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

*Thu, 16 Sep 1999 10:38:21 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**WDEshleman@aol.com: "[time 768] Re: Noumenon and Phenomenon"**Previous message:**Matti Pitkanen: "[time 766] Re: [time 765] Re: [time 763] Noumenon and Phenomenon, Question for Matti"**Next in thread:**Stephen P. King: "[time 769] Re: [time 767] Generalizing the concept of integer, rational and real, etc.."

The discussions with Stephen about lexicons inspired the attempt

to define precisely the concepts of generalized integer, rational

and real. Rather surprisingly, infinite integers and generalized

reals can be regarded as infinite-dimensional vectors spaces

having ordinary integers and reals as coefficients. Thus

the tangent space of configuration space of 3-surfaces might

perhaps be regarded as the space of generalized octonions!

The concept of generalized lexicon lends support for the idea

that TGD:eish Universe has enough subjective memory to discover

laws of physics and consciousness.

I glue latex file about addition to the chapter

'Infinite primes and consciousnes' in 'TGD inspire theory

of consciousness...' at

http://www.physics.helsinki.fi/~matpitka/cbook.html

Best,

MP

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%55

\documentstyle [10pt]{article}

\begin{document}

\newcommand{\vm}{\vspace{0.2cm}}

\newcommand{\vl}{\vspace{0.4cm}}

\newcommand{\per}{\hspace{.2cm}}

\subsection{How to generalize the concepts of integer,

rational and real?}

The allowance of infinite primes forces to generalize also the concepts

concepts of integer, rational and real number. It is not

obvious how this could be achieved. The following arguments

lead to a possible generalization which seems practical (yes!) and

elegant.

\subsubsection{Infinite integers form infinite-dimensional vector space

with integer coefficients}

The first guess is that infinite integers $N$ could be

defined as products of the powers of finite and infinite primes.

\begin{eqnarray}

N&=&\prod_k p_k^{n_k}= nM\per , \per n_k\geq 0\per ,

\label{product}

\end{eqnarray}

\noindent where $n$ is finite integer and $M$ is infinite integer

containing only powers of infinite primes in its product expansion.

\vm

It is not however not clear whether the sums of infinite integers

really allow similar decomposition. Even in the case

that this decomposition exists, there seems to be no way of

deriving it. This would suggest that one

should regard sums

$$ \sum_i n_iM_i$$

\noindent of infinite integers

as infinite-dimensional linear space spanned by $M_i$

so that the set of infinite integers would be analogous

to an infinite-dimensional algebraic extension of say p-adic numbers

such that each coordinate axes in the extension corresponds to single

infinite integer of form $N=mM$.

Thus the most general infinite integer $N$ would have the form

\begin{eqnarray}

N&=& m_0+ \sum m_iM_i\per .

\end{eqnarray}

\noindent This representation of infinite integers

indeed looks promising from the point of view of

practical calculations. The representation looks also

attractive physically.

The integers $m_0$ and $m_iM_i$ are formally analogous to a many-boson

states consisting of $n_k$ bosons in the mode characterized by finite

or infinite prime $p_k$. Therefore this

representation is analogous to a quantum superposition

of bosonic Fock states with binary, rather than complex valued,

superposition coefficients.

\subsubsection{Generalized rationals}

Generalized rationals could be defined

as ratios $R=M/N$ of the generalized integers. This works nicely

when $M$ and $N$ are expressible as products of powers of finite

or infinite primes but for more general integers the definition

does not look attractive. This suggests that one should restrict

the generalized rationals to be numbers having the expansion

as a product of positive and negative primes, finite or infinite:

\begin{eqnarray}

N&=&\prod_k p_k^{n_k}= \frac{n_1M_1}{nM}\per .

\label{rational}

\end{eqnarray}

\subsubsection{Generalized reals form infinite-dimensional

real vector space}

One could consider the possibility of defining

generalized reals as limiting values of the generalized rationals.

