10 ms·
How to Learn Advanced Mathematics Without Heading to University
- analognoise 11y agoIt's cute that quants think what they do is advanced, isn't it?
- orbitingpluto 11y agoIt's cute that the article is littered with Amazon referral links to textbooks.
- davidivadavid 11y agoI'm curious why people care about articles with referral links. It's not like you're paying more if you end up buying something, is it?
- fabulist 11y agoIt is always good to know when someone is profiting financially from a recommendation they're giving you.
- davidivadavid 11y agoSure, I'll buy that in principle. But wouldn't you agree that it mostly matters if someone is recommending some shady financial product or otherwise high-ticket item? What's the worse that could happen when someone makes (text)book recommendations? Affiliate links are also a good way to compensate people for curating valuable information. So the sort of "it goes without saying affiliate links are bad" attitude is kind of a mystery to me.
- whatok 11y agoI don't know any successful quants that claim what they do is "advanced" in any fashion.
- EliRivers 11y agoA year or so ago a chum asked me to take a look at a paper that some of her quants were basing their work on. When I dug through it and pulled out all the plagiarism in it (and I do mean that; it had literally been hacked together out of other papers by some chancer in a middle-eastern university) it turned out to be a Crank–Nicolson method, and then when I saw what they were doing with it, I reckoned they could have just used a crayon and drawn over the graph they were interested in to get their smoothed graph. I do know that soon after that she got a job somewhere else. You can find hacks and chancers in every walk of life. [1] In the above text, "hack" and "hacker" is used in the pejorative sense, similar to a "hack journalist".
- jasode 11y agoThe word "advanced" is relative to the intended audience and it's obvious how he meant that word to be applied. The author wrote: ", chances are if you are considering studying advanced mathematics you will already have formal qualifications in the basics, particularly the mathematics learnt in junior and senior highschool (GCSE and A-Level for those of us in the UK!). " The author is not a Fields Medalist writing to other PhD mathematics graduate students. In such a case, the adjective "advanced" would have a different meaning.
- steQ 11y agoI know that many learn programming themselves but I'm curious if anyone have learn advanced mathematics this way.
- namelezz 11y agoWhat are the best ways for a non-genius, normal person to learn advanced mathematics?
- shogunmike 11y agoThat depends primarily on your favoured learning style. MOOCs are certainly changing the available resources and lecturers are publishing freely available notes for particular courses on their home pages. 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!
- bmer 11y agoIt is very easy to get most mathematics textbooks online, through slightly unsavoury means. It is far more difficult to figure out which textbooks are worth reading---this is a process that requires trial and error, and browsing through recommendations (math.stackexchange and mathoverflow.net have many good textbook recommendation questions, with many excellent answers). 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.
- kafkaesq 11y ago"Unsavoury", as in?
- forgetsusername 11y ago
- graycat 11y agoAn early place where the OP dropped the ball: > and some good experience manipulating continuous functions and their derivatives. Nope. If a function is differentiable, then it is continuous, but continuity is not sufficient for differentiability. So, we can't talk in general about the derivatives of continuous functions. E.g., each sample path of Brownian motion is almost surely differentiable nowhere. Just f(x) = |x| is continuous but not differentiable at x = 0. Maybe the OP advice is okay in England, but here in the US I would advise people wanting to learn to f'get about the OP and get better advice. For job opportunities in quantitative trading, I tried that on Wall Street here in NY, and got nowhere. I came with a Ph.D. from a world-class US research university with my dissertation research on stochastic optimal control, which should have put my resume near the top of any stack. My favorite prof was a star student of E. Cinlar at Princeton and, thus, about the best there is for mathematical finance. I came with a long, solid background in software, peer-reviewed publications in mathematical statistics, optimization, and artificial intelligence. I had a good background in second order stationary stochastic processes, power spectral estimation, and the fast Fourier transform -- no interest. Got nowhere. That was before I heard about James Simons. One interview was by a guy who recruited for Goldman Sachs, and he didn't have a clue about my background. Another interview was at Morgan Stanley: The interview was in their computer group, but I indicated that I'd like to get into quantitative trading -- they acted like they had never heard of any such thing. I got the impression that only a very tiny fraction of the people on Wall Street had good backgrounds in measure theory and stochastic processes based on measure theory, that my resume never got in front of any such person, and that the other people didn't know measure theory, anything about Brownian motion, power spectra, time series analysis, etc. and were looking to hire people like themselves. Candidate Lesson: Study all the math you want, but don't expect Wall Street to be interested.
- shogunmike 11y agoThank you for pointing out my mistake! Indeed, that was a slip of mine to write about continuous functions, assuming they had derivatives.
- nmrm2 11y ago>> and some good experience manipulating continuous functions and their derivatives. > Nope 1. "good experience manipulating differentiable functions and their derivatives" sounds weird in prose. 2. Some continuous functions are differentiable. Those ones have derivatives you can manipulate. In fact knowing when a function is not differentiable is a pretty useful skill. > The interview was in their computer group, but I indicated that I'd like to get into quantitative trading -- they acted like they had never heard of any such thing. Sounds like you interviewed for position X and talked about wanting position Y, and were rightly rejected as "not a good fit; likely to leave at first opportunity". > only a very tiny fraction of the people on Wall Street had good backgrounds in measure theory and stochastic processes based on measure theory Probably true. Think of this as "calculus for engineers vs. analysis", and imagine how well a civil engineering interview would go if you talked about different types of integrals instead of talking about how to use the basic stuff to build good bridges. Fact is most people in industry are looking for "calculus for engineers" levels of formal understanding. Enough to be useful and make money while avoiding expensive mistakes. > and were looking to hire people like themselves. Also probably true. This is a good assumption not just on Wall St but everywhere.
