No natural process computes. When observing a physical dynamic, in
to associate it with a computation, one has to fix a "measure" (an
association of a state of the dynamics to a number, at the chosen
time), then let the process go. Finally, measure again the result at
chosen output moment and produce a number, according to the given
approximation. As a matter of fact, by principle, physical measure is
always an interval (even of conjugated variables, in Quantum
it does not intrinsically produce a number in a discrete 0-1
Moreover, most natural processes are not "laplacian", as Turing calls
his own Discrete State Machine in 1950: most of them are subject to
"exponential drift", which breaks any approximation. This is one of
novel features he observes (and mostly cares of) in the "continuous
systems" for Morphogenesis, in 1952.
A computation instead is a matter of alpha-numeric sequence matching
sequence replacement; it is a re-writing procedure handled by a human
(or by a suitably programmed Universal Turing Machine). As a matter
fact, in order to have a physical device compute, we had to invent:
- the alphabethic coding of language (from Mesopotamia, IVth
BC, to the Greek alphabet)
- the coding of the alphabeth by numbers (Gödel, 1931)
- a purely mathematical Logic Computing Machine (Turing, '36) for
alpha-numeric terms re-writing and, then, its physical implementation
a Discrete State Machine (Turing '46-50).
This makes Turing's Logical and Discrete State Machine a unique
system, when materialized, an extraordinary invention for its
and perfect iterability, away from of the natural world, like the
alphabet, a dualistic device based on syntax and semantics (or
References (see http://www.di.ens.fr/users/longo )
- Francis Bailly, Giuseppe Longo. Mathematics and Natural Sciences :
Physical Singularity of Life, 333 pages, Imperial College Press,
Giuseppe Longo, Thierry Paul. The Mathematics of Computing between
Logic and Physics. Invited paper, "Computability in Context:
Computation and Logic in the Real World ", (Cooper, Sorbi eds)
College Press/World Scientific, 2010.
Giuseppe Longo. Incomputability in Physics and Biology. Invited
Lecture, Proceedings of Computability in Europe, Azores, Pt, June 30 -
July 4, LNCS 6158, Springer, 2010.
Giuseppe Longo. Critique of Computational Reason in the Natural
Sciences, In "Fundamental Concepts in Computer Science"
and J.-P. Kahane, eds.), Imperial College Press, pp. 43-70, 2009.