18 ms·

How to Learn Advanced Mathematics Without Heading to University – Part 3

480 points by shogunmike 10y ago

bitchy 10y ago
These books will kick your teeth in if you're not prepared. You either get a teacher who'll hold your hand or you need to gear up for fight(develop math maturity and learn all the tricks and tips). To the latter end, you can check out the Book of Proof by Richard Hammack[0] and Discrete Math by Susanna Epp[1]. [0] http://www.people.vcu.edu/~rhammack/BookOfProof/ http://www.people.vcu.edu/~rhammack/BookOfProof/ [1] https://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328/ref=sr_1_1?ie=UTF8&qid=1473084498&sr=8-1&keywords=susanna+epp https://www.amazon.com/Discrete-Mathematics-Applications-Sus...
Hnrobert42 10y ago
Am I missing something? This just appears to be a link to buy some guy's ebooks.
- jnbiche 10y ago
  Did you read the post? He's discussing the typical coursework needed for a math degree. I've not bought his book and am not interested in becoming a quant, yet I've found his site to be a useful source of tips and textbooks for me as I self-study mathematics.
- mathgeek 10y ago
  There are about a half dozen ads (and a fade-in popover) above the actually content. It's a terrible site design from the perspective of someone wanting to read the content on mobile, but the article is there. I also just left without reading it, due to the ads.
  - shogunmike 10y ago
    Thankyou for the necessary feedback! I'm actually in the process of overhauling the design of the site, particularly with regards to mobile, as the current Bootstrap-derived design pushes all sidebar content to the top on mobile/table.
- Chris2048 10y ago
  He lists ebooks on his blog, but this is a blog post. That said, I'm not sure of the value of linking to part 3, unless the idea is it is the latest and already links back to parts 1 & 2.
  - shogunmike 10y ago
    Indeed, this is the latest article and it does link back to parts 1 & 2. Each part of the series is designed to cover the "typical" modules on a UK four-year undergraduate Masters of Mathematics degree. However, it can be challenging in the latter two years to include a broad enough set of modules to cover all interests, so I have had to stick to those "core" modules likely to be found in many degrees, as well as those more specifically related to quant finance and machine learning. However, it is my hope that individuals will be motivated to look at other areas as well, even if they're not directly related to career paths!
biofox 10y ago
I have found the Chicago undergraduate mathematics bibliography useful for directing self-study: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm https://www.ocf.berkeley.edu/~abhishek/chicmath.htm Previously discussed on HN: https://news.ycombinator.com/item?id=9927909 https://news.ycombinator.com/item?id=9927909
- mrkgnao 10y ago
  There's a GitHub version that is better-updated. https://github.com/ystael/chicago-ug-math-bib https://github.com/ystael/chicago-ug-math-bib
- tgb 10y ago
  As a math grad student, I can say that this Chicago list has primarily books that mathematicians know. The one posted here has primarily books that I am unfamiliar with. If one were to follow that one, other people trained in math would have a hard to judging what you've done (it's common to say things like "I've learned algebra at the level of Dummit and Foote" but this only works if people know the book you're referring to). On the other hand, books written for mainstream math majors and graduate students are not necessarily the ones best suited for an autodidact. Perhaps the author of this post has selected those that are more appropriate, but I can't judge. Also, Springer is a great name in math books and you generally don't go too far wrong by sticking with them, but I've never seen their undergraduate series before. Perhaps they're more common in the UK than the US?
  - ryanmonroe 10y ago
    An added benefit to learning from a book that's more popular is that if you hit a wall and have a specific question about the material as it's presented in your book, you're more likely to find your question answered online. You might even be able to find course material that follows the book, whether from an official online course or just because the professor at some university didn't bother to make the course page blocked off from non-students.
    - chestervonwinch 10y ago
lordnacho 10y ago
Seems pretty comprehensive to me. As a career quant trader I'd say it's a matter of doing the advanced stuff so that you understand the simple stuff. Especially in statistics, there are a number of simple principles, but they need to be learned by incorporating them into some complicated lessons. There's also programming. That's a whole can of worms in itself. There's both theory and practice, where I'd say the practice is far more important than it seems. You really have to have bashed your head against a wall (of your own making) to appreciate how to code in a sensible, maintainable way.
- Chris2048 10y ago
  I'm aiming to become a quant dev. I'm already a dev, but trying to catch up wrt the math at the moment :-O I'm finding there are a whole bunch of skills unrealted to most dev concerns. looking at this: http://quantjob.blogspot.com/2011/12/how-to-avoid-quantdeveloper-black-hole.html http://quantjob.blogspot.com/2011/12/how-to-avoid-quantdevel... I think a lot of dev skills are "housekeeping" - VC, commenting, testing, agile, automation, standards etc. The quant dev stuff seems to be a lot more concerned with performance, correctness and accuracy, and the last two in particular are somewhat specialist dev skills I think - I've found code derived from mathematical equations be be a little different to other code.
  - lordnacho 10y ago
    But housekeeping is important! I've seen supposed quants who didn't know how version control worked. It caused productivity to plummet when people just did what they thought quants do. If anything it's knowing the plumbing that makes you productive as a dev, of any kind. You just can't get around understanding how branching works, or having some unit tests. Very little of the work ends up being the bit you think you're there for. I suspect it's the same in many industries. My parents ran a restaurant, and there's a lot of cooking, but there's also a lot of driving to the wholesaler, picking out vegetables, cleaning surfaces before and after a day, doing the plates, accounts, and so forth.
    - Chris2048 10y ago
ziedaniel1 10y ago
To get the key ideas behind many of these topics, you could try reading Evan Chen's Infinitely Large Napkin: http://www.mit.edu/~evanchen/napkin.html http://www.mit.edu/~evanchen/napkin.html
- jacobolus 10y ago
  Concept: undergraduate with no teaching or writing experience who was an International Math Olympiad gold medalist writes a draft book explaining all of undergraduate mathematics (or rather, most of the topics from his coursework in random-ish order) to slightly younger IMO participants. It’s great that he’s making the effort, but I’m not sure this is the most useful resource for a typical autodidact.
ekm2 10y ago
There is another guide online: " http://hbpms.blogspot.com/2008/05/stage-1-elementary-stuff.html?m=1 http://hbpms.blogspot.com/2008/05/stage-1-elementary-stuff.h...
mathgenius 10y ago
