10 March 2006

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Silent, tasteless, odourless and intangible predictability in oriented shades of grey

 

Abstract

 

It is generally agreed that it helps to know something about the past and the present such that successful predictions may occur. How else could the result of a prediction be tested? Four classes of predictability are defined and described and these are summarized as the physical, psychical, their union and the exception classes.

Initially, the physical class allows system predictability, as the function p(t), to be monitored classically as time increases from a reference. The time derivative of a system’s predictability is modeled by the complex function dp/dt = β² e^–iωt and examples are shown that describe its non-linear behaviour.

In a necessarily limited experience of psychic predictability there is the indication that both the total number and rate of occurrence of events, ω, may well be transfinite and we are chaotically doomed in all time – who or what can tell?

An appendix on primitive quantum gravitation indicates that predictability in hyper-dimensional space is a possible route of further investigation.

 

 

Introduction

 

There are many types of prediction but fortunately, they may be grouped and placed into one of at least four classes of predictability. With no particular reasoning the first covered here is the physical class and the collection of statements anent this physics is presented in three sections: Past and present, Future(s) and as an appendix on primitive quantum gravitation.

 

The next section concerns psychical prediction and is necessarily a small collection of statements.

 

Thirdly, there exists the class that contains all combinations or the union of the physical and psychical classes.

 

Fourthly, there is the exception class[1]. This class is not physical, psychical, neither their union nor any other knowable or unknowable class. The information content of this class is paradoxically both null and void yet some of its class members are stunning.

 

In the figure below, both of the physical and psychical classes are represented by lightly shaded regions, their union is the darker segment that relates directly to the amount of overlap and some of the exception class is apparent as the remainder of the image container[2].

 

Figure 1: A diagram to represent the four classes of predictability.

 

0) Physical predictability

 

The class of physics contains procedures, functions, methods and events that do work upon the physical attributes or properties of objects. Physical predictions are numeric such that psychical interpretation may be minimized: the output from the UK Meteorological Office weather forecasting computers initially is a set of numbers that contains for example the surface-level dry-bulb temperature in Kelvin degrees[3]. Since ‘warmth’ is a psychical construct then no concept of ‘warmness’ is attributable to this number at this stage. Indeed, such concepts are only moulded when an individual forecaster attempts to convey their understanding of the overall forecast detail to a more psychic audience i.e. someone other than ‘self’.

 

Physical predictability is a property of time dependent systems that may be expressed as the function p(t) and it is a measure of the likelihood of a system’s next state. The range of physical predictability is -1 <= p(t) <= 1. This range indicates that when p(t) = -1 the system will arrive at a state with opposite characteristics to those predicted; p(t) = 0 means that the system will not arrive at the predicted state; p(t)= ± 0.5 each mean that the system will arrive at one of two predicted states in equal proportion given enough attempts and p(t) = 1 means that it is certain that the system will arrive at the predicted state. In physics, a dimensionless number that changes in time quantifies the property of predictability. The direction of time’s arrow arises in both thermodynamic and quantised systems and Stephen Hawking has described mechanisms to predict the evaporation of rotating black holes[4].

 

a) Past and present

An origin in physics obeys the tenets of General Relativity theory[5], is an arbitrary but convenient point of reference and is, therefore, movable: if it is apparent that the view is made better, clearer or simpler from ‘over there’ then go there.

Each prediction on the outcome of an event leads to a state that appears in one of three sets: affirmations, y; refutations, n; other outcome, o[6].

After many events and sometimes paradigm shifts[7], the outcomes distribute something like:

y ~ ½ the times.

n ~ ½ the times.

o ~ 3rice in a fiddler’s blue moon season ticket holder i.e. randomly, chaotically and intermittently.

The statistics for questions like: “Given a set, how many of type x exist within?” are modeled well by the moments of lognormal or N distributions if both the set and type x are well defined and there are few and sparsely populated illegal states. The dynamic of such statistics was described first by Karl Freidrich Gauss (1777-1855) and the analysis techniques have been refined, applied and adapted by many others since[8].

Time series analysis[9], [10], [11], [12], [13] indicates how N-statistics and predictions may relate. The properties of the required function are first described and a suitable candidate is sought.

The results from many and repeatable observations appear to show that although some events occur in cycles, the predictability of a given outcome decreases with both increasing forecast time and the complexity of the system. It is as if the predictability in a given system, X() is in inverse proportion to its thermodynamic entropy[14], [15].

At a reference time, t_ref, the predictability p(t_ref) is taken as β² and hence a real and positive value: branes, space-times &c with complex or other referential predictabilities are found somewhere near Oz, beyond the Looking Glass, within the halls of Gormenghast &c and are beyond the realm of this analysis (see below). X() will remain with this predictability until an event occurs to change its state. Let f be the rate of occurrence of events in X() per unit time. It is convenient and useful to define ω (omega) = 2πf that incorporates the circular symmetry or periodicity of geometric analysis here.

