The forget-restore principle: a paradigmatic example

The aim of this note is to give a simple but instructive example of the forgetrestore principle, conceived by Giovanni Sambin as a discipline for a constructive development of mathematics and first appeared in print in the introduction of Sambin and Valentini 1997. The best way to explain such a philosophical position is to quote from that paper: “To build up an abstract concept from a row flow of data, one must disregard inessential details . . . this is obtained by forgetting some information. To forget information is the same as to destroy something, in particular if there is no possibility of restoring that information . . . our principle is that an abstraction is constructive . . . when information . . . is forgotten in such a way that it can be restored at will in any moment.” The example we want to show here refers to Martin-Löf’s intuitionistic type theory (just type theory from now on). We assume knowledge of the main peculiarities of type theory, as formulated in Martin-Löf 1984 or Nordström et al. 1990. Type theory is a logical calculus which adopts those notions and rules which keep total control of the amount of information contained in the different forms of judgment. However, type theory offers a way of “forgetting” information, that is, supposing A set, the form of judgment A true. The meaning of A true is that there exists an element a such that a ∈ A but it does not matter which particular element a is (see also the notion of proof irrelevance in de Bruijn 1980). Thus to pass from the judgment a ∈ A to the judgment A true is a clear example of the forgetting process. We will show that it is a constructive way to forget since, provided that there is a proof of the judgment A true, an element a such that a ∈ A can be re-constructed. Of course the simple solution of adding only the rule