This stuff is brutally difficult to learn from books. Sigh. Maybe in the years since I studied this as an undergraduate things have changed with youtube and so-on. But there is nothing quite like talking to a real mathematician. One minute you are asking a question about some little thing you are stuck on, and the next minute the master is levitating and bending spoons! That's when you start to feel the real depth behind the concepts. It doesn't come from books.
- Chris2048 10y ago
  You want an amazing, 21st century math tutorial? I was astounded by this: https://acko.net/blog/how-to-fold-a-julia-fractal/ https://acko.net/blog/how-to-fold-a-julia-fractal/ Amazing intro to complex numbers.
  - Chris2048 10y ago
    also: https://acko.net/files/gltalks/toolsforthought/ https://acko.net/files/gltalks/toolsforthought/
  - iorrus 10y ago
    Thanks that was incredible.
    - Chris2048 10y ago
      I actually emailed the author, asking if they had a tip jar since I was so impressed. They replied "I'd be equally happy if you made a charitable donation this holiday to someone or some organisation that can use it more." :-)
- ryanmonroe 10y ago
  I'd say the biggest benefit of being in the present year when learning math from a book is the availability of online resources like Stack Exchange where you can ask experts about the material you're learning if you get stuck. It's also nice that most books are available in pdf form so you can pick up multiple books on each subject you learn and easily switch back and forth between them.
yodsanklai 10y ago
> How to Learn Advanced Mathematics Without Heading to University I wonder if it's even possible. Learning maths requires much work, time and dedication. Doing so alone must be very difficult. There are several things universities provide that are hard to replicate alone: a degree, which gives you access to a job, motivation, learning environment, and "peace of mind". What I mean by peace of mind is that, when you're a student, your job is to study, that's what you're expected to do and normally your degree will give you access to a job (esp. if your university is reputable). Now suppose someone learns advanced maths on their own. There's a huge opportunity cost. Not only it takes a lot of time, and the few lucrative jobs that make use of maths are in finance. I suspect financial institutions are very conservative and rarely recruit someone without a proper academic background. An other thing when learning things alone, is that your job is twofold. You must be teacher and student at the same time. You need to find the material, impose yourself some pacing, decide when it's ok to move on etc... It may be ok when you want to learn a new technique in a field you already know, but something as broad as "learning advanced mathematics" seems impossible.
- elorant 10y ago
  You could learn math as a hobby, not because you're looking for a job. At least that's what I did. I like them for what they are and the immense satisfaction I get once I realize how a certain theory works. Other than that I don't expect any material reward from that knowledge.
  - sotojuan 10y ago
    This is a good point. Many of us learn a new language or tool as a hobby after work. Why not consider math like that?
  - FabHK 10y ago
    Wonderful book, btw, for maths as a hobby is "Proofs from THE BOOK" [1]. One of Erdős's quirky notions was THE BOOK, in which God collected the most elegant and wonderful mathematical proofs. He said "You don't have to believe in God, but you should believe in THE BOOK." The book above collects some wonderful proofs that could have made it into THE BOOK. [1] https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK
deleted 10y ago
[deleted]
paulpauper 10y ago
On question is: why? The examples in differential geometry can be difficult and time-consuming , unlike simple calculus, and are best done with computer, not by hand. A single tensor, as found in general relativity, may have dozens of components...writing them out would be taxing. My question is, what do want to do with this knowledge. There is value in learning complicated, abstract math to signal intellect and thus become more popular online, and maybe get consulting work. But in terms to practical applications, a lot of it is done by software programmed by large teams (not just one person), although learning the rules is always helpful. If you want to be a professional researcher who makes original findings in pure mathematics, it will presumably require full dedication, and one can't be both a quant trader and pure researcher at the same time (even someone as smart as James Simmons, founder of Renaissance Capital, was forced to choose between one or the other; he chose the former). It seems as of late ,especially since 2013, there is huge demand for learning complicated mathematics, coding, and trading algorithms. It's like the AP-math class of high school, but as of 2013 expanded to include almost everyone, not just a dozen students lol. This recent obsession with math and finance is described in more detail in . People observe, read headlines about high-IQ founders, venture capitalists, and coders making tons of money in Web 2.0 (Uber, Pinterest, Snaphat, Dropbox, etc.); STEM people getting tons of prestige, status, and global notoriety for their finding (Arxiv physics and math papers frequently go viral); and how the economy, especially as of 2008, rewards intellectualism and STEM in terms of higher wages and surging asset prices (like stocks (the S&P 500 has nearly tripled since the 2009 bottom), web 2.0 valuations (Snapchat is worth $15 billion, on its way to $50 billion), and real estate (Palo Also home prices have doubled since 2011)), and, understandably, many people want a piece of the wealth pie. They see that intellect - which includes STEM, finance, and also quantitative finance - is the path to both riches and social status (as embodied by wealthy geniuses like Musk, Thiel, Zuckerberg, Shkreli), which is why there is so much interest in these technical, difficult subjects, unlike decades ago when only a handful of people were interested. But another question is: Does algorithmic trading work? I don't know for sure, but I think a lot money is made in market making (Citadel Capital comes to mind), which tends to full under the umbrella of algorithmic trading - the two are closely related. And the math in involved has much less to do with differential geometry and number theory and more to to do with statistics and linear algebra (such as analyzing correlations between data). This involves a lot of trading and paying constant attention to order books - it's a full time job. I don't think it's as glamorous as many think it is, and I'm not sure if the returns are worth the effort. There are simpler methods, based on mathematics such as the ETF decay, that an also generate very good returns and don't require full-time trading. Here is one http://greyenlightenment.com/post-2008-wealth-creation-guide/ http://greyenlightenment.com/post-2008-wealth-creation-guide...
karma_vaccum123 10y ago
People don't go to university to learn, they go to university to get degrees...or to be more blunt...to buy degrees.
kikishortler 10y ago
On the face of it this looks more like studying than learning. The distinction is not meaningless. I studied French for six years and passed an exam at the end. However despite all this activity I have never been able to converse in French. By a variant of Gell-Mann Amnesia effect, I conclude that I cannot do mathematics either.
