Princeton University Math Major Course Guide
blogs.princeton.edu
blogs.princeton.edu
"The introductory courses for math majors are MAT 215: Single Variable Analysis, MAT 217: Linear Algebra, and MAT 218: Multivariable Analysis. Like the great majority of math courses at Princeton, these three courses are theoretical and proof-based. [...] These three are usually the first math classes that math majors take at Princeton. However, the math department is very flexible in allowing advanced freshmen to skip some or all of these courses."
MAT 215 is classical analysis (epsilon-delta, differentiability, etc.). Its own course page (https://www.math.princeton.edu/undergraduate/course/mat215) advises regular students considering taking the course:
"Typically students have a 5 on the BC calculus exam together with a math SAT score of at least 750."
The concept of being so advanced as a freshman that you skip this class is pretty amazing to me. I know those folks are out there, but skipping honors analysis really brings it home.
I struggled through another Ivy's version of Honors Analysis for math majors (out of baby Rudin) as a beginning grad student.
But my surprise isn't just relative to my own preparation (such as it was).
If you've learned Calculus, and you've learned introduction to proofs (either on your own, or in summer nerd camp, or by visiting a local college), you are ready to learn new material (like linear algebra) in a proof-focused curriculum.
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.
That being said, I think you raise a lot of valid points, but I think a lot of them are the same valid points that - particularly this community - are very aware of and are already oft repeated, that is, you can still learn material effectively outside of university, much of the curricula are outdated or need to be re-designed (hello, calculus, diff. eq), and that there are some very rigorous, harsh, antiquated practices used to put students down that we need to be rethinking.
On a broader system-wide level though, do you have any thoughts on how to propel this shift?
It appears that Princeton has concluded that their long emphasis on "The analytic, algebraic topology of the locally Euclidean metrization of infinitely differentiable Riemanian manifolds" (after Tom Lehrer) had Princeton math, often regarded as the best math department in the world, their professors, and students, missing out on good opportunities for research, grants, jobs, money, startups, money for donations back to Princeton, etc.
Princeton won't have any trouble often doing the math of ML better than the computer sciences. So, the computing people and ML, etc., should be providing some new career opportunities, including for Princeton math students. So, the Princeton math department can respond by being more welcoming to students, get more students, etc.
As it is, I have to suspect that in recent years the Princeton math department has hoped that their best students would extend the path plowed by A. Wiles and his proof of Fermat's last theorem, etc., but now with ML, and maybe more, maybe they are welcoming the new opportunities.
That, then, might be an answer to your question. For more, there's the Internet and the ability of students to watch videos or just download good textbooks as PDF files.
It would be easy to guess that Princeton math has long been stuck in the most pure of math, algebraic geometry, algebraic topology, and number theory, but actually history shows that Princeton math has been quite open to new topics. So, there was Ford and Fulkerson and network flows. There was J. Nash and game theory. There was H. Kuhn and A. Tucker and the Kuhn-Tucker conditions. There was J. Tukey and a wide variety of topics in statistics and signal processing. And likely E. Cinlar in stochastic processes, operations research, and financial engineering worked closely with the math department.
But making the contents of Baby Rudin difficult looks not so good. Gee, guys, the main point of Baby Rudin is just to do a good job on the Riemann integral, especially have sufficient conditions for that integral to exist. For that Rudin wants to define uniform continuity. For that he wants to define compactness. So, the first chapters are about compactness. So, if a function is continuous on a compact set, then it is uniformly continuous, the Riemann sums converge, and the Riemann integral exists. Compact? Sure, as Rudin proves, closed and bounded in R^n implies compact. Then nicely enough near the end of the book Rudin does just enough in measure theory to define a set of measure zero and shows that a function has a Riemann integral if and only if the function is continuous everywhere except on a set of measure zero -- so, right, here he gets both necessary and sufficient conditions. Really nice. So, that's two ways to know that the Riemann integral exists. There is more in Baby Rudin, e.g., Fourier series, but IMHO I mentioned the high points. Those points don't have to be super difficult.
Really, the place to learn integration from Rudin is the first half of his Real and Complex Analysis. There he defines the integral by partitioning on the Y axis instead of the X axis -- right, that's the key to measure theory, for theorem proving, a better way to define an integral. Right, the integral of x^2 on [0,1] is still 1/3rd. And he also does the Fourier integral and the basics of Banach space and Hilbert space.
Note: By partitioning on the Y axis, don't have to be able to partition the domain of the function are integrating. So, the domain of the function can be some unseen, abstract measure space, in particular, a probability space. Then real valued (measurable) functions with that probability space as domain become the random variables of grown up probability and statistics. Darned nice! It appears from the Princeton materials, their courses don't get to this material until graduate courses -- so, there Princeton is not moving a lot faster than a lot of other math programs.
