Matti Pitkanen (matpitka@pcu.helsinki.fi)
Tue, 17 Aug 1999 09:04:41 +0300 (EET DST)
Dear Stephen et all,
the article about Zeno's paradox was really stimulating.
Some additional comments below.
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.
**Comment: The finite rationals in the lexicon could code the information
about quantum jumps as experienced by finite p self: pinary cutoff
would make everything rational. Since almost every real is lexicon one
could conclude that infinite sequences of quantum jumps experienced
by beings characterized by infinite primes would give complete
statistical description of the subjective universe: all possible quantum
jumps would appear infinitely many times in the sequence!
**Comment: Rationals are crucial in the construction of lexicon. If
one allows infinite primes one must allow rationals which are ratios
of infinite integers. This obviously would generalized that idea about
rationals as numbers characterizing sensory experience. In particular the
phase preserving canonical identification playing key role in quantum TGD
would generalize: Pythagorean triangles characterized by infinite
integers would appear.
**Comment: Could one interpret this by saying that at higher levels of
existence (infinite primes) it is *generalized rationals*, which are
directly accessible to sensory experience? Does the theorem about lexicons
generalize to the context of generalized reals in the sense that all
generalized rationals appear infinitely many times in the expansion of
a generic generalized real?
**Question: We perform something like 10^(49) quantum jumps during
lifetime. Could it be really possible to construct the number theory
and quantum physics on basis of this tiny subjective experience (actually
actual wake-up periods are of order few seconds). Could it be
that we actually entangle now and then (during sleep) with spacetime
surfaces having infinite value of p and thus infinite subjective and
geometric memory and remember what happened as funny ideas about infinite
primes and things like that?
**Comment: Ramajunan told that he was in contact with some God(dess?)
during sleep and got his miraculous number theoretic formulas
from this Godd(dess?). Could it be that some higher level
being indeed friendly drilled a small hole in his/(her?) cognitive
spacetime sheet of infinite but bounded duration for
Ramajunan to get attached to its boundary by join along boundaries bond
and get mathematically enlightened?(;-)
******
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 the 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.
****
**Comment: One can write the expansion of finite generalized real in the
form
x= xa+xb
xa= SUM x(n)P^(-n)
xb= SUM(N,n) xc(n)p^(-n*N)
where N is infinite integer having *only* infinite primes in its
product decomposition to powers of primes. xa is the finite part
and xb is the infinitesimal part. I would guess that competing
generalizations of reals do not allow this kind of explicit representation
of generalized reals: for instance, surreals are defined with the help
of some kind of intervals. This representation in principle makes
it possible to multiply and sum formally infinite reals (I hope
that no one asks me to do it!).
**Comment: An interesting question is are these extensions *independent
of prime p as base*? Could it be that extensions are mathematically
nonequivalent so that every prime would give rise to slightly different
reals? Could this be the ultimate reason for why p-adics are so
fundamental?
**Comment: The existence of p-adic pseudoconstants, functions having
vanishing p-adic derivative, is a purely
p-adic phenomenon. If one however allows in real context infinitesimals
then all analytic functions of the form
f(x) = g(xa),
where xa is the finite part of x and g is analytic function,
have vanishing derivative with respect to extended reals. With respect
to ordinary reals derivative is nonvanishing. In fact, one can construct
infinite hieararchy of functions of this kind by cutting off xb in
suitable manner.
**Comment: For surreals integral calculus does not exist: am I correct. I
do not however see any obvious reason for why the integral calculus could
not exist for functions defined in the proposed manner. These numbers
are well ordered: this is crucial. The concept of integral function
defines definite integral and one can construct integration theory for
piecewise analytic functions. Apart from infinitesimals integral is
completely determined by f(xa): is this true?
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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