Observational analysis often shows that the rate of occurrence of events is sometimes overwhelming: it is not discounted that ω à infinity (∞) when the predictability collapses to zero immediately. (Tough luck.) If ω is limited or merely whelming, however, then constraints may be placed on forecast ranges and limits.

Other observations include that each event has its contribution to the outcome: ‘small’ events sometimes have ‘large’ consequences e.g. sustained nuclear fusion, volcanic eruptions, heart failure, straws on the backs of camels and other impact events[16] and vice versa. A statement related to this causal partition is that the slope or gradient in time of the required function should be of an equivalent nature to the function itself.

From the above characteristics it is possible to show that from time t to time t + δt, we may derive p(t) such that in the limit δt à 0, δp/δt à dp/dt and

dp/dt = β² e^–iωt

A result set from this function is shown below as Figure 2. The modulus of the complex predictability function, |p| = Real(p)² + Imaginary(p)², is plotted against the forecast time. For the case shown (ω diminishing as time increases) the predictability is seen to approach zero increasingly rapidly as the forecast time increases beyond one second.

 

a

b

Figure 2: Two views of |p| against forecast time for the case ω à 1 as t à ∞. a) logarithmic ‘x’ and ‘y’ axes; b) logarithmic ‘x’ but linear ‘y’ axis. 

That is, if ever there was a time when a given state was the predictable outcome of an individual event then that time is long since gone, probably.

Some outcomes of paradigm-shifts[17] e.g. inflationary epochs in the standard cosmological model[18] and strange attractor flips[19] include that: interdependence increases with understanding; the picture is complex and non-linear but may be approachable and that although constraints are arbitrary, restraints are plentiful.

Late 20^th century cosmology had the last inflationary period ending ~10^-100 seconds after the beginning of space-time (t=0) and the current maximum future time (t_{thermodynamic heat death}) is ~ 10¹⁰⁰ years after this beginning. Although it is acknowledged that this range is close to nothing when compared with eternity, it is the best guess that that this physics makes, currently.

Other current physical research[20] seeks a mechanism for the observed accelerating expansion of the intergalactic vacuum. The best candidate appears to be membrane theory (M-theory) and the mathematics of eleven-dimensional geometries[21].

 

b) Future(s)

1) Apply Fourier/Gauss analysis, General Relativity, QCD and object-oriented techniques (combined as M-theory) to whatsoever is met with next. For example, a current mania concerns the sustainable production of electrical power.

2) An obvious test of the analysis described is to predict the outcome of an event from many starting points and observe what happens.

For example: What will be the reading, in Kelvin degrees, of an exposed but shaded from direct insolation, dry-bulb thermometer located 10m above sea level near a remote island at a given future epoch?

Before any predictions can be made, the original question must be refined (to reduce the degrees of freedom and, therefore, ω) somewhat. On which planet/island is the experiment to be made? When is the specified date and time? Is the thermometry located over vegetation, concrete, water &c or, remote? Further, the forecasters must be given sufficient time to ensure that they ‘feel confident’ in the statistics of their predictive schemes. Some dry runs scattered over a few years before the actual event, say.

Next, the measurements are taken and the comparisons may begin.

The analysis above suggests that for an earthly island situated between Tropic and Circle latitudes, e.g. South Uist in the Outer Hebrides:

Predictions made more than 10 years before the event will be almost unrelated to the actual conditions encountered.

Forecasts produced a couple of months before the event will correspond to a 1% accuracy level, i.e. within ± 1.5K, or so.

24-hour forecasts will be within 0.1%, ± 0.15K.

Forecast periods of less than 1 second will produce increasingly accurate results.

3) Who cares? If ω is limitless then the universe is at once infinite, chaotic and pandaemonic: all predictions come true since everything exists in an infinite number of places, simultaneously.

 

1) Psychical prediction

 

?

 

1

 

Love.

 

Religious beliefs.

 

This sentence no verb[22].

 

You ain’t seen nothin’, yet.

 

We are all mad about something.

 

The bigger they come, the harder they fall.

 

History is that which one relates to another.

 

Politics is that which can be got away with, without starting a revolution.

 

Because physics isn’t everything, something that travels faster than light in a vacuum may be the cause of it.

 

Trickery is that given the same knowledge of seemingly key events some outcomes are more predictable than others are.

 

A nowhere-near-logarithmic experience of an individual conscious mind, gained through interactions with a few resources from a very small volume of space over a very short period of time, shows that these listed attributes, their counterparts and all combinations thereof exist, apparently simultaneously, within the collective human psyche.