Really, IMHO, there isn't anything more difficult about the measure theory approach to integration than the Riemann approach in Baby Rudin. So, sure, maybe go light on Baby Rudin, that is, for sure don't try to see how difficult can make it, and rush to measure theory, probability, stochastic processes, mathematical statistics, and apply those to theory/applications in ML, do some good original work, call it a dissertation, get a Ph.D., get a good job or do a startup, make money, make babies, and send them to Princeton! Then everyone is happy!
In fact the historical record shows that Princeton has an Applied Math department ( http://www.pacm.princeton.edu ) and does a lot of math-like things in their ORFE department ( https://orfe.princeton.edu )
For ORFE, I mentioned Cinlar. My best prof was one of Cinlar's star students.
And I mentioned Ford and Fulkerson, Kuhn and Tucker, Tukey, and you mentioned still more. Good.
Obviously, the tutorial system at Oxford was a great benefit though and I don't know how that compares to teaching at Princeton. (This was also long enough ago that I didn't have to pay for tuition so it was a no-brainer decision to go there rather than study independently!)
Also while in college, I taught myself electronics and computer programming, to a level where I was employable in those areas. In music, I combined self learning with formal lessons that provided a much needed critique of my physical approach. I am employable in music at a journeyman level.
Working under a mentor in physics research taught me to be a better scientist.
What you got for free through self-study, my parents had to pay for. That's life. ;-)
Lesson: Use school to learn the stuff that you wouldn't learn outside the classroom. Learn additional topics on your own. I don't know many people who are comfortable working with math, physics, electronics, and programming. Those who are, tend to call themselves physicists.
And here at HN, my guess is that 75+% of what the people know that is technical they taught themselves.
So, for yet another programming language, sure, write "Hello World" and get it to work. Then see how that language handles if-then-else and do-while, elementary data types, arrays, trees, objects, functions, memory allocation, threads, locks, exceptional conditions, file I/O, direct access files, interacts with SQL database, gets to TCP/IP, sends Web pages, etc. what libraries and APIs are available, etc. Lots of people at HN have done that for several langages. Can do much the same for undergraduate math such as described in the Princeton materials.
Perhaps in any field, you get past a "learning how to learn" stage, so the early learning prepares you for independent learning later on. I'm at a stage in music where I can advance my abilities pretty well on my own, having developed the foundation of a correct physical approach to the instrument from a teacher.
That was a good start in teaching myself that did me a lot of good. Maybe the best part was I was able to do publishable research. So, in grad school I took a problem I'd seen in a course, a problem with no obvious solution, and asked for a "reading course" to attack it. Well, I did more, got a clean solution and also proved a nice, surprising theorem. That was two weeks, and I was done with the "reading course" and the last I needed for a Masters. It looked publishable and it was -- later I published it. Doing that research gave me a nice halo that served me well for the rest of my grad school time. So, that independent study/research ability totally saved my tail feathers.
For music, once I heard a little Beethoven, I fell in love with classical music. Later at Indiana University, with its terrific music school, I took a beginning course in violin. So, I had to start with how to hold the violin and the bow, tune the violin, etc. I got through the A major scale in first position!
Later I continued on my own. I got through about 3/4 of the Bach E major Partita, in time and in tune! Then I got through most of the Bach Chaconne also in time and in tune. Really liked the central D major section. Then I went for some lessons: Conclusion, I needed more practice! But the self teaching seemed to work again!
And, yes, at one time I taught several sections of computer science at Georgetown University. Taught it? Yes. Had ever taken it? No. Sure, like no doubt nearly all of the HN audience, I had taught the stuff to myself.
I never had any desire to be a college professor. I took the Ohio slot to be better able to care for my wife during her long illness.
I'm pursuing the certificate right now and the courses have been great so far. Princeton's known for having a rather theory-heavy approach in their quantitative classes but I've found a good balance with applications in some of the classes (COS 424, COS 402).
The algebra introductory courses have different numbers, and are now 345-346 instead of 322-323. There is now an advanced graph theory class (477) that follows the introductory one. Also, there is a theoretical machine learning class (COS 511) and the algorithms/complexity graduate sequence (COS 522-523) may be relevant as well.
I will probably update these changes on the website sometime in the near future...
https://gowers.wordpress.com/2011/09/23/welcome-to-the-cambr...
a dumb question if I may - What is an "example sheet"?
> Lastly, don’t forget the ever-present Rule of 12!
> (That’s “twelve,” not “twelve factorial.”)
I actually never really liked study groups so to me studying math is the same as studying any other subject, but everyone is different. Considering you do well with study groups it seems wise to first try to find others who also want to study so you guys can study together.
Anyways, my point is that I bet you can do it, you should try, and it might feel hopeless at first. Don't be surprised or discouraged if it takes you something like an hour per page (or more) to work through a textbook. The good news is that's probably all you really need to do: work through it.
But I would Terrance Tao's notes on analysis: http://www.math.ucla.edu/%7Etao/resource/general/131ah.1.03w...
Before you start with Tao go through all the calculus series on Khan academy & get an introductory book on proof writing.