The Difference between Clocks and Turing Machines

Abstract

During several centuries the prevailing metaphor for the human brain was the “clock” This metaphor became precise with Descartes. For Descartes, there is a mechanical body and a brain, a « statue d’ automate», similar to the fantastic automatic devices of his time and ruled by cog-wheels, gears and pulleys. Separated from it, but ruling it, is the « res cogitans ». The physical connecting point of this dualism was provided by the pineal gland; the philosophical compatibility was given by the fact that the res cogitans too is governed by rules, logical/deductive ones. «Non evident knowledge may be known with certainty, provided that it is deduced from true and known principles, by a continually and never interrupted movement of thought which has a clear intuition of each individual step». Knowledge is a sequence or « a chain » where « the last ring… is connected to the first» [Descartes,1619–1664; rule III]. In view of their role in the mathematical work of Descartes, it may be sound to consider these remarks as the beginning of Proof Theory; mathematics is no longer (or not only) the revelation or inspection of an existing reality, where “proofs” (mostly informal and incomplete, often wrong) were only meant to display “truth”, but it is based on the manipulation of algebraic symbols and stepwise deductions from evident knowledge. Descartes’ Analytic Geometry, an algebraic approach to Geometry, brought the entire realm of mathematics under the control of formal deductions and algebraic computations, as distinct from geometrical observation. In Algebra and, thus, in Algebraic or Analytic Geometry, proofs are sequences of equations, formally manipulated, independently of their (geometric) meaning. Three centuries later, Proof Theory will be the rigorous (mathematical) description of these deductions.