jimhefferon 10y ago
> it is essential that we study topics such as Measure Theory and Linear Functional Analysis I love this sentence. I'm not sure it is true, but it is nonetheless a great sentence.
partycoder 10y ago
Unless you work on graphics, physics, signal analysis, sound, trading, data science, computer vision, machine learning, etc... it's hard as a software engineer to be exposed to math past the basics, meaning that you can survive without having to go beyond arithmetic. You might still get some exposure to discrete mathematics once in a while. Statistics is always there to help you, some people avoid it, some others embrace it.
- kkarakk 10y ago
  >data science how do you survive as a software engineer without data science?
hal9000xp 10y ago
As I a software developer with no degree (working since 2009 in 3 countries), I can share my experience of attempts to learn advanced math. In 2010, I was very interested in foundations of mathematics, an extremely abstract math branches: https://en.wikipedia.org/wiki/Foundations_of_mathematics https://en.wikipedia.org/wiki/Foundations_of_mathematics In particular I spent huge amount of time on: https://en.wikipedia.org/wiki/Nicolas_Bourbaki https://en.wikipedia.org/wiki/Nicolas_Bourbaki (Set theory) https://en.wikipedia.org/wiki/Principia_Mathematica https://en.wikipedia.org/wiki/Principia_Mathematica https://en.wikipedia.org/wiki/The_Foundations_of_Arithmetic https://en.wikipedia.org/wiki/The_Foundations_of_Arithmetic http://www.jhtm.nl/tudelft/tw3520/Introduction_to_Mathematical_Logic.pdf http://www.jhtm.nl/tudelft/tw3520/Introduction_to_Mathematic... What attracted me is that these books doesn't require any specific knowledge of classical math. I.e. they are self-contained. It was fun and ... the experience to delve into highly abstract view on entire math. The big problem is that while I read that for more than a year, I had no experience in problem solving and just ignored exercises (thinking that concept is everything). As a result of that, my entire knowledge is completely evaporated and I literally can't solve any of exercises. After that year, I dropped math till recently. Now, I have completely different approach. I learning elementary olympiad style math and most importantly solving problems all the time. Currently, I'm into series of books: https://www.artofproblemsolving.com/store https://www.artofproblemsolving.com/store These books made for math olympiad preparation. While I solving exercises, I feel how solid my knowledge is. So if you want to learn advanced mathematics, learn elementary olympiad-style math first. It will give you solid background to start learning advanced math (not just knowledge background but most importantly problem solving skills).
- adrianm 10y ago
  I don't agree with your suggestion about olympiad math since it often has little relationship to applying advanced mathematics, but there definitely is merit to the idea that you might need to put theory into practice in order to gain insight about these things. I recommend (surprise surprise) programming. Implement fast fourier transform in C and then Common Lisp. Write a finite difference PDE solver. Try solving actual problems to motivate you. Signals analysis can be a fun way to exercise your knowledge. Try analyzing your favorite songs and figuring out what makes them sound the way they do. Maybe implement some audio filters. If that's not your cup of tea, write physics or chemistry simulations instead. Then use OpenGL to visualize them. Then make them interactive. I can go on and on, but I'll just leave two book recommendations for those who might enjoy programming advanced mathematics. Structure and Interpretation of Classical Mechanics : https://mitpress.mit.edu/classical_mech https://mitpress.mit.edu/classical_mech Functional Differential Geometry : https://mitpress.mit.edu/books/functional-differential-geometry https://mitpress.mit.edu/books/functional-differential-geome...
lumberjack 10y ago
This is stupid. The hard part about a Math degree is the number of hours you have to put in. If you cannot go to university full time, go part time. If you cannot go part time, you don't have enough time to actually learn any of these topics on your own. I've done these classes. It's typically 150 hours per class and it's not something you do after coming exhausted home from work either. After those 150 hours you'll get a basic understanding of the topic. You won't be an expert by any means. That will require more exposure, more time. The lectures themselves are not that useful, I find. The lecturers are mostly useful in guiding you along, telling you which aspects of the theory to focus on and weeding through the study material to deliver you the best bits. The problem sets are indispensable. Exams make sure you actually know the basics in depth instead of just knowing about them. My advice: enrol part-time, take one class at a time, catch up on the lectures and do the problem sets and the homework over the weekend.
- bpchaps 10y ago
  Sometimes the issue isn't time - it's money and access to schooling. Just because you've had the access, doesn't mean others do - and there are many, many who don't.
  - throwaway729 10y ago
    > Just because you've had the access, doesn't mean others do... I don't think that the author of the post you're responding to is making any sort of normative claim. > Sometimes the issue isn't time - it's money and access to schooling I think the author is suggesting that if you have that kind of free time, then you either a) must be independently wealthy, or else that b) trading some of that free time for cash you can use to buy credit hours is a net positive because the guidance is worth it (again, assuming your goal is to learn math as opposed to eg obtain a uni degree). I agree that's not necessarily a good assumption globally, but it's probably a decent assumption for almost everyone in the west, both in terms of self-learning capabilities and in terms of available funds.
    - clarry 10y ago
Tycho 10y ago
What do people think of this idea. Let's say you want to casually improve your maths knowledge in your spare time. Let's say you find it frustrating how you generally can't just google concepts as you come across them, because the material is usually presented so obtusely and you need to be able to ask questions and have things explained in different ways. Let's say you live in a university town. Let's say you pay a graduate student to just spend a couple hours per week answering your questions, on whatever concepts you're having difficulty with. Do you think this would work well? Obviously it costs money but I'm guessing the rate wouldn't need to be too high to make it worth their time. They wouldn't need to do any preparation, just have a good grounding in the language of maths.
- ctchocula 10y ago
  That was the idea with this business [1]. [1] https://news.ycombinator.com/item?id=12268362 https://news.ycombinator.com/item?id=12268362
DavidWanjiru 10y ago
"Why are you wanting to learn mathematics?" The author is called Michael Halls-Moore. Why would a person called Michael Halls-Moore write a sentence like this? I'm genuinely curious, coz I'd assume this is a native English speaker.
IvanK_net 10y ago
The hard part of learning anything without heading to university is not learning that subject, but persuading yourself, that you realy do need to learn each specific area of the subject thoroughly, even if you have never heard of it in your previous "career".
