Matti Pitkanen (matpitka@pcu.helsinki.fi)
Mon, 5 Apr 1999 08:47:12 +0300 (EET DST)
On Sun, 4 Apr 1999, Ben Goertzel wrote:
>
> Thinking about the possible pregeometric meaning of prime numbers...
>
> In my pregeometric explorations with Tony Smith/Kent Palmer/Onar Aam, I
> made the following
> philosophical observations...
>
> ADDITION = COEXISTENCE
>
> MULTIPLICATION = INTERPENETRATION
>
> Suppose we take two sets, e.g.
>
> X = {a,b}
> Y = {c,d,e}
>
> Then
>
> X + Y = {a,b,c,d,e} with a cardinality of 5
>
> this represents drawing a common boundary around X and Y,
> considering X and Y to live in the same "local system" so to speak
>
> X * Y = {a*c,a*d,a*e,b*c,b*d,b*e} with a cardinality of 6
>
> this represents X interpenetrated through Y, each
> element of X allowed to transform (occupy the same local space as)
> each element of Y
>
> The idea is that coexistence and interpenetration are basic philosophical
> operations, reflected in the quantitative domain by + and *
>
> Prime numbers enter in in an obvious way: a set with a prime number of elements
> can ~never~ be obtained by interpenetrating two sets different from itself..
>
> Pregeometrically, prime numbers seem to correspond to "minimal systems"
> in some sense.
>
> These are obviously just some half-formed ideas. However they are the
> first inkling I have
> had as to what could form a philosophical foundation for Matti's elaborate
> mathematical improvisations on the theme of p-adicity and physics
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
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