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?

Too many great books out there obviously but 2 easily made up for for self study plus clearly immediately applicable in many fields i have seen out there are the 2 classics;

http://www.amazon.com/Basic-Complex-...8680710&sr=8-1

http://www.amazon.com/Introduction-P...8680856&sr=1-1

You may have read them or have them already, if not its nice to have even if you still have other plans.


Of course ignore the outrageously expensive prices in amazon and find them a lot cheaper in many other places.

Maybe these 2 examples are not what you are looking for (maybe if you gave specific books you have read as examples it would establish a level for you and make such suggestions irrelevant) but they are both good to have read and apply the arsenal developed from them and also often have proofs that help you understand deeper issues. Maybe you are looking at something a lot more abstract/formal but then how does one go from the characterization of applied and necessary evil to pure theory. It would instantly drive someone insane to go and do that if they started from necessary evil positions lol. Hence the much less tough examples above that still will make anyone interested in science a lot stronger in many ways. I used these 2 as a starting example because both have proofs in them but also many real life applications that may further motivate someone and keep them interested long enough to also grasp the formal content a lot easier eventually in other books. They are friendly enough so to speak.

Maybe another two more also interesting/classic and not complicated are;

http://www.amazon.com/Introduction-T...5&sr=1-1-spell


http://www.amazon.com/An-Introductio...8681758&sr=1-1


Lets us know how trivial and already known/read or interesting those look for example to establish some base.

Go also over all classic and cheap books by Dover on Mathematics and Geometry.

eg one of many dozens literally;

http://www.amazon.com/Introduction-D...8682058&sr=1-1



What all these books have in common is that they are easy to read without taking a class in them necessarily but still of course will require your full attention and will benefit everyone until a more advanced more focused/specialized direction is selected.

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")

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).

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.

It will help also to have access to say JSTOR from an electronic magazines database at local university or even register for limited free usage. They have lots of classic math papers past 100-200 years.

http://en.wikipedia.org/wiki/Jstor

http://www.jstor.org/action/showJournals#43693411

Most definitely need to add also books on Algebraic Geometry and also Group Theory.

At the risk of sounding like a criminal (lol) your best bet in any self study that you do not have friends or advisors to suggest you books is to download from usenet about 10000 math texts (lol real full books in pdf) in all fields of interest and then browse them and the ones that you find friendly and rich and readable based on your background you can go out and buy, then delete the downloaded files (same as visiting a huge library). Or its up to you to delete it lol. I am suggesting how one can achieve the same result as having used the advice of 100 people about 20 math topics without doing anything unethical if they then go out and find the books they settle in reasonable cheap prices. On the books you settle you can then go to say amazon and read reviews that often describe well the quality of the book and the audiences it fits best. Basically if you have nobody to help you use your mind and modern technology that makes things possible like nowhere near what the case was for our parents etc.

You can also visit online many math departments of various universities and look at the classes they offer and relevant textbooks or notes if available online. Obviously most departments will agree on the classic top books. Some of the time you may be able to build a class for your self even if really motivated.

As you also noted you cant go wrong getting this for starters anyway;

http://www.amazon.com/Princeton-Comp...8754624&sr=8-1

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.

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.

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.

Griffth's QM book has an appendix with all of the LA that you need to get through the book.

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.

I think you'll enjoy and benefit from studying Abstract Algebra, starting with group theory.

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

+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.

Here's the books I would recommend for self study broken down by topic:

Abstract Algebra:
The best book on this subject in my opinion is Dummit and Foote's "Abstract Algebra." It has a ton of useful exercises and examples and covers a broad range of material. I suppose an undergraduate class in linear algebra would be a prerequisite, but it has a 0th chapter which goes over most of the material you'll need.

http://www.amazon.com/Abstract-Algeb.../dp/0471433349

Real Analysis:
As an undergraduate we used Lay's book "Analysis: with an introduction to proof." I think this would be good for self study, it has some good exercises and hints to exercises in the back of the book, however it does not cover a lot of material. After mastering the material in this book you would of course move on to Rudin's "Principles of Mathematical Analysis."

http://www.amazon.com/Analysis-Intro.../dp/0131481010

http://www.amazon.com/Principles-Mat.../dp/007054235X

Complex Analysis:
Marsden, as previously suggested is a good introduction to complex analysis. After Marsden maybe Alfhors or Rudin.

http://www.amazon.com/Basic-Complex-.../dp/071672877X

http://www.amazon.com/Complex-Analys...dp/0070006571/

http://www.amazon.com/Complex-Analys...dp/0070542341/

Number Theory:
Hardy and Wright is a classic, but not really a good place to start. I used "Elements of the Theory of Numbers" by Dence and Dence in an undergraduate course and I think this would be a great book to start with.

http://www.amazon.com/Elements-Theor...dp/0122091302/

Set Theory:
As an undergrad I used "Naive Set Theory" by Halmos. I recall thinking it was a good book at the time, although its been quite a while.

http://www.amazon.com/Naive-Theory-U...dp/0387900926/

Topology:
Munkres!

http://www.amazon.com/Topology-2nd-J...dp/0131816292/

Manifolds:
After you have a firm grasp of the material covered in Munkres you could move on to more advanced topics. The books by Lee, Tu or Spivak are generally used in first year graduate courses on smooth manifolds. I used Lee in a first year graduate course on manifolds. I can recommend Tu as good for self study, as its exercises are fairly easy and the writing is clear. The exercises in Lee are however much more worthwhile, albeit some are quite challenging. Lee is also brilliantly written, but I think it might be hard to work through on your own. I haven't used or read Spivak but it seems to be quite popular.

http://www.amazon.com/Introduction-M...dp/1441973990/

http://www.amazon.com/Introduction-S...dp/0387954953/

http://www.amazon.com/Comprehensive-.../dp/0914098705

These topics should cover most of the foundational material you would cover as an undergraduate through your first year as a graduate student (had you been a mathematics major). Abstract Algebra, Real Analysis and Topology would be considered part of the core curriculum in most universities so you may want to start there.

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