Stephen P. King (stephenk1@home.com)
Tue, 22 Jun 1999 20:16:05 -0400
Dear Matti,
Matti Pitkanen wrote:
snip
> Matti Pitkanen wrote:
> >
> > Dear Matti,
> snip
> > [Stephen]
> > About dissipation and consciousness: I believe that the two processes
> > are the reverse of each other! This notion is inspired by Roger
> > Penrose's argument in The Emperor's New Mind were he shows that the
> > annihilation of information by black holes is balanced by the creation
> > of "flow lines" in a phase space container representing the entire
> > universe.
> >
> > [MP]
> > Yes. Dissipation and consciousness are reverse in the sense that
> > gain of conscious information means loss of unconscious information
> > defined as information of quantum history. Hence dissipation is direct
> > measure for consciousness. The question is only about length scales:
> > in which length scales dissipation occurs. Only in atomic or perhaps
> > also in longer length scales.
>
> [SPK] There is an equation in information theory where dissipation is
> measurered in terms of free energy and information entropy that I need
> to look up...
From John R. Pierce's "An Introduction To Information Theory: Symbols,
Signals and Noise." Second, revised edition, Dover Books, 1980. pg.
205-206
"Suppose that a physical system has at a particular temperature a total
of m states. Suppose that we divide these states into n equal groups.
The number of states in each of the groups will be m/n.
Suppose that we regard the specification as to which one of the n
groups of states contains the state that the system is in as a message
source. As there are n equally likely groups of states, the
communication-theory entropy of the source is log n bits. This means
that it will take n binary bits to specify the particular group of
states which contains the state the system is actually in. To transmit
this information at a temperature T requires at least
.693 k T log n = k T log_e n
joule of energy. That is, the energy required to transmit the message is
proportional to the communication-theory entropy of the message source.
If we know merely that the system is in one of the total of m states,
the entropy is
k log_e m
If we are sure that the system is in one particular group of states
containing only m/n state (as we are after transmission of the
information as to which state the system is in), the entropy is
k log_e m/n = k (log_e m- log_e n)
The change in entropy bought about by information concerning which one
of the n groups of states the system is in is thus
-k log_e n
The corresponding increase in free energy is
k T log_e n
But this is just equal to the least energy necessary to transmit the
information as to which group of states contains the state the system is
in, the information that has lead to the decrease in entropy and the
increase in free energy.
We can regard any process which specifies something concerning which
state a system is in as a message source. This source has a certain
communication-theory entropy per message. This entropy is equal to the
number of binary digits necessary to transmit a message generated by the
source. It takes a particular energy per binary digit to transmit a
message against a noise corresponding to the temperature T of the
system.
The message reduces our uncertainty as to what state the system is in,
thus reducing the entropy (of statistical mechanics) of the system. The
reduction of entropy increases the free energy of the system. But, the
increase in free is just equal to the minimum energy to transmit the
message which lead to the increase of free energy, an energy
proportional to the entropy of communication theory.
This, I believe, is the relation between the entropy of communication
theory and that of statistical mechanics. One pays a price for
information which leads to a reduction of the statistical-mechanical
entropy of a system. This price is proportional to the
communication-theory entropy of the message source which produces the
information. It is always just high enough so that a perpetual motion
machine of the second kind is impossible.
We should note, however, that a message source which generates messages
concerning the state of a physical system is one very particular and
peculiar kind of message source. Sources of English text or of speech
are much more common. It seems irrelevant to relate such entropies to
the entropy of physics, except perhaps through the energy required to
transmit a bit of information under highly idealized conditions.
There is one very odd implication of what we have just covered.
Clearly, the energy needed to transmit information about the state of a
physical system keeps us from ever knowing the past in complete detail.
If we can never know the past fully, can we declare that the past is
indeed unique? Or, is this a sensible question?"
> Wed, 03 Feb 1999 10:15:40 -0500
> [SPK]
> > http://www.math.washington.edu/~hillman/postings.html
> >
> > "The Causal Symmetry of Shannon's Entropy. My claim that Shannon's
> > entropy exhibits a ``causal symmetry'' proved to be a bone of
> > contention. This file strings together several postings of mine
> > defending that claim, beginning with the post which contained my
> > original statement of the symmetry."
> [RF]
> This is easy to show. John H. Karl does a good job of it in
> "An Introduction to Digital Signal Processing" in the pages on
> maximum entropy. Essentially the Shannon entropy is the average
> information per unit of time.
