Which Math Books Should I Study Next?

[whyevengothere]

I'm an english major with a vivid interest in mathematics,I've read and enjoyed What Is Mathematics? by Courant and Robbins (does this count as some background?),and I've decided to begin a serious study of mathematics, I've looked up many books and judged them by the reviews on ams ,maa and zbmath,and narrowed it down to the following:

Calculus/Analysis: Mathematical Analysis I,II By Zorich(what are the exact prerequisites for this one?) ,Problems in Mathematical Analysis by Boris Demidovich .

Abstract/linear algebra: A Course in Algebra by E. B. Vinberg,Algebra by Michael Artin.

Topology:O.Ya. Viro, O.A. Ivanov, Kharlamov and Netsvetaev, Elementary Topology: Textbook in Problems.

Methods in physics:A Course in Mathematics for Students of Physics by Bamberg and Sternberg.

Do you have any comments on the choice of books ? any better suggestions? any suggestions for other subjects (diff geometry,complex analysis...)?Do you have any comments on the choice of books ? any better suggestions? any suggestions for other subjects (diff geometry,complex analysis...)?

 

[homeomorphic]

Well, I would ask what your goal is.

If it's mainly for entertainment, lighter books like Visual Complex Analysis or Geometry and the Imagination would give you the best bang for the buck in terms of entertainment, if your taste is anything like mine. I also find graph theory to be pretty amusing and it doesn't take a lot of prerequisite knowledge to get started on it.

If it's to understand more about science and be more generally educated, I don't know that you would want to dig into all that pure math stuff so much, though linear algebra, calculus, differential equations, and prob/stat are good to know. Unless maybe you are interested in knowing about more specific things, such as some topics in physics or cryptography or something like that or just in understanding math as a subject for its own sake.

If you have more philosophical goals, I would suggest studying logic. To some outsiders who don't know what logic is about, it might seem kind of boring (I say that because it seemed that way to me before I knew what it was about), but stuff like Godel's theorem, Turing machines, and computability are kind of mind-blowing (not so much the more recent stuff that people are doing in logic). Godel, Escher, Bach is a popular book on that subject that I've been meaning to read.

 

[whyevengothere]

Well, I would ask what your goal is.

If it's mainly for entertainment, lighter books like Visual Complex Analysis or Geometry and the Imagination would give you the best bang for the buck in terms of entertainment, if your taste is anything like mine. I also find graph theory to be pretty amusing and it doesn't take a lot of prerequisite knowledge to get started on it.

If it's to understand more about science and be more generally educated, I don't know that you would want to dig into all that pure math stuff so much, though linear algebra, calculus, differential equations, and prob/stat are good to know. Unless maybe you are interested in knowing about more specific things, such as some topics in physics or cryptography or something like that or just in understanding math as a subject for its own sake.

If you have more philosophical goals, I would suggest studying logic. To some outsiders who don't know what logic is about, it might seem kind of boring (I say that because it seemed that way to me before I knew what it was about), but stuff like Godel's theorem, Turing machines, and computability are kind of mind-blowing (not so much the more recent stuff that people are doing in logic). Godel, Escher, Bach is a popular book on that subject that I've been meaning to read.

I don't think I have any goal in particular,but I do enjoy math a great deal ,even more so when it take a great deal of effort to understand . I plan to read those two books by Hilbert/Cohn-Vossen and Needham,and I love books laid out in visual intuitive manner with strong connections to physics,any comment on Zorich or Artin's book?

 

[homeomorphic]

any comment on Zorich or Artin's book?

I'm not familiar with those. As far as analysis goes, I would just say A Radical Approach to Real Analysis gives better motivation, along with Lanczos' book, Discourse on Fourier Series, for understanding what analysis is good for. And partial differential equations--Farlow is not a bad place to start for that, and V I Arnold for something a bit more advanced.

Really, if you're just learning it on the side, I think Visual Complex Analysis alone ought to keep you busy for a while. You probably don't want TOO many books.

 

[whyevengothere]

I'm not familiar with those. As far as analysis goes, I would just say A Radical Approach to Real Analysis gives better motivation, along with Lanczos' book, Discourse on Fourier Series, for understanding what analysis is good for. And partial differential equations--Farlow is not a bad place to start for that, and V I Arnold for something a bit more advanced.

