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> Of course someone who gets paid to teach would highly stress learning tidbits slowly and excruciatingly. That's their economic incentive...

Well, I can see that's a factor, but not necessarily an overriding one. As someone who's taught at uni myself (not pure maths) we are not usually that cynical or fond of serving up "detritus". It's not as if we lack for valuable and interesting stuff to teach if we go through foundations too quickly, and we aren't paid just for teaching. Anyway, I think most do benefit from a quite painstakingly incrementalist approach to maths; me taking that too far and sometimes getting stuck is a personal failing.

(I recall a quote by a colleague of the group theorist Simon Norton, who famously suffered a career collapse/hiatus after a series of brilliant results, something along the lines of him having opened a doorway into a wondrous realm of new mathematics, but ending up stuck there, at the doorway, obsessed by the details of the doorframe.)

If I was teaching a linear algebra course, I'm not going to say "by the way you can skip this subject entirely because it will just fall out of your working backwards through Wiles' proof of FLT". For those of us without a once-in-a-generation mind I think the traditional approach is the right one. I was only trying to say, I personally sometimes get stuck and it will be interesting to try the opposite approach.

> Frankly, professors would be more useful to me, if I could purchase their time by the hour

If you go to a good uni, at least by postgrad level you do have that kind of access, and, if you get along, you retain it for free after you leave.

> Well, I can see that's a factor, but not necessarily an overriding one. As someone who's taught at uni myself (not pure maths) we are not usually that cynical or fond of serving up "detritus". It's not as if we lack for valuable and interesting stuff to teach if we go through foundations too quickly, and we aren't paid just for teaching. Anyway, I think most do benefit from a quite painstakingly incrementalist approach to maths; me taking that too far and sometimes getting stuck is a personal failing.

Possibly so, but I never went in any of the grad programs. Most of the lower classes are taught by AI's and contract-based "instructors" paid by the uni on a per credit-hour basis. And much of the time, the department shovels the syllabus and required areas to them for the students.

And unfortunately, this avenue of teaching very much shows. You have instructors who have some semblance of caring, but not terribly much. They teach weeder classes with the intent of failing much of the class. Whomever is teaching isn't always able to explain what's going on in an area - they can do the process, but can't explain why their actions work.

Perhaps it is a jaded viewpoint. But after spending way too much money in "Higher Ed", along with working at an institution, I know the game. And I'm sure it's better if you're a post-doc with prestige or on that track. But the rest of us are spoon-fed bland crumbs these days, and pumped-and-dumped for excessive scholastic loans to get a job to pay the loans back with.

> If you go to a good uni, at least by postgrad level you do have that kind of access, and, if you get along, you retain it for free after you leave.

Yep, and if you're not on that track, the access isn't there. I'd be willing to pay for it directly. Google had a program quite a while back, of paying experts for direct guidance in specific fields. Too bad they cancelled it.

I did have bad experiences with bad lecturers as an undergrad (hello, here is a handout, now I will project the handout on a screen, now I will read what's on the screen without any elaboration, goodbye), but they were the exception. Obviously this depends hugely on the exact institution in question. And yes, many are now increasingly functioning as blandly corporate battery student farms...

> Yep, and if you're not on that track, the access isn't there.

Actually I'd also be interested in such a scheme, now that I'm exploring ideas far away from my original research area. Although if you have a bona fide interest to discuss something technical and specific with an academic who has the relevant expertise, I've found they can be pretty approachable, even if you email them out of the blue to ask for a chat... but I do have the right sort of background to do that I suppose.

How do you identify the "holes" when you reach them? How do you even know whether you've reached one?

"Aha! This is clearly a situation in which a monad would be the best approach. Time to go learn about monads!" Just doesn't seem like a reasonable method to me. Some things you just to to learn well before you can even recognize when you need to remember them later.

That's not a valid criticism if you have intent and will to learn.

As for your monads example, getting into functional programming via things like CLisp, Erlang, Haskell, and the like will expose you to lambda calc pretty darn quick. And learning how monads work is near the beginning of that path.

And just being inquisitive leads to a whole lot of areas that give indications on what to learn. For example, doing computer vision enforces you to learn how linear algebra works. Machine learning teaches a great deal of how statistics works. Finite State Machines have their own really interesting niches to work with. Working a crummy operator job teaches how to do automation (on the sly!).

It really depends on how you approach learning. If you're just slowly grinding away because you have to, going through a 4 year BS degree is probably better.

Not all knowledge sits on the frontier of yours.

What's that supposed to mean?

If one wants to learn anything in the sciences, we things like MIT Courseware, Arxiv, Libgen, SciHub, and "canihaspdf" on twitter. Yes, these are primarily pirate options - so what?

I can publicly see the course projections for any arbitrary degree, along with class titles. And many have book lists linked, so I can hunt for the books online using less legal methods. The only difficulty with some STEM learning paths is they require laboratories - those are hard/impossible to do at home and thusly necessitate academic environments. Computing, on the other hand, is easy to learn even at a Starbucks with a laptop and a phone.

What's stopping people from learning what they wish is primarily time and the will to (and the fact that school does a great job at beating the will to learn out).

> I also have learned compsci with his similar methods of finding interesting areas and digging in

I don't think those things are similar. His method -- working backward for Wiles' proof of FLT to Linear Algebra -- is not really analogous to teaching yourself some undergrad CS. It'd be more analogous to deciding you want to understand Mulmuley's latest results from nothing and discovering while loops along the way.

There's a difference in kind. Maybe this is splitting hairs, and at some level they're both "self teaching", but the huge chasm in relative difficulty/impressiveness still irks me :)

> Frankly, professors would be more useful to me, if I could purchase their time by the hour over issues I don't understand

Find a decent university and go to office hours / ask for independent studies.

Paying tuition is quite literally purchasing their time. Only a small amount of official instructional time is spent in lecture halls. And most professors spend more time on teaching than they're technically required to. At decent universities that prioritize teaching, maybe 80%+ of teaching time is spend in one-on-one or small group interactions.

IME most people who dislike formal higher education never learned how to use it properly in the first place. Or attended undergraduate at colleges primarily known for the attached research institutes, not their undergraduate program.

> Of course someone who gets paid to teach would highly stress learning tidbits slowly and excruciatingly. That's their economic incentive.

Believe me: Most TAs and professors would prefer to teach much more advanced stuff in a much faster pace - but for (good?) reasons they are not allowed to do.