Matti Pitkanen (matpitka@pcu.helsinki.fi)
Tue, 17 Aug 1999 09:44:09 +0300 (EET DST)

Hi Matti,

Matti Pitkanen wrote:
>
> Dear Stephen et all,
>
>
> 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).

[SPK] 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?

[MP]
Physics forces this. The entire physics starting from Newtonian physics
and ending to GRT and quantum field theories relies on geometric time.
The essential property of geometric time is that it is metric
just like space. Subjective time has not metric. The only structure in
the set of quantum jumps is that they are ordered. Psychological time
provides approximate statistical metrization for the set of quantum jumps
but would not be possible unless one had geometric time.

Second reason is that in the world without geometric time we could
not have any emotions: a world without possibility to get
frustrated now and then would be a dull place. The reason is that most
emotions result from the comparison of geometric time development
providing expectation for what will happen with what happened, that is
subjective time development.
*******
[SPK]

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?

[MP] M^4_+xCP_2 is completely static. Spacetime surface
X^4 is dynamical in the sense of classical physics and is absolutely
necessary but from spacetime view point could be said to be static.
Quantum average X^4 changes in each qjump.

Selves are observers and correspond geometrically to cognitive spacetime
the macroscopic spacetime changes in each quantum jump (the location
of cognitive spacetime sheet of self on it changes). Therefore selves
'see' different temporal cross section of the *material* spacetime
sheet in each quantum jump. This material spacetime sheet does not
change much in quantum jump: cognitive spacetime sheet shifts its
temporal position.

Only subjective experience tells differences
and these temporal differences are on the averages of order 10^4
Planck times if the simplest guess is correct.
I hope this is answer to your question: I am not quite sure whether
I understood the question properly.

The problems are created if I assume that entire time development
is computing. I however assume that classical physics, classical
spacetime and unitary time evolution U belong to hardware, the
quantum computer itself: only the subjective time development involves
activities like computation.

I believe that classical computation is one of the most recent
evolutional developments in biosphere. If this is the case
then it is wrong to assume that basic cognition, and even more, entire
physics would rely on classical computation.

> 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... :-)

lexicons, generalized reals and generalized rationals.

> 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

[SPK] Can we encode a description of an arbitrary material system with
them?

[MP]
I would say that it is impossible to code all information about say
real valued function on real axis to single real. The only manner to
achieve this is to perform discretization. Pinary cutoff would
be this discretization in TGD. Reals would be replaced by subset
consisting of rationals and function would have values in subset
of rationals. In this case single real would code the whole function.

Interesting possibility is that generalized rationals defined
as ratios of generalized integers having infinite primes in their
decomposition to powers of primes could help somewhat.

> 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"?

[MP] You certainly have some practical problems if you believe that
universe does not contain any hardware, which is just given. Even Turing
machine fails to be pure software: its reading head cannot be a part of
program and thus corresponds to 'tacit' information (I know that you say

Personally I am convinced that geometry, both finite- and
infinite-dimensional and hence physics apart from quantum jump
in TGD framework is what might be called 'tacit' information: it this the
quantum computer itself, the hardware of it. Quantum jumps define
subjective time development having computational aspects, in particular,
pinary cutoff and (generalized?) rationals emerge at this level.
Universe as a mere program seems impossible for me: also
hardware is needed.

> 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?
http://www.cs.auckland.ac.nz/CDMTCS//researchreports/032crisRR.pdf

I will look.

Best,

MP

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