A more practical definition of the generalized reals is

based on the generalization of the pinary expansion of ordinary

real number given by

\begin{eqnarray}

x&=& \sum_{n\geq n_0} x_np^{-n}\per ,\nonumber\\

x_n&\in& \{0,..,p-1\} \per .

\end{eqnarray}

\noindent It is natural to try to generalize this expansion

somehow. The natural requirement is that

sums and products of the generalized reals

and canonical identification map from the generalized

reals to generalized p-adcs are readily calculable.

Only in this manner the representation can

have practical value.

\vm

These requirements suggest the following generalization

\begin{eqnarray}

X&=& x_0+ \sum_{N} x_N p^{-N}\per ,\nonumber\\

\end{eqnarray}

\noindent where $x_0$ and $x_N$ are ordinary reals, $N$ denotes

infinite integer.

Note that generalized reals can be regarded as infinite-dimensional

linear space such that each infinite integer $N$ corresponds

to one coordinate axis of this space.

The sum of two generalized reals can be

readily calculated by using only sum for reals:

\begin{eqnarray}

X+Y&=& x_0+y_0 + \sum_{N} (x_N +y_N)p^{-N}\per ,\nonumber\\

\end{eqnarray}

\noindent The product $XY$ is expressible in the form

\begin{eqnarray}

XY&=& x_0y_0 +x_0Y+Xy_0 + \sum_{N_1,N_2} x_{N_1} y_{N_2}

p^{-N_1-N_2}\per ,\nonumber\\

\end{eqnarray}

\noindent If one assumes that

infinite integers form infinite-dimensional vector space in the

manner proposed, there are no problems and one can calculate

the sums $N_1+N_2$ by summing component wise manner

the coefficients appearing in the sums defining

$N_1$ and $N_2$ in terms of

infinite integers $M_i$ allowing expression as a product

of infinite integers.

\vm

Canonical identification map from ordinary reals to p-adics

$$x =\sum_k x_kp^{-k}\per \rightarrow \per x_p= \sum_k x_kp^{k}\per ,$$

\noindent generalizes to the form

\begin{eqnarray}

x&=& x_0+ \sum_{N} x_N p^{-N}\per \rightarrow \per

(x_0)_p+ \sum_{N} (x_N)_p p^{N}\per ,

\end{eqnarray}

\noindent so that all the basic requirements

making the concept of generalized real calculationally useful

are satisfied.

\vm

There are several interesting questions related to generalized reals.

a) Are the extensions of

reals defined by various values of p-adic primes mathematically

equivalent or not? One can map

generalized reals associated with various choices of the base $p$

to each other in one-one manner using the mapping

\begin{eqnarray}

X&=& x_0+ \sum_{N} x_N p_1^{-N} \per \rightarrow \per x_0+

\sum_{N} x_N p_2^{-N}\per .\nonumber\\

\end{eqnarray}

\noindent The ordinary real norms of

{\it finite} (this is important!) generalized reals are

identical since the representations

associated with different values of base $p$ differ from each other only

infinitesimally.

This would suggest that the extensions are

physically equivalent.

It these extensions are not mathematically equivalent then p-adic

primes could have a deep role in the definition of the generalized

reals.

b) Could one perhaps regard

infinite-dimensional configuration space of 3-surfaces

or rather, its tangent space, as a realization of the

generalized reals?

Or could one perhaps regard the tangent space of

the configuration space

as infinite-dimensional algebraic extension

of octonions as the 8-dimensionality of the imbedding

space suggests? This kind of identification could perhaps reduce

physics to generalized number theory.

\subsection{Can TGD:eish Universe discover the laws of physics

and consciousness?}

Any theory of consciousness should predict the possibility

of its own discovery. It seems that TGD:eish universe could

indeed discover itself.

\subsubsection{Geometric and subjective memories and self hierarchy}

TGD:eish Universe has the following tools to discover itself.

a) Geometric memory resulting from the fact

that the contents of conscious experience is determined by

the entire initial and final quantum {\it histories},

makes possible conscious simulations of the geometric

time development determined by the deterministic laws

of physics. What results are prophecies of future and

past: that is conscious information about

how universe would evolve in subjective future and

how it would have

evolved in subjective past assuming that quantum jumps had no

effect on the time evolution. Classical physics can be regarded as

a good example of geometric memory.

b) Subjective memory is memories about conscious experiences.