>
> [Matti]
> Information density in time direction.
[SPK]
We need to also model spatial extensions in information terms. My
recommendation of Amiri's book in [time 414] A possible text to use to
work out Interaction model. An analogy that I use in discussing the idea
of constructing space-times is the process that occurs in a conversation
between persons that have differing native languages and environments.
At first, it is very difficult to exchange meaningful information, there
is only the commonality of the interface establishing a connection.
Meaningful information has been historically defined in terms of
similarities in the repertoires of behavior of the conversationalists.
The ideas of mutual information or cross-correlation and mutual entropy
are thus useful.
The key idea is how an extended conversation is used to generate models
of the environments of the conversationalist. More below.
> [RF]
> It really refers *only* to non-deterministic data:
>
> h=-K \sum_i^n P_i ln P_i
>
> Shannon derived the following based Norbert Wiener's work in solving
> the problem of predicting time-series in the frequency domain
> (Weiner-Khintchine theorem)
>
> h ~ \int_{-\pi}^{\pi} ln Phi(w) dw
>
> over the Nyquist interval (from -PI to PI), where Phi(w) is the
> power spectrum so that h is proportional to this integral. He did this
> because a "stationary time series" has no Fourier transform but it does have a
> power spectrum! Stationary time series are ones which have invariant
> statistical properties that hold up across ensembles (the collection of all
> possible realizations).
>
> But, this integral will diverge to negative infinity if the power
> spectrum is zero over any finite bandwidth.
>
> So the necessary condition is that all non-deterministic information
> in a time-series must obey h > -\infty. This is the Paley-Wiener
> condition.
>
> Looking now to the convolution theorem, the power spectrum is
> expressible via the convolution symmetries in terms of the Fourier
> transform of the the autocorrelation:
>
> FFT(f* x f) = F* F = |F|^2
>
> where FFT(x) is the Fourier transform of x. and f* x f is the
> autocorrelation of f in the time-domain, and F is the Fourier transform of f.
>
> This, as you noted earlier, has the time reversal built into it in
> terms of the complex conjugates f*.
>
> The cross-correlation:
>
> f* x g
>
> is similar to "convolution" but the difference is that the ordering
> of the time-series is not reversed as in convolution. Convolution is
> therefore commutative, but cross-correlation is not. They are
> time-reversed versions of each other !
>
> This should give you a good idea of the link between time-reversal
> and Shannon entropy.
[SPK]
> This is very important to understand carefully because it applies
> to so much of what we are discussing.
> [MP] Let me try to translate this to TGD framework. What is possibly
> nontrivial is related to multiverse nature of quantum history.
>
> The information in question is nonlocal and about spacetime.
> In quantum TGD frameowork one can certainly construct information
> measures giving information about spacetime surface (involving, say,
> autocorrelations of classical fields). In quantum
> context one would consider quantum expectation values
> for this kind of information measures.
[SPK]
This 'nonlocality' aspect of spacetime information, is it nonlocal in
the sense that the information encoding the topology of a manifold
and/or the orientation of a Penrose tiling is non-local, e.g. not
determinable from a survey of the infinitesimal neighborhood of a point.
[MP]
> Time reversal is with respect to the geometric time associated with
> the spacetime surface. At quantum TGD level this reversal is only
> indirectly related to the fundamental psychological irreversibelity
> is with respect to subjective time ticking quantum jump by quantum jump:
> this time only approximately corresponds to the increasing value
> of geometric time.
[SPK]
Umm, if we assume many almost disjoint spacetime surfaces there would
be associated many almost disjoint geometric times. Frankly, I find the
idea of a geometric time to be a construction, not an a priori
synthetic. Thus, I am calling into question the entire idea of a priori
space-times that has been advanced in physics for years.
I think that Descartes' introduction of his co-ordinate system with its
mapping of the Real numbers to idealistic points in a pre-existing
"space" and the subsequent idealization by Newton et al of time as a
mapping of R^1 to the parameters of change of mechanical systems is the
idealism that I am speaking against. The fact is that such an
idealization is merely a model, it is not an a priori synthetic fact.
