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To be really pedantic, "algebraic functions" are functions that can be defined as the roots of polynomial equations, I.e. solutions of equations of the form f(x_1, x_2, ..., x_n) = 0. There are other, more general notions of "function" in mathematics, though. I'm not sure how pedantic the author of the original post meant to be! Most generally, a function is not a number, it is a relation between two sets. One can define such things in a number of ways, but one way is to describe the relation via a formal language such as the lambda calculus. The lambda calculus is not technically a "programming language", it is a formal system of meaningless symbols and a set of rules for forming, manipulating, and eliminating them. Although there is no fixed relation between reduction rules in the lambda calculus and physical resources in a computer, one can still compare the number of reduction steps required to determine which form in the co-domain of a function defined in the lambda calculus corresponds to a specific form in its domain, and this will provide a reasonable stand-in for comparisons between resource usage of similar computer programs. So, really, computation is not completely foreign to mathematics, and mathematical functions and "computing functions" are not completely different animals, just... distantly related? Some languages are more mathematical in their approach than others. |