Any Mathematics Book Recommendations?

[someperson05]

(Tried moving this thread from the general academic guidance thread. If someone has the power and time, can you delete the thread in the academic guidance subforum under my name?)

Hello,

As the title would suggest, I was wondering if someone could suggest a book for me to start working on over Winter Break? (I don't think I will finish over winter break of course.)

I was thinking about getting started on Little Rudin, but I keep reading conflicting opinions on Little Rudin. Most troubling one is that Little Rudin does not build any sort of conceptual understanding or intuitiveness.

My current "formal" background is the usual calculus sequence, differential equations, and "discrete mathematics." Beyond that, I've done a crap load of stuff on my own, which has mostly been a jumbled mess. I think I am ready to get serious, and I really want to dig my teeth into something challenging and substantial. (I've dabbled in linear algebra, group theory, number theory and other random stuff that escapes me. I've also bored myself to death with countless books on how to prove stuff.)

I am looking for a book with proofs, I really want to work on my proving abilities, but I also want a book that will help develop my mathematical intuition. I like having the picture in my mind, and asking what a theorem means and what a proof teaches me about the problem and theorem. I am willing to work as hard as possible, and would prefer a challenge.

I don't care about the field really. But from my limited experience, I like the ideas of linear algebra, and functional analysis sounds cool though I can't honestly say I get what it is about. Vector spaces appeal to me, and we talked about in one of my classes the concept of function space and that sounds positively fantastic. Topology also seems kind of cool, but most books seem to require analysis. I suck at Combinatorics and combinatorial proofs, but this past Putnam exam the two problems I submitted solutions for, one of which I am reasonably confident in, I used combinatorial proofs and enjoyed it. I think in the very very far future I want to do research on L-functions and Langland's program, but this is idle speculation without much justification.

Next semester I am signed up for our "Introduction to Proof" class, reasonably redundant with Discrete Mathematics. Also Linear Algebra and History of Mathematics. I am also planning on auditing combinatorics since I wouldn't be able to take it again until my senior year. (I'm a sophomore.)

Brief Summary:

Rigorous, proof-based.
Something "juicy". (If possible at my level.)
A good challenge! (I kind of want it to make me lose sleep over the problems.)
Appeals to intuition, meaning and understanding.
Really some beautiful mathematics. (Well all mathematics is beautiful, but honestly how to prove books have winded me.)

Thanks for your help in advance! I greatly appreciate any suggestions.

 

[davesface]

While this might not be exactly what you're looking for, I love reading books about the history of math. Check your library and they'll probably have some books on the history of important mathematical concepts (I know I've seen books on everything from the development of the current number system to the history of the number 0). Often times these will contain a very elementary introduction to the concept of proofs and other important mathematical concepts along the way.

Just a thought.

 

[Fredrik]

(Tried moving this thread from the general academic guidance thread. If someone has the power and time, can you delete the thread in the academic guidance subforum under my name?)

The proper way to make such requests is to use the report button.

Most troubling one is that Little Rudin does not build any sort of conceptual understanding or intuitiveness.

That's true, but I don't think I've seen any book that does that.

(I've dabbled in linear algebra, group theory, number theory and other random stuff that escapes me. I've also bored myself to death with countless books on how to prove stuff.)

If you have only dabbled in linear algebra, maybe you should choose to study that instead?

functional analysis sounds cool though I can't honestly say I get what it is about.

It's the mathematics of linear operators between vector spaces. So linear algebra is functional analysis on finite-dimensional vector spaces. However, everything that "feels" like analysis (limits and stuff) become irrelevant when the vector spaces are finite-dimensional. So functional analysis is very different from linear algebra. I'm trying to learn some functional analysis myself, and it's really hard. The books on the subject assume that you know topology perfectly. I don't, so I have to spend most of my time working through proofs in topology. There's no way you can study functional analysis without studying topology first, and you should study Rudin before you study a topology book. (That may not be necessary, but it's the way almost everyone does it).

Unfortunately I don't have any recommendations for you, except Rudin, and Axler for linear algebra. (In another thread recently, some people preferred Hoffman & Kunze over Axler).

 

[someperson05]

davesface:
Well it sounds like the History of Mathematics class is going to use a book that does exactly what you mentioned. It will contain proofs and problems. I like the History of Mathematics, but I don't think enough to revisit it twice. Thanks for the suggestion.

Fredrik:
Thanks for the response. I reported the other thread per your suggestion. Didn't know that was the protocol.
Yea, I suppose conceptual understanding is probably something that really has to be developed from within my own framework.
Thanks for the description of functional analysis, and a plan of study to get there. That is greatly appreciated.
I wanted to grab the Axler book a while back, but from my understanding it is for a second course in Linear Algebra. Or can I jump into it without any extensive formal background?