Kant and other philosophers, especially Bergson, recognized this, but
alternatives to the Cartesian model have only recently been in
development. David Bohm is one of the pioneers of this endeavor. I am
trying to carry it further in my limited way. :)
The idea of a quantum jump I find very useful to work out the details
of a praxic model, as an alternative to the mechanist naive realism that
is pervasive in theoretics. The main point that I am advancing is that
the "subjective time ticking quantum jump by quantum jump" is "clocked"
by the scattering propagator of the LS. This makes it truly "subjective"
in that the subject is the finite Local System. It is an "observer" in
the sense that each has a partially ordered set of observations
associated to it. I am identifying a quantum jump with an act of
observation and an observer with a partially ordered set of quantum
jumps.
[MP]
> Irreversible time development at spacetime
> level emerges when one replaces sequence of quantum jumps between quantum
> histories with single dissipative quantum history: in particular, when
> one replaces irreversible time developments of classical fields
> on spacetime level by single dissipative time development of classical
> field in fixed average spacetime. This is effective description
> replacing family of spacetime surfaces with their dissipative
> 'almost envelope'. Fundamental description assings
> dissipation to the sequence of quantum histories.
[SPK]
I see this "replacement" with the averaging over many sequences of
quantum jumps. It is an idealization, but a useful one. My thinking of
our common world is constructed by such an averaging procedure! It
creates the illusion that we all exist in a single finite space-time
with a black-body background radiation and an apparent event horizon
that is assumed to be caused by a Big Bang singularity.
In the same way, I see "classical" notions as convenient fictions used
in our modeling of the world. Umm, I need to better understand your
meaning of the word "dissipation", you do talk more about it below...
snip
> > [MP]
> > You are quite right.
> > For me understanding of pinary cutoff was pleasant surprise since it is
> > forced by real to p-adic quantum TGD correspondence and I regarded
> > the concept as unsatisfactory. It is counterpart of length scale cutoff
> > of QFT:s and there are unpleasant associations about infinities.
> > It seems that quantum TGD proper and TGD inspired theory of cs are
> > now converging to single theory.
>
> [SPK] They should converge! :) An essential notion of any "quantum"
> theory must have an explanation of what an observer is. :) I am very
> interested in the nature of "length scale"!
> Say we have a huge number of observers, each with their own
> standard of length and duration: a clock of their own. Assuming that it
> takes a non-zero duration for each observer to compare their standard to
> all of the objects on their environment and that the comparasons
> (measurements) can not be done simultaneously without some cooperation by
> the observers with each other. But there is a difficulty!
> It is usually thought that the standard of one observer can be
> transformed via a number of finite steps into one identical to that of
> another, thus we can think of this procedure as an "algorithm". We then
> rephrase the above story in terms of Peter's new computer theory...
>
> [MP] The idea of observer makes sense as a high level abstraction
> (like clock and computer or dissipative dynamcis) but I am sceptic about
> observer as *fundamental* concept. I am happy with observation/moment of
> consciousness as a fundamental concept. But this is not a news
> anymore(;-).
>
> This comparison of standards: is it really needed? Or does our brain
> perform something like this?
[SPK]
The very definition of meaningful conversation implies a comparison of
standards! I would say that our brains perform this comparison as
"pattern recognition"! Assuming that all observers have identical
standards is as unrealistic as assuming that there is a fundamental
language for all observers. (I define observers as posets of
observations associated (mapping) with posets of material particle
configurations (thus dualistic!) in a mapping unique to the observer.
The "consciousness" of the observer is identified with the mapping, not
the information or material structure, I apologize for not explaining
this earlier... :) This is analogous to how a language qua acts of
communication is an agreement between persons, not a "thing in-itself".
This tacitly implies that those involved in the communication acts have
similar conventions and similar enough framings ("clockings"). This idea
is why I am asking to discuss Weyl's original gauge theory!
The notion that intervals are not integrable for all possible geodesics
should not be so controversial! Then I say that space-times are
constructions and not a priori structures, it is implied tacitly, that
there is a choice in the metric, interval and and inner product used to
define the manifold in question. This is a property that is unique to
3-dimensional manifolds! To say that there is a Universal standard of
length is a nice idealistic assumption, but it is a holdover from
Classical physics. Gravity is defined in terms of the non-intergrability
of angles, and Weyl attempted to model electromagnetism by a
non-integrability of the interval. Thus does generate a "smearing" of
the spectra of atoms, but if we consider the possibility that LSs, as
quantum systems, can only absorb or emit discrete quantities, thus the
smearing would not be observable directly, but it would have observable
effects!
