Giuseppe Longo (CNRS, Paris)
Computing? A Machine Imitating a Man in an Alphanumeric Game

No natural process computes. When observing a physical dynamic, in order 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 input time), then let the process go. Finally, measure again the result at the 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 Physics): it does not intrinsically produce a number in a discrete 0-1 universe. Moreover, most natural processes are not "laplacian", as Turing calls his own Discrete State Machine in 1950: most of them are subject to the "exponential drift", which breaks any approximation. This is one of the 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 and sequence replacement; it is a re-writing procedure handled by a human (or by a suitably programmed Universal Turing Machine). As a matter of fact, in order to have a physical device compute, we had to invent:
- the alphabethic coding of language (from Mesopotamia, IVth millennium 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 as a Discrete State Machine (Turing '46-50).
This makes Turing's Logical and Discrete State Machine a unique physical system, when materialized, an extraordinary invention for its stability and perfect iterability, away from of the natural world, like the alphabet, a dualistic device based on syntax and semantics (or hardware and software).

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

  1. Francis Bailly, Giuseppe Longo. Mathematics and Natural Sciences : the Physical Singularity of Life, 333 pages, Imperial College Press, London, 2011.
  2. 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) Imperial College Press/World Scientific, 2010.
  3. Giuseppe Longo. Incomputability in Physics and Biology. Invited Lecture, Proceedings of Computability in Europe, Azores, Pt, June 30 - July 4, LNCS 6158, Springer, 2010.
  4. Giuseppe Longo. Critique of Computational Reason in the Natural Sciences, In "Fundamental Concepts in Computer Science" (E. Gelenbe and J.-P. Kahane, eds.), Imperial College Press, pp. 43-70, 2009.