December 2, 1993.

Time seems to be the one aspect of Reality that is most puzzling to any that would inquire into its workings. Many different hypotheses have been entertained to explain it, and just as many problems have arisen from them.

The one key ingredient common to all is an attempt to model and thereby explain Change in some form of quantity. (It could well be that time qua time is not a quantity and maybe it is our paradigm that is a problem!) All ideas concerning time involve a means to deal with something that does not yield easily to the rigid systems of taxonomy that investigators employ to classify aspects of experience.

As sentient and self-aware beings, we are very interested in the basic workings of the Objective universe so that we may discover the best methods of causing change in conformity with will. Time is a key facet of the world that I have been investigating for some time now and have reached a point a which I have accumulated sufficient material to put a new hypothesis forward. I will cover only the basic concepts involved so as to introduce the idea.

I will begin by contrasting the two most prevalent theories of time and briefly outline their failings.

The linear or "dimensional" model of time: This model uses a geometric paradigm to construct an explanation. It assumes the traditional ideas of causation such that a single effect always follows a single cause. This is used to generate a unidirectional chain of events one after the other, much like the points making up a line, defining unique dimension along which consciousness is bound to follow.

This model has become fully integrated into all prominent scientific theories today. It has lead to marvelously precise explanations of everyday dynamic processes that have been used to accomplish activities ranging from going to the moon to the construction of weapons of mass destruction.

Unfortunately the linear model of time can not account for our perception of change since it models time as a static a priori chain of events that all "exist" at once beyond our limited sensory perception. It is interpreted to state that change is an illusion and this is used to completely undermine any claims of free choice in our experience.

This model's key logical failing is its inability to reconcile the physical laws with reversible dynamics it supports and the empirical evidence that we have of a fundamental irreversibility that we observe in the nature of time.

The cyclical model of time stems from ancient observations of the seasonal changes in the environment. The idea of endless recurrence of events gave rise to the model of time in which the circle plays a leading role. We find this model very vividly displayed in the calendars of many ancient civilizations such as the Mayan and Indian.

Unfortunately, this model suffers the from the same pathos as the linear model which it precedes. It is unable to account for the irreversibility that is observed in events and it does not allow for alteration in the cycles, thus no room for free-will or choice.

The many resent mathematical studies of entropy have lead to some very clear findings. Resent work has shown that reversible dynamics will not give rise to irreversible increases in the entropy of systems since they would be already at their entropic maximum. The world that we observe does present us with a clear case of a system that is not at maximum entropy, thus we are lead to the conclusion that irreversible processes are at the foundation of the behaviors we see in nature.

To wit: "... There it is shown that for there to be a global evolution of the entropy to its maximal value ... it is necessary and sufficient that system have a property known as exactness. ... Though providing totally clear criteria for the global evolution of system entropy, at the same time these criteria suggest that all currently formulated physical laws may not be at the foundation of thermodynamic behavior we observe every day of our lives. This is simply because these laws are formulated as (invertible) [e.g. "reversible" S.P.K. note] dynamical systems, and exactness is a property that only noninvertible systems may display. [ from Mackey (1992) pg. xi ]

But how to we construct a model of time that is fundamentally irreversible and exhibits the "flow" that we sense? If we use the tools of mathematics which have given us the amazing degree of control over Nature we enjoy today, we find fractals and Chaos theory as likely candidates to the task.

Fractals are a relatively new concept and unfortunately are unknown to most. They are derived from ideas in geometry where integers are used to enumerate the degrees of freedom of movement - the quantities of dimension. Fractals represent non-integer dimensions and have been found to describe best the curves that we observe in objects such as coast lines, trees, clouds, etc.

Chaos theory stems from the study of systems whose behavior is not able to be simplified into the neat linear package deals as offered by conventional mechanical models. The key difference is that they can not be broken down into separate parts, they must be modeled as wholes. If we take two identical chaotic systems and start them up with initial conditions that vary only slightly, their subsequent behaviors will rapidly diverge such that they end up very different. This makes for extreme problems in any attempt to predict their behavior; witness the weather, the economy, the trends of fashions, the typical conversation; all are examples of chaotic systems.

I propose that time be modeled in terms of a fractal curve, much like what is know as a "strange attractor." This model incorporates both the dimension-like qualities of the linear time model and the cyclic-process-like qualities of the cyclical time model. This model of time also attempts to offer a time that "flows" as it were, where the rate of "flow" (its "flux-gauge") is to be determined . This stems from considering that a period of change is determined against a frame of reference that is not changing at the same rate or of the same type.