It would, on the other hand, create the appearance of a loss of energy
(conservation of energy violation!) in particles that I think is
geometrically equivalent to a Robertson-Walker metric, and thus the
so-called doppler shift of the spectra of light proportional to distance
is an effect of the Weyl gauge invariance. If this idea is wrong, I
would like to know! It is just a crazy notion that I would like to
dismiss if it is impossible. :) My "crazy idea" does contradict the
usual way that conservation laws are defined, but I firmly believe that
Noeter's theorems need to be reexamined! Since time is not a Universal
property, the idea that time symmetry use in the formulation of the
conservation laws is suspect.
In the definition of time, following Hitoshi's model, we also see that
the notion of conservation laws using the notion of time also need to be
redefined! Everything must be re-examined! I suspect this is why
Hitoshi's model has received such a cold response! :(
> > [Stephen]
> > I highly recommend Michael C. Mackey's book Time Arrows. If you can't
> > get it from the library let me know any I will send you copies of the
> > relevant parts. His "God Theorem" is very important for your ideas
> > relating dissipation to consciousness. It proves that an invertible
> > system U can have subsystems U_i that are not invertible and are thus
> > dissipative and irreversible. The way that mapping between the Real
> > valued states and p-adic valued states occurs is indicative. I do not
> > know how to represent this mathematically....
> >
> > [MP]
> > It would be interesting to learn about Mackey's thoughts: my basic
> > philosophy is of course somewhat different: entire U is invertible in
> > my approach. In your and Hitoshi's approach situation is different. In
> > any case, dissipation is for me a direct experimental proof for quantum
> >jumps between quantum histories concept.
>
> [SPK] We agree with Mackey on this: U as a whole is invertible! It is
> the finite subsets of U, the Local Systems that are not invertible! I see
> your quantum jumps as instances of interactions among the LSs. The key
> is to work out a model of how these "jumps" behave to generate the
> illusions that we communicate about: our "common reality".
>
> [Matti]
> Really sorry to say this: we do not!!(;-) It was only a typo! My
> intention was to write non-invertible but 'in-' somehow led me to think
> that 'non-' is not needed. It would be wonderful to agree with someone at
> least once in life!
[SPK]
So you say that U (the Totality of Existence) is not invertible? How do
you argue this? Existence is by definition tenseless, it has no
associated time, or spatial extension for that matter! It just exists,
it has no properties in it-self (a priori synthetics). *All* properties
(other that *existence*), are derived from the interactions of U's
finite subsets. The "consistency implies existence" postulate affirms
this, as it is inconsistent to postulate that "Existence non-exists" or
"existence is nothingness" unless we recognize the fundamental dichotomy
involve with the were idea of Existence: in-it-self Existence contains
its own opposite. I identify it with the point of maximum fuzziness at
the center of the fuzzy hypercube, that is framed at its vertices by the
infinity of crisp Boolean variables. For more on this I recommend Bart
Kosko's books and papers...
[MP]
> But I could perhaps agree (with certain preservations of course!(;-)) that
> quantum jumps can be seen as interactions between LS and its complement.
[SPK]
Yes! :) But we need to look carefully at the relationship between an LS
and its complement! The complement of a given LS is identified by
Hitoshi with a set of center of mass "classical particles" with are the
exteriors of other LSs. We do need to explore the details of this! I
have been saying that there is a non-Hausdorfness among the sets of
complements of a given LS. This is the mathematical representation of
the idea that our common world is constructed by the overlapping of our
individual subuniverses, i.e. finite subsets of U that are the
complements of an LS.
> [MP]
> > There is also interesting connection with self organization. Self
> > organization can be understood as iteration. Iteration creates fractal
> > like fixed points: for instance dissipation without energy feed leads
> > rapidly to the state in which nothing moves.
>
> [SPK] Look at the discussion of fixed points in Peter's work! Is this
> "dissipation without energy" called "adiabatic"?
>
> [Matti] Interesting question. 'Dissipation without energy feed' was
> meant for situation in which there is no external energy feed
> to the subsystem so that system dissipates its energy and 'comes
> at rest'. "Adiabatic" in thermodynamics means "no entropy generation".
> What would this mean in quantum context?
