.51> zero(x) j 0 2. successor: defined by succ(x) j x + 1 3. projection: defined by proj n k (x 1 ; : : : ; xn ) j x k and two basic operators for constructing new functions: 1. composition: denoted by h k = comp( f n ; g k 1 ; g k 2 ; : : : ; g k n ) 2. primitive recursion: denoted by h n+1 = prec( f n ; g n+2 ) Here all arguments are natural numbers and the superscripts on the functions f , g, and h denote their "arities"; that is, the number of arg

Computer science inherited its present conceptual foundations from a branch of pure mathematics that, historically, had been exploring the fundamental nature of mathematical computation since before the turn of the century. It is argued that the conceptual concerns of computer science are different from the conceptual concerns of mathematics, and that this mathematical legacy, in particular the notion of the algorithm, has been largely ineffective as a paradigm for computer science. It is first necessary to understand the role of the algorithm in mathematics.

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