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
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
ben
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