The contribution of Jean-Louis Krivine was to show that these same processes are to be found at the base of all reasoning and mathematical structures. As the mathematician ascends into abstraction, these superposed strata, which become more and more complex, can always be reduced to the programming rules of lambda-calculus. In this context, the mathematician demonstrated that the non-fulfillment theorem of Godel* could be reduced to a simple computer program expressed in lambda-calculus. A program, which "would resemble a file repair program", implemented by a computer operating system in case of an unexpected breakdown of the system. The program itself, according to a hypothesis by Krivine, could be similar to what happens in the brain when, during sleep, all the previous evenings programs are shut down to allow the restoration of the cognitive contents in order to face the hazards of the following day, maintenance processes of which dreams could be vague echoes.

The presentation of Jean-Louis Krivine's research shows amply that the work he accomplished was important, arriving at the apparently simple conclusion- even if revolutionary- that all human language, diverse languages like cognitive construction, including mathematics, could have been built by the brain using lambda-calculus instructions. From this perspective, the most famous mathematicians and scientists are therefore only, if one can so say without violating their grandeur, PC computers exploring the world.

In a Darwinian, evolutionary hypothesis which seems indispensable to the coherence of this grandiose vision, it will be right from the first appearance of the nervous and nevraxes cells, in species of multi-cellular origin, that the lambda-calculus logical supports will be constructed. It is a survival tool, which allows them to adapt as quickly as possible to the constraints of the environment.

The appropriateness, more or less marked, between mathematics and the world, which has always surprised philosophers, will no longer be more surprising than the appropriateness, for example, of retina cells to the source of light waves. In all organisms characterizing living beings such as they are today, including the brain, one thus finds a co-evolution between the organism and the environment, which never ceases to exist.

This co-evolution will effectively continue in the future, thanks to the development of mathematics in partnership with computer science, producing more and more autonomous information systems: be it in the field of artificial life (with a progressive ascent towards the nano-technologies) such as those of artificial intelligence.

But one can also think that bringing the elementary processes of brain function to the fore would better allow us to understand, the smallest and most complex languages and symbolic exchanges of animals. One could well end up accepting that all animals are capable of algebraic or geometric calculations carried out in lambda-calculus, even if these calculations cannot be expressed for us in mathematical formulae. Which goes to show that animals could also well have a seat of consciousness not yet perceptible by us, such as still incomprehensible mathematical calculations - much less, that all this will not be compiled in the common terms of lambda-calculus. For a theoretician of artificial consciousness such as Alain Cardon, however, such an approach finds its limits in fact even if it comes from information technology, for which all is calculable to the nth degree. The lambda-calculus brought us Turing's machine. Thought is something different. To understand it, one has to call upon other approaches, such as René Thom's continuum.

One could conclude this brief discussion by taking the conclusion proposed by Jean Petitot, director of the Centre de Recherche en Epistomélogie Appliqué de l'Ecole Polytechnique, questioned by Science et Vie: a vast and urgent interdisciplinary research program is essential in order to really understand how the brain thinks consciously, as much in the human as in the animal. All this leaves us to suppose that we are waiting to see "the emergence" of revolutionary concepts around the concept of this subject grasping the idea that this is a subject - a problem that our "sameist" friends return to when they ask themselves what has happened when "a same" realizes that he is "a same" - which happens to us daily when we ask ourselves about the sameness of the composition of our thought.

* Demonstrated in 1931, this theorem affirms that truths always exist that mathematics cannot demonstrate.

© Automates Intelligents 2002