Tony wrote: > I was wondering what the reason is for having multiple levels of operator > precedence? > > Phrased another way, why is it that we don't just evaluate everything from > left to right?
Your use of "just" suggests that you think that there's already a rule that unparenthesized expressions are evaluated from left to right in the absence of overriding precedence rules. Actually there's no such rule. The expression "x - y + z" is a shorthand for "x + (-y) + z", where (-y) denotes the additive inverse of y. The additions may be done in any order because addition is associative. A mathematician would never write an expression like (x # y # z) if # was not associative, because that would be ambiguous.
The notion of "left associative" and "right associative" infix operators is peculiar to computer science, as far as I know. It shows up mostly in programming language grammars, but also in the lambda calculus, where xyz means (xy)z, which is different from x(yz). If there are examples of this kind of thing outside of computer science, I'd be interested in hearing about them.