**Hitoshi Kitada** (*hitoshi@kitada.com*)

*Thu, 15 Apr 1999 22:30:59 +0900*

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Dear Ben,

Just a miscellaneous remark on the parallel between math and Big Bang

theory... (I hope to understand your paper, but the notion of boundary you

mention in the paper is difficult for me to understand...)

----- Original Message -----

From: Benjamin Nathaniel Goertzel <ben@goertzel.org>

Sent: Sunday, April 11, 1999 11:23 AM

snip

*> X-Sender: ben@goertzel.org
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*> X-Mailer: QUALCOMM Windows Eudora Pro Version 4.1
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*> Date: Sat, 10 Apr 1999 22:20:21 -0400
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*> To: "Hitoshi Kitada" <hitoshi@kitada.com>,"Time List" <time@kitada.com>
*

*> From: Ben Goertzel <ben@goertzel.org>
*

*> Subject: multiboundary algebra
*

*> Cc: "Lancelot R. Fletcher" <lance.fletcher@freelance-academy.org>
*

*> In-Reply-To: <002501be8358$ee98bae0$9450a3d2@kitada.com>
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*> References: <4.1.19990409234045.00bef210@goertzel.org>
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*> Mime-Version: 1.0
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*> Content-Type: text/plain; charset="iso-8859-1"
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*>
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*>
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*> Just a half-baked thought that will be more fully baked later...
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*>
*

*> I was thinking about how to make a good formal algebraic formulation of
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*> Hitoshi's theory
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*> of physics and my theory of consciousness, and my mind traveled back to an
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*> unfinished
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*> mathematical exploration that is on the Web in the form of some rough notes
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*>
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*> http://goertzel.org/ben/Multi.html
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*>
*

*> (this is not a research paper, it is notes on about the level of rigor of=
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*> our
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*> e-mails ;)
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*>
*

*> A brief excerpt, describing the notion of multiboundary algebra, is=
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*> extracted
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*> here, preceded by
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*> some brief comments on how it might be relevant
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*>
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*> The basic idea is that you can have different types of parentheses, and
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*> different algebraic
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*> laws applying to entities depending on the kinds of parentheses they live
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*> between.
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*>
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*> I was using this idea to explain how you get the algebraic structures of the
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*> Standard Model
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*> (quaternions, octonions, clifford algebras) out of nothingness. I.e.
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*> pregeometry.
*

During lecturing to freshmen about real numbers as Dedekind's cuts today, I

noticed that mathematics itself starts with nothingness as in your treatment

of "Standard Model." I.e. in the Peano type construction of natural numbers,

we assume 0 (zero) is defined as an empty set, and define 1 as {0}, 2 as

{0,1}, 3 as {0,1,2}, ..., for example. Of course this may be natural because

you are treating numbers: quaternions, octonions, ... . What I felt

interesting is that this seems the same spirit as Big Bang theory which starts

with nothingness in "constructing" the universe. I suspect that here seems to

be a modern "constructionism" spirit that began in the 19 century along with

Peano, Frege, Gentzen, Cantor, Hilbert, Zermelo, etc. in mathematics.

If Big Bang can be thought as one of those streams, is it not possible to say

Big Bang is one of the fashions of the modern age after the 19 century?

Best wishes,

Hitoshi

*>
*

*> But this seems to capture Hitoshi's theory too, in a slightly different
*

way.=

*> =20
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*> He is saying that
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*> different algebraic rules apply inside the boundary of the LS, than
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outside.=

*> =20
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*> I.e. he is introducing
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*> a boundarizing operator that defines an LS. Inside the LS, calculations are
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*> done with amplitudes;
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*> outside they are done with probabilities.
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*>
*

*> What the best formal way to capture this is, I'm not sure. I'm thinking=
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*> that
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*> the LS-parenthese
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*> is possibly representable formally as an operator that transforms amplitudes
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*> into probabilities...
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*> But this doesn't work in the crudest sense, I need to think about it more...
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*>
*

*> Still this is very exciting to me. I see the vague possibility of building=
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*> a
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*> unified physics in terms of
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*> a hierarchy of boundary operators, each one associated with an algebra. The
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*> lowest level
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*> boundaries give the emergence of quantum structures from the void; then the
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*> highest level boundary
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*> gives the emergence of classical reality from quantum structures via
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*> observation.
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*> (Finally there is a boundary operator that takes a the quantum state of a
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*> classical structure,
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*> and makes it impenetrable, thus rendering it a basic indecomposable=
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*> particle,
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*> and closing the
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*> loop? -- dreaming again ;)
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*>
*

