I was reading a book on Database foundations, recently. There were three foundational notions presented: a Relational Algebra, a Tuple Calculus, and a Domain Calculus. Each of these could be viewed as a formal language for expressing items in the database and for doing calculations on them like queries, joins, intersections, and what not. A sketch was given of a proof that all three are “equal” in power. I'd like to do some riffs on that theme in the near future, but today's topic, really, is “why call one of them an algebra and the others calculi?”
I don't know whether there is a consensus “diagnostic” that can unambiguously tell whether a thing is a calculus or an algebra. It could just be up to “inventor's discretion:“ if you invent something, you can call it anything you like. But I do know some examples of each thing. Things like lambda calculus, pi calculus, predicate calculus, and even college calculus are called “calculi.” Things like groups, rings, fields, vector spaces, and Clifford algebras are called “algebras.” Both of them consist of axioms and rules for transforming expressions. Both of them insist on “closure,” so that if you do some calculations (transformations) with items “in” the calculus or algebra, you end up with items “in” the calculus or algebra.
I can sense, possibly, a little difference in focus. Seems like the focus of a calculus is on the expressions. You have rules for making them longer (composition, for instance) and rules for making them shorter (reduction, for instance). The things that expressions represent (like procedures) are of secondary interest. In an algebra, the focus might be on “theorems” rather than on “expressions.” Now, really, there is little difference between a “theorem” and an “expression,” at least formally. A theorem is something you get when you apply the transformation rules to an expression correctly. But in an algebra, you're really looking for theorems, often using intuition, which is more like magic or spiritual inspiration than it is like calculation.
Well, I guess all I've done here is either confuse myself more or ramble on something that doesn't amount to a hill of beans. I do think that if there IS anything interesting to say about the difference, it will be found in Category Theory, an branch of metamathematics that reasons about things like sets and algebras and calculi from a higher perch in heaven. I have a nice book by Benjamin Pierce on Category Theory. I'll be reading some of it if time allows while I'm at the race track this weekend testing out my old restored Corvette race car.
Cheers till Tuesday.