Without subjective memory it would not be possible to even

discover the notion of quantum jump! Thus there seems to

be no reason, which would not allow

TGD:eish Universe to discover also quantum theory and theory

of consciousness.

c) Self hierarchy and summation hypothesis imply an infinite

hierarchy of abstractions and at the top of the hierarchy

is the entire Universe possessing subjective memory

about all quantum jumps occurred.

In order to avoid the question ``What was the initial

quantum history?'', one must assume that the number of these

quantum jumps is infinite. The cardinality

for the number of the quantum jumps occurred could quite

well be larger than the cardinality of natural numbers.

This means that Universe has huge life experience and

it would not be wonder if it could construct rather precise

theory of EveryThing!

\subsubsection{Does a generic quantum jump sequences of the Universe

contain each possible quantum jump infinitely many times?}

A natural question is whether the infinite sequence

of quantum jumps could give complete information

about the probabilities about all possible quantum jumps so that

quantum statistical determinism would be exact at

the level of the entire universe and Universe could

be able to discover the precise quantum laws dictating

its behaviour. This would require

that the infinite sequence of quantum jumps

contain every possible quantum jump infinitely many times!

This requirement is probably too strong.

The presence of the pinary cutoff however provides a natural manner to

classify quantum jumps to equivalence classes such that members

in same equivalence class are identical in the accuracy

provided by the pinary cutoff. Hence the question is

whether each possible equivalence class appears infinitely many times

in the sequence: that is, whether universe has experienced all

possible moments of consciousness infinitely many times.

\vm

Pinary cutoff presumably means that each

class of quantum jumps for each self can be coded by a

a finite integer. Thus it would seem

that each quantum jump of the Universe could be characterized

by an infinite sequence of finite integers, one for each self

characterized by p-adic prime. Thus the information about

the time evolution of the Universe could be coded to an

infinite sequence of reals with each real being obtained

by simply lumping the integers characterizing invididual

quantum jumps to a sequence using some convention to distinguish

between the integers characterizing different quantum jumps.

Thus the question has been transformed to the following

form:

\vm

{\it Does each real in this infinite sequence of reals

contain each integer infinitely many times in the generic case?}

\noindent The concept of lexicon introduced by Calude \cite{lexicon,Zeno}

suggests that this might be the case!

\subsubsection{The concept of generalized lexicon does it!}

Lexicons are

real numbers, whose expansion contains infinitely many times

the binary (or pinary) expansion of any finite integer. Furthermore,

almost all reals are actually lexicons

\cite{Calude}! The generalization of the concept real number

suggests a generalization of the concept of lexicon.

Generalized reals correspond to the points of an

infinite-dimensional real vector space with coordinate axes

labelled by infinite integers $N$. Generalized lexicons

correspond to generalized reals such that each

real component of the infinite-dimensional vector characterizing

generalized real, is lexicon. But this kind of number is expected

to describe

the quantum jump sequence associated with the entire universe!

*>From the properties of the lexicons it would thus follow that each
*

equivalence class of possible sub-quantum

jumps characterized by pinary cutoff

appears infinitely many times in the sequence of quantums jumps

of entire Universe. This would

mean that statistical determinism is realized exactly and that TGD:eish

Universe has good changes to discover its the laws of physics

and consciousness.

\vm

\begin{thebibliography}{99}

\bibitem[Calude and Zamifrescu]{lexicon}

C. Calude and T. Zamifrescu (1998), {\em The Typical Number is a Lexicon},

New Zealand Journal of Mathematics, Volu. 27, 7-13.

\bibitem[Meyerstein]{Zeno}

F.W. Meyerstein (1999), {\it Is movement an illusion?: Zeno's paradox

from a modern point of view.}\\

http://www.cs.auckland.ac.nz/CDMTCS//researchreports/089walter.pdf.

\end{thebibliography}

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