So I decided to try to learn more rigorous mathematics, and read Apostol's Calculus and Higher Algebra by Barnard and Child.
Problem is that Higher Algebra is so fucking dry. It reads like the Elements. It's just 600 pages of definitions, propositions, and proofs. Apostol's book was way, way, easier to understand even though the subject is arguably more complex. Are you really supposed to and be expected to learn a subject like that? It's hard to explain, but there's no motivation for the propositions I guess. No stuff "in between" like in Apostol.
It might be that I'm just not used to rigour, but Apostol's book is supposed to be rigorous, but even that has motivations and some intuition behind its formality.
Do rigorous mathematics textbooks really read like that? Just definitions, propositions, and proofs, and nothing else?
>>7737327
Read Hall and Knight or G. Chrystal
>>7737327
A rigorous book usually follows the Definition, Proposition, Theorem, Discussion structure, yeah.
Also what the fuck is this book you're reading? Why don't you use something more modern for elementary algebra like Khan academy or something? And elements? What the fuck are you doing bro? Please, take a look at a real mathematics curriculum and learn following the syllabi if you're really interested in doing math.
If you already did a chunk of apostol's calculus you're ready for linear algebra, I like Hoffman & Kunze but Axler's good too. If you want rigor you can do analysis, lectures from youtube searching Real Analysis are pretty nice.
>>7737344
I guess it lacks the discussion part then.
>Why don't you use something more modern for elementary algebra like Khan academy or something?
I was told rigour was important. Khan Academy hardly has a rigorous approach to algebra. I already know the subject, but not rigorously.
The elements I just read as a historic thing, not to learn Geometry.
The book itself might be the issue, but I'm having a bit of issue seeing what "counts" as rigour and formality and what does not.
Read Homotopy Type Theory.
>>7737344
What syllabus would you recommend then?
>>7737327
get used to it op all of algebra is that dry
>>7737920
Any! See http://math.mit.edu/academics/undergrad/major/index.php for instance.
>>7737433
You don't need rigor for elementary algebra. It's just not going to work, which is why that book you're citing is so weird. If you want rigor in algebra then start learning abstract algebra already, I like "A First course in Abstract Algebra" by Rotman
>>7738165
That makes sense, considering I can handle rigour pretty well in more complex subjects, but I just break when it's this elementary. I feel like the elementary things need to be intuitive first, mostly because they came about that way, and because there needs to be some motivation for them, instead of seeming like an arbitrary system of rules and constraints, does that make sense?
>>7738165
Also, I'm looking through their site, and while it does list subjects and classes, the textbook they use isn't listed, which would be helpful.
Second, depending on the college, the recommendations aren't always the best, e.g. very many choose Stewart over Spivak or Apostol, which would be bad for someone trying to learn mathematics rigorously.
>>7737327
Try Algebra, Chapter 0 by Aluffi instead.
I've never heard of the book in your OP, it must be shit.
>>7738188
http://ocw.mit.edu/courses/mathematics/
Anyway, you want to take Linear Algebra and Real Analysis 1 first in my opinion (Linear is usually taken before abstract).
For Linear Algebra I love Huffman & Kunze. You'll find non-rigorous linear algebra classes for engineers EVERYWHERE and books for them as well. So stay clear, learn from the book. No need to do all of it, just after determinants should do.
For Real Analysis, Rudin's principles is a standard, renowned book. But it's terrible for self study. So you've got two options here. Find an easier book and learn from it, or learn from https://www.youtube.com/watch?v=sqEyWLGvvdw&list=PL0E754696F72137EC + Rudin. These lectures are pretty great.
After that, there's plenty of stuff you can take. I already mentioned Abstract Algebra (from Rotman's A First Course) and there's also Measure theory & Probability theory (K.Lai Chung / Athreya Lahiri are good), Numerical Analysis (from Burden), Operations Research if you like (look for linear optimization books) or a second course in Analysis (functional, complex, real 2) or Algebra (Galois Theory).
You don't need to read ALL of each book, the preface of them usually includes tips on how to use them for equivalent one-semester classes, so the books usually contain extra material that you can optionally read (you can get a lot of extra stuff from all of the books I listed for instance).
Oh, before I forget, you probably want to read a book on introduction to proofs before you start learning. "How to prove it, a Structured approach" is a great, accessible book. All the exercises on the books should be done, and you should write them formally.
Can someone put this post in the sticky? People come here asking how to learn math really frequently and the sticky is kind of a mess, especially the "First year undergrad" that has you read Baby and Papa rudin on your own, what a fucking joke.
>>7737344
Please stop posting here if you don't know what Euclid's Elements is.
>>7738165
You seriously need to stop posting here.
>>7738188
MIT uses Artin's Abstract Algebra for Algebra I and II
>>7738172
Yeah me too, its so exhausting to think formally on low level stuff