[SPK]
Your "dissipation" idea seems to be strongly related to "dissipative
structures", thus my ideas stated above need some qualification! :) What
you state here appears to me to be the situation in classical
thermodynamics were a systems is disturbed and they allowed to
transition to equilibrium. I have read a book that tries to correlate
all irreversibility, especially that of time, to this situation. It is
interesting! :)
I would be interested in the quantum context definition. Perhaps it
would have something to with the difference between pure and mixed
states.
[MP]
> If subsystem, which is sufficiently entangled with the surrounding world
> develops without participating in quantum jumps, it would not have
> moments of consciousness and no dissipation: same for its subsystems.
> Entanglement would shield subsystem from the occurrence of dissipation.
> Could this be a synonym for adiabaticity? Time development would roughly
> mean iteration by U.
[SPK]
Invertible or reversible computation would have this property!
> Could the system end up to a fixed point of time development operator
> U==U_a, a --> infty and would not generate entanglement entropy anymore.
> Just a thought....
[SPK]
I do think so! Umm, I think I am understanding your idea here
symbolized by "U==U_a, a --> infty"
Is the "==" your symbol for equivalence usually show as three parallel
horizontal line segments?
This definition of U is different from the one I have been using. You
are using the idea of a potential or asymptotically approached infinity
in your definitions of the Universe and I am using an actual infinity.
This is an important difference! It distinguishes the notion of "the
Universe as its experiences of its subsets" in the former that you us
and my definition which is abstract metaphysics :) for the Universe in
it-self. Umm, this dichotomy may be used to define the subject-object in
fundamental terms! :)
> [MP]
> > The informational time development operator U==U_a, a--> infty indeed acts
> > as iteration in good approximation on subsystems which do not suffer
> > quantum jump. N quantum jumps corresponds to U^N in good approximation.
> > This would mean that dissipation of energy leading to fixed point, limit
> > cycle etc.. would indeed be iteration basically. I am beginning to look
> > for more details related to this.
[SPK]
I think that this kind of idea is actually used in attempt to construct
a "time operator" for a QGR theory. Most physicists do not like the idea
of discontinuities inherent in the notion of quantum jumps, except for
Penrose! I see no problem, if and only if the discontinuities arise from
the fact that observations are by their very nature discontinuous, since
they are mappings between finite subsets of the Whole U which is not
discontinuous. The fact that reals are used to identify the points in
co-ordinate systems and in topology, instead of p-adics, I believe, is
the reason that the error prevails. :(
This notion that "N quantum jumps corresponds to U^N in good
approximation." to me, corresponds closely with Hitoshi's definition of
uncertainty! :) Since finite systems have an upper bound on the accuracy
with which that can be mapped to infinite sets of systems, it follows
that our models must take this asymptotic approximation notion into
account.
> > [Stephen]
> > The irreversibility of quantum jumps as an action of collapsing
> > the many possibilities down to one actuality is in a fundamental way the
> > reverse of a dissipation (like the thermodynamic evolution of a system's
> > phase space) that maps one flow to many. Perhaps "flow" is the wrong
> > word... The key is that we have dual semigroups of dynamics, one
> > semigroup representing the evolution of consciousness and the other the
> > evolution of dissipation. Both involve a "time" but they "flow in
> > opposite directions".
> >
> > [MP] I think the best manner to say this is to say that information
> > of quantum state is transformed to conscious information and destroyed
> > as unconscious information. But there is paradox involved: entropy
> > is potential information and it seems to be a matter of taste whether
> > to speak of entropy or potential information. This is like creativity.
> > Discoveries are not possible without diverging period when one generates
> > counter arguments and is totally lost in fog!(;-)
>
> [SPK] Umm, I am reminded of how the key irreversibility (and heat
> generation!) in a computer is the act of errasing the memory... There is
> work being attepted to minimize, if not eliminate, this heat by never
> errasing the memory. The idea is called "reversible computing". I had a
> brief correspondence with a nice fellow at MIT that was working on this.
> I told him that he was chasing a chimera! :)
>
> [MP]
> In quantum jump informational time evolution U could be regarded as
> quantum computation lasting for infinitely long lightcone proper time
> (but only a single moment of subjective time).
[SPK]
Yes, that follows. ;) In order to compute the Least Energy
configuration of a finite concurrent FC system's spacetime we must allow
for an arbitrary number of discrete quantum jumps, and this for every
moment of subjective time. So we are associating to each "moment" of
subjective observations a potentially infinite quantum computation which
involves the entanglement of of the subsets of the FC observing system,
which I am saying is an LS.
