Matti Pitkanen (matpitka@pcu.helsinki.fi)
Mon, 16 Aug 1999 21:46:00 +0300 (EET DST)
Dear Stephen et all,
the article about Zeno's paradox was fascinating.
Some comments.
1. In TGD context quantum jump Einsteinian solution to Zeno paradox
holds is modified somewhat. With respect to geometric time
there is no motion: tortoise becomes 4-dimensional
geometric object. With respect to subjective time
the observed motion of tortoise is discretized with
average time increment of about 10^4 Planck times per quantum
jump: cognitive spacetime sheet jumps by this temporal distance in
each quantum jump and sees new t=constant section of 4-dimensional
tortoise (in good approximation).
2. I only now realized that every infinite prime, whose inverse is
infinitesimal is smaller than 1/0, the largest possible infinity.
3. The concept of lexicon is phantastic but I could not understand
the notion of rational as novelty and subsequent claim that motion
is illusion.
4. I realized a nice manner to represent surreals (or whatever TGD
version about extension of reals is). Consider definition of a finite
real as pinary expansion:
x= SUM(n>n0) x(n)p^(-n)
a) For ordinary reals all *finite* integers n
are present in series
b) For extened reals also infinite integers
n are present. Certainly infinite values of n correspond to infinitesimal
contributions in the expansion of x in negative powers of p.
c) How should one define the part of expansion for which the values of n
are infinite? One can make the expansion unique by following trick: sum
over all n expressible as products of finite and infinite primes!
If one can construct *all* infinite primes (I have constructed quite
many good candidates!) one can make sense of this expansion
at least formally.
5. Riemann zeta function contains product over factors over
all primes. An interesting question is whether one could understand
something about zeta function by allowing
also infinite primes in the product formula
Z(s) = Prod(p prime) [1/(1-p^s)].
Best,
Matti Pitkanen
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