> "Typically students have a 5 on the BC calculus exam together with a math SAT score of at least 750."

Okay, I went to the Princeton site and looked at the course descriptions and contents.

I noticed a surprising theme: It looked like there was a big intention to make the courses difficult. Gee, guys, a student is paying a lot of money to go to Princeton. To get in, they had to do a lot of preparation and, apparently, have a lot of aptitude. For such a student, the material listed is not so difficult that the courses have to be difficult.

So, a question: Why the heck should a good student who wants to know some math put themselves through such difficulties just to learn some material that is not really difficult?

There's an alternative: Essentially every topic in the courses is in beautifully polished textbooks. The students are being expected to work really hard outside of class, anyway. So, just save the money, the stress, and the difficulties, skip those Princeton courses, get 2-3 shelves of appropriate books, and study independently. "Look, Ma, no expensive Princetion tuition, expensive cost of living, etc.".

I did notice that in places the course materials were lectures and notes. Gads: For that material, there is no excuse for other than beautifully polished textbooks since so many of them exist.

Why? Why, what the heck is the goal of such a Princeton undergraduate math major? Sure, the goal is to go to graduate school in math or something closely related.

So, get the undergraduate material by independent study, save the botheration of Princeton, and go to graduate school.

With high irony, at least at one time, the Princeton math department's Web site stated that graduate students are expected to prepare for the math Ph.D. qualifying exams on their own, that graduate courses are introductions to research and given by experts, and no courses are given for preparation for the qualifying exams. Okay, so the attitude is that the students should learn by independent study. Right.

Likely the main challenge of a Princeton math Ph.D. is the usual one -- the research. But, now, there is a largely new approach, likely also permitted at Princeton: Get a problem from the non-academic, real world, derive some math likely at least somewhat new for a good solution, and let that research be the Ph.D. dissertation. From those Princeton materials, with a lot of emphasis on machine learning (ML), it looks like such a dissertation approach would be acceptable.

My experience can serve as an example: In high school I took 1st and 2nd year algebra, plane and solid geometry, and trigonometry but took no calculus. I went to a college selected because I could live at home and walk to it! It was not a good college! They wouldn't let me take calculus as a freshman and, instead, forced me into some course beneath what I'd done in high school, so I got a good calculus book and dug in. For my sophomore year, I went to a good college and started on their sophomore calculus from the
same text Harvard was using. I did fine.

Lesson: It's possible to teach this stuff to yourself. Big buck tuition, big challenges, etc. are not necessary.

Another Lesson: Don't need AP calculus. Instead, just get a good calculus book popular for college freshmen and dig in. Indeed, the time I looked at the AP calculus material, it seemed to be written by people who didn't understand calculus well. Likely the used book collections are awash in good college calculus books -- we've had very highly polished calculus texts for decades, and the subject hasn't changed much.

Note: At some point, might want to learn vector analysis with Stokes theorem. Okay. The high end approach is via Cartan's exterior algebra, but really that should be for a second pass. Besides, if you want vector analysis for Maxwell's equations, potential theory, fluid flow, etc., then the exterior algebra material will likely not yet be popular. I suggest

Tom M. Apostol,
Mathematical Analysis:
A Modern Approach to Advanced Calculus,
Addison-Wesley,
Reading, Massachusetts,
1957.

It's great fun to read, especially after sitting through physics courses where the professors struggle with this material and the math departments don't want to teach it.

Get the book used. It's just the right compromise for, say, Maxwell's equations. Leave the high end versions with differential geometry,
exterior algebra, emphasis on manifolds, measure theory, etc. for later.

So, in college, I got a math major. Okay, I found that after the first theorem proving course, I could teach myself material that was about proving theorems.

Lesson: Take a course that tries to teach theorem proving; then you should be able to continue on with independent study of the math that is all about theorem proving.

So, right, my undergraduate math major had a theorem proving course from Baby Rudin (W. Rudin, Principles of Mathematical Analysis, hasn't changed much in a very long time). I did well enough in the course, but at the end I neglected it to finish my math honors paper. Later on my own I took a second pass through Baby Rudin and learned it much better. That second pass was slow, careful, where I chewed on the proofs to try to understand why they worked, took the time to develop intuition, etc. Then I took a careful pass through Fleming, Functions of Several Variables and, in my career, studied a wide variety of related topics. Then on my Ph.D. analysis qualifying exam, those passes through Rudin and Fleming got me the best score in the department.

Lesson: That analysis material at the level of Baby Rudin or a little more, you can teach yourself plenty well enough.

The Princeton materials put a lot of emphasis on linear algebra. Okay. I'm not sure that quite so much emphasis is crucial, but in the end I did have that much emphasis. For linear algebra, I learned a lot, and nearly all of it I taught myself from a stack of mostly quite good books. Then later I got pushed into a advanced course in linear algebra taught by a world expert guy. I told the faculty that likely I didn't need the course. Yup, the course was intended to filter students, and that was sad because some of the filtered students were quite good and should have done well. I was right: I didn't need the course: The course was carefully graded, and on all the grading I totally blew away all the other students, without so intending, effortlessly. I felt sorry for the other students I made to look bad -- only near the end of the course did I understand I was blowing away the other students.

Lesson: Linear algebra really is important, but you really can teach it to yourself to a quite high level.

My Ph.D. dissertation was in stochastic optimal control. I had the main, intuitive ideas on an airline flight. Then for the careful theory, I taught that to myself from various sources and derived the rest myself. Really, I wrote my dissertation essentially independently.

Lesson: it's possible to do that.

Overall Lesson: You can teach that math to yourself and do good research with it, all with very few courses and without paying big bucks for tuition and living in Princeton, putting up with rough class notes, having people trap you into some very stressful situations for no good reasons, etc.

Yes, at one point I did get accepted to graduate school by the Princeton math department. I didn't go to Princeton. I also got accepted to Cornell, Brown, and more.

Finally, why learn all that stuff to A+++ level, have memorized all the difficult proofs, can work right away all the tough exercises from all the relevant texts in the library?

Here's the main answer: There is no really good reason. Maybe for some student there is a good reason, but IMHO mostly there is not. So, how do the professors know the material so well? Usually from having taught a course in it a few times! In the end, both in academics and outside, for any math you do or apply, you will get paid for what you can create that is powerful and valuable. Or, you won't get paid for carrying the library around between your ears. The respect is for what you can create, not what you know. Yes, for good creation, do need to know the prerequisites well enough, but you still can use the books on your bookshelf and the ones in the library. Sure, the more stuff, even just 100 challenging exercises, you know, the better, but it's rarely necessary to have A++++ in Baby Rudin, etc. to do the needed research.

I will say, it does appear that learning the material well enough to have built some good intuitive models, that let you make good guesses, is worthwhile. Professors who are willing to pass out such models should be listened to.

Yes, the Princeton quote above mentioned doing better than 750 on the SAT Math. Okay, I confess, I did, both times I took it. But I don't believe that such an SAT Math score is necessary. For me, likely the secrets were just that I liked the material and wanted to learn it.

Finally, I fear that the Princeton Math Department is hurting a lot of good students, maybe creating some sadistic professors, all for no good reason.