Whew..

I'm afraid you'd have to spend a bunch of time digging through math textbooks, starting with algebra-the-theoretical-subject (as opposed to what most of us sat through in 7th grade). Dig around for information on 'ring' and 'group' theory, then just keep reading the same things over and over until it all starts to sink in.

Broadly speaking, a space is a set of entities that support some kind of operation. the type of operation defines the type of space. Officially, a rational space is one that supports an 'algebra': a commutative operation known as 'addition' and a 'multiplication' operation that's transitive across addition. And just for the record, there are about a zillion other kinds of spaces that I didn't mention (like Banach spaces), because the operations in question don't have any immediate connection to programming.

algebra-the-theoretical-subject (as opposed to what most of us sat through in 7th grade)

Important distinction. :)

I am currently reading a book called "Mathematics: The Science of Patterns" by Keith Devlin (amazon) and starting to open my mind up to mathematics beyond the high-school learning-by-rote level.

I imagine it would be interesting to have a closer look at what one can learn about programming by studying algebra. Thanks for the pointers anyhow.