[vitaminj]

I certainly agree that logic, set theory, etc are the formal bases of mathematics, but I wouldn't say they are the basics. It'd be like learning the syntactic rules of grammar before learning words and constructing simple sentences by rote. Sometimes it's better to have an appreciation of the goals (which are easier to learn) before embarking on the fundamentals (which are rigorous but abstract).

When I first started learning set theory, I wondered why this wasn't taught first since it was so fundamental. It took me a while to realise that I wouldn't have understood any of it, because you need some measure of number sense and a moderately well-formed abstract reasoning to appreciate this stuff.

Throughout my experiences in learning, I've always found that it is a zig-zag path - learning the superficial or applications, before drilling down to the fundamentals, and then going back to applications with a new sense of appreciation and so on. Going from the bottom up sounds to me like a recipe for losing interest in the subject very quickly.

[asdflkj]

I agree about the zig-zag path, but I don't think it necessarily means you should study calculus before set theory (for example). Only that you shouldn't delve too deep on your fist pass over set theory. Obviously, AOC independence proof would not be suitable for high school students. But it doesn't take much effort to understand what a function is in terms of sets. And don't they teach Venn diagrams to kids in grade school? That's already set theory.

[smopburrito]

along these lines check out: Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by Hubbard and Hubabrd

http://www.amazon.com/Vector-Calculus-Linear-Algebra-Differe...

[vitaminj]

Yeah fair enough