# [time 165] Prime numbers in pregeometry

Ben Goertzel (ben@goertzel.org)
Sun, 04 Apr 1999 16:38:27 -0400

Thinking about the possible pregeometric meaning of prime numbers...

In my pregeometric explorations with Tony Smith/Kent Palmer/Onar Aam, I
philosophical observations...

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

ben

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