Can some kind /sci/entists here recommend the GOAT books for

>>9144689
Algebra by Gelfand to re-do elementary algebra. It's in the /sci/ wiki. There's also Thomas' Calculus w/Analytic Geometry (3rd edition). It's an old 1960s book that is absolutely amazing, the 4th edition is very good too if you can find it.

There's also "What is Mathematics" by Courant and is pretty much GOAT for covering everything elementary in Topology, Number Theory, Algebra, Calculus ect. You can get an excellent and well motivated introduction to reasonably rigorous calculus using limits in this book in less than a 100 pages, up to the point of understanding basic differentiation and integration, the exponential function, power series etc.

A mathematician I work with is adamant that anybody interested in math should not start with Calculus and instead should do Halmos' Naive Set Theory which is a totally rigorous development of ZF set theory with discussion on the meaning and equivalent statements of AC. They should then try Introduction to Calculus and Analysis I by Courant so it will finally make sense to them.

>>9144862
> Halmos' Naive Set Theory which is a totally rigorous
Wrong. Halmo's is a great starter book, and one of the first I read when beginning my undergrad, but it is not rigorous since he applies formal logic in a hand wavy way (specially problematic when you think of what propositions are acceptable within the specification axiom etc)

Analysis: Baby Rudin
Algebra: Herstein or Lang's undergraduate text

I don't care about geometry at all so can't help you there. After reading these three books you will be done with all undergrad material in these areas.

>>9144689
some advice for you and undergrads in general: there's no one textbook that will teach you everything you need to know for the "next level", and there's no point starting at the bottom. this is what immature students often try to do. you will never be able to learn everything, so don't try.

instead, if you know you want to work in math, find a paper you want to understand (preferably with guidance of a professor) and work towards learning what you need in order for you to understand it. start at the top and work down

>>9144862
So the order is?
>Halmos' Naive Set Theory
>What is Mathematics by Courant
>Algebra by Gelfand
>Thomas' Calculus w/Analytic Geometry
>Calculus and Analysis I by Courant

>>9144889
Thanks but Rudin, Herstein and Lang all have multiple books on these topics. Which ones and in what order?

Just to put things in perspective I'm currently doing Basic Mathematics by Serge Lang.

>>9145575
Well, you asked about undergrad texts, afaik the only author which has more than one aimed at undergrads is Herstein (get the Topics in Algebra, he has another simplified book if I'm not mistaken).

Explicitly:
Rudin's PoMA
Lang's Undergraduate Algebra
Herstein's Topics in Algebra

>>9145113
IMO this is a good approach after skimming a few books on at least abstract algebra/analysis (so after about half of your undergrad).

For instance, my motivation to understand homology in algebraic topology was Jordan's and Brouwer's theorems, I wouldn't even know about them/why they are impressive results if I hadn't done a bit of the legwork in point-set topology.

>>9148501
Thanks. In that order? As I said in >>9145579 I'm currently almost done with Basic Mathematics and I'm planning to start Thomas' Calculus w/Analytic Geometry that >>9144862
recommended after that, so can I start these books after I'm finished with it? Also, can you recommend something for Calculus too? I'm undecided between Spivak, Apostol ad Courant right now. After I'm done with these 5 books will my knowledge be on par with an undergrad's in these 4 subjects?

>>9148564
Spivak is good, but there's a text titled "Honors Calculus" written by a professor at University of Georgia. I haven't worked out of it that much, but it looks like it might be better than spivak. Honestly just use those two texts as your guide to Calculus - use Apostol's text for multivariate calculus.

>>9148564
>>9148642
Not me btw.

This really depends on your goals. If you want a undergrad in pure math you really don't need any appreciable amount of analytical geometry except if you want to study geometry itself.

Calculus is only as useful as it motivates real analysis/lets you practice integration/differentiation so if you had some calculus course already (even if non rigorous) I would try to jump directly to Baby Rudin. If it is a bit too much for you get Apostols Volume 1, IMO its style is much closer to Rudin than Spivak and Courant, Rudin will be quite "normal" after Apostol though you will stay days at an exercise page. Multivariable calculus (Apostol Vol 2) is a waste of time at this point, the proofs require way to much work with the tools you have available, look at analysis on R^n after Baby Rudin or even latter.

As an anon said before, you shouldn't feel the need to finish all these books nor read them in sequence, do not make the mistake of pausing your study of analysis to focus only on algebra or the other way (at least in the beginning when you have so many areas to learn).

Most people agree that Baby Rudin goes downhill after chapter 7 (Rudin himself only puts ch 1-7 in Papa Rudin's prerequisites) and IMO you should read groups in Lang's and then proceed to rings in Herstein's.

Don't forget Halmo's Naive Set Theory, it will be very handy for cardinality proofs and using Zorn's lemma and equivalents. I read it in my commute so it is an easy book to just keep with you and read a bit at a time.

Obs: nothing above applies to other undergrads as I have no idea what physics/applied fags use

Bump for more suggestions.