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

*Sun, 4 Apr 1999 17:25:40 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Ben Goertzel: "[time 140] Re: [time 81] Discreteness and p-adics"**Previous message:**ca314159: "[time 138] Re: [time 136] Re: [time 123] RE: [time 69] Spacetime & consciousness"**In reply to:**Matti Pitkanen: "[time 136] Re: [time 123] RE: [time 69] Spacetime & consciousness"**Next in thread:**Ben Goertzel: "[time 140] Re: [time 81] Discreteness and p-adics"

Below an observation stimulated by Ben Goertzel's claim about discreteness

as something fundamental. I think Ben is right in a well defined sense!

p-Adic numbers with norm bounded by some upper bound

are of from x= p^rk , where k is possibly *infinite* integer

having pinary expansion k= SUM(n>=0) k_np^n. These numbers form a

discrete set! The real counterparts of these numbers in canonical

identification SUM x_np^n --> SUM x_np^(-n) however form a continuum!

In fact, Fourier analysis for p-adic planewaves providing representation

for translations is possible and p-adic planewaves are orthonormal with

*Kronecker delta normalization* rather than Dirac delta renormalization.

The real counterparts of p-adic momenta with fixed

upper bound form discrete set whereas their

p-adic counterparts correspond to possibly infinite integers. Thus p-adic

planewaves are more beautiful mathematically since infinities are avoided

in the definition of inner product and there is no need to introduce

distributions.

Since the coordinates of the infinite-dimensional configuration space

are proportional to possibly infinite integers one has discrete in

this sense also at the level of configuration space.

Best,

MP

**Next message:**Ben Goertzel: "[time 140] Re: [time 81] Discreteness and p-adics"**Previous message:**ca314159: "[time 138] Re: [time 136] Re: [time 123] RE: [time 69] Spacetime & consciousness"**In reply to:**Matti Pitkanen: "[time 136] Re: [time 123] RE: [time 69] Spacetime & consciousness"**Next in thread:**Ben Goertzel: "[time 140] Re: [time 81] Discreteness and p-adics"

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