[time 558] Re: [time 556] Modeling change with nonstandardnumbers

Stephen P. King (stephenk1@home.com)
Mon, 16 Aug 1999 15:35:16 -0400

Hi Matti,

Matti Pitkanen wrote:
> 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).

        Ok, but do you see that we have to allow for the existence of an
infinite (unenumerable!) number of geometric "turtles"? One question I
have is: Why do we have a geometric time at all?! We obviously have a
subjective time, but why postulate an "geometric" one? For geometry, we
only need a 3+1 manifold, M^4. As it is, as you say, static, it has no
"change" related to it. Time is a subjective measure of change.
        My problem is that you seem to assume the existence of an "outside"
observer that can tell the difference between a Planck length of
duration h and \infinitesimal + h. What does this entity use to measure
the difference?
> 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.

        Let us talk about it further... :-)
> 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

        Can we encode a description of an arbitrary material system with them?
> 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.

        How long does it take the Universe to "do" this summation operation?!
Consider the problem of deciding if a given number is prime. Does the
Universe have a look up table? If it does, "where" is it "written" and
how is it "accessed"?
> 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)].

Does this paper give you any ideas?


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