Seeking education
This will not be a long post; I have a simple question to ask: if you wanted to educate yourself to graduate level in mathematics, but didn’t actually want to go to university, what would you do? I would ask for text-book recommendations, but I don’t want to limit your responses (however, bear in mind that the wikipedia articles on, say, cardinality or well-ordering go over my head – they may skim my hairline, but over they go). Also bear in mind that while I personally have A-levels (British qualifications) in both Maths and Further Maths (which is to say, I know some calculus at least), there are probably plenty of people on lesswrong who don’t and who desire the same information – so assume as much ignorance as you feel necessary (it’s a shame, actually, that there isn’t a sequence here on lesswrong for maths). What do you advise (if you think the query ill-defined, I would like to know that as well)?
This isn’t easy. Look for something to steady your motivation. Question your reluctance to go through the formal process (i.e. go to a university), weigh it against the substantially higher risk of “dropping out” of self-directed study.
If you live near a decent university, look for ways to use it without attending it formally. In many places, you can audit the courses freely without anyone caring that you’re not a paying student. Use the library, not just for the books, but to force yourself off internet and to study, if you have a problem with that. Knock on the door of a math professor during their visiting hours and ask them to advise you how to prepare a self-study program (don’t trust any of their recommendations blindly). Or do the same with a PhD student. Don’t feel you’re intruding. Mathematicians as a rule feel that anyone without a college-level background in math is missing out terribly in life. They will nearly always be sympathetic to someone trying to gain that knowledge.
Try different ways of acquiring knowledge; look for the one that clicks with you. Some people are very strong at studying from a textbook. Others learn much better from a lecture, and there’re many excellent university courses available as video online. If you’re studying from a textbook, be sure to try and do the exercises. There are some very few people who can learn without doing exercises; you’re almost certainly not one of them. Most mathematicians, including most brilliant mathematicians, aren’t either.
Treat any textbook recommendation, here or elsewhere, as a weak hint. For whatever reason, even though the contents of undergraduate math textbooks on a given subject are almost completely determined by tradition (and trivial from a professional point of view), different textbooks seem to resonate strongly with different people. Even the most famous texts don’t work for everyone. If you hate the text several people told you is the best, it may be wrong for you; try a different text (w/o compromising on content).
Go to the website of a top university. Study the degree requirements for a math major. They will have a number of required courses; find their course pages, look through the syllabi, note the textbooks the instructors have chosen. Compare with 2-3 other top universities. Use this data to design your own course of study based either on detailed lecture notes/videos of the actual courses, or on the textbooks they are based on, in the latter case make sure to cover everything in the course’s syllabus.
Example: start with http://math.berkeley.edu/programs/undergraduate/major/pure Examine the lower division required courses in detail: http://math.berkeley.edu/courses/choosing/lowerdivcourses For any given course, google its name and “berkeley” to find an even more comprehensive course page for a particular version in a particular year, e.g. http://math.berkeley.edu/~zworski/math54.html It might have the complete list of textbook chapters to study, exercises to do, homework/midterm examples, etc. Make use of those. You now have enough information to structure the study of this particular course for yourself, if you can muster the necessary mental discipline.
You cannot skip (basic) things you don’t like. For example, you can’t skip calculus, and just go on to study other things you like. You won’t have the flexibility of abstract thinking that comes from running the epislon-delta formalism a few hundred times in your head, in different variations, and you’ll run into a mental wall later on in any other subdiscipline of math. Basically all the material in the required courses list in the example above is something that is a) blindlinly obvious to any professional mathematician; b) should be, if not blindingly obvious, thoroughly familiar to you by the time you’re done with your course of study. All of it is real core stuff, something that either lies directly in the foundation of most advanced math material, or has been shown by time to help you develop thought patterns necessary for most advanced math material.
Very important points, I didn’t get them right until less than a year ago, despite having gone through college (applied math/physics) and maintaining technical interests since then.
Why? It’s a long and difficult road you want to travel, and in my albeit limited experience, people without something to protect don’t travel it very long.