saretired 10y ago
What bothers me about this article is the hook: if you can learn this stuff you can get a quant job on Wall St. Realistically, very very few people can (truly) learn this amount of material on their own, and even so, you will be competing with top people with advanced math degrees, so if you're not the second coming of Gelfand, the goal of getting this kind of job is completely unrealistic. On the other hand, if you have the time (and ability) to learn some of this material on your own, for a purpose other than competing for a highly paid job as a mathematician, great.
fantispug 10y ago
Learning advanced mathematics without going to university would take an extreme amount of dedication, focus, and effort, but it's certainly possible. It's much easier with the resources available on the internet, and being able to connect with people through forums and stack exchange. John Baez's recommendations: http://math.ucr.edu/home/baez/books.html http://math.ucr.edu/home/baez/books.html For theoretical physics 't Hooft's recommendations: http://www.staff.science.uu.nl/~gadda001/goodtheorist/ http://www.staff.science.uu.nl/~gadda001/goodtheorist/
- jwdunne 10y ago
  Those are some awesome recommendations. Half way through a bookmarking so paused to give a big thank you. I don't care if I need 2 lifetimes to learn advanced mathematics. I might not even scratch it. It's the journey that counts to me - if I can learn one new tool, one new perspective of looking at problems and the world, I'm a very happy man. The only person who loses out is my poor wife who must listen to my excitement and then has to go lie down for a bit because it's too much to digest.
ronald_raygun 10y ago
My two cents - I got a BS in math and an MS in stats. A ton of this stuff is really hard and takes a lot of time to understand, and it really does help to have a bunch of time to dedicate to it, a professor to guide you, and friends to try problems with. It also really helps to be exposed to a good order to learn stuff (for example I'd suggest tons of functional analysis, then prob, then stats, then finally ML) But once this stuff clicks it becomes very easy to teach yourself. I've been learning stuff like quantum algorithm, network analysis, etc.
echelon 10y ago
Would it be possible to do this for quantum mechanics, chromodynamics, etc. (to the point where I can follow the primary literature)? I have an undergraduate understanding of physical chemistry, but that was the closest exposure I got to the subject. My mathematics background is petty weak, too (only a modest background in PDE, no ODE, and a faint memory of linear algebra). I'd pay handsomely for a personal tutor / teacher.
graycat 10y ago
Somewhere in the OP or its links is a statement that in 1997 or so the world of finance was really hot for quants. Net: What I found was not "hot" but ice cold. In contrast, early in my career around DC, for applied math for US national security and NASA, in one two week period I went on seven interviews and got five offers. In four years, my annual salary increased by a factor of 4 to six times what a new, high end Camaro cost. That was "hot". When I went for my Ph.D. in applied math, I'd read E. O. Thorpe who had, basically an early but basically correct version of the Black-Scholes option pricing model. In the back of his book, he mentioned measure theory. So, I dug into Royden's Real Analysis, and in grad school I got a really good background in measure theory, probability, and stochastic processes from a star student of E. Cinlar, long in just those topics and the mathematics of operations research and mathematical finance at Princeton. In more detail, about 1992 to 2000, after my Ph.D., I tried to get into finance in NYC as a quant. My Ph.D. dissertation research was in stochastic optimal control, with careful attention to measure theory and the relatively obscure topic of measurable selection and with a lot of attention to real world modeling, algorithms, and software. I had a good background in multivariate statistics and time series techniques, an especially good background in advanced linear algebra and numerical linear algebra (e.g., numerically exact matrix inversion using only double precision machine arithmetic and based on number theory and the Chinese remainder theorem), double precision inner product accumulation and iterative improvement, etc. So, I sent nicely formatted resume copies, in total 1000+. I have held US Federal Government security clearances at least as high as Secret; never arrested; never sued; never charged with worse than minor traffic violations; never bankrupt; good credit; physically normal; healthy; never used illegal drugs or used legal drugs illegally; married only once and never divorced; etc. Results: (1) I got an interview at Morgan Stanley, but all they wanted was software development on IBM mainframes (where I had a good background at the time) with no interest in anything mathematical. (2) I got a lunch with some guy who claimed to be recruiting for Goldman Sachs, but, except for the free lunch and what I had to pay for parking in Manhattan, that went nowhere. (3) I had a good background in optimization, published a nice paper in JOTA that solved a problem stated but not solved in the famous paper in mathematical economics by Arrow, Hurwicz, and Uzawa. So, for mathematical finance, I got a reference to Darrell Duffie, Dynamic Asset Pricing Theory, ISBN 0-691-04302-7, Princeton University Press, Princeton, New Jersey, 1992. and dug in: The first chapters were heavily about the Kuhn-Tucker conditions, that is, the topic of my JOTA paper. By the end of the chapter, I'd found counterexamples for every significant statement in the first one or two (IIRC) chapters. I had to conclude that Duffie was not a good reference for anything good! (4) Headhunters: I tried them, especially the ones claiming to be interested in technical work, computing, etc. They were from unresponsive down to insulting. It wasn't clear they had any recruiting requests. (5) In those days, getting names and mailing addresses of hedge funds was not so easy. But I did get some lists and mailed to them. Got next to nothing back. I didn't hear about James Simons until well after year 2000. (6) Right, there was Black-Scholes. Well, of course, that was Fisher Black at Goldman Sachs. So I wrote him and enclosed a copy of my resume. I got a nice letter back from him saying that he saw no roles for applied mathematics or optimization on Wall Street. So, I gave up on trying to be a quant on Wall Street! So that was 1992-2000, 8 years, 1000+ resume copies, and zip, zilch, and zero results. Curious that the OP thinks that 1997 was a "hot" year for applied math on Wall Street. Now I'm an entrepreneur, doing a startup based on some applied math I derived, computing, and the Internet! To heck with Wall Street: If my startup is at all successful, I will make much more money than I could have on Wall Street. And I don't have to live in or near the southern tip of Manhattan and, instead, live 70 miles north of there in the much nicer suburbs! Lesson: Take the OP with several pounds of salt!
naveen99 10y ago
Sympy can help get over the fear of following scary looking equations or just speed you up. http://minireference.com/static/tutorials/sympy_tutorial.pdf http://minireference.com/static/tutorials/sympy_tutorial.pdf

