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.
Words cannot express how grateful I am. This seems to be the very thing I looked for the long time and missed somehow! Seems not to cover really All the Mathematics I Missed however (nothing on algebra and number theory for example, which is admittedly my weak spot), but I'm already thrilled to start reading.
All major universities have their course catalog and requirements online, e.g. MIT, and with OCW, you can see the textbooks and syllabus for almost every class.
Eh, I was thinking about something else, actually. What you listed are taught in every CS program, aren't they? It isn't what I imagined when I heard "rigorous" at all.
Topology, number theory, abstract algebra (I mean, real one, not CS-course basics), statistics, tensor analysis? Isn't that "undergrad math"?
For things like Set theory/combinatorics/logic basics I'd recommend Rosen's "Discrete Math and Applications"[1]. CS oriented, simple, interesting, broad. Covers all the basic stuff.
Linear algebra — two books, "Linear algebra done Right" and "Linear algebra done Wrong". Second one more math-oriented, the first one — pretty simple, pretty clear, fun to read.
Real analysis ("calculus" you mean?) — I personally learned from different sources and probably the most concise book I read is Fichtengolz's "differential and integral calculus", but I don't know if it's available in english. I guess, almost any book on topic is fine.
Geometry & Probability theory — not sure what to recommend, because books on topic vary in depth dramatically, I would appreciate myself if somebody would outline the borders for what to cover first. Anyway, most of what I read and found useful is in russian, unfortunately. But still, what do you mean by geometry and prob. theory? Differential geometry, Riemannian geometry, erlangen program covered or only basic euclidean/analytic geometry stuff? Same goes for probability. If you care only for very basics — Khan's academy (or any random youtube videos) is fine. Any intro book on statistics covers it as well.
I actually just bought Rosen's 'Discrete Math and Applications' on amazon. My college education was in art, so needless to say, not much mathematical training in my undergrad education. In my current job I'm a web developer and I've been trying to cover a CS knowledge base through self study. Through reading reviews about discrete mathematics books I was pointed towards Epps' 'Discrete Math with Applications' http://amzn.com/0495391328 and Rosen's book. I was told Epps' book doesn't go into as much depth, but doesn't assume any prior knowledge from the student. This is great for someone like me with an art degree and the will to learn discrete math. All this is tangentially related to why I'm posting. All of these books mentioned are really expensive, mostly because they are used in college classes. A trick I've started using is buying previous additions of these books for pennies on the dollar. I got Rosen's fifth edition for $7 as opposed to ~$160 for the seventh. After reading the update pages online, I don't think I really miss all that much doing this either.
Thanks. I have already covered those topics during my CS undergrad. But I'm trying to redo some things at a much higher level. Hence the comparison with SICP, which is far from a regular introductory book. In the same vein, a course with Axler & Rudin is far more advanced than a standard one.
http://ocw.mit.edu/courses/mathematics/ find a course(s) that covers what you're looking for, write a blog and call it Structural Interpretation Of Undergrad Math and pick the book chapters and lectures that go into depth like SICP does (:
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.