William Stein

A lot of people say this, but if I were to sit next to you and tell you what the notation means whenever you asked, would you instantly understand the concept? I don't think so. You still need to do work on your own with the concept before you can get to any understanding.

Say, you come across this symbol: ζ(s)

And you asked me, what does it mean?

Oh, that's probably the Riemann zeta function (it could be other things, but that's probably its most famous meaning).

Would you be like, ah, now I understand it all!

I sincerely doubt just knowing the English words for the symbols (and all mathematical notation is just shorthand for English, or whatever other natural language; it can all be read out loud) would get you that much further along. I could give you one of several possible equivalent definitions, but you still wouldn't really know much of anything. I could start explaining some things about primes and how they relate to the Riemann zeta function, and you'd still be in the dark. I'd have to talk about the prime number theorem and estimates and complex analysis and a whole bunch of other things before the Riemann hypothesis even started making sense. In fact, people have written entire books just trying to explain in the most accessible way possible what that one symbol is:

https://www.wstein.org/rh/

I believe what's going on is that a lot of people who find notation to be the first stumbling block are trying to learn mathematics in isolation, without the benefit…

[Disclaimer: I'm part of that 1% she is talking about, since I'm tenured faculty in Mathematics at University of Washington, with "Harvard" on my resume due to my first job being there. http://wstein.org]

The author says: "The most prestigious universities – the Ivy League, University of Chicago, Stanford University, the University of California system – tend to lie in the most expensive parts of the country. Even with full funding, it is nearly impossible to live in such costly cities without incurring debt, given that stipends tend to be $25,000 or less."

Such entitlement. I got into UC Berkeley (1995-2000) where I received $13K/year to live on from them (for TA'ing), and I made it work by having a roommate, being frugal, and getting some student loans (the minimum I needed), which I paid off within 2 years of graduation. It sucked, but I had to just be lean and deal with it, and reminded myself that I was a student.

Also, there is a naive and incorrect (in math at least) assumption in this article that the only difference between a graduate student from a top-10 university and a top-100 university is the name of the university where they graduated. There may be some truth to this for undergraduates, but for graduate students---who spend between 4 and 10 years busting their ass full-time on their research area---the situation is completely different.

The author thus seems to discount the possibility that the very best students who go to graduate school at…