480 points by shogunmike 10y ago

These books will kick your teeth in if you're not prepared. You either get a teacher who'll hold your hand or you need to gear up for fight(develop math maturity and learn all the tricks and tips). To the latter end, you can check out the Book of Proof by Richard Hammack[0] and Discrete Math by Susanna Epp[1]. [0] http://www.people.vcu.edu/~rhammack/BookOfProof/ http://www.people.vcu.edu/~rhammack/BookOfProof/ [1] https://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328/ref=sr_1_1?ie=UTF8&qid=1473084498&sr=8-1&keywords=susanna+epp https://www.amazon.com/Discrete-Mathematics-Applications-Sus...

Hnrobert42 10y ago

Am I missing something? This just appears to be a link to buy some guy's ebooks.

jnbiche 10y ago
Did you read the post? He's discussing the typical coursework needed for a math degree. I've not bought his book and am not interested in becoming a quant, yet I've found his site to be a useful source of tips and textbooks for me as I self-study mathematics.
mathgeek 10y ago
There are about a half dozen ads (and a fade-in popover) above the actually content. It's a terrible site design from the perspective of someone wanting to read the content on mobile, but the article is there. I also just left without reading it, due to the ads.
- shogunmike 10y ago
  Thankyou for the necessary feedback! I'm actually in the process of overhauling the design of the site, particularly with regards to mobile, as the current Bootstrap-derived design pushes all sidebar content to the top on mobile/table.
Chris2048 10y ago
He lists ebooks on his blog, but this is a blog post. That said, I'm not sure of the value of linking to part 3, unless the idea is it is the latest and already links back to parts 1 & 2.
- shogunmike 10y ago
  Indeed, this is the latest article and it does link back to parts 1 & 2. Each part of the series is designed to cover the "typical" modules on a UK four-year undergraduate Masters of Mathematics degree. However, it can be challenging in the latter two years to include a broad enough set of modules to cover all interests, so I have had to stick to those "core" modules likely to be found in many degrees, as well as those more specifically related to quant finance and machine learning. However, it is my hope that individuals will be motivated to look at other areas as well, even if they're not directly related to career paths!

biofox 10y ago

I have found the Chicago undergraduate mathematics bibliography useful for directing self-study: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm https://www.ocf.berkeley.edu/~abhishek/chicmath.htm Previously discussed on HN: https://news.ycombinator.com/item?id=9927909 https://news.ycombinator.com/item?id=9927909

mrkgnao 10y ago
There's a GitHub version that is better-updated. https://github.com/ystael/chicago-ug-math-bib https://github.com/ystael/chicago-ug-math-bib
tgb 10y ago
As a math grad student, I can say that this Chicago list has primarily books that mathematicians know. The one posted here has primarily books that I am unfamiliar with. If one were to follow that one, other people trained in math would have a hard to judging what you've done (it's common to say things like "I've learned algebra at the level of Dummit and Foote" but this only works if people know the book you're referring to). On the other hand, books written for mainstream math majors and graduate students are not necessarily the ones best suited for an autodidact. Perhaps the author of this post has selected those that are more appropriate, but I can't judge. Also, Springer is a great name in math books and you generally don't go too far wrong by sticking with them, but I've never seen their undergraduate series before. Perhaps they're more common in the UK than the US?
- ryanmonroe 10y ago
  An added benefit to learning from a book that's more popular is that if you hit a wall and have a specific question about the material as it's presented in your book, you're more likely to find your question answered online. You might even be able to find course material that follows the book, whether from an official online course or just because the professor at some university didn't bother to make the course page blocked off from non-students.
  - chestervonwinch 10y ago

lordnacho 10y ago

Seems pretty comprehensive to me. As a career quant trader I'd say it's a matter of doing the advanced stuff so that you understand the simple stuff. Especially in statistics, there are a number of simple principles, but they need to be learned by incorporating them into some complicated lessons. There's also programming. That's a whole can of worms in itself. There's both theory and practice, where I'd say the practice is far more important than it seems. You really have to have bashed your head against a wall (of your own making) to appreciate how to code in a sensible, maintainable way.

Chris2048 10y ago
I'm aiming to become a quant dev. I'm already a dev, but trying to catch up wrt the math at the moment :-O I'm finding there are a whole bunch of skills unrealted to most dev concerns. looking at this: http://quantjob.blogspot.com/2011/12/how-to-avoid-quantdeveloper-black-hole.html http://quantjob.blogspot.com/2011/12/how-to-avoid-quantdevel... I think a lot of dev skills are "housekeeping" - VC, commenting, testing, agile, automation, standards etc. The quant dev stuff seems to be a lot more concerned with performance, correctness and accuracy, and the last two in particular are somewhat specialist dev skills I think - I've found code derived from mathematical equations be be a little different to other code.
- lordnacho 10y ago
  But housekeeping is important! I've seen supposed quants who didn't know how version control worked. It caused productivity to plummet when people just did what they thought quants do. If anything it's knowing the plumbing that makes you productive as a dev, of any kind. You just can't get around understanding how branching works, or having some unit tests. Very little of the work ends up being the bit you think you're there for. I suspect it's the same in many industries. My parents ran a restaurant, and there's a lot of cooking, but there's also a lot of driving to the wholesaler, picking out vegetables, cleaning surfaces before and after a day, doing the plates, accounts, and so forth.
  - Chris2048 10y ago

ziedaniel1 10y ago

To get the key ideas behind many of these topics, you could try reading Evan Chen's Infinitely Large Napkin: http://www.mit.edu/~evanchen/napkin.html http://www.mit.edu/~evanchen/napkin.html

jacobolus 10y ago
Concept: undergraduate with no teaching or writing experience who was an International Math Olympiad gold medalist writes a draft book explaining all of undergraduate mathematics (or rather, most of the topics from his coursework in random-ish order) to slightly younger IMO participants. It’s great that he’s making the effort, but I’m not sure this is the most useful resource for a typical autodidact.

ekm2 10y ago

There is another guide online: " http://hbpms.blogspot.com/2008/05/stage-1-elementary-stuff.html?m=1 http://hbpms.blogspot.com/2008/05/stage-1-elementary-stuff.h...

mathgenius 10y ago

This stuff is brutally difficult to learn from books. Sigh. Maybe in the years since I studied this as an undergraduate things have changed with youtube and so-on. But there is nothing quite like talking to a real mathematician. One minute you are asking a question about some little thing you are stuck on, and the next minute the master is levitating and bending spoons! That's when you start to feel the real depth behind the concepts. It doesn't come from books.

