Giuseppe Longo (CNRS, CREA, Ecole Polytechnique b(& CIRPHLES, Ecole Normale
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
"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).