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...wow, OK, that's a lot. This feels more like a Medium post than an HN comment. That being said, I think you raise a lot of valid points, but I think a lot of

...wow, OK, that's a lot. This feels more like a Medium post than an HN comment. 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oft repeated, that is, you can still learn material effectively outside of university, much of the curricula are outdated or need to be re-designed (hello, calculus, diff.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?

graycat 10y ago
It appears from the Princeton Web pages in the OP that the Princeton math department is taking machine learning (ML) seriously. 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!
- GFK_of_xmaspast 10y ago
  > It would be easy to guess that Princeton math has long been stuck in the most pure of math ... but actually history shows that Princeton math has been quite open to new topics In fact the historical record shows that Princeton has an Applied Math department ( http://www.pacm.princeton.edu http://www.pacm.princeton.edu ) and does a lot of math-like things in their ORFE department ( https://orfe.princeton.edu https://orfe.princeton.edu )
  - graycat 10y ago
    We are in full agreement, and you added evidence to the evidence I gave: 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.

graycat 10y ago

It appears from the Princeton Web pages in the OP that the Princeton math department is taking machine learning (ML) seriously. 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!

GFK_of_xmaspast 10y ago
> It would be easy to guess that Princeton math has long been stuck in the most pure of math ... but actually history shows that Princeton math has been quite open to new topics In fact the historical record shows that Princeton has an Applied Math department ( http://www.pacm.princeton.edu http://www.pacm.princeton.edu ) and does a lot of math-like things in their ORFE department ( https://orfe.princeton.edu https://orfe.princeton.edu )
- graycat 10y ago
  We are in full agreement, and you added evidence to the evidence I gave: 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.