You mean why do I want to learn maths? Do you mean to imply that it could be more useful to learn something else (or indeed that it’s a better strategy to learn nothing at all)? Maths is important. I haven’t studied it seriously in a few years, and being part of the lesswrong community I come across things moderately often which elude my understanding. Is that not reason enough? I should say that whether or not I get advice from lesswrong (some of our fellow contributors have already been quite instructive), I will learn this stuff anyway—it just seemed more efficient to ask lesswrong first before setting off alone.
Sure, that’s reason enough, but there are benefits associated with having something to protect. Math is huge, and I don’t see going into it directionless ending well. Further, generic advice probably isn’t going to be more helpful than specific advice.
Related thread.
Well, I’m not going to pretend that my sister’s life depends on learning maths - at the moment it’s not that important to me, and I’m going to have to muddle through on lesser motivations. That is a very useful thread, though (I realise now that I have been thinking of maths as hierarchical—thinking that either you learned it in a particular order, each tier supported by the previous, or you didn’t learn it at all) - thanks very much.
Math is sort of hierarchical in that advanced maths depend on earlier math learning. But the overall map is more a huge confused network, with topics splitting and feeding into each other all over the place. I am going to repeat my regular recommendation, if you want to study maths on your own, do some reading about math. My recommendation, as always, is Philip Davis and Reuben Hersh, The Mathematical Experience. It is the broadest discussion of maths that I know of. I haven’t seen the new edition which has been enlarged and with exercises for studying, but that is mostly intended (from the reviews) for college students’ only exposure to math, and if you intend to continue studying math, I don’t really think it is necessary.
Is there any particular sort of math you’re interested in?
If you want to really learn the stuff, and you’re not outlandishly talented, it’d still take you at least a couple of years of full-time effort, likely more (and correspondingly even more in part-time effort), assuming you don’t trip up and go in a wrong direction (e.g. form shallow understanding of advanced topics instead of building reliable foundation). This is unlikely to be achieved if you aren’t led step-by-step by a college system, don’t enjoy the process, and don’t have something to protect.
MIT OCW has quite a few mathematics courses listed. Many of them rely on textbooks, but one might be able to find the books used or borrow them from libraries if one can’t afford or just doesn’t want to purchase them.
Khan Academy might be useful if one wants to build up a more basic foundation, but it looks like the highest level of their offerings consists of differential equations and linear algebra.
Thank you.
My first inclination is to say Khan Academy. Their stuff tends to be quite high quality, at least compared to the curriculum at a middling American university. Which really isn’t saying much.
Thank you—I know of KA and from what I remember it doesn’t go beyond my level. But then, as I said, there are probably others on lesswrong who are starting at a lower level and want the same information.
Khan goes up to Calc II (If Calc II involves Taylor Series. Basically almost everything before Multivariable Calc). It contains Diff EQ, intro stats and probability, as well as Linear Algebra but he doesn’t have anything on the Junior Undergraduate level. I’m not sure of his plans to expand to it either. From videos I’ve seen, I’m given to understand he wants to do more topics on an intro level rather than delving into the deep dark depths of say, Fourier Transformation.
If you need the basics though, Khan is your man. You’ll learn how entertaining the phrase “switch to magenta” is as well.
As I’m a bit further on in this path, so the details of the beginnings (introductions to proofs and logic) are a bit blurred to me, but I’ll present a few books that I think are good for moving from basics to graduate level topics.
In number theory to move from basics to graduate level topics would be Hatcher’s Topology of Numbers into Silverman and Tate into Koblitz.
For linear algebra I’d recommend Axler’s book, for complex analysis I liked Churchill and Brown in that I could basically read through and do the exercises very quickly as an undergrad. I didn’t have a great time with any of my real analysis texts so I don’t think I can give a good recommendation for that.
Of course your mileage may vary but these are the books that I’ve enjoyed the most, in that they stay grounded in computations, give decent intuition with good geometric pictures, and tend to have approachable exercises compared with other books I’ve read.
