[time 533] Re: [time 530] Surreal numbers

Stephen P. King (stephenk1@home.com)
Fri, 13 Aug 1999 03:23:39 -0400

Dear Matti,

Matti Pitkanen wrote:
> On Thu, 12 Aug 1999, Stephen P. King wrote:
> > Hi All,
> >
> > Perhaps this might spark a discussion!
> >
> > http://www.innerx.net/personal/tsmith/surreal.html
> >
> > "Surreal Numbers are just sequences of binary choices, and
> > constructing them is something of a game. It begins with the
> > simplest surreal number, an empty sequence made up of nothing
> > at all: this is written as 0, and is the starting place of what
> > mathematician Martin Kruskal calls the Binary Number Tree."
> >
> > It is this notion of "contruction by games" that I am proposing is
> > occuring when I say "LSs interact with each other by bisimulating each
> > other". Here we think of a bisimulational action as a mutual labeling of
> > properties. This, I tenatively propose is that happens in an
> > observation...
> [MP] This is attractive idea. One variant is structure recognition.
> Features of pattern are recognized as familiar or nonfamiliar.
> What about replacing Binary Number Tree with Pinary Number Tree:
> does anything change?

        I see "structure" or pattern recognition as a general type of
bisimulation. I believe that this is modeled by Pratt then he discusses
Chu spaces with K > 2 values. For K = 2, we have a binary situation like
you ask: "familiar" or "unfamiliar", but this is easily seen to be vague
as it tacitly assumes an absolute "standard" to distinguish the two...
He proposes that QM can be represented by the behavior of Chu_K = C (C
being the set of complex numbers)...
        I think that the Pinary tree is a better model since it captures
phylogenetic relations that a binary tree can not! Thus the "history" of
a particular player has an effect on the possible moves it will make in
a particular game... We skew (?)or weigh the moves as a function of the
number of times that a particular move was successful for that player!?
Since Chu spaces represent both the games and the players of the games,
we can identify a player with a set of games and a game by an antiset of
players. (I think, I may be misinterpreting Pratt!?)

> > I am thinking of how it is that we can think about labeling "points" in
> > our spaces with numbers; how is it that the orderings are manifested?
> > Are orderings ontologically a priori, or are they ex post facto defined
> > by the interactions of the subsets of the Universe, or something else? I
> > do not think that "objects" exist a priori with labels attached. It
> > seems that the act of labeling is implicit in any observation and that
> > the particular order of labels is "subjective"...
> [MP]
> I wish I had time to get some intuition about surreals to
> participate discussion. Just a comment relating to our earlier
> discussions, what you said above, and information in general. Dennet
> classifies representations of information is to explicit (file
> of numbers), potentially explicit (two lines of code producing Mandelbrot
> set) and tacit. For instance, in case of Turing machine everything else
> can be explicit except for the mechanism of reading head. It cannot be
> part of program and must remain tacit and given by the laws of 'raw'
> physics.

        Surreals are a very new idea. There is an article about them in a
science magazine, I think Discover, that explains them. I will try to
find a reference for it. I remember it as being very informative! :-) (I
found it: Discover Magazine. Shulman, Polly; 12-01-1995 "Infinity plus
one and other surreal numbers". Also:
http://www.maa.org/mathland/mathland_3_18.html and
        You make a very good point here, but I believe that Dennett is mistakes
in thinking that the difference between, say "files of numbers" and the
"mechanism of the reading head" are only differences in *degree* and
that this is all there is to be said of the situation. If we are
strictly talking about the information content of the physical
embodiments of these informational structures, we can see that the
difference in only in degree, but Dennett's material monism blinds him
to the categorical difference in *kind* that exists between the "mental
object" (information about) A* and the physical object A.
> Isn't the situation same in physics? To take example relating to previous
> discussions. Could it be that spacetime geometry is tacit information?
> The dynamics of spacetime surface defined Kahler action as dynamical
> principle is tacit information not allowing representation in terms
> of LS interactions: simply because it defines these interactions!?
> Same would apply to unitary time evolution U: it would also represent
> 'raw physics'. Explicit (DNA, short term memory?) and potentially
> explicit (motor program in my brain realized as cascades of selves, long
> term memory realized in terms of self hierachy and communication between
> levels of hierarchy?) information would emerge only at the level when
> selves emerge.

        Given my comment above, I agree with you here! :-) (Does Dennett allow
for "implicit" as the complement of "explicit"? I have read his book,
but I can't remember...)
        In Hitoshi's LS theory, the "outsides" of LS are "physical" and the
"insides" are "mental", I think!? We could categorize the information
involved in the external behavior of LS in the way you describe here.



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