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 06-02-2012, 02:26 PM #1 Carpal \'Tunnel     Join Date: Jun 2006 Location: 39, 46, 56, 59, 191 Posts: 39,730 Self study of mathematics - help/thoughts Hi, I'd like to improve my mathematics skills. Quick background: I did some college level math in high school including some basic proofs (Leistungskurs for the Germans) and more or less did the same stuff in college again (slightly more advanced). I have what is comparable to a masters in a mixture of economics, CS and business (DII Diplom Wirtschaftsinformatiker for the Germans). What I've done so far has been - (Euclidean) Linear Algebra...I'd sum it up as vector stuff, planes and matrix stuff - Analysis...I'd sum it up as curves, functions, differentiation, integration etc. - Statistics...pretty much the big tour but mostly applied stuff i.e. not too deep - Some logic and related stuff (Gödel etc.) as part of AI and science theory For the most part it has just been applied stuff with a more or less deep understanding of the underlying theory. I mostly concidered mathematics a "neccessary evil" I'd like to develop a better "mathematical mindset" and am mostly interested in axiomatic systems and proofs. I've already started relearning logic and would also like to relearn set theory. I'm currently reading through Tarski's "Introduction to Logic" and some German book on set theory that seems pretty good. At the same time I'm reading "How to prove it" which has been pretty fun so far. What I'm lacking is a big, birds eye overview of how different subfields of math interact and so forth, a friend recommended the Princeton Companion to Mathematics for that. Is my approach of starting with logic, set theory and I guess "proof theory"+some birds eye overview material yet to be determined a decent idea or should I take a completely different route?
 06-03-2012, 10:40 AM #3 Carpal \'Tunnel     Join Date: Jun 2006 Location: 39, 46, 56, 59, 191 Posts: 39,730 Re: Self study of mathematics - help/thoughts I want to make the jump from neccessary evil to pure theory. I never really had the time to properly study mathematics and mostly did ad-hoc...know some algorithms, have some basic clues here and there type of stuff. When I thought of it as a neccessary evil it just seemed too complicated to ever grasp on a deep level and thus I just "gave up". I think I'm a bit wiser now :P Always thought proper math is pretty beautifull. Looked at the books, thanks for the reply. The topology and number theory ones look pretty neat. Still looking for a good high level overview though. I guess if I'd have to design a study course for myself with a concrete goal it would be eventually understanding basic quantum mechanics or general relativity on a mathematical level. Note that I know pretty much nothing about physics other than having a super high level overview. I dunno which one would be the easier to pick nor do I have any clue what "path" I'd need to take Edit: I guess for QM I'd need mostly linear algebra so I might follow your suggestion and pick up some Dover book on that (unless there's some other LA "bible") Last edited by clowntable; 06-03-2012 at 10:56 AM.
06-03-2012, 01:38 PM   #4

Join Date: Aug 2010
Posts: 888
Re: Self study of mathematics - help/thoughts

Quote:
 Originally Posted by masque de Z Too many great books out there...
After picking up a book on the millennium problems (that was written for the layperson), I have also had an interest in increasing my math knowledge. I'd like to be at a point where I at least understand the math behind each of these questions, not just the oversimplifications that the book presents. I also want to better understand the math used in modern physics.

My college degree was in math/comp sci, so I have some background: basic abstract algebra and number theory, linear algebra, real analysis, etc. I have had no courses in complex analysis or topology though, and my statistics class left a lot to be desired with respect to probability theory.

Anyway, just hoping this thread produces some more responses (agreements on the complex analysis/probability theory texts? these are pretty expensive).

06-03-2012, 04:25 PM   #6
old hand

Join Date: May 2012
Posts: 1,405
Re: Self study of mathematics - help/thoughts

Quote:
 Originally Posted by pocketzeroes After picking up a book on the millennium problems (that was written for the layperson), I have also had an interest in increasing my math knowledge. I'd like to be at a point where I at least understand the math behind each of these questions, not just the oversimplifications that the book presents. I also want to better understand the math used in modern physics.
That's not a bad goal....but keep in mind that the difficulty in understanding the statement of the problem is radically different depending on which Clay problem you are talking about. P<>NP and Navier Stokes can be well understood by a reasonably bright undergrad in a technical field willing to put in a bit of time. For the Hodge Conjecture, some Field's Medalists will admit to not really having a good understanding of what the conjecture even means.

 06-03-2012, 10:11 PM #7 veteran   Join Date: Mar 2005 Location: USA Posts: 3,003 Re: Self study of mathematics - help/thoughts I find logic and set theory a bit too dry. I think learning that is more useful when you've had a better notion of math first. I would suggest studying abstract algebra. Algebra affects essentially every field of mathematics. You can't do current research in analysis, topology, number theory or combinatorics without having a good grasp on algebra. It also allows you to start thinknig more generally and might open the door to make logic easier to understand. My research is mostly in analysis (I am an analytic number theorist), but I think abstract algebra was a big step in my mathematical formation. Another big step (but this I feel is more personal) was working on competition style problems, which a books like "How to solve it" would deal with.
06-03-2012, 10:30 PM   #8
old hand

Join Date: May 2012
Posts: 1,921
Re: Self study of mathematics - help/thoughts

Quote:
 Originally Posted by masque de Z Yes as i said ignore the prices they make no sense they are at least 3-4 times cheaper usually as paperbacks or in university bookstores or even cheaper as slightly used etc. These prices are probably very high because the books are older, hardcover, maybe harder to find, who knows, certainly i never paid as much when i did years ago.
abebooks.com. When I want a book and I want to heft it around, I go there.