Since there is a computation is involved in the construction of the
Least Energy configuration it follows that extremal geodesics, what
define the space-time, are also constructed by the same action! Thus I
say that space-time are constructions! :)
> Ideal quantum computation would mean single U followed
> by quantum jump for the entire computer.
> The problem is that quantum computer contains several subs-system
> complement pairs besides the desired one and some undesired pair can win
> entanglement entropy generation race and make quantum jump.
[SPK]
This notion of a "race" I like! It is very similar to my thinking,
inspired by Penrose's "one graviton criterion", that the quantum jump is
a "tournament" where the competitors are the configurations of the LS. I
have been working on an idea based on "gossiping on graphs" that, from
conversations with graph theorist, are equivalent to tournaments. The
ideas involved sequences of pairings such that information or energy
differences are equilibrated or evenly distributed in a step or
piece-wise discrete manner.
[MP]
> One must be very careful with 'reversible'. Time development
> operator U_a for entire universe is reversible just like Schroedinger
> time evolution. In general it however changes the information content
> (say increases information about the position of Universe=3-surface in
> configuration space).
[SPK]
You really must read Mackey's book! He shows that invertible dynamics
can not, in them-selves give rise to irreversibility! But, "traces"
which are a type of subset of an invertible Whole, can have irreversible
dynamics. Mackey also discusses some of the same mathematical objects
that Freiden uses!
Hitoshi uses a variation of the Schroedinger Operator to derive the
propagator of an LS, which defines its "clock" and scale gauge. But, I
am not sure that his propagator is f* exact, as required by Mackey's
proof! I am corresponding with Mackey about this.
[MP]
> Question: What could be the role of informational time development in the
> understanding of memories? Information currents are involved with
> memories.
[SPK]
This is a very interesting question! :) My dyslexia is caused by a
short term memory persistence problem (I think), so I am very interested
in this! One of my reasons for advocating a matter/information dualism
is that matter acts as a persistent "medium" in which to encode
information. It is very difficult to explain how information is merely
an "epiphenomena" of matter and it is just as hard to explain how matter
is an 'epiphenomenon' of information (or mind). If there is an
"epiphenomenalism" at all, it is easily seem as a detail of the dualism,
we say {Information is epiphenomena of matter} <=> {Matter is
epiphenomena of Information}, where "<=>" is the duality symbol. Either
mode of thinking is correct, they are "complementary"!
The main and most important point is that the dualism only hold between
the subsets of the Universe. The Universe is neither matter nor
information! These are properties that are meaningless in a tenseless
context, thus only apply to finite subsets of the Universe, e.g. LSs and
their complement Space-times.
> > [Stephen]
> > The unity of the two is realized at the Grundlagen level of the
> > Totality, which is one. It has no dynamics in itself as seen by the fact
> > that it has no time associated. Mackey's proof that invertible systems
> > have no time (he does not say this exactly but it is implied) while
> > non-invertible systems will have time.
> >
> > [MP]
> > This is realized very precisely in TGD. Without the classical
> > nondeterminism of Kaehler action classical dynamics would be invertible.
> > There would be no time. There would be no consciousness since in von
> > Neumann inspired scenario since only quantum entanglement between
> > cognitive and material spacetime sheets can be reduced by quantum jumps
> > in this scenario.
>
> [SPK] I wish I understood Kaehler action better. :( I don't know what
> this term means! I also do not comprehend what "cognitive and material
> spacetime sheets" are. The ideas that I have that seem to correlate to
> them are the dual aspects of a Chu space: SET and anti-SET, Mind and
> Body, Information and Matter. The pictures in my mind of these looks
> very similar, but the devil is in the details!
>
> [MP]
> Perhaps you are trying to see Kaehler action from
> too computational point of view. Chu spaces seem to be rather
> non-geometric concepts: closely related to formal logic, symbolic systems
> and these kind of things: unfamiliar to me. Kaehler action, cognitive
> spacetime sheets, etc. are rather concrete geometric concepts. There
> are 2-dimensional illustrations on my homepage: they might help.
[SPK]
I am trying to discuss these ideas in a complementary fashion. The
computational view is like the abstract aspect of praxic (process)
algebra, the geometry speaks to concrete aspects like concurrency and
space-times. The key is that they complement each other. :) I will read
more of your papers. :)
Onwards to the Unknown,
Stephen
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