Re: Infinite Computing

From: scerir (
Date: Sun Apr 29 2001 - 11:26:59 MDT

> wrote:
> To be honest with you, I was surprised that Smolin said anything truly,
> novel, such as what I had quoted before. I do remember searching Lanl for
> some work with the math of it, and such, but I don't remember if I obtained
> any hits.

Freeman Dyson wrote
< Lee Smolin gave the longest and most substantial response. He describes a
third possible form of information processing which is neither analog
(because it is based on discrete rather than continuous components) nor
digital (because it cannot be simulated by a digital computer algorithm).
His information storage is based on the topological structure of finite
graphs in three-dimensional space. This illustrates the general statement
that the categories of analog and digital are too narrow to cover the range
of possible machines and organisms. It is possible that Smolin's topological
information processing may actually exist, both in living cells and in the
fine-structure of space-time.>

Manfred Requardt wrote
< We base our own approach on what we call `cellular networks', consisting
of cells (nodes) interacting with each other via bonds (figuring as
elementary interactions) according to a certain `local law'. Geometrically
our dynamical networks are living on graphs. Hence a substantial amount of
the investigation is devoted to the developement of various versions of
discrete (functional) analysis and geometry on such (almost random) webs.
Another important topic we address is a suitable concept of intrinsic
(fractal) dimension on erratic structures of this kind. In the course of the
investigation we make comments concerning both different and related
approaches to quantum gravity as, say, the spin network framework. It may
perhaps be said that certain parts of our programme seem to be a realisation
of ideas sketched by Smolin some time ago. >

Lee Smolin and Stuart Kauffman wrote
< We may note that if the Hilbert space is not constructible, we cannot ask
if this procedure is unitary. But we can still normalize the amplitudes so
that the sum of the absolute squares of the amplitudes to evolve from any
spin network to its successors is unity. This gives us something weaker than
unitarity, but strong enough to guarantee that probability is conserved
locally in the space of configurations.
To summarize, in such an approach, the amplitude to evolve from the initial
spin network W_0 to any element of S^N [W_0] , for large finite N is
computable, even if it is the case that the spin networks cannot be
classified so that the basis itself is not finitely specifiable. Thus, such
a procedure gives a way to do quantum physics even for cases in which the
Hilbert space is not constructible.
We may make two comments about this form of resolution of the problem.
First, it necessarily involves an element of time and causality. The way in
which the amplitudes are constructed in the absence of a specifiable basis
or Hilbert structure requires a notion of successor states. The theory never
has to ask about the whole space of states, it only explores a finite set of
successor states at each step. Thus, a notion of time is necessarily
Second, we might ask how we might formalize such a theory. The role of the
space of all states is replaced by the notion of the successor states of a
given network. The immediate successors to a graph Gamma_0 may be called the
adjacent possible. They are finite in number and constructible. They replace
the idealization of all possible states that is used in ordinary quantum
mechanics. We may note a similar notion of an adjacent possible set of
configurations, reachable from a given configuration in one step, plays a
role in formalizations of the self-organization of biological and other
complex systems.
In such a formulation there is no need to construct the state space a
priori, or equip it with a structure such as an inner product. One has
simply a set of rules by which a set of possible configurations and
histories of the universe is constructed by a finite procedure, given any
initial state. In a sense it may be said that the system is constructing the
space of its possible states and histories as it evolves.
There are further implications for theories of cosmology, if it turns out to
be the case that their configuration space or state space is not finitely
constructible. One is to the problem of whether the second law of
thermodynamics applies at a cosmological scale. If the configuration space
or state space is not constructible, then it is not clear that the ergodic
hypothesis is well defined or useful. Neither may the standard formulations
of statistical mechanics be applied. What is then needed is a new approach
to statistical physics based only on the evolving set of possibilities
generated by the evolution from a given initial state. It is possible to
speculate whether there may in such a context be a ³fourth law² of
thermodynamics in which the evolution extremizes the dimension of the
adjacent possible, which is the set of states accessible to the system at
any stage in its evolution. >

The first use of "Topos Theory" by Chris Isham
is, perhaps, more rigorous than the one
"Three roads to quantum gravity, Weidenfeld & Nicolson, 2000".
by Lee Smolin, which is more speculative.
"Topos Theory" has been invented, independently,
by Grothendieck, in the field of agebraic geometry,
and Lawvere, in the field of the foundation of mathematics.

For a different approach, see also
Jürgen Schmidhuber
"Algorithmic Theories of Everything"
"A Computer Scientist's View of Life, the Universe, and Everything"

(scerir compiler)

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