If consider that we are only aware of interactive events in the world and not the world "in itself," and we note that these are changes to some degree and kind; we should find that it is change that we perceive not stasis. We do not live in a world of static Being, but in a flux of Becoming.

I propose that time, as currently thought of in physics, is not a global Kantian category but that it is a local property of events. Each event has its "own time."

The Nobel prize winner Ilya Prigogine has been exploring something similar in his "internal time" model relating to the number of transformations needed to perform to go from the generating partition to the present partition. (see Prigogine 1984 pg.272-)

It is assumed that the components that make up a changing system show some form of oscillatory behavior. The flux-gauge of these systems is determined in the open systems by the statistical average of change (the period of oscillation), "weighted" by the degree of interaction (the coupling strength), in the systems that it is interacting with; in the closed systems it is arbitrary - which is to say that there are many possible time flux-gauges for such systems, thus many possible "times," each corresponding to a distinct series of events (or in phase spase terms: a different sequence of states).

The time of a system is the flux-gauge against which change is defined in that system. Since it has been proven that a closed system having reversible dynamics can not deviate from equilibrium [e.g. maximum entropy], and thus is incapable of change; only open systems and closed systems with irreversible dynamics may under go change.

Within any system there will be many different oscillations possible, depending on the oscillatory periods of the components and, given a complex closed system, there will be many different flux-gauges depending on the strength of the coupling among them. Another aspect of this hypothesis that I must point out is that "moving" from one flux-gauge to another is not possible unless the coupling parameters of the system are altered, each is in essence separate from all others such that only one set of synchronous flux-gauges are experienciable at a time.

The time flux-gauge variability is initially different from the variability of the time parameter found in the theory of Special Relativity. In it the time parameter variations are solely due to variations in relative velocities and is not dependent on the oscillatory dynamics of the system being examined. But, if I may be so bold to speculate, it could be that the Relativistic dilations, etc. are due to an averaging over - a rounding off --as it were of the many interacting events that make up the world.

Interestingly, a set of open systems that mutually interact is identical to the closed system with irreversible dynamics so that only from the frame of reference of an individual open system is the time gauge not arbitrary.

The fractal nature of this time model rises out of the fact that the oscillatory periods of all of the individual systems do not given rise to the identical repetition of states as idealized in the cyclical time model, each oscillation will end up in a slightly different final state because of alterations of the system and its components during each oscillation since there is feedback in most systems. This gives rise to an irreversability since the amount of alteration is determined by the chaotic properties of the complete system and as illustrated by the "Butterfly Effect," the two state that start out differing a very small amount can end up very different.

The idea of multiple time flux-gauges is not entirely new, it was explored by J. W. Dunne in his book An Experiment with Time and many science fiction writers. In this paper we are only discussing the basic ideas of variable "flux-gauge time."

The essence of this hypothesis is that time as a measurement of change is not due to some chronometer outside a system, it is such that only the components of systems can play the role by providing a background of change having a specific value against which a portion of the system gauges its rate of time flow, e.g. all of the system that it is not determines its flux gauge.

If the entire system's components are synchronized with each other, a components flux-gauge will be the same as any others but that does not imply that it is one unique time, it only appears so under that special condition. If some of a system's components are, shall we say, out of synch, then there will be some ambiguity as to the selected background - kind-of-like a relativity of time. Interestingly this last aspect models the psychological sense of time very well.

I will not attempt to construct a mathematical model for this hypothesis here or get more into detail at this juncture, but I hope that this brief outline will be a catalyst for thought and action.

Addendum: February 25, 1994

Recent work on this time theory has lead me to new developments, to be specific, those relating to the mathematic properties of the periodic systems and the means by which the time flux-gauge is defined.

(EDIT) The flux-gauge or "rate of time" of each system is given by the ratio of the "densities" (Hausdoff dimensions) of the "Origin" system (that system whose flux-gauge we are determining) versus the "reference frame" system which is the average of the densities of the systems that the origin system is coupled to, weighted by the "coupling strength."

Now, how do we define the densities? Well, if we assume that the systems, defined as Poincaré maps of a fractal manifolds ("strange attractors") are en toto isomorphic, the densities are given by the Hausdoff dimensionality of the Poincaré map of the manifolds. These, if considered as "traces" - as discussed by M.C. Mackey - generate irreversible geometric structures that would create the necessary irreversability.