 

[qspeechc]

I don't know why everyone keeps saying that Axler is for a second course. Axler would make an ideal first course in linear algebra. Axler says in his preface that usually the material in his book is left for a second course, where in the first course such nonsense as inverting matrices and Gauss reduction are studied. Such first courses are, in my opinion, a waste of time, for the mathematics student at least. Those things can be incorporated in a course from Axler if needed, and with a background from Axler they will be simple(r) to learn and understand. By all means get Axler as a first book.

As an alternative to baby Rudin there is the book by Pugh: Real Mathematical Analysis, which is far superior to Rudin, in my humble esteem. Pugh does precisely what you speak of, he builds intuition, among other excellent things.

 

[someperson05]

qspeechc:
Pugh sounds excellent. I looked at some of the reviews on Amazon, and I believe it is in the library here so I will have to take a look at it. Thanks for the suggestion.
Now I am torn between Analysis or Linear Algebra. I am thinking I will start learning some analysis since I am taking linear algebra next semester, and it would be nice to do Axler along with that course. (I have a heavy suspicion the course I am taking will be pretty applied, so doing Axler during the course may keep me sane.)

 

[Fredrik]

You can definitely use Axler even if you have never heard of a vector space before. Some of those "first" courses on linear algebra spend a lot of time on topics that aren't really relevant to the things the theory is really about, for example, how to find the distance between a point and a line in ℝ³. Axler just tells you what you need to know about vector spaces, linear operators, matrices and so on.

 

[jasonRF]

For someone interested in learning math for fun, I really like "vector calculus, linear algebra, and differential forms" by Hubbard and Hubbard. I'm familiar with the 2nd edition which is really interesting. Ties together the topics in the title (I still need to learn forms I must confess) as well as other topics and is really interesting. Doesn't "tie you down" to a single subject. Has proofs. Helps you understand how math works.

Free books you may want to look at include the linear algebra book by Heffron, and "linear algebra done wrong". I like Axler a lot, but I know I could not have handled it my first time through the subject - folks smarter than me will disagree.

Good luck,

jason

 

[someperson05]

Ah thank you very much for your suggestions jasonRF.

The first book sounds pretty interesting. I will have to see if I can get my hands on it.

I will check out the linear algebra texts you recommended as well.

Many thanks!

 

[deluks917]

Michael Artin's Algebra is a hard book to top. If and only if you like to learn by doing problems then I suggest Spivaks Calculus on Manifolds. You actually probably could finish spivak or Rudn over winter break if you worked about 8 hours everyday except Christmas (I'll be doing something similar).

Another suggestion since you seem smart is to just skip over "Undergraduate Analysis." Serge Langs real and Functional Analysis is perfectly readable with just calculus and linear algebra (the book is not strongly recommended but its the only grad real analysis book I've seriously looked at, however it has a reputation of being stuff so I'd imagine something like Folland or Stein would be both more accessible and more of what you are looking for in terms of explanation). Also look at Stein's book on Complex Analysis from the Princeton Lectures, I've never read it but its got a lot of very beautiful mathematics.

 

[mathwonk]

Have you considered a rigorous calculus book such as the ones by Spivak or Apostol? That way you relearn calculus correctly and at the same time learn proofs.

 

[discrete*]

Have you considered a rigorous calculus book such as the ones by Spivak or Apostol? That way you relearn calculus correctly and at the same time learn proofs.

This seems pretty sensible to me. I did just this last year and it was well worth it. Relearning the calculus via proof-based mathematics is challenging, rewarding and extremely useful.

In the event that you want to learn a different area of mathematics, I can make a few recommendations as I do quite a bit of self-study. Here's a short list:

1) Vanden-Eynden's Elementary Number Theory. This book is excellent, and was my first foray into the theory of numbers. I recommend it to anyone who's interested.

2) Suppes's A First Course in Mathematical Logic. Learning symbolic logic was something that I did out of sheer interest, but it turned out to be one of the most beneficial things that I've ever done. Learning proper logic really helps when doing proofs and this book provides an excellent foundation in logic. I plan on studying the subject further when I have the time.

3) Coddington's Ordinary Differential Equations. I'm not sure how rigorously DEs were taught to you, but this book helped me a lot in understanding the theory and proofs behind ODEs. If you're think you'd like to see ODEs done from a different perspective, give this book a try.

As I mentioned, I plan on studying logic and related fields a bit more deeply and have purchased Stoll's Set Theory and Logic. I plan on working through it over winter break. If you're interested, I'll let you know how it is.

 

[FutureScience]

Have you considered a rigorous calculus book such as the ones by Spivak or Apostol? That way you relearn calculus correctly and at the same time learn proofs.

I've no personal preference, but would you recommend Spivak or Apostol over a more recent book like Thomas or Stewart?
If so, why? Do you think that calculus is better explained on those "older" books?