**Matti Pitkanen** (*matpitka@pcu.helsinki.fi*)

*Mon, 5 Apr 1999 08:47:12 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 172] Re: [time 167] Re: [time 164] Question"**Previous message:**Matti Pitkanen: "[time 170] Re: [time 161] Re: [time 160] Re: [time 157] tangent-cotangent; spaces and algebras that is!"**In reply to:**Stephen P. King: "[time 161] Re: [time 160] Re: [time 157] tangent-cotangent; spaces and algebras that is!"**Next in thread:**Ben Goertzel: "[time 174] Re: [time 171] Re: [time 165] Prime numbers in pregeometry"

On Sun, 4 Apr 1999, Ben Goertzel wrote:

*>
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*> Thinking about the possible pregeometric meaning of prime numbers...
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*>
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*> In my pregeometric explorations with Tony Smith/Kent Palmer/Onar Aam, I
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*> made the following
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*> philosophical observations...
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*>
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*> ADDITION = COEXISTENCE
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*>
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*> MULTIPLICATION = INTERPENETRATION
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*>
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*> Suppose we take two sets, e.g.
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*>
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*> X = {a,b}
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*> Y = {c,d,e}
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*>
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*> Then
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*>
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*> X + Y = {a,b,c,d,e} with a cardinality of 5
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*>
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*> this represents drawing a common boundary around X and Y,
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*> considering X and Y to live in the same "local system" so to speak
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*>
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*> X * Y = {a*c,a*d,a*e,b*c,b*d,b*e} with a cardinality of 6
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*>
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*> this represents X interpenetrated through Y, each
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*> element of X allowed to transform (occupy the same local space as)
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*> each element of Y
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*>
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*> The idea is that coexistence and interpenetration are basic philosophical
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*> operations, reflected in the quantitative domain by + and *
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*>
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*> Prime numbers enter in in an obvious way: a set with a prime number of elements
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*> can ~never~ be obtained by interpenetrating two sets different from itself..
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*>
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*> Pregeometrically, prime numbers seem to correspond to "minimal systems"
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*> in some sense.
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*>
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*> These are obviously just some half-formed ideas. However they are the
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*> first inkling I have
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*> had as to what could form a philosophical foundation for Matti's elaborate
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*> mathematical improvisations on the theme of p-adicity and physics
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The idea about primes as 'minimal systems' resonates with my own

thinking. I have also played with the idea about prime numbers as minimal

systems, minimal system regarded now as elementary particle. Prime number

decomposition for integer n= prod_k p_k^(n^k) is like many particle state

containing n_k elementary particles (bosons) of type p_k.

In the construction of infinite primes this idea comes very concrete

and a direct connection with the state construction of supersymmetric of

QFT:s emerges. For instance, the following numbers are rather general

infinite primes (or good candidates for them):

P = m*X/S + n* S

X= prod_kp_k is product of *all* finite primes. It is like *Dirac sea*

with all fermion states labelled by primes p_k filled.

S= is product of finite primes of form S= prod_{i in S}p_i. Only *first

powers of p_i* appear: S corresponds to n-fermion state and first powers

mean exclusion principle! What has been done is to kick some fermions

from sea so that holes are formed (formation of X/S and S).

m = prod_k p_k^(n^k) is integer *not divided* by primes dividing S.

n= prod_ip_i^(n^i) is integer not divided by primes dividing X/S.

m*X/S and n*S have no common (prime) divisors and this in fact implies

primeness since P mod finite prime is always nonvanishing.

m*n can be interpreted as many boson sates: n_r bosons of type p_r.

Thus P corresponds to many particle state of super symmetric theory with

finite primes labeling various fermionic and bosonic modes.

P has two parts. Since n*S can be finite it has interpretation

as a subsystem (local system!) and m*X/S is interpreted as its

complement so that subsystem complement separation crucial for strong NMP

and TGD inspired theory of consciousness appears naturally.

[I end up with a concrete translation of this analogy to real

decomposition at the level of spacetime surface: the hypothesis is

that the infinite prime characterizing the asymptotic p-adic

effective topology of spacetime surface codes the information about

various p-adic regions contained by the spacetime surface of infinite

size.]

Primeness follows from the fact that any finite p divides either m*X/S

or n*S but never both of them. This, if there are no

infinite primes with representation which is something which I have not

being able to imagine. The beautiful analogy with supersymmetric QFT:s

makes me believe that primes are indeed in question.

MP

**Next message:**Matti Pitkanen: "[time 172] Re: [time 167] Re: [time 164] Question"**Previous message:**Matti Pitkanen: "[time 170] Re: [time 161] Re: [time 160] Re: [time 157] tangent-cotangent; spaces and algebras that is!"**In reply to:**Stephen P. King: "[time 161] Re: [time 160] Re: [time 157] tangent-cotangent; spaces and algebras that is!"**Next in thread:**Ben Goertzel: "[time 174] Re: [time 171] Re: [time 165] Prime numbers in pregeometry"

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