[time 535] Re: [time 533] Re: [time 530] Surreal numbers

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Fri, 13 Aug 1999 12:30:23 +0300 (EET DST)

On Fri, 13 Aug 1999, Stephen P. King wrote:

> 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)...

Interesting situation occurs when binary numbers Z_2 are replaced
with G(p,1): finite field with p elements. What one obtains
by the construction taking it to infinite is p-adic numbers with
norm not larger than one: also p-adic numbers infinite as
ordinary integers are included. Binary case would give 2-adics.
What differentiates between this construction and construction of surreals
is presumably that p-adic topology is introduced.

> 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!?)
> [SPK]
> > > 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
> http://www.askdrmath.com/problems/robert.7.15.96.html)
> 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.

Yes, I understand you point.

> > 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...)

Probably: I think he divided implicit to potentially explicit and tacit.
(I read a paper about implicit learning yesterday and found the
definitions there).

> 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.
> :-)

I read the rathmech in train and I think I understand the general
idea and philosophy. Although the basic philosophy is quite different
from my stubborn beliefs, I find the mathematical idea beautiful. I hope
I could apply it in my own thought constructions. To put
it mildly, I am still far from any concrete model for cognitive
representations: in any case, cognitive spacetime sheets and material
spacetime sheets could replace mind and matter in TGD framework.
Perhaps the models provided by cognitive spacetime sheets for the
behaviour of material ones could be formulated in terms of
Chu pair somehow defined by cognitive and material spacetime sheet forming
self and K valued mapping |= would characterize the simulation
provided by cognitive spacetime sheet for the behaviour of material
one. Something like this...

What troubles me that that the causation from mental to material
was replaced by a K-valued function. And interpretation
of the values of |= as complex time or logical value.
If K=Z_2 this everything is ok but
K=C? I did not quite understand the construction of left and
right residuations in case of QM.
I understood right residuation in general case.

Chu spaces involve the
assumption about *given* spaces A and X: isn't this assumption
very similar to the assumption 'spacetimes are 4-surfaces
of 8-dimensional H', which assumption in turn induces
the concept of configuration space and its spinor structure
crucial for quantum theory?


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