I had a bit of (*cough*) spare time yesterday and had a quick scan around the Clay Mathematics Institute links ... all in the aim of solve "some" of the millennium problems (buahaha!) ... but it kept me thinking all the way throughout my work out ... dinner and even whilst trying to sleep ... basically I cannot find a type of mathematics that would make calculations in a "batch" way; to get actual solutions within sets, mathematics has just operations that are iterations across a set. What appears to be required is "parallel" operations that operate on the sets themselves. Abelian groups are described in a "sequential fashion" - discrete - (as the theory of element X is member_of set Y "for each" X1...Xn that behaves in a certain way, blah blah, all that can work in one operation, but when you have the actual "Xi" that is a different story!).
My question is: "union" is a no brainier, as it is just one iteration to obtain the desired result (oversimplified - OK, but hope you get my point); but what about "intersection" and "difference" (which are the principles required by all NP problems), is there any field in mathematics that is working on "parallel" execution ? I assume it requires a a totally new branch of mathematics - and a total new way of thinking ? - as our brain is used to go throughout each element in the input side to generate the corresponding result on the output side - as opposed to get a whole set of elements, apply one single operation "a la quantum computing" and get back all the solutions required. (OK it was more of a paragraph than a question :))