Also: learn to program in python, and use SAGE to solve project Euler problems.
There is a great 1st real analysis book that would work from HS level: “S. Abbott (2001). Understanding Analysis. Undergraduate Texts in Mathematics.” (For comparison, Baby Rudin would be way more advanced than that, I’d schedule it even after Axler’s Linear Algebra, which itself should go after a more matrix-y introduction to linear algebra, like Strang.)
Note however that this type of statement is strongly dependent on individual personality. For me, the correct order was definitely Axler first, then matrix-y later. (A position not to be confused, by the way, with “Axler only”, which would indeed be a mistake, even for me.)
Don’t know about matrix algebra books in general, but Strang is mostly elementary and incomplete, there are few proofs and the focus is on simple examples. It builds intuition for basic things like bases, rows/columns in matrix multiplication, subspace, kernel/image of a matrix and its transpose (not even thought of as adjoint, and the complex case is reduced to a syntactic analogy with real one), action of elementary transformations and change of basis. Axler then provides a detailed explanation of what’s really going on (which builds on the intuition formed by Strang) and extends the picture (enough to train intuition about things like the complex case, invariant subspaces, polar decomposition, generalized eigenvectors).
This seems like a natural order to me. How does the other way around work (taboo “personality”)?
I prefer to be told “what’s really going on” before practicing computations; this is both more intrinsically pleasant (I find) and aids in memory. See my post on Bayes’ theorem, where I contrast my abstract approach with the usual one, which starts with concrete examples.
Perhaps an even more extreme example would be multivariable calculus: I was never able to properly remember, let alone understand or apply, the theorems of Gauss, Green, and Stokes until I had learned the formalism of differential forms and the generalized theorem (itself arbitrarily called Stokes’ theorem, although either of the other names could also have been used). Presumably this is avoided in a first course because the idea of a multilinear map varying from point to point is considered too abstract—a completely spurious reason in my case.
There is a widespread false assumption that computational facility is a prerequisite for theoretical understanding, when in reality they are independent skills. An unfortunate consequence of believing this falsehood in the case of someone to whom theoretical understanding comes easily (e.g. me) is that you can get the impression that you don’t need to bother training computational facility (after all, you’re “already” doing the thing for which it is supposedly a prerequisite), and even eventually end up thinking that doing so would be beneath your status.
What would be ideal would be a larger version of Axler that supplemented the theoretical exercises with large numbers of computational ones (Axler itself contains a few, but not enough).
Hmm.
I think learning styles differ here. The thing that helped me understand multivariable calculus was taking it concurrently with electromagnetism. I really liked having a concrete example to have in mind whenever we talked about vector fields—most of the theorems about general vector fields have specific physical consequences, and I find remembering the physics helps me remember the theorem.
One good trick in addition to book study is to make friends with some mathematicians, and then prompt them to talk about mathematical topics. Mathematicians/scientists/programmers/etc usually LOVE to hold forth about technical topics and love to have an audience that will listen.
A tutor of some kind wouldn’t hurt. Now, understand: I live with a bio-chem/chemistry major who loves math and physics and two of my friends are in mathematics. So, I know that I would have an easier time than most finding someone already with knowledge of the subject.
That said, it wouldn’t hurt to look and see if you could find anyone to help. I feel you can learn any subject under the sun on your own with the right motivation and resources, but it would be easier if you had someone to correct you in mathematics because simple mistakes or misunderstandings build. It could be someone online you get to know or someone you meet (not an actual paid tutor, god no. Then you might as well just go to school).
This is how I would do it at least. Find someone to help ensure that my understanding of the material is right. I could research the material on my own, but a mental safety net never hurts.
I disagree. Say you pay a grad student $45 to meet with you for one hour every week. They can make sure you aren’t making any mistakes, point you in a new direction, and generally lead the way for your studies. This would cost a total of $450 over the course of a 10-week quarter—Less than half the tuition for a math class. In return you get personalized attention. If you don’t have any friends who can tutor high-level math, I would say paying for a tutor might at least be worth trying.
