TITLE: Difference between a 'calculus' and an 'algebra'
QUESTION [23 upvotes]: What is really the conceptual difference between a calculus and an algebra. 
Eg. Is SKI combinator calculus really a calculus?
A friend claims that free variables are fundamental for a calculus, and as such that SKI is not a calculus, but an algebra.

REPLY [2 votes]: a calculus is a symbolic system for computation where computation can most generally be seen to be a spatial reorganization of symbols. any kind of numerical computation can be described in terms of recursion which can fundamentally be seen as a symbolic manipulation process. logic calculi and the calculus that set theory uses is also describable as a symbolic computation system.
an algebra is a mathematical structure in the informal sense which turns out to be a vector space with the added ability to multiply the actual vectors together. the complex numbers with addition and multiplication is an algebra over the complex numbers; http://en.wikipedia.org/wiki/Algebra_over_a_field