your schedule, rather than sitting them out in pain if you don't like them. </p>
<p>I think the number of people who just love abstract math and nothing else is not huge anyway, so most likely the OP will be interested in math as a means of getting a higher perspective on other topics, like advanced economics. And yes, logically a strong math mind could probably do pretty well on LSAT's and such tests; so being an actuary, doing some sort of job with lots of emphasis on statistical analysis, doing some CS, going to law school, going to econ grad school -- all options beyond the standard world of math academia.</p>
<p>Again, the only way in my opinion to know for sure is to take a look at the math courses and curriculum yourself. Best of luck.</p>
<p>So my post above kind of suggests that most math majors consider either a minor in something more practical or at least some study of classes in other fields. Now, a specific practical background is not always necessary; I'm sure actuaries could be hired without economics courses, but I think someone who's not very keen on a research career would do well to get involved in statistics classes and such things.</p>
<p>I guess my point is, be sure you're gearing your education towards what you generally aim to do with your math major, because there are lots of things you can do. You're also free to forget about all this and figure out what you want to do after college is done, but that seems less prudent to me.</p>
<p>I read an interesting article that lots of people who're hired for their math backgrounds aren't hired because of their SPECIFIC knowledge. They're hired because succeeding in college level math is certainly not something everyone can do, and does actually signal that someone's pretty smart. </p>
<p>Referring to "(although an actuary friend told me the hardest math he has used on the job is a weighted average)." </p>
<p>Realistically, if you want to actually USE your abstract algebra and such knowledge, consider a career in pure math. But be warned that you'll have to reach for the sky in terms of your knowledge if you want to do that!</p>
<p>"A major in nothing but abstract mathematics suggests aims for a research career, and the fact is very few people can make it in such a career." </p>
<p>I agree 100%. I loved math for its own sake and hoped for that very thing but after I roomed with the Putnam winner my junior year and saw how much better he was than I was, I trimmed my sails. It's still great training for law school!</p>
<p>But mathboy98, I wish you all the success I didn't have!</p>
<p>PS - My Putnam-winning roommate is now a psychiatrist.</p>
<p>Another friend of mine -- also a much better mathematician than me -- started as a pure mathematician but ended in a field of applied math that is nonetheless fascinating for its own sake:</p>
<p>You can understand how I might have decided I was outgunned! But I could never have explained things so cogently as you've done in your posts here and elsewhere, so I think you'll go much further than I did.</p>
<p>Johnshade, thanks for your kind encouragement. You seem quite modest, and I imagine you don't give yourself enough credit for your own math endeavors =] </p>
<p>I try to look at it in as little of an elitist fashion as possible, i.e. without the attitude that "so and so isn't good enough to do research math" -- I really think abstract math keeps changing in nature as you go higher up the levels, significantly from undergrad to grad. And to an extent I think people will just naturally drop out as they find it's not for them. Maybe I will too. I mean oh well, at least I've been enjoying what I've done thus far!</p>
<p>"I try to look at it in as little of an elitist fashion as possible, i.e. without the attitude that 'so and so isn't good enough to do research math'"</p>
<p>That's an excellent attitude and one I wish I'd had the good sense to have when I was an undergrad. I think I bailed too early (though I did finish my major).</p>
<p>haha okay im sorry but i dont really understand.. what exactly is "abstract math".. im taking precalc h right now and really like it because i like solving problems etc.. and what are these proofs in college math that everyone talks about</p>
<p>Abstract math. Well, if one were to ask what a mathematician does, it's to ask certain important, natural questions, such as "If you take all elements of form a + b\sqrt(3)" for a, b integers, which of them are going to be roots to a polynomial with integer coefficients? "What are the solutions to this polynomial equation like" is a loaded question, and one that is just sort of natural to ask. Other questions will be about geometric objects. Other questions will be more of a calculus sort of focus. </p>
<p>The thing is that when we ask these questions, we can't always answer them with the machinery we have. Take this problem: you, a precalculus student, can understand it, but the solution won't come without some machinery -- say we have a function \phi, which takes in a positive integer, and spits out the number of positive integers smaller than it, but sharing no common factors but 1 with it. If you take a positive integer of form a^n - 1 for a, n integers, phi(a^n - 1) is divisible by n. Looks really hard? It's easy with some machinery, and can be solved in a few lines!</p>
<p>Now, it turns out problems get so hard to solve that you'll need whole classes of people devoted to advancing subjects which were even developed as a response to the difficulty of problems in others! I.e. subjects that sort of came out to develop means of thinking about certain classes of problems. The level of abstraction goes considerably up as you keep on developing machinery. That is, theorems and little facts that let you narrow down cases, and show you when things are and aren't possible, but not directly. </p>
<h2>And that is a bit on abstract math.</h2>
<p>Math of a more applied flavor will frequently deal with questions on statistics, and often will talk about efficiency of certain processes. How to analyze certain real life situations with mathematical models? Stuff like that. You can tell I have less to say on this ;) I'd yield for someone with more to say to take the platform.</p>
<p>By the way, the actual questions research mathematicians ask are considerably more abstract than the ones I listed. The ones I mentioned are ones you could see in basic abstract algebra or algebraic number theory.</p>
<p>It's the same word used in your high school geometry class. Did you enjoy doing proofs when you took geometry (assuming precalculus follows it)? Proving theorems in college is, of course, a lot more sophisticated and creative than high school stuff, but the process of deductive reasoning is similar. </p>
<p>Although enjoying high school math does not imply the same in college, it's rare to find a mathematician who does not like proofs. So if you don't like them, then a math major is probably not for you.</p>
<p>Here's a good sample requiring little background to solve. Say a set S has |S| elements. Show that the set of all subsets of S has 2^|S| elements. Include, in the set of all subsets, the null set and the set itself.</p>
<p>Ny0rker - get hold of one of Martin Gardner's or Ian Stewart's mathematical recreation books or Raymond Smullyan's books of logic puzzles. If you read a bit in them and think "What fun! I don't necessarily understand it all on first reading but it's cool!" then temperamentally, at least, you have the makings of a math major. One clarification of mathboy's problem, which is a good one: the "null set" may be known to you as "the empty set" (or if that's not a term you know, just think of it as a set with no elements).</p>
<p>Or, you could try to prove that there is only one empty set, using the definition that sets X and Y are the same set if, for every w, w is an element of X if and only if w is an element of y. (This is a little more abstract than mathboy's example.)</p>
<p>Here's a nice succinct explanation (starting from ground zero) of one of the most famous proofs ever, Euclid's proof that there are infinitely many prime numbers:</p>
<p>It's straightforward to understand the question. It's also straightforward (once you get a little used to the techniques involved) to understand the proof. And if you start trying to list primes you find that you can just keep going and going, so it appears likely that there are infinitely many. But just listing them doesn't prove it either way because you would either (a) have to keep listing them forever (if there are in fact infinitely many); or (b) be sure that the last one you listed was the last one period, and you could only be sure of that by the listing technique if you tested all numbers *after *that and verified that they were non-prime. You need a trick and Euclid found one (or maybe one of his predecessors did and he just wrote it up). </p>
<p>If you think finding tricks like that would be fun, then math may be for you; if you look at it and say "Whoop-de-do, who cares?" you should not major in math.</p>