Also, as someone who self-retaught UP to calc level, I recommend the “X for Dummies”, “X Demystified” and “Idiots Guide to X” books. They go higher than people think (but not grad school level), and include books on: Diff EQ, Linear Algebra, and Discrete Math. (I know you said that you personally have already studied these, but for recommendations for others, I think they are still useful.)
If you don’t want to pay for tuition, you probably would prefer not to pay for a super-expensive text either. Don’t forget to check with your local library, and if your state has an interlibrary loan. It’s not uncommon that at least ONE library in your state will have the text you need.
Libraries routinely turn down ILL requests for textbooks or just academic works, as I’ve discovered multiple times. (Even academic libraries will do this if they think you’re requesting too many papers, as I discovered while using a university library for Wikipedia & anime research.)
Scattered thoughts:
Use multiple textbooks if you can. Often a definition or theorem doesn’t quite gel until I’ve read multiple presentations of it.
Google is your friend. For example, this blog post is the best explanation of normal subgroups I’ve ever seen. When you combine it with this Math Exchange thread, this web page from John Baez, and Wikipedia, you can actually learn a lot about normal subgroups without even opening a textbook.
This is probably exclusive to how I learn, but I find that going through a text book linearly is, in general, horribly boring. You are typically given a collection of defintions, which you are supposed to memorize and trust that they will eventually become important. Then you prove theorems about the properties of these minimally-motivated definitions, and if you persist long enough you’ll eventually make it to an interesting result and see that the definitions were indeed useful. I can’t learn like that; my eyes glaze over. Instead I skim places like Wikipedia for topics that seem interesting and not too far out of reach, and then I learn only the things that I need to understand the topic. Essentially, I try to learn like a package manager, installing only the depencies I need for the package I’m currently interested in. I mention this only because I spent a lot of time trying and failing to make my way systematically through math textbooks before I found this method, which seems to work for me.
It might be worthwhile to get to the level of something like “mathematics for electrical engineers”, which is reasonably easy and would cover all the math presented or mentioned on this site, including QM. It is unreasonable to shoot for the graduate level in math, unless you actually have the goal of doing a PhD in math, or unless you are a math genius. The difference in effort between the two is at least an order of magnitude (there are many more topics to cover, and each topic is harder than the one before), while the payoff difference is negligible. Well, unless (another unless) your goal is to research AGI/FAI.
Given that, have you thought through the utility of learning all this math vs doing something else equally hard and long, and possibly boring?
I’m somewhere a long this path, so I can share my experiences. Something worth playing with is Coq, I enjoyed doing the most of course here. One benefit is that it uses a consistent notation to represent things and not too many unfamiliar symbols. It introduces things like the Peano axioms and other interesting stuff. It will also nail down commutativity, transitivity and other useful concepts. Not sure how useful it is, if you haven’t done some programming.
For learning important symbols and terminology, I would look at Set theory. Knowing the difference between “member of” symbol and “subset of” symbol of is fairly basic stuff, I just picked it up by osmosis. Predicate calculus also crops up a fair bit, I got taught it in my computing degree. I’ve got Conceptual Mathematics because I found a fair amount of the maths around functional programming languages was expressed in terms on Category Theory. It was good for the first few chapters but my interest waned at some point and I find it hard to get back into,
I’m interested what other people would recommend.
As far as programming goes, pretty much anything that can be done with Game Maker 7, I can do, and I know a bit of Python. Formal logic-wise, I have Greg Restall’s “Introduction to...” (I have a B.A. in Philosophy, on of the modules of which was formal logic). Thank you for the advice.
You should be able to cope with coq, I think. I think just going up to the logic section would be useful, the rest is only really relevant for heavy program proving. Let us know how you get on and what you find useful.
If you are in London I can lend you Conceptual Mathematics, if you want.
I’m not in London, I’m afraid, or even close to it. Thank you anyway. I’ll reply again to this post or send a PM when I’ve gotten through it.
wink wink nudge nudge
Try not to go broke buying textbooks.