Turing-Church thesis, constructve mathematics and intuitionist logic

TL;DR

This paper critically examines the foundations of the theory of computation, highlighting its reliance on potentially misleading notions and proposing a rational reconstruction aligned with constructive mathematics and intuitionist logic.

Contribution

It offers a critique of the traditional theory of computation's reliance on classical logic and potential infinity, and suggests a new rational reconstruction based on constructive and intuitionist principles.

Findings

Current theory uses doubly negated propositions and ad absurdum proofs.

The notion of the thesis in computation theory may be misleading.

A sketch for a rational reconstruction of the theory is proposed.

Abstract

At a first glance the Theory of computation relies on potential infinity and an organization aimed at solving a problem. Under such aspect it is like Mendeleev theory of chemistry. Also its theoretical development reiterates that of this scientific theory: it makes use of doubly negated propositions and its reasoning proceeds through ad absurdum proofs; a final, universal predicate of equivalence of all definitions of a computations is translated into an equality one, and at the same time intuitionist logic into classical logic. Yet, the last step of this development of current theory includes both a misleading notion of thesis and intuitive notions (e.g. the partial computable function, as stressed by some scholars). A program for a rational re-construction of the theory according to the theoretical development of the above mentioned theories is sketchy suggested.