For really old books, the gutenberg project is awesome. Most of the books I and Zeno recommend can be found there for free.

06-04-2012, 01:45 PM   #9
Carpal \'Tunnel

Join Date: Sep 2005
Location: Berkeley
Posts: 13,702
Re: Self study of mathematics - help/thoughts

Quote:
 Originally Posted by clowntable I want to make the jump from neccessary evil to pure theory. I never really had the time to properly study mathematics and mostly did ad-hoc...know some algorithms, have some basic clues here and there type of stuff. When I thought of it as a neccessary evil it just seemed too complicated to ever grasp on a deep level and thus I just "gave up". I think I'm a bit wiser now :P Always thought proper math is pretty beautifull. Looked at the books, thanks for the reply. The topology and number theory ones look pretty neat. Still looking for a good high level overview though. I guess if I'd have to design a study course for myself with a concrete goal it would be eventually understanding basic quantum mechanics or general relativity on a mathematical level. Note that I know pretty much nothing about physics other than having a super high level overview. I dunno which one would be the easier to pick nor do I have any clue what "path" I'd need to take Edit: I guess for QM I'd need mostly linear algebra so I might follow your suggestion and pick up some Dover book on that (unless there's some other LA "bible")
Griffth's QM book has an appendix with all of the LA that you need to get through the book.

 06-06-2012, 08:24 AM #10 Carpal \'Tunnel     Join Date: Jun 2006 Location: 39, 46, 56, 59, 191 Posts: 39,730 Re: Self study of mathematics - help/thoughts Nah the physics would just be an application, it's really the math I'm interested in. I worked through the first half of Tarski's "Intorduction to Logic" now, looking forward to doing the second half today which constructs a basic theory of arithmetic from axioms and deduction. I enjoyed it so much that I ordered Enderton's "Mathematical Introduction To Logic" (for a good price, too). Hopefully after working through both of these I'll have a decent grasp of formal logic. After that I want to move on to set theory because it relates to logic quite a bit but mostly because Cantor was a pretty interesting historical figure. Since they are relevant in my field I'll try to work through Gödel's completeness theorem and incompleteness theorems and hopefully will be able to follow them. I'll also have a look at Turing (and/or Church) regarding the halting problem. I suspect Cohen may be a bit over my head but I guess it can't hurt to take a peek :P Looking at wikipedia logic,set theory, theory of computation and category theory are grouped as "foundations". Technically I should know more about the theory of computation than I currently do so I may read up a bit on the side (hope I'm not confusing it with something else but that would be "computer science stuff" i.e. O(), PvsNP and so forth). I feel like I don't want to dive into foundations too much so I'll probably just leave it at logic+set theory+the broad stuff I know about the rest (skip category theory altogether) I concider that my base curriculum and after I'm done with it I think I'll dive into abstract algebra but I'm still a bit undecided, will have to research what seems like the most interesting area Random gut feeling: I feel like statistics is the area I'd least like to do, followed by geometry.
 06-06-2012, 06:47 PM #11 adept     Join Date: Sep 2009 Location: location ,location. Posts: 1,086 I think you'll enjoy and benefit from studying Abstract Algebra, starting with group theory.
 06-06-2012, 07:23 PM #12 Carpal \'Tunnel     Join Date: Aug 2004 Posts: 21,429 Re: Self study of mathematics - help/thoughts Princeton Companion is a great buy. Lots of short readable articles with good references to further study. It's also a badass coffee table book for impressing guests
 06-07-2012, 05:46 PM #13 Carpal \'Tunnel     Join Date: Feb 2005 Posts: 13,973 Re: Self study of mathematics - help/thoughts +1 on Princeton Companion. I'd recommend that before almost anything else to help get more ideas about what else intrigues you and to start figuring out what else you need to learn. If you like axiomatic systems for themselves, that's cool, but you could learn enough set theory to do anything you're likely to do for the next few years in other ways. Trying to build a completely solid foundation before moving on to the good stuff, if that's how you're viewing it, seems likely to result in fizzling out, because there's an awful lot of stuff down there. (In other words, I agree with Enrique.) Barring that approach, I think you're on the right track by choosing something technical that you'd actually like to understand better. If you're not interested in geometry, though, you might want to pick something other than GR. As you say, linear algebra will get you plenty far in understanding quantum mechanics. But you can make things sufficiently complicated that you start running into more ornate stuff quite rapidly. Stochastic processes and functional analysis are things I'd like to learn more about for QM related reasons, for example.
 06-14-2012, 08:07 AM #15 Carpal \'Tunnel     Join Date: Jun 2006 Location: 39, 46, 56, 59, 191 Posts: 39,730 Re: Self study of mathematics - help/thoughts Bohr8346 I looked over those books and it seems to be an excellent list. I guess I'll be bumping this thread in a year or so because it's time to dive in

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