Set theory - gotta love it

One of my Uni friends, Ben, was much better at this sort of thing than I was, he could talk very enthusiastically about why people introduced set theory at all (it was something like a rigerous approach to mathematics a few decades ago, when previously mathematicians had been more like physicists, "it works, it's good enough).

Anyways, set theory is way of representing maths, but not in the most intuative ways. They don't define how to do a function, such as addition, they define it's relationship. Why I like it is that it's logically sound, but a very counter intuative way of looking at maths. For example, addition

Addition: Addition is a function from a pair of numbers to a number. So it's a set of ordered pairs, whose first element is an ordered pair. So it's a set of pairs of the form {((x,y),z) where:

1. if x=0, then z = y
2. if y=0, then z=x
3. otherwise, let x' = successor(x) and let y' = predecessor(y) then z = Add(x',y').

This statement, by the relationship between z, x' & y' defines the function that gives z from x & y, which can be done because all numbers have known predecessors and successors (numbers were defined earlier, in similar ways). So it fully defines addition - if you think about it, which is probably worth some thought.

Of course, if this was just a counter intuative way of solving maths, then it'd be a bit silly (well, in my opinion), but the way of defining these fuctions and sets is really powerful - if applied to physics, things like rotations in Newtonian space and quantum mechanics all relate to each really impressivly, if I can't quite remember how.

Learning it for Undergrad was bizarre, fun and challenging. Utterly impratical, but it was good :)