*> ben
*

*>
*

*> ***
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*> snip from rough paper on multiboundary algebra
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*>
*

*> ****
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*> The development here will remain within the framework of boundaries that
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*> interact in two different ways; the Laws of Form algebra will be enriched,
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*> however, by the addition of more types of boundaries. This development is
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*> eccentric with regard to conventional mathematics, which contains only one=
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*> type
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*> of parenthese for grouping entities, but numerous operations for combining
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*> groups and ungrouped entities. However, introducing several types of=
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*> boundary
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*> is a perfectly viable alternative to introducing several types of operator,=
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*> and
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*> is in some ways, we shall see, a more powerful alternative.
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*> Notation for multiple boundaries is fairly difficult to come by, though the
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*> typewriter keyboard provides a number of alternatives, e.g. ( ) , [ ], { },=
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*> < >
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*> . For talking about multi-boundary algebra in the abstract I will use the
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*> notation (k k) to refer to a boundary of type k, but this notation is not=
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*> good
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*> for working out concrete examples.
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*> The interior of a boundary will be called a "space," and the operator
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*> (k k) * b
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*> will be understood to mean that the entity b is placed in the space=
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*> demarcated
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*> by the boundary (k k). Multi-boundary algebra envisions a universe in which
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*> various simple and composite entities coexist and interpenetrate within=
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*> various
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*> spaces, and in which the results of coexistence and interpenetration depend=
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*> on
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*> the composition of the entities involved, and the space in which the=
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*> entities
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*> exist.
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*> Formally, a general multi-boundary algebra consists of=20
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*> =B7 A collection of boundary types, (k k), k=3D1,=85,n; a "boundary"=
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*> is an
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*> instance of a boundary type=20
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*> =B7 A collection of operators * i, which are used to build composite
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*> entities called "forms" out of boundaries=20
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*> =B7 A rule set R which determines the interaction of boundary forms=
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*> via the
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*> * i operators=20
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*> =B7 A collection of "local" rule sets Rk ,k=3D1,=85,n; where Rk=
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*> determines the
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*> interaction of boundary forms (via the * i operators) which lie within the
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*> boundary (k k)=20
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*> =B7 A function r which maps the set of boundaries into the set of rule
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*> sets;
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*> i.e., it assigns individual rule sets to specific boundaries
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*> The local rule sets must be consistent with the global rule set R. The
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*> individual rule sets must be consistent with the global rule set and with=
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*> the
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*> local rule sets. The local rule set applying to the interaction of two=
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*> boundary
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*> forms is determined by the the boundary that they most locally belong to;=
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*> e.g.
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*> if we have [ < a + b> ] then the meaning of the operation + is determined by
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*> the rule set for < > and not the rule set for [ ].=20
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*> Multiboundary algebra, in general, is an extremely general framework, much=
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*> like
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*> Universal Algebra (but more general). The specific multiboundary algebras
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to=

*> be
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*> discussed here will involve the standard two operators, + and *, plus an
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*> additional operator ^ dealing with temporality. Furthermore, in all the
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*> specific
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*> algebras to be discussed here, the + operator will be commutative, and will=
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*> be
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*> associative in regard to the ordinary parenthese ( ), meaning that + can be
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*> commuted across an arbitrary number of arguments, e.g. a + b =3D b + a, a +=
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*> b + c
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*> + d =3D d + c + b + a =3D d + b + a + c, etc. In more formal language, we=
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*> will be
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*> dealing with +-polycommutative (+,*,^) multiboundary algebras (where "*i
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*> -polycommutative means, in general, that all rule sets in the multiboundary
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*> algebra hold the operator * i to be commutative and associative with regard
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*> to (
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*> ) ).
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*> From the definition of multiboundary algebra, we see exactly what the
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*> difference is between the standard mathematical technique of having one
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*> boundary (the parenthese) and numerous operators, and the present=
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*> alternative
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*> of having few operators and multiple boundaries. The notion of=
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*> space-dependent
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*> rule sets has no parallel in standard mathematics, and breaks down the=
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*> barrier
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*> between algebraic rules and algebraic formulas in a very interesting way,=
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*> which
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*> is only explored here very partially.=20
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*> ******
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*>
*

*> ben
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*>
*

*>
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*>
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*>
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*>
*

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