Really, if you're just learning it on the side, I think Visual Complex Analysis alone ought to keep you busy for a while. You probably don't want TOO many books.

I've already bought Needham's book and browsed through it ,it's a treat ,and I'm really curious about Zorich's books because of Vladimir Arnold's review of it on the back cover,what do I need to know to read it? can I try doing it equipped only the calculus knowledge from Courant's ''what is mathematics?'' ?

 

[homeomorphic]

If Arnold recommends it, it's probably not too bad. I heard of the "What is Mathematics?" book, but never read it, so I don't know. You could always attempt it. Strictly speaking calculus is the only prerequisite for real analysis. If you get stuck, you could go back and read some naive set theory to get warmed-up with proofs.

 

[verty]

There are videos by Benedict Gross that use Artin's book, they should make it easier to learn from. So Artin is surely a good choice (for Abstract Algebra I mean).

 

[whyevengothere]

There are videos by Benedict Gross that use Artin's book, they should make it easier to learn from. So Artin is surely a good choice (for Abstract Algebra I mean).

Doesn't Artin's book teach abstract and linear algebra concurrently?

 

[bolbteppa]

How well does this essay sync your goals

http://pauli.uni-muenster.de/~munsteg/arnold.html

 

[fisicist]

It should be possible to find a way between Arnold and Bourbaki. As to his PDE book, there is to say that it is intentionally written with minimal generality and maximal intuition in mind. He blames more "modern"/Bourbaki-style/axiomatic books for obscuring fundamental mathematical concepts by striving for needless generality and making PDE just a playground for functional analysis, and their authors for being scholastic, ignoring applications and not being willing to learn about anything than their beloved "algebraic non-sense".
Personally, I feel that he is right in a way. I believe that the correctness and usefulness of a mathematical argument is independent of how abstract it is formulated. Many long proofs reduce to few concepts and ideas that one can replace by pictures. So does Arnold. However, I value his books rather as a good and beneficial complement to more formal ones, rather than an alternative.

 

[verty]

Doesn't Artin's book teach abstract and linear algebra concurrently?

Apparently it does, chapters 2-5.

 

[porcupine137]

You want to try try Apostol for Calc I/II if you you plan to do a really pure mathematics take on Real Analysis, that gets you used to building everything up proof by proof from way below the level of calc.

 

[fisicist]

For linear algebra, I recommend Serge Lang's book.

 

[whyevengothere]

For linear algebra, I recommend Serge Lang's book.

I know some his books ,including this one, but whenever I learn something from them,I forget it quickly afterwards.

 

[whyevengothere]

How well does this essay sync your goals

http://pauli.uni-muenster.de/~munsteg/arnold.html

It entirely does, I really like Arnold ,do you know any similar mathematician?

 

[whyevengothere]

Well, I would ask what your goal is.

If it's mainly for entertainment, lighter books like Visual Complex Analysis or Geometry and the Imagination would give you the best bang for the buck in terms of entertainment, if your taste is anything like mine. I also find graph theory to be pretty amusing and it doesn't take a lot of prerequisite knowledge to get started on it.

If it's to understand more about science and be more generally educated, I don't know that you would want to dig into all that pure math stuff so much, though linear algebra, calculus, differential equations, and prob/stat are good to know. Unless maybe you are interested in knowing about more specific things, such as some topics in physics or cryptography or something like that or just in understanding math as a subject for its own sake.

If you have more philosophical goals, I would suggest studying logic. To some outsiders who don't know what logic is about, it might seem kind of boring (I say that because it seemed that way to me before I knew what it was about), but stuff like Godel's theorem, Turing machines, and computability are kind of mind-blowing (not so much the more recent stuff that people are doing in logic). Godel, Escher, Bach is a popular book on that subject that I've been meaning to read.

By the way,what do I need to know to read hilbert's book?

 

[homeomorphic]

By the way,what do I need to know to read hilbert's book?

Probably just high school math, I think. It's not super-advanced or anything.

 

[HajarB]

I see this is an old thread..I'm curious to know how far you did go since that time.

By the way, I'm currently studying A Course in Algebra by E.B. Vinberg and I must say I'm enjoying it a lot