10 years ago I was an early employee of a risk analytics firm. I got pushed into a quant roll because I had an engineering background and could solve PDE's.
My first meeting I remember the head of the quant team hand-waved over a bunch of math that took me two weeks to go through on my own.
I had to spend 2-3 nights a week for 1 year going over my old statistics texts, Differential equations text and Knuth's Concrete Mathematics before I moved onto Stochastic Calculus.
It would have been almost impossible without having a team of people who were willing to help me get through sticking points. That was 8-10 years ago. I still spend atleast 1 night a week learning new math, though now its mostly non-linear regression and chaos theory.
And to be honest I still get alot of quants looking down on me, "A guy who dropped out of his masters, trying to claim he's our equal?". That does bother me a bit, and it means I probably couldn't get a quant job at Goldman, but I'm very proud of where I am now and how I got there!
TL/DR It can be done, but it gets done over the course of years, not months or weeks. You won't get there without applying what you learned to real world applications, reading the text's aren't enough.
I fully agree with you that one needs to think in terms of years and not days, weeks or even months. I do make this point in the article, as well.
I did have some very good (and very patient) instructors early on. But at some point when I had the chance to read and understand the basics of discrete math and intermediate calculus in-between my semesters (undergraduate was a bit broken up for me, I was a deployed military reservist), I guess I found it interesting enough at that point to go deeper and change my major.
[0] http://www.amazon.com/gp/product/0495391328?keywords=susanna...
Second, it should be something you really, really like doing. Like music is to a musician (or an audiophile), cooking (and watching people get off on your creations) is to a chef, sports training is to an athlete, etc.
And third, like anything else of true value in this life -- it will take a significant amount of time; in particular devoted to practice (and very importantly, play), especially solving (often obscure-seeming) problems on your own, just to scratch an itch, or to know that you can.
Easily a few thousand hours to attain what's called "mathematical maturity"[1], and probably somewhere on the order of the fabled 10,000 to obtain what might be called true expertise in the field. Which should (by itself) be no obstacle, if it's something you're really, really, really into.
Just like any other field, basically.
I started with Basic Mathematics by Lang, Eccles book on Mathematical Reasoning, A Course of Pure Mathematics by Hardy combined with the lecture notes of MITs honors single variable calc and Polya's How To Solve It, currently doing Advanced Calculus by Loomis & Shlomo.
I would imagine if you're interested in advanced math you would go to free university seminars from visiting professors and network with whoever is there as a self learner.
I get up 3hrs before work everyday and read a chapter then do as many exercises as I can. Repeat until done, or I get stumped and skip that exercise then come back to it later.
Unfortunately some textbooks can be expensive, but some are more reasonably priced. Unfortunately, the more "niche" the mathematical area becomes, the harder it becomes to find freely available sources.
I personally prefer a mix of video lectures and textbooks. Being able to watch video lectures, with the ability to pause and rewind, is a very useful feature that is not available in live lectures!
Also, it is very easy to audit courses at universities! Get out there, ask the professor if you can audit the courses (make friends with them too!), and enjoy yourself a stress-free, and money-free, quality education.
Though, between price fixing and booksellers going under and not being able to guarantee that you will retain access to the books you bought, I'd say that this form of piracy is morally ambiguous.
https://www.reddit.com/r/math/comments/3ixyw3/how_can_i_begi...
https://www.math.ualberta.ca/~xinweiyu/117-118.14-15/Math117...
It's not quite entirely by yourself, as in the webpage linked, but it worked like this:
1) Get sent problem sheets and a list of what chapters
in a textbook to work through.
2) Read textbook.
3) Solve the problems in the textbook.
4) Watch a one-hour live webcast to you (and a dozen
other people) four times.
5) Send off completed problem sheets; if you make the
pass mark, your prize is a seat in the exam.
6) Optionally pay for a weekend of direct instruction to
you (and a dozen others) in person somewhere (I did
this 3 times out of six, I think).
7) Sit three hour exam in supervised exam hall.
8) Go home, open next textbook.
9) Repeat four more times, and then write a dissertation.
Also not the same; a formal, full-on three hours exam in a formal exam hall. This sets the bar pretty high and is a really good indicator to yourself. I think this makes a real difference; someone else validating that yes, you really do know what you're talking about (or not, as the case may be - I think that of everyone who starts this, less than fifty percent pass the first exam, and if I had to guess, I'd say that it's not so much that the maths is hard; making yourself study it to the level required month after month is hard).
I estimated that the time I spent on this was roughly equivalent to working full-time for 6-8 weeks per year. Interestingly, if I'd been given an actual 8 week block, I don't think I could have done it. I couldn't spend 8 hours a day reading a maths textbook and get the same from it as if I'd done it in four two-hour blocks spread over a week.
Coming back from an exam, sitting down at the table, putting all the notes and papers and books from the previous subject to one side and then sliding the virginal textbook and brand new notes and problem sheets onto the table was brutal. It can be done, but you have to want it. I found that in terms of taking in new knowledge, I couldn't do more than a couple of hours with the textbook at a time. For practising and solving problems, I could sometimes sit down at lunchtime on Saturday to just have a go at one quickly and stand up again eight hours later, table covered in notes and the problem at hand having been thoroughly explored and answered (typically, sadly, spread over several pieces of paper which I would come back to in the future and condense into a smaller paper space).
The study habit it forced into me is enormously valuable in itself. I can now pick up a high-level maths text book and simply start learning. I won't understand much of it, but I know that if I keep at it, I will. Grind, grind, grind. Not just mathematics, either. Someone gave me a Learn Japanese book for Christmas and now it's in me. I ground through it, and now I've got another one and I'm going through that. It's as if the subject doesn't really matter, so much as the studying.
Hopefully, they do a good story with the most recent movie on Ramanujan's life:
http://deadline.com/2015/09/the-man-who-knew-infinity-exclus...