11–20 of 92 posts
Unfortunately I have to hold my own tongue this time, because books I found useful are mainly in russian and they surely aren't like SICP. And I still lack the whole understanding of the area anyway.
Sheldon Axler's "Linear Algebra Done Right" has my highest recommendation if you want expertise in linear algebra. As a followup, Paolo Aluffi's "Algebra: Chapter Zero" is the best synthesizing text for abstract algebra for a beginning graduate student. The thing that makes it so amazing is the writing style: it introduces and demystifies category theory, and then discusses groups, rings, modules, linear algebra, fie…
I was looking for a real analysis companion, perhaps baby Rudin. I was also wondering whether it'd make sense to proceed directly to real analysis, or to step down a bit and read something like Spivak's Calculus.
I'd also like to hear about recommendations at undergrad level in the fields of logic and set theory, geometry, combinatorics, and probability theory. Those would complete my basic math curriculum.
It's not nearly as easy to work through as SICP is but Principles of Mathematical Analysis by Walter Rudin (sometimes referred to as little Rudin) is a great place to start if you're interested in analysis. It's a hard book, but it's pretty much the standard for undergrad analysis.
Echo the above. Pairing little Rudin with a more user-friendly book like Understanding Analysis by Abbott is a great way to learn the material if you get stuck.
I share exactly the same interest. Can anybody suggest some outline for what topics (more or less) should somebody cover to be on "undergrad-math" level? Unfortunately I have to hold my own tongue this time, because books I found useful are mainly in russian and they surely aren't like SICP. And I still lack the whole understanding of the area anyway.
* Set theory * Linear algebra * Geometry * Real analysis * Combinatorics * Probability theory
Sheldon Axler's "Linear Algebra Done Right" has my highest recommendation if you want expertise in linear algebra. As a followup, Paolo Aluffi's "Algebra: Chapter Zero" is the best synthesizing text for abstract algebra for a beginning graduate student. The thing that makes it so amazing is the writing style: it introduces and demystifies category theory, and then discusses groups, rings, modules, linear algebra, fie…
Thanks. Axler's LADR is indeed fantastic, I have already worked through 1/2 of it. I'm really happy to see my opinion seconded. It's really short and precise. Some people seem to prefer Halmos' Finite Dimensional Vector Spaces, but I found the presentation less didactic. Perhaps it's also more of an upper division text, and I'm not there yet. I was looking for a real analysis companion, perhaps baby Rudin. I was also…
I'd actually like to hear about alternatives to Rudin. My undergrad analysis class used it, but it's lack of diagrams was particularly bothersome. I'd spend an hour digesting a rat's nest of a paragraph only to discover that the underlying concept was simple enough that even a rough sketch ought to be able to get the gist of it across in seconds.
Math is a much, much broader field than the part of Computer Science than SICP covers, so it's really hard to search for "consistent and broad knowledge throughout all areas". Do you really need to learn analytic number theory? category theory? axiomatic set theory? tensor calculus? My suggestion would be to narrow it down to a specific field you're interested in like abstract algebra and ask for suggestions about th…
Apart from that I wanted to expand on basic set theory, geometry, combinatorics and probability theory to have a well-rounded basic education.
My ultimate goal is to be able to digest advanced probability and statistics books.
Sheldon Axler's "Linear Algebra Done Right" has my highest recommendation if you want expertise in linear algebra. As a followup, Paolo Aluffi's "Algebra: Chapter Zero" is the best synthesizing text for abstract algebra for a beginning graduate student. The thing that makes it so amazing is the writing style: it introduces and demystifies category theory, and then discusses groups, rings, modules, linear algebra, fie…
Thanks. Axler's LADR is indeed fantastic, I have already worked through 1/2 of it. I'm really happy to see my opinion seconded. It's really short and precise. Some people seem to prefer Halmos' Finite Dimensional Vector Spaces, but I found the presentation less didactic. Perhaps it's also more of an upper division text, and I'm not there yet. I was looking for a real analysis companion, perhaps baby Rudin. I was also…
Likewise I essentially learned all the probability theory and combinatorics I know from people and scraps, so I can't recommend a synthesizing text.
Undergraduate geometry can be a mess, so you should know what you're looking for. There are three kinds of undergraduate geometry classes: 1. Euclid's Elements (ugh), 2. The hyperbolic version of Euclid's Elements (meh), and 3. The "Erlangen Programme" style, which involves studying geometry via group theory and linear algebra. As you can probably tell, my recommendation is to study the last, because you already know group theory and with the other two you'll spend a lot of time wondering whether you can apply some basic obvious fact to prove some other basic obvious fact. The Erlangen style also allows you to describe projective and hyperbolic geometry via linear algebra (as well as the Euclid way), which is far more useful. See, for example, my post on projective geometry for elliptic curves [1]. I went through all three styles, but the last unfortunately had no textbook.
I'm not a huge fan of logic/set theory, but again the best treatment I can see for basic logic is to view it as algebra. In that vein, Halmos's "Logic as Algebra" was all I needed, and the prose is superb. This book does not contain any real set theory (say, about higher cardinals), but it's nice and short.
[1]: http://jeremykun.com/2014/02/16/elliptic-curves-as-algebraic...
https://mitpress.mit.edu/sites/default/files/titles/content/...