# [time 770] Re: [time 769] Re: [time 767] Generalizing the concept of integer, rational and real, etc..

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Thu, 16 Sep 1999 17:53:11 +0300 (EET DST)

On Thu, 16 Sep 1999, Stephen P. King wrote:

> Dear Matti et al,
>
> Just to let you know, there is a plug-in for Netscape and MS Explorer
> available to view LaTeX and Tex at:
> I interleaved a few questions and comments...
>
> Matti Pitkanen wrote:
> >
> > 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.
>
> Is this equivalent to saying that the TGD:eish Universe allows for its
> own self-awareness? This speaks to the idea that "we are the Universe
> experiencing itself". :-)

Precisely. Universe is the highest self in the hierarchy of selves
and is continually awake since it cannot entangle with anything larger
and thus lose its 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.
>
>
> Could you elaborate on this last point a bit more?
>

One can simply replace m_0 and m_iM_i with |m_0> and |m_iM_i>
in above formula to see the analogy. The usual complex coefficients c_m
of basis states are now either 1 or 0. But this is only
linear space. I do not no natural inner product.

> > \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:
>
> COuld we think of the ratios $R=M/N$ as the harmonics or what ever the
> quantities that occur when p-adic oscillators interact? I am trying to
> examine an old idea of how oscillators behave when coupling is allowed
> between them...
>
Difficult to say. Point is however that finite generalized rationals
differ only by infinitesimals from ordinary ones.

> > \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.
>
> So you are using infinite integers to act as basis vectors? How would
> the rules of vector algebra be change by their use, such as cross, dot,
> and scalar products? Could we define Hilbert spaces with them?

X can be regarded as linear space with real coefficients.
One could define formally inner product as

x_0y_0 + SUM_N x_Ny_N

which would make extended reals real Hilbert space. Same applies
to extended complex numbers.

One can extend also quaternions and octonions.
The tangent spaces of spacetime surface and imbedding
space could be regarded as extended quaterions and octonions
and it could be that the tangent space of configuration space of
3-surfaces could be regarded as extended octonions!

I do not know whether inner product is useful or not. There
is real inner product equal xy equalto ordinary real inner product
apart from infinitesimals for *finite* generalized reals.

Note that the infinitisemality of p^(-N) can be
forgotten in algebraic manipulations totally. These objects are just
symbols.

> > 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.
>
> Could we relate Bill's idea to this?
>
About infinite products. Difficult to say. I however think we are
speaking about very different things.

> > \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}
>
> Would there not be an uncountable number of choices?
>
Yes since also infinite primes are possible.

> > \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.
>
> If the extensions are not equivalent, would there exist a
> transformation to make the equivalent when applied?
>

The transformation is written above explicitely: one just
replaces p_1 by p_2 in the expansion of *generalized real*.
The equivalence is obvious for finite generalized reals.
But for infinite generalized reals situation is not same
since the ratio for infinite norms of p_1 and p_2 real
is zero or infinite.
One might however argue that in predictions for S-matrix
elements only *finite* reals appear and therefore physically
p_1 and p_2 generalized realized are equivalent.

> > 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.
>
> That is very interesting! Could you elaborate on extended octonions?
>
One just takes ordinary finite octonsions o and writes the
previous expansion as

O= o_0 +SUM_N o_N p^(-N)

Sum is performed componentwise and product is

OP = 0_0p_0+o_0P+Op_0 + SUM_NM o_Np_M p^(-N-M)

Everything reduces to the multiplication of finite octonions.
Order of multiplication is important and associativity is of course
lost.

> > \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.
>
> To me this implies that subsets of the Universe (totality of existence)
> can model the whole is some way. I think of this an information
> compression scheme...

Pinary cutoff is involved also now.
>
> > \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.
>
> Could we think of "deterministic laws" as the result of mapping real
> valued statistical dynamics into the pinary dynamics? How do you model
> the thermodynamic entropy generated by the computation required for
> constructing the "prophesies"?
>
In my approach prophecy is not a result of modelling! Prophecies are
induced by the fact that quantum jump must be determined by the initial
and final quantum histories. Of course, one could try to model
prophecies computationally. This would mean solving field equations
associated with the absolute minimization of Kaehler action.

> > 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.
>
> Does not this "subjective memory" of the entire Universe "exist" only
> in the limit of infinite time? We need to look carefully at the
> cardinality of the infinities that we are playing around with! Which are
> countable, uncountable, undecidable, etc. We know from Cantor that there
> exist an infinity of different cardinalities!
>

The memory of universe has infinite span with respect
to both geometric time and subjective time. Infinite primes
and infinite reals give grasp to the what infinite time intervals
means. Time interval which is infinite but bounded!

Infinite primes define infinite hierarchy of cardinalities.
The entire cosmology at given level of hierarchy is
literally infinitesimal moment of Big Bang for the
next hierarchy.

What is really amusing that the asymptotic cosmology and very early
cosmology are both string dominated
as if our asymptotic cosmology would know that it
must smootly join to very early cosmology at the next
level of infinity!

> > 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?}
>
> I see this as saying that each generic quantum jump contains all
> possible descriptions of a quantum jump. This is like a message that is
> in all possible languages simultaneously!

Something like this!

>
> > 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.
>
> We must remember that we must include all possible representations of
> those "experiences", the "many languages" notion speaks to this. Not all
> observers have the same language, even if they agree on what numbers are
> prime!
>
p-Adic numbers provide preferred 'language' for self characterized
by p-adic effective topology p. One must express
integers characterizing individual quantum jump in pinary basis:
n= SUM_n x_np^n. Everything is simple in
pinary basis. Note that p-adic counterparts of
extended reals are certainly *not* equivalent!

> > \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:
>
> We should be able to construct a diagonalization over these finite
> integer codings, thus their number must be nonenumerable!

The number of reals is infinite and nondenumerable:
infinite integers label the coordinate axis of infinite-dimensional
linear space defined by extended reals and there
is huge number of infinite integers. Something totally unimaginable!

>
> > \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.
>
> Are the reals that are not Lexicons "regular"?

I cannot say: I do not know the definition of regular lexicon.

>
> > \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}
>

Best,
MP

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