The remaining "big question" is: do the relative angles of the traces to each other play a role; perhaps in the coupling strengths? This would make sense if we consider the coupling strengths in terms of the degree of similarity or isomorphism of the traces (playing the role of the periodic systems) to each other.

Addendum July 31, 1994

One of the questions that I have, up to now, been unable to address was how are the attractors altered so that a static state is not reached eventually; such that we get a continuously varying dynamics. This (slightly modified) quote from Jack Cohen & Ian Stewart's book The Collapse of Chaos may shed some light on this question:

Say we have two dynamical systems each with its own phase space and attractors; if "the two spaces have a very different geography , their individual attractors don't match up nicely , so the feedback between the spaces has a creative effect. It changes both, usually in a rather unpredictable way. Feedback between spaces with different geographies tends to produce new types of behaviour that are seriously different from anything that you will find in either system alone.

For example, suppose the combined system tries to sit on an attractor in ..." space A "that does not match up which any attractor in ..." space B "The ..." space B "... dynamic will try to change the state of the combined system by altering ..." system B "...; the ..." space A "... dynamic will try to preserve the state because in ..." space A "... we have a state that - were it not for the feedback from ..." space B "... - is an attractor. But, because of the feedback, both dynamics are trying to operate, and each is trying to influence the other." pg. 420-1

This modifies the above statement about using isomorphic attractors. If we consider the interaction and feedback between different traces having different dynamics (the "geography") of initially isomorphic attractors we eventually will get very different attractors. The feedbacking process, coupled with a optimization algorithm [which is a form of feedback itself!] would continuously generate new attractors and every one and a while new spaces would form via convergence and divergence of attractors, each new space representing a CIS.

Addendum August 28, 1994

My current model turns out to resemble a neural network. So I have considered that an optimization algorithm may play the role of a Principle of least action in combination with a method to "select" the experienced actuality of a measurement. In The Creative Loop: How the Brain Makes a Mind by Erich Harth, the author discusses how such an algorithmworks in the pattern recognition devices.

His ALopex Optimization Algorithm: xi' = xi + Ù + (last change in x1)(last change in S) (where Ù is a random step and S is the global synchonization amoung the components of a CIS) fits this time model if we take the ramdom step variable to be determined by the weak interactions of systems "outside" the CIS. The xi variables would be the various weighted coupling strengths within the CIS.

Onward to the Unknown

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Bibliography:

- Dunne, J. W., An Experiment with Time (London: Faber, 1934)
- Azeni, Anthony F. Empires of Time: Calendars, Clocks and Cultures (New York: Basic Books, 1989)
- A good survey of the way time was and is currently measured and partitioned by many differing cultures and its repercussions.
- Mackey, Michael C., Time's Arrow: The Origins of Thermodynamic Behavior (New York: Springer-Verlag, 1992)
- For those with the interest in the mathematics of entropy, etc. this is the book to read.
- Gleick, James., Chaos (New York: Penguin Books, 1987)
- The premier book explaining fractals, strange attractors, etc.
- Harth, Erich., The Creative Loop: How The Brain Makes a Mind (Reading, Mass. ...: Addison-Wesley Publishing Comp., 1993)
- Cohen, Jack & Stewart, Ian, The Collapse of Chaos: Discovering simplicity in a complex world (New York: Viking, 1994)
- This most interesting book explores for the lay reader how all of the complexity observed in the world and beyond can arise from the interaction of simple systems obeying very simple rules.
- Heinleim, Robert A., The Number of the Beast (New York: Ballantine Books, 1980)
- This work of science fiction presents an excellent discussion of "multidimensional time."
- Prigogine, Ilya and Stengers, Isabelle., Order out of Chaos (New York: Bantam Books, 1984)
- Strogatz, Steven H. and Stewart, Ian., Coupled oscillators and Biological Synchronization (in Scientific American, Dec. '93 Vol. 269 #6 pg. 102-109)
- This article lays out the all of the basic ideas involved in the behaviors of coupled oscillatory systems and outlines the wide range that this phenomena covers.
- Per Bak, The Devil's Staircase (in Physics Today, Dec. '86. pg. 39-45.)
- This article outline's the connections between fractals, such as the Devil's staircase and coupled oscillating systems and long-range spatially periodic solid structures.