Giuseppe Longo (CNRS, CREA, Ecole Polytechnique b(& CIRPHLES, Ecole Normale Supérieure, Paris)
Turing, from the "Discrete State Machine" to the "Continuous Systems" for morphogenesis

Turing's Logical Computing Machine of 1936 grounds its effectiveness (and success) on a fundamental split: the distinction of hardware and software. This idea was originally proposed as a purely mathematical abstraction. It later became the core structure on which the universality and the effectiveness of computing is grounded: a "rigid" hardware allows a "soft" (immaterial) program to run. After World War II, when dealing (again) with physics, Turing stresses the key physical aspect of his machine: it is a Discrete State Machine (a DSM, as he says). He then works also with completely different physical structures, for his analysis of morphogenesis (1952): the "continuous systems", as he calls them. And his action/reaction/diffusion systems will pave the way to a new insight into some physical (and biological) processes, where the "computation" lies entirely in a continuous dynamics of forms: no software, just plastic and ever changing "hardware" and its unpredictable evolutions, subject to the "exponential drift". By his two inventions, Turing set the basis for an understanding of the core ideas of the Theory of Computability vs. Bio-physical phenomena in two opposing ways, that I would like to summarize as follows: computing as an "always identical iteration of alpha-numeric re-writing rules" (1936) and of Biology as a "never identical iteration of a morphogenetic processes", (1952) far away from the Computable. How does the "exponential drift" relate to incomputability, which is always proved by reduction to Gödel's and Turing's diagonal methods, even in computability over continua?

References (see http://www.di.ens.fr/users/longo )

Francis Bailly, Giuseppe Longo. Mathematics and Natural Sciences : the Physical Singularity of Life, 333 pages, Imperial College Press, London, 2011.

Giuseppe Longo. Incomputability in Physics and Biology. Invited Lecture, Proceedings of Computability in Europe, Azores, Pt, June 30 - July 4, LNCS 6158, Springer, 2010.

Marcello Buiatti, Giuseppe Longo. Randomness and Multi-level Interactions in Biology. Ongoing work. (arxiv.org/abs/1104.1110v1).