Chris2048 10y ago
You want an amazing, 21st century math tutorial? I was astounded by this: https://acko.net/blog/how-to-fold-a-julia-fractal/ https://acko.net/blog/how-to-fold-a-julia-fractal/ Amazing intro to complex numbers.
- Chris2048 10y ago
  also: https://acko.net/files/gltalks/toolsforthought/ https://acko.net/files/gltalks/toolsforthought/
- iorrus 10y ago
  Thanks that was incredible.
  - Chris2048 10y ago
    I actually emailed the author, asking if they had a tip jar since I was so impressed. They replied "I'd be equally happy if you made a charitable donation this holiday to someone or some organisation that can use it more." :-)
ryanmonroe 10y ago
I'd say the biggest benefit of being in the present year when learning math from a book is the availability of online resources like Stack Exchange where you can ask experts about the material you're learning if you get stuck. It's also nice that most books are available in pdf form so you can pick up multiple books on each subject you learn and easily switch back and forth between them.

yodsanklai 10y ago

> How to Learn Advanced Mathematics Without Heading to University I wonder if it's even possible. Learning maths requires much work, time and dedication. Doing so alone must be very difficult. There are several things universities provide that are hard to replicate alone: a degree, which gives you access to a job, motivation, learning environment, and "peace of mind". What I mean by peace of mind is that, when you're a student, your job is to study, that's what you're expected to do and normally your degree will give you access to a job (esp. if your university is reputable). Now suppose someone learns advanced maths on their own. There's a huge opportunity cost. Not only it takes a lot of time, and the few lucrative jobs that make use of maths are in finance. I suspect financial institutions are very conservative and rarely recruit someone without a proper academic background. An other thing when learning things alone, is that your job is twofold. You must be teacher and student at the same time. You need to find the material, impose yourself some pacing, decide when it's ok to move on etc... It may be ok when you want to learn a new technique in a field you already know, but something as broad as "learning advanced mathematics" seems impossible.

elorant 10y ago
You could learn math as a hobby, not because you're looking for a job. At least that's what I did. I like them for what they are and the immense satisfaction I get once I realize how a certain theory works. Other than that I don't expect any material reward from that knowledge.
- sotojuan 10y ago
  This is a good point. Many of us learn a new language or tool as a hobby after work. Why not consider math like that?
- FabHK 10y ago
  Wonderful book, btw, for maths as a hobby is "Proofs from THE BOOK" [1]. One of Erdős's quirky notions was THE BOOK, in which God collected the most elegant and wonderful mathematical proofs. He said "You don't have to believe in God, but you should believe in THE BOOK." The book above collects some wonderful proofs that could have made it into THE BOOK. [1] https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK

deleted 10y ago

[deleted]

paulpauper 10y ago

On question is: why? The examples in differential geometry can be difficult and time-consuming , unlike simple calculus, and are best done with computer, not by hand. A single tensor, as found in general relativity, may have dozens of components...writing them out would be taxing. My question is, what do want to do with this knowledge. There is value in learning complicated, abstract math to signal intellect and thus become more popular online, and maybe get consulting work. But in terms to practical applications, a lot of it is done by software programmed by large teams (not just one person), although learning the rules is always helpful. If you want to be a professional researcher who makes original findings in pure mathematics, it will presumably require full dedication, and one can't be both a quant trader and pure researcher at the same time (even someone as smart as James Simmons, founder of Renaissance Capital, was forced to choose between one or the other; he chose the former). It seems as of late ,especially since 2013, there is huge demand for learning complicated mathematics, coding, and trading algorithms. It's like the AP-math class of high school, but as of 2013 expanded to include almost everyone, not just a dozen students lol. This recent obsession with math and finance is described in more detail in . People observe, read headlines about high-IQ founders, venture capitalists, and coders making tons of money in Web 2.0 (Uber, Pinterest, Snaphat, Dropbox, etc.); STEM people getting tons of prestige, status, and global notoriety for their finding (Arxiv physics and math papers frequently go viral); and how the economy, especially as of 2008, rewards intellectualism and STEM in terms of higher wages and surging asset prices (like stocks (the S&P 500 has nearly tripled since the 2009 bottom), web 2.0 valuations (Snapchat is worth $15 billion, on its way to $50 billion), and real estate (Palo Also home prices have doubled since 2011)), and, understandably, many people want a piece of the wealth pie. They see that intellect - which includes STEM, finance, and also quantitative finance - is the path to both riches and social status (as embodied by wealthy geniuses like Musk, Thiel, Zuckerberg, Shkreli), which is why there is so much interest in these technical, difficult subjects, unlike decades ago when only a handful of people were interested. But another question is: Does algorithmic trading work? I don't know for sure, but I think a lot money is made in market making (Citadel Capital comes to mind), which tends to full under the umbrella of algorithmic trading - the two are closely related. And the math in involved has much less to do with differential geometry and number theory and more to to do with statistics and linear algebra (such as analyzing correlations between data). This involves a lot of trading and paying constant attention to order books - it's a full time job. I don't think it's as glamorous as many think it is, and I'm not sure if the returns are worth the effort. There are simpler methods, based on mathematics such as the ETF decay, that an also generate very good returns and don't require full-time trading. Here is one http://greyenlightenment.com/post-2008-wealth-creation-guide/ http://greyenlightenment.com/post-2008-wealth-creation-guide...

karma_vaccum123 10y ago

People don't go to university to learn, they go to university to get degrees...or to be more blunt...to buy degrees.

kikishortler 10y ago

On the face of it this looks more like studying than learning. The distinction is not meaningless. I studied French for six years and passed an exam at the end. However despite all this activity I have never been able to converse in French. By a variant of Gell-Mann Amnesia effect, I conclude that I cannot do mathematics either.

