sounds like you like making assumptions.

> Take an actual graduate course in pure mathematics, very little hand holding.

i have taken many. and you missed my point. i didn’t mean having one’s hand held through the actual material. that isn’t true at all of course, in part due to poor teaching and in part due to the difficulty of the material. i meant that one is forced to take x amount of courses. i believe, past a certain level, that this becomes extremely limiting. you spend all this time and time effort doing coursework and going to class rather than researching. learning within a context is much better than learning without context, aka most courses. and many schools hardly take any coursework transfers. there aren’t many good reason for this.

and of course i have a bias because that’s what personal experience yields. but there’s no reason for you to insult me. i would wager my bias is shared by many who have been through ph.d. programs. and of course it’s not necessarily true. there are excellent advisors. but they are rare.

To offer another point of contrast - while it's true that you often need to spend a year taking courses, in most graduate programs they're very very flexible on what courses to take. And those courses themselves are very flexible on how much work you want to do.

I don't find this too troubling - it's important to have a shared fundamental knowledge so you can communicate effectively and not reinvent the wheel, and I don't think most programs are giving you busy work (that's really against the whole ethos).

>> you spend all this time and time effort doing coursework and going to class rather than researching

Is it possible the problem is particular to the US postgraduate system? In UK PhD programmers you are not required to follow any courses. That said, at least at my university as a PhD student you have the right to follow any courses you want for the duration of your PhD (obviously at no extra fee), which I'm absolutely taking advantage of.

I should also point out that a PhD is not just a time to do research- it's an opportunity to become an expert in your chosen field. And you don't do that just by inventing new knowledge. You need to also become familiar with the work others have contributed before you- and by "familiar" I mean "learn it very, very well". Perhaps the US universities are just trying to make sure you don't spend all your time with your nose in your own research, while ignoring everything others are doing around you?

just as a point of optimistic contrast, i've taken only three required courses over my PhD. some of my peers (who came in with better undergrad prereqs) have taken less, and in a few cases, none

You know I’d actually pay the crazy fees if only I could find a place that’d reteach me advanced maths “in context”. I find often that adding context means subtracting the “advanced” part :)

Study Physics. I'd take the math prerequisites and not understand what this is useful for, and then the first week of physics class we'd review the math and suddenly it would become quite practical.

Solid advise. I think finance massively recruits maths aptitude from physics for precisely this ability to connect hard core maths to models of the world.

I wouldn’t expect a pure mathematician to be good at this nor to care about it either.

Did you ever read “How to Solve it?” Polya says he became a mathematician because he was too clever to be a philosopher and not clever enough to be a physicist :)

Not all math has applications; some of it is just interesting for its own sake.