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I guess I should jump into arguments earlier, to stop this from happening.

I taught myself calculus from a book called, well, "Teach Yourself Calculus." It basically said things like, "This function is the derivative of this function. This is how the chain rule works. This is how integration works." Apparently, it is entirely possible to create an entire calculus course without giving analysis-style proofs of *anything*. Hell, I learned chain rule as, "The dx's cancel" in the equation dy/dt = dy/dx * dx/dt. Same with chain rule for integration. Surprisingly few calculus textbooks I've seen, all the way up through multivariable calculus, have proofs - full or partial - for all the material they contain. Hell, I'm a grader for a summer class in "basic calculus" right now, and while their book has *some* epsilon-delta proofs, the kids aren't required to be able to understand the notation or recreate a proof. Such is the way of calculus in the sciences.

(Like Layra, I'm making a distinction between calculus and analysis.)

My point is - as Layra's was - that being able to solve basic problems with theorems doesn't require *any* knowledge of why the theorems work or how they're proven. Most calculus books I've seen, and every non-math-major calculus class I've peeked into, is based on this principle. Most people who get into Princeton, math major or not, will never have seen a rigorous analysis proof (or definition, probably). C'est la vie.

Would that make for a good definition? Analysis is about proofs, definitions, and concepts, while calculus is just abusing the results of analysis for problem-solving. It seems pretty accurate to me, at least.

So the only point where Layra and I differ is that she went out into the world (well, the cyberworld) to look for proofs. I didn't know such things *could* be proven; I didn't think there was any math beyond calculus until I got into it. I thought that after multivar, linear algebra, ODEs and PDEs, I'd be done with math...and even after taking/teaching myself all of these, all I had learned were methods of problem-solving. Of course, that made my physics classes pretty trivial, but my math classes kinda ripped me a new one for a while...

Until now. I'm finally caught up to everyone else! Well, except the Romanian students, and those two kids who were taking grad. math classes freshman year. My point is that I moved from the bottom rung to the middle of the pack in math, and to the top of the morning in mixing metaphors.

High school steered me back towards math, probably only because it started being trivial (i.e. purely problem-solving methods). I mean, I was *really* bad in math until I got to algebra. Hell, I didn't know how to work with fractions until I got into algebra. But from then on, since all the classes asked for was formula manipulation and word-problem-solving, all it took was a bit of memorization and bam! Math was trivial and an easy A.

I find it strange that I started getting interested in math only so I could do more interesting physics. I taught myself calculus not to learn calculus, but to take the AP physics C tests, and to be able to work with some equations in chemistry that I didn't understand. In college, then, physics was still trivial - all just problem solving and equation manipulation - but math wasn't. It was rigorous, proof-based, challenging, interesting. Now the physics that I'm taking isn't trivial anymore, and is starting to get more mathematically rigorous, but...I'm hooked for life. I'm officially a math major with interests in physics problems.

That sounds like a really great program, Layra. I wish I had known about something closer to equivalent in Los Angeles. But I also didn't come from anywhere near the same background you did. Neither of my parents graduated high school, and while they did value education, they didn't have any themselves. Nor did they have the resources - either the money to pay for such programs (we've been officially below the poverty line for my entire life) or the contacts to know such programs existed. I keep joking that, as far as Princeton (and Ivy league schools in general) go, I'm a token: poor, uneducated, from a big city. Class differences > race differences.

*gives virtual hugs to everyone*

I don't want to hijack or dwell on it, but that's a very true point

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I think the cost of education is a double-edged sword. While university education in England and Wales used to be free, that only worked when it was extremely elitist. In trying to open it up to make it more inclusive has been met with the problems of financing the number of students. So while the cost of tuition is still very low in comparison with the US — if you compare the "elite" UK universities with the "elite" US — the UK counterparts are nigh-on bankrupt.

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*hugs everyone*

Man, the UK sounds so awesome. It just makes me want to go there more.

On a different note, it's true that in general, colleges in the US are extremely expensive, and loans suck - hard. However, Princeton was my top choice for a pretty big reason: no-loan policy. I may have to work a little during the school year (or, preferentially, just a lot over the summer) but it means that I leave college with no loans. Even in most European education systems, that's not possible. It also allows for blind financial aid packages, where the scholarships awarded are based solely on income available to the student, and not on previous grades, skills, sports, etc.

The biggest downside to the system is that it doesn't account for students who don't receive help from their parents. My girlfriend's from Norway and there (so she tells me) parents don't support the child through education. So her financial aid package is based on her parents' middle-class income, and not on her personal income of ~0$ net per year. C'est la vie...the system's pretty good, otherwise.

Yeah, I know...plus, without some sort of ridiculously competitive scholarship or fellowship (and I'm simply not that smart or experienced), I'd need a Master's from the States before doing my PhD in the UK. Maybe I could hope for a postdoc there, but that's waaaay too far in the future for me to be thinking about. I just really, *really* wanted to do a PhD in mathematical physics at Cambridge.

Universities play a very tricky, very political game of supply and demand when it comes to tuitions. On the surface, it seems like they would just want to balance out the supply-demand curve, charging a moderate amount of money and getting a moderate amount of students, and maximizing their income. But when that income isn't enough; when there are no government subsidies, or that doesn't cover the rest; when raising the price creates a much higher backlash, socially and economically, than is predicted by simple supply-and-demand; when competition with other universities and the sake of reputation is taken into account. Postsecondary education is not in a very nice position.

(I've heard about the UK-US exchange rate. I don't expect your position to weaken anytime soon, either. Somehow I figured that the pitiful state America's in now would help pretty much everyone else.)

Yeah, there are disappointingly few girls in the math department here, although amongst the professorship all of the females are Caucasian of some variant.
Actually, there seem to be more Jewish males than anything else.