31–40 of 88 posts
I'm going to second Arun2009's recommendation for How to Solve It by Polya. And then I'm going to recommend Haskell. I recently realized how much my understand of set theory improved and was solidified purely by programming Haskell a lot.
You might like "To Mock a Mockingbird" by Raymond Smullyan. It's a book of logic puzzles with a focus on combinatory logic (a model of computation related to the lambda calculus). Aside from the logic/math/CS nature of the content, the progression of the exercises should help with mathematical-style reasoning in general.
If you have some prior exposure to the SKI combinators, one big aha-moment may come from the book's exploration of several alternative axiom systems that, while preceding the SK system chronologically, are often left out of modern CS-oriented treatments of combinatory logic.
Also, while I think the exercises on symbol manipulation are not very enlightening on their own, the syntactic rewriting point of view is a useful complement if your view of combinatory logic has been mostly semantic. Proving completeness in the syntactic framework becomes a matter of showing that any combinatory rewriting rule can be expressed in terms of your base combinators. This in turn can be done if your combinators allow arbitrary permutations to be expressed in addition to duplication and dropping of symbols (cf. lengthening and shortening rules in GEB). For example, K is a dropping combinator, and rather than viewing S as a substitution operator on the semantic level (as is done when translating lambda calculus to SK calculus), it can be viewed as a hybrid permutation and duplication combinator at the syntactic level. If these dual roles are split into separate combinators, you get alternative axiom systems.
So what if you really suck at algebra, and need to start over? Geometry always made much more sense to me, as I could see it, touch it, it was real!
Then I started applying it into programming step by step which made it possible for me to look at every little calculation.
Then I made a tiny physics simulation: http://www.youtube.com/watch?v=ud8NirjyLAA
I haven't found many other books that explain calculus as well as that one. If you get that and then a book of exercises from Schaum's outlines, you'll be able to get pretty good at it. Also, the book is a really good read for a math book.
Start with these topics. Work your way through them, and make sure you understand everything you have read so far before progressing. Do the exercises, or it will be a waste of time. 0) Algebra, Trigonometry, Calculus Make sure you have a decent grasp over high school level math topics. You might not need to use these topics frequently (though trig comes up a surprising amount), but they are necessary to establish a…
I'm going to second Arun2009's recommendation for How to Solve It by Polya. And then I'm going to recommend Haskell. I recently realized how much my understand of set theory improved and was solidified purely by programming Haskell a lot.
Computer Science: http://aduni.org
Another vote for khanacademy. I did this myself, and worked out a plan to put every piece of information into long term memory. When you're motivated and not limited by century-old university bureaucracy, you can move very quickly, and run circles around people with a "real" degree, most of whom have forgotten everything but the general concepts.
Udacity is also coming up as an excellent resource, but the important thing to realize is that every resource adds a different perspective on the matter that is extremely valuable.
For CS-specific information, aduni.org is a glorious resource. It's from 2001, but the CS basics never change. I'd recommend getting a handle on all the math from khanacademy so that you understand everything they're referencing.