Programs as Mathematical Objects

According to John Backus, in order for computing to become more of a science or an engineering discipline and less of an art, it should first become a mathematical discipline and for that, it would be important to make the program space is a mathematical space with respect to program-forming or functional operations, and to distinguish it from the value space with its specific value-construction operations.

Informally, a space S is a set S_v of values and a set S_o of the operations on S_v ranging from (S_v)ⁿ in S_o and are interrelated by algebraic properties.

The most useful of these are those that display some kind of symmetry, such as the distributive property that relates two operations O_A and O_B such that the combination of O_A with values formed with O_B can be expressed as O_B combining values formed with O_A.

The more useful the properties of the elements of S_o have, the stronger the algebraic structure and the mathematical structure of the space S will be stronger.