jimhefferon 10y ago

> it is essential that we study topics such as Measure Theory and Linear Functional Analysis I love this sentence. I'm not sure it is true, but it is nonetheless a great sentence.

partycoder 10y ago

Unless you work on graphics, physics, signal analysis, sound, trading, data science, computer vision, machine learning, etc... it's hard as a software engineer to be exposed to math past the basics, meaning that you can survive without having to go beyond arithmetic. You might still get some exposure to discrete mathematics once in a while. Statistics is always there to help you, some people avoid it, some others embrace it.

kkarakk 10y ago
>data science how do you survive as a software engineer without data science?

hal9000xp 10y ago

As I a software developer with no degree (working since 2009 in 3 countries), I can share my experience of attempts to learn advanced math. In 2010, I was very interested in foundations of mathematics, an extremely abstract math branches: https://en.wikipedia.org/wiki/Foundations_of_mathematics https://en.wikipedia.org/wiki/Foundations_of_mathematics In particular I spent huge amount of time on: https://en.wikipedia.org/wiki/Nicolas_Bourbaki https://en.wikipedia.org/wiki/Nicolas_Bourbaki (Set theory) https://en.wikipedia.org/wiki/Principia_Mathematica https://en.wikipedia.org/wiki/Principia_Mathematica https://en.wikipedia.org/wiki/The_Foundations_of_Arithmetic https://en.wikipedia.org/wiki/The_Foundations_of_Arithmetic http://www.jhtm.nl/tudelft/tw3520/Introduction_to_Mathematical_Logic.pdf http://www.jhtm.nl/tudelft/tw3520/Introduction_to_Mathematic... What attracted me is that these books doesn't require any specific knowledge of classical math. I.e. they are self-contained. It was fun and ... the experience to delve into highly abstract view on entire math. The big problem is that while I read that for more than a year, I had no experience in problem solving and just ignored exercises (thinking that concept is everything). As a result of that, my entire knowledge is completely evaporated and I literally can't solve any of exercises. After that year, I dropped math till recently. Now, I have completely different approach. I learning elementary olympiad style math and most importantly solving problems all the time. Currently, I'm into series of books: https://www.artofproblemsolving.com/store https://www.artofproblemsolving.com/store These books made for math olympiad preparation. While I solving exercises, I feel how solid my knowledge is. So if you want to learn advanced mathematics, learn elementary olympiad-style math first. It will give you solid background to start learning advanced math (not just knowledge background but most importantly problem solving skills).

adrianm 10y ago
I don't agree with your suggestion about olympiad math since it often has little relationship to applying advanced mathematics, but there definitely is merit to the idea that you might need to put theory into practice in order to gain insight about these things. I recommend (surprise surprise) programming. Implement fast fourier transform in C and then Common Lisp. Write a finite difference PDE solver. Try solving actual problems to motivate you. Signals analysis can be a fun way to exercise your knowledge. Try analyzing your favorite songs and figuring out what makes them sound the way they do. Maybe implement some audio filters. If that's not your cup of tea, write physics or chemistry simulations instead. Then use OpenGL to visualize them. Then make them interactive. I can go on and on, but I'll just leave two book recommendations for those who might enjoy programming advanced mathematics. Structure and Interpretation of Classical Mechanics : https://mitpress.mit.edu/classical_mech https://mitpress.mit.edu/classical_mech Functional Differential Geometry : https://mitpress.mit.edu/books/functional-differential-geometry https://mitpress.mit.edu/books/functional-differential-geome...

lumberjack 10y ago

This is stupid. The hard part about a Math degree is the number of hours you have to put in. If you cannot go to university full time, go part time. If you cannot go part time, you don't have enough time to actually learn any of these topics on your own. I've done these classes. It's typically 150 hours per class and it's not something you do after coming exhausted home from work either. After those 150 hours you'll get a basic understanding of the topic. You won't be an expert by any means. That will require more exposure, more time. The lectures themselves are not that useful, I find. The lecturers are mostly useful in guiding you along, telling you which aspects of the theory to focus on and weeding through the study material to deliver you the best bits. The problem sets are indispensable. Exams make sure you actually know the basics in depth instead of just knowing about them. My advice: enrol part-time, take one class at a time, catch up on the lectures and do the problem sets and the homework over the weekend.

bpchaps 10y ago
Sometimes the issue isn't time - it's money and access to schooling. Just because you've had the access, doesn't mean others do - and there are many, many who don't.
- throwaway729 10y ago
  > Just because you've had the access, doesn't mean others do... I don't think that the author of the post you're responding to is making any sort of normative claim. > Sometimes the issue isn't time - it's money and access to schooling I think the author is suggesting that if you have that kind of free time, then you either a) must be independently wealthy, or else that b) trading some of that free time for cash you can use to buy credit hours is a net positive because the guidance is worth it (again, assuming your goal is to learn math as opposed to eg obtain a uni degree). I agree that's not necessarily a good assumption globally, but it's probably a decent assumption for almost everyone in the west, both in terms of self-learning capabilities and in terms of available funds.
  - clarry 10y ago

Tycho 10y ago

What do people think of this idea. Let's say you want to casually improve your maths knowledge in your spare time. Let's say you find it frustrating how you generally can't just google concepts as you come across them, because the material is usually presented so obtusely and you need to be able to ask questions and have things explained in different ways. Let's say you live in a university town. Let's say you pay a graduate student to just spend a couple hours per week answering your questions, on whatever concepts you're having difficulty with. Do you think this would work well? Obviously it costs money but I'm guessing the rate wouldn't need to be too high to make it worth their time. They wouldn't need to do any preparation, just have a good grounding in the language of maths.

ctchocula 10y ago
That was the idea with this business [1]. [1] https://news.ycombinator.com/item?id=12268362 https://news.ycombinator.com/item?id=12268362

DavidWanjiru 10y ago

"Why are you wanting to learn mathematics?" The author is called Michael Halls-Moore. Why would a person called Michael Halls-Moore write a sentence like this? I'm genuinely curious, coz I'd assume this is a native English speaker.

