par Marchal, Bruno 
Référence Physics of life reviews, 2, 4, page (251-289)
Publication Publié, 2005-12
Référence Physics of life reviews, 2, 4, page (251-289)
Publication Publié, 2005-12
Article révisé par les pairs
| Résumé : | I present some fundamental theorems in computer science and illustrate their relevance in Biology and Physics. I do not assume prerequisites in mathematics or computer science beyond the set N of natural numbers, functions from N to N, the use of some notational conveniences to describe functions, and at some point, a minimal amount of linear algebra and logic. I start with Cantor's transcendental proof by diagonalization of the non enumerability of the collection of functions from natural numbers to the natural numbers. I explain why this proof is not entirely convincing and show how, by restricting the notion of function in terms of discrete well defined processes, we are led to the non algorithmic enumerability of the computable functions, but also - through Church's thesis - to the algorithmic enumerability of partial computable functions. Such a notion of function constitutes, with respect to our purpose, a crucial generalization of that concept. This will make easy to justify deep and astonishing (counter-intuitive) incompleteness results about computers and similar machines. The modified Cantor diagonalization will provide a theory of concrete self-reference and I illustrate it by pointing toward an elementary theory of self-reproduction-in the Amoeba's way - and cellular self-regeneration - in the flatworm Planaria's way. To make it easier, I introduce a very simple and powerful formal system known as the Schoenfinkel-Curry combinators. I will use the combinators to illustrate in a more concrete way the notion introduced above. The combinators, thanks to their low-level fine grained design, will also make it possible to make a rough but hopefully illuminating description of the main lessons gained by the careful observation of nature, and to describe some new relations, which should exist between computer science, the science of life and the science of inert matter, once some philosophical, if not theological, hypotheses are made in the cognitive sciences. In the last section, I come back to self-reference and I give an exposition of its modal logics. This is used to show that theoretical computer science makes those "philosophical hypotheses" in theoretical cognitive science experimentally and mathematically testable. © 2005 Elsevier B.V. All rights reserved. |
