>. there’s so much hand holding through required coursework (and homework) that one can’t hardly think deeply about anything actually original.

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

>there’s basically no time or space for one to explore one’s own thoughts and program unless it specifically matches up with what someone else (namely an advisor) wants to do.

Not necessarily true. Find an adviser that will support YOUR interest (they exist). Sounds like you have some biases against PhD programs without actual experience of being in a good one.

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.

Come to Europe. Or at least to some European countries. In mine, Spain, Ph.D. programs now have no courses at all (let alone homework). They are exclusively research-oriented, your goal is to produce good research (papers) and then your thesis. Also, the duration is capped at 4 years, and the tuition fee at my university is around €250/year. Other European countries have similar systems.

That is not the whole picture. Most Ph.D programs require you to successfully complete 2 years in a research master program which is based on classes and homework assignments. It's not easy.

After that you enter into the Ph.D. for 3 years(max 4) and you are right, you can almost do whatever you want. Just bear in mind that either you publish in top journals or your academic career ends here.

This is true, but in practice admission in most Ph.D. programs in Spain is not very selective as there are more slots than applicants, so any master's degree will do, even if it's not research-oriented. And for foreign students, many Ph.D. programs have a rule that "if you can prove that your degree gives access to Ph.D. programs in your own country, then you can enter" so it's even possible to be admitted with e.g. a US bachelor's degree (YMMV per university though).

You don’t need a masters in the UK. Three years bachelors and then straight into a PhD which is just research - no classes or teaching. And we produce more than our share of top science so it must work.

You'll have to have a first in order to be able to skip the masters, and since entry to funded programs is competitive, you have to be an exceptional candidate for this to actually happen – it's quite rare.

Then there's the program itself. DTCs (doctoral training centres) now mostly run 3 + 1 programmes, where your first year is a master's, involving taught classes and a research project. This is to get everyone up to speed on how to actually do independent research, and to build core skills. While it's true that teaching doesn't always form a mandatory part of a doctoral degree, working as a teaching assistant for at least one semester is often mandatory.

In short, you've mischaracterised doctoral education in England and Wales (Scotland may be different) to quite an extraordinary degree. I see no reason to comment upon the "more than our top share" claim, for reasons which are, I hope, obvious.

> You'll have to have a first in order to be able to skip the masters

Well I'm not sure that many people without a first would seriously consider doing a PhD, so this isn't relevant in most cases.

> DTCs (doctoral training centres) now mostly run 3 + 1 programmes

Right, which is a big disadvantage of them. DTCs are just one way to do a PhD and they're not the traditional approach in the UK.

> In short, you've mischaracterised doctoral education in England and Wales

I don't agree! This matches from I see of students going into and coming out of the system today.