On Tue, Aug 23, 2011 at 10:58 AM, Ezra Cooper <e...@ezrakilty.net> wrote:
> An algebra is a specific kind of structure which is itself formalized > mathematically. I've never seen a formalization of the notion of "a > calculus" and I believe it to be a looser term, as KC defined it. > > Specifically, an algebra consists of a set (or several "sorts" of sets) and > operations that reduce pairs of elements from that set (or the pairs can be > triples, etc.) back into the set. Usually that set corresponds to the > "semantics" of the algebra, and syntactic equations like xy = yx exist in a > different realm from the operations and their actions. > > Lambda calculus differs from an algebra by having a construct (lambda > abstraction) that only makes sense if you know the syntactic structure of > the term it applies to. That is, it has a binding construct. You could > define lambda calculus as an algebra by taking the underlying set to be the > syntax of the calculus itself, but that would require infinitely many > operations (a lambda-binder for each variable) and equations, so perhaps > that would be awkward. > > Pi calculus, like lambda calculus, has binders, while "process algebras" > are usually defined via operations on processes. I believe this to be a > general trait of things described as "calculi"--that they have some form of > name-binders, but I have never seen that observation written down. > > I'm sure that an algebraist could give a more definite answer about this. > Aside from "the calculus", a calculus is just a language with syntactic rules of inference/deduction. Indeed, "the calculus" was a calculus in this sense way back when, before Riemann formalized the subject with deltas and epsilons (that is, arguments about sequences and limits). I am referring to the syntactic rules for manipulating derivatives and integrals which Liebniz invented, such as "formal cancellation of derivatives": dy/dx * dz/dy = dz/dx An algebra is a model for a language that quantifies over objects and function symbols on objects. For example, the "symmetric group S4" is an algebra that models the symmetric group axioms.
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