From Bruno:
> But the word "mechanism" cannot have the same sense before and
> after the discovery of the universal machine and its limitation. As my
> work illustrates in detail, universal machine have already two
> internal aspects which conflict with each other, and are close to the
> analytical/intuitive distinction.
This suggests the need to transcend the Universal machine as articulated by
Turing.
It suggests the need for us to articulate a ultracomplex mechanism that is of a
different variety.
Otto Rossler in conversation with Seaman suggests the employment of transfinite
numbers ---
Cantor and transfinite accuracy
This paper may be of interest:
The motives behind Cantor's Set Theory - Physical, biological, and
philosophical questions.
http://personal.us.es/josef/Cantor.pdf
Analogue mechanisms (or their highly parsed emulation in binary machines) might
be one approach.
See also:
Neural Networks and Analog Computation: Beyond the Turing Limit
Author: Hava T. Siegelmann
The theoretical foundations of Neural Networks and Analog Computation
conceptualize neural networks as a particular type of computer consisting of
multiple assemblies of basic processors interconnected in an intricate
structure.
Examining these networks under various resource constraints reveals a continuum
of computational devices, several of which coincide with well-known classical
models. What emerges is a Church-Turing-like thesis, applied to the field of
analog computation, which features the neural network model in place of the
digital Turing machine. This new concept can serve as a point of departure for
the development of alternative, supra-Turing computational theories. On a
mathematical level, the treatment of neural computations enriches the theory of
computation but also explicates the computational complexity associated with
biological networks, adaptive engineering tools, and related models from the
fields of control theory and nonlinear dynamics.[i]
Segelmann states: “The surprising finding has been that when analog networks
assume real weights, their power encompasses and transcends that of digital
computers.”[ii] She goes on to say “our model captures nature's manifest
“computation” of the future physical world from the present, in which constants
that are not known to us, or cannot even be measured, do affect the evolution
of the system.”[iii]
[i] Siegelmann, H. (2007), Neural Networks and Analog Computation: Beyond the
Turing Limit, http://www.cs.umass.edu/~hava/advertisement.html, Accessed 1
December 2009. See also Siegelmann, H (1999), Neural Networks and Analogue
Computation, Beyond the Turing Limit, Boston, MA: Birkhäuser.
[ii] Ibid.
[iii] Ibid.
Best
Bill Seaman
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