Any thorough discussion of computing machines requires the examination of rigorous concepts of computation and is facilitated by the distinction between mathematical, symbolical, and physical computations. The delicate connection between the three kinds of computations and the underlying questions "What are machines?" and "When are they computing?" motivate an extensive theoretical and historical discussion. The relevant outcome of this discussion is formulated at the beginning of section 3.

The paradigm of the first kind of computation is given when a human calculator determines, by finitely many and mathematically meaningful steps, the values of number-theoretic functions for particular arguments. The informal concept of such effectively calculable functions is thought to be captured by Kurt Gödel's concept of general recursive functions. The latter notion was introduced in 1934 and arose in an intellectual context that includes the contemporaneous development of David Hilbert's program as well as earlier steps toward...