**Next message:**J. R. Molloy: "Re: Infinite Computing"**Previous message:**GBurch1@aol.com: "Re: The simputer project"**In reply to:**Spudboy100@aol.com: "Re: Infinite Computing"**Next in thread:**scerir: "Re: Infinite Computing"**Reply:**scerir: "Re: Infinite Computing"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

*> Spudboy100@aol.com 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

http://www.edge.org/documents/archive/edge83.html

< 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

http://xxx.lanl.gov/abs/hep-th/9806135

< 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

http://www.edge.org/3rd_culture/smolin/smolin_p2.html

http://xxx.lanl.gov/abs/gr-qc/9703026

< 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

introduced.

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.

<snip>

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

http://xxx.lanl.gov/abs/gr-qc/9910005

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"

http://www.idsia.ch/~juergen/toesv2/

"A Computer Scientist's View of Life, the Universe, and Everything"

http://www.idsia.ch/~juergen/everything/html.html

(scerir compiler)

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