- ska 11y agoThe main thing to remember is that nobody learns mathematics to any significant depth by reading. The only way to learn it is by doing it. Doing it carefully, and in full detail, not falling into the trap of "and I understand it from there". So the biggest barriers to doing it on your own aren't source material (there is lots of that) it's a good source of correction. There is also the usual problem of self study, in that you don't have a roadmap and can waste time easily. That said though, one advantage if you are diligent is that you probably by necessity learn techniques of checking your work (formally and informally) earlier than typical students, which is a good thing.
- riprock 11y ago> So the biggest barriers to doing it on your own aren't source material (there is lots of that) it's a good source of correction. Does anyone have any recommendations for where I can find a "good source of correction" (private tutor) online or locally for discrete math/algorithms/college-level mathematics? I looked at WyzAnt which gave me no results for my area -- most of the tutors seem to be for SAT/high school prep.
- chatmasta 11y agoThis is probably the reason why there are so many more autodidact programmers than mathematicians. When programming, you know if you made an error as soon as you try to run your code. The program, by definition, must be "correct" in order to execute. The "source of correction" is the error detection built into any programming language runtime or compiler. No such "source of correction" exists for mathematics, and that makes it an inherently more difficult subject to teach yourself, because any errors you make will "fail silently" unless you are capable of detecting them yourself, which by definition you cannot do without experience. This is why a mentor/professor makes learning mathematics so much easier; he/she plays the role of mathematical compiler.
- thegenius2000 11y agoGood point and great analogy, though I think EWD might be turning over in his grave. While the error detection built into compilers and the testing process are, in general, indispensable tools for developing stable software, a program that executes without any evidence of error is a great distance from a program that has been shown to be correct.
- goc 11y agoAny one know any books or resources that concentrate on invariant programming? Translating recursive code into properly tail recursive or iterative code can be pretty difficult.
- kenjackson 11y agoI recall that Concrete Mathematics has a pretty section on solving recurrence relations. That's probably as close as I can get you to this.
- danharaj 11y ago`Pearls of Algorithmic Design` by Bird - Beautiful little book. It's a series of problems that are solved by first writing the naive program and then transforming it rigorously to make it more efficient. `Algebra of Programming` by Bird and De Moor - This a treatment of the theory that is implicit in the methods of the previous book.
- mathattack 11y agoAmazing how thorough the Math curriculum is in the UK! I got a minor at my undergrad just by doing stats, linear algebra and calc through multivariate. I think I was just 2-3 courses shy of a BA in the subject. I'm curious how many people have met self-trained mathematicians in real life. I know there are storied examples, but is it feasible? (Compared to say - self-trained programmers or writers)
- h1srf 11y agoWhere did you go to school? Those were just the prerequisites for my engineering degree.
- duaneb 11y ago> I got a minor at my undergrad just by doing stats, linear algebra and calc through multivariate. I wouldn't call this a good thing; this just seems like a sane basis for any maths work (including Comp Sci).
- mathattack 11y agoThat's my point. It was pretty much what was required for the CS degree. The UK standard is much higher.
- mng2 11y agoI don't think it's just the UK. At Berkeley, for example, the Math minor requires 4 lower division courses and 5 upper div. https://math.berkeley.edu/programs/undergraduate/minoring-mathematics https://math.berkeley.edu/programs/undergraduate/minoring-ma...
- mathattack 11y agoYes - 9 versus 5. Is Berkeley quarterly or semester? Either way, it looks like a much more rigorous program. (I've generally been very impressed with Berkeley Math and CS grads)
- danharaj 11y agoLearn it in a group. Mathematics is a social activity. Meeting even once a week to discuss the material you're trying to learn will vastly improve the learning rate for everyone involved.
- Patient0 11y agoBrilliant suggestion!
- noir_lord 11y agoI've been getting into math recently and found https://www.youtube.com/user/professorleonard57 https://www.youtube.com/user/professorleonard57 Professor Leonard's videos are excellent, clear and he injects just enough humour to keep it from been too dry.
- lumberjack 11y agoThere is no reason to shun university. If you want to learn mathematics but do not have the time for a full time degree do it part time. You are most definitely not actually going to reach the same level of mastery of mathematics as even the most mediocre graduate by studying alone. It requires a huge amount of commitment and staying focused and discipled enough is very hard when you don't have a set goal and good continuous guidance and motivation. It is also very boring and much less productive to study alone. I'd estimate that 70% of my learning comes from the tutorial sessions. Reading the literature is just the starting point. It only counts for 10% if that. Homeworks and exams are also very useful because they make sure you have the prerequisites to move on and start learning something new.
- tanker 11y agoI disagree, at least at the undergraduate level. There is no shortage of problems and solutions to check your knowledge for much of undergraduate mathematics. As to the value of tutorial sessions for you, it is important to remember that people learn differently. Speaking with an expert is valuable, but not required in my opinion. To really learn a subject, you have to go beyond the minimum required to get a good grade on homework and tests. You can absolutely do this in university, but I found that I was rarely willing to do so. Studying on my own, I find it easier to reach a deeper level of understanding.
- totony 11y ago>it is much less productive My annecdotal evidence differs. I miss a lot of university class because I find either the pace or the content lacking, in which case I just read the book and go to exams. It takes me way less time and I usually learn a lot more.
- amai 11y agoA single book is enough to learn mathematics: Riley, Hobson, Bence: Mathematical Methods for Physics and Engineering: A Comprehensive Guide It has a whopping 1300 pages, but it has everything you need. And if that is not enough for you get Cahill: Physical Mathematics This will give you advanced topics like differential forms, path integrals, renormalization group, chaos and string theory.
- diymaker 11y agoThanks for the suggestion. Like you mentioned though getting to the end of 1300 pages will take ages. Hope I can find that sort of time.