IvanK_net 10y ago

The hard part of learning anything without heading to university is not learning that subject, but persuading yourself, that you realy do need to learn each specific area of the subject thoroughly, even if you have never heard of it in your previous "career".

saretired 10y ago

What bothers me about this article is the hook: if you can learn this stuff you can get a quant job on Wall St. Realistically, very very few people can (truly) learn this amount of material on their own, and even so, you will be competing with top people with advanced math degrees, so if you're not the second coming of Gelfand, the goal of getting this kind of job is completely unrealistic. On the other hand, if you have the time (and ability) to learn some of this material on your own, for a purpose other than competing for a highly paid job as a mathematician, great.

fantispug 10y ago

Learning advanced mathematics without going to university would take an extreme amount of dedication, focus, and effort, but it's certainly possible. It's much easier with the resources available on the internet, and being able to connect with people through forums and stack exchange. John Baez's recommendations: http://math.ucr.edu/home/baez/books.html http://math.ucr.edu/home/baez/books.html For theoretical physics 't Hooft's recommendations: http://www.staff.science.uu.nl/~gadda001/goodtheorist/ http://www.staff.science.uu.nl/~gadda001/goodtheorist/

jwdunne 10y ago
Those are some awesome recommendations. Half way through a bookmarking so paused to give a big thank you. I don't care if I need 2 lifetimes to learn advanced mathematics. I might not even scratch it. It's the journey that counts to me - if I can learn one new tool, one new perspective of looking at problems and the world, I'm a very happy man. The only person who loses out is my poor wife who must listen to my excitement and then has to go lie down for a bit because it's too much to digest.

ronald_raygun 10y ago

My two cents - I got a BS in math and an MS in stats. A ton of this stuff is really hard and takes a lot of time to understand, and it really does help to have a bunch of time to dedicate to it, a professor to guide you, and friends to try problems with. It also really helps to be exposed to a good order to learn stuff (for example I'd suggest tons of functional analysis, then prob, then stats, then finally ML) But once this stuff clicks it becomes very easy to teach yourself. I've been learning stuff like quantum algorithm, network analysis, etc.

echelon 10y ago

Would it be possible to do this for quantum mechanics, chromodynamics, etc. (to the point where I can follow the primary literature)? I have an undergraduate understanding of physical chemistry, but that was the closest exposure I got to the subject. My mathematics background is petty weak, too (only a modest background in PDE, no ODE, and a faint memory of linear algebra). I'd pay handsomely for a personal tutor / teacher.

graycat 10y ago

Somewhere in the OP or its links is a statement that in 1997 or so the world of finance was really hot for quants. Net: What I found was not "hot" but ice cold. In contrast, early in my career around DC, for applied math for US national security and NASA, in one two week period I went on seven interviews and got five offers. In four years, my annual salary increased by a factor of 4 to six times what a new, high end Camaro cost. That was "hot". When I went for my Ph.D. in applied math, I'd read E. O. Thorpe who had, basically an early but basically correct version of the Black-Scholes option pricing model. In the back of his book, he mentioned measure theory. So, I dug into Royden's Real Analysis, and in grad school I got a really good background in measure theory, probability, and stochastic processes from a star student of E. Cinlar, long in just those topics and the mathematics of operations research and mathematical finance at Princeton. In more detail, about 1992 to 2000, after my Ph.D., I tried to get into finance in NYC as a quant. My Ph.D. dissertation research was in stochastic optimal control, with careful attention to measure theory and the relatively obscure topic of measurable selection and with a lot of attention to real world modeling, algorithms, and software. I had a good background in multivariate statistics and time series techniques, an especially good background in advanced linear algebra and numerical linear algebra (e.g., numerically exact matrix inversion using only double precision machine arithmetic and based on number theory and the Chinese remainder theorem), double precision inner product accumulation and iterative improvement, etc. So, I sent nicely formatted resume copies, in total 1000+. I have held US Federal Government security clearances at least as high as Secret; never arrested; never sued; never charged with worse than minor traffic violations; never bankrupt; good credit; physically normal; healthy; never used illegal drugs or used legal drugs illegally; married only once and never divorced; etc. Results: (1) I got an interview at Morgan Stanley, but all they wanted was software development on IBM mainframes (where I had a good background at the time) with no interest in anything mathematical. (2) I got a lunch with some guy who claimed to be recruiting for Goldman Sachs, but, except for the free lunch and what I had to pay for parking in Manhattan, that went nowhere. (3) I had a good background in optimization, published a nice paper in JOTA that solved a problem stated but not solved in the famous paper in mathematical economics by Arrow, Hurwicz, and Uzawa. So, for mathematical finance, I got a reference to Darrell Duffie, Dynamic Asset Pricing Theory, ISBN 0-691-04302-7, Princeton University Press, Princeton, New Jersey, 1992. and dug in: The first chapters were heavily about the Kuhn-Tucker conditions, that is, the topic of my JOTA paper. By the end of the chapter, I'd found counterexamples for every significant statement in the first one or two (IIRC) chapters. I had to conclude that Duffie was not a good reference for anything good! (4) Headhunters: I tried them, especially the ones claiming to be interested in technical work, computing, etc. They were from unresponsive down to insulting. It wasn't clear they had any recruiting requests. (5) In those days, getting names and mailing addresses of hedge funds was not so easy. But I did get some lists and mailed to them. Got next to nothing back. I didn't hear about James Simons until well after year 2000. (6) Right, there was Black-Scholes. Well, of course, that was Fisher Black at Goldman Sachs. So I wrote him and enclosed a copy of my resume. I got a nice letter back from him saying that he saw no roles for applied mathematics or optimization on Wall Street. So, I gave up on trying to be a quant on Wall Street! So that was 1992-2000, 8 years, 1000+ resume copies, and zip, zilch, and zero results. Curious that the OP thinks that 1997 was a "hot" year for applied math on Wall Street. Now I'm an entrepreneur, doing a startup based on some applied math I derived, computing, and the Internet! To heck with Wall Street: If my startup is at all successful, I will make much more money than I could have on Wall Street. And I don't have to live in or near the southern tip of Manhattan and, instead, live 70 miles north of there in the much nicer suburbs! Lesson: Take the OP with several pounds of salt!

naveen99 10y ago

Sympy can help get over the fear of following scary looking equations or just speed you up. http://minireference.com/static/tutorials/sympy_tutorial.pdf http://minireference.com/static/tutorials/sympy_tutorial.pdf