 

2

 

Most of us spend most of our time in close proximity to a massive and rotating spherical magnet that spins in the fields of an ultra-massive, magnetic and rotating collection of planetary and stellar objects that is embedded in the fields of dreams.

 

There exists the shortest, thin walled and parallel-sided tube that connects the poles of any magnet.

 

Statements concerning the human psyche are necessarily psychic.

 

The only Golden Rule is that there are no Golden Rules.

 

There is continuous debate on the property north-ness.

 

On reflection, I could be isolated in this view.

 

C’est si n’est pas une sentence Française.

 

The old ones are the best.

 

Ich bien ein Bin-liner.

 

Rock’n’roll

 

I know.

 

!

 

3

 

         

Conclusions

 

          When some learn to count they start from zero, nothing, <NULL> or 0: others start from 1 and this leads demonstrably and quickly to confusion within their logical psyche. It is unknown to the author if any consciousness learns to count starting from –1, the square root thereof or any other number although it seems most probable.

 

 

Appendix

 

Another test of the derived predictability function is to apply primitive quantum gravitation[23] to X().

 

For large objects that move at non-relativistic speeds, (i.e. << c[24]), Newtonian[25] mechanics provides useful and satisfactory approximations to the motions of objects under the influence of inertial and gravitational forces.

 

This view is extended with a primitive quantum mechanical approach to see gravitation as an attractive force that acts across space-time at speed c by the exchange of virtual (i.e. non-interacting) gravitons. At this early stage, non-relativistic speeds for the masses are maintained. A Feynman[26] diagram for such a quantum gravitational action between the masses Mi and Mj looks, essentially, like:

 

Mi      G       Mj

 

There are two null exchange event cases:

 

0)       For the case that Mi = Mj = G = 0 then only exception events can occur and the system dynamic remains void until a Planck time[27] after such an occurrence.

 

1)       The case that mass Mi (or j) > 0 but Mj (or i) = G = 0 is also trivial[28] and continues until an exception event.

 

If the centres of mass of Mi & Mj are co-located in space-time then we may replace this combined mass with Mk = Mi+Mj that degenerates to the second case.

 

The simplest non-null exchange has events that affect the system X(Mi, Mj, G) where the separation from Mi to Mj is near to a Planck length[29]. This defines a minimum event sphere for X() that is L_p metres in radius and has volume, V = 4πL_p³/3 ~ 10^-103 m³. From a human perspective, V is a very, very small volume.

 

The events that occur in X() are:

 

• G leaves Mi heading towards Mj or conserving parity vice versa.
• G is midway and at all other separations between Mi and Mj.
• G arrives at Mj (Mi).
• An exception event affects X().

 

If exception events are disallowed then we may set the initial predictability p(t=0)  = 1 and the predictability function result remains thus as t à ∞. The dynamic of such systems is described completely by the two-body time dependent Schrödinger equation[30]. The results from the application of this physics are measurable to the highest accuracy.

 

An occurrence of an exception event leads to chaos through bifurcation[31]. The predictability in X() vanishes for a Planck time and the system state is dependent upon both the energy of the exception event and its dispersal throughout the system. That is, both ‘How hard?’ and ‘Where hit?’ need to be known or resolved.

 

The frequency of occurrence of events is proportional to the total graviton frequency, ω_tot = Fn(f_G, f_G’, f_H) per unit time.

 

The initial predictability, p(t₀) is thus inversely proportional to ω_tot and we may set p(t₀) = H/( ћ ω_tot τ) where H is the Hamiltonian (kinetic + potential) energy of X(), ћ is Planck’s constant divided by 2π and τ is the age of X() in units of seconds. If we substitute this for β² in the ‘classical’ predictability function derived earlier, we may rewrite it as:

 

p(t) = H e^-iωt/( ћ ω_tot τ), for ω_tot and τ > 0.

 

This shows that we need to know of the existence of each and every event in X() in order to make successful predictions on future outcomes. That is, predictive outcomes are certain (i.e. they will occur) if and only if all of the previous events occurred in sequence[32].

 

This definition of physical predictability includes, therefore, that any outcome is ‘unknowable’ until after a time when all events are accounted for.

A specific exception event.

 

What if an exception event affecting X() were to lead to the instantiation of a third mass within?

 

To the system X(Mi, Mj, G) introduce Mk at separation L_p from Mi and Mj to take X à X’(Mi, Mj, Mk, G’, Gi, Gj)

 

The sequence of events that takes X to X’ is tabulated below and is as follows:

                  

t = 0, Mk makes contact with Mi through Gi and Mj through Gj.

t = t_p, X(Mi, Mj, G) à X’(Mi, Mj, Mk, G’, Gi, Gj).

 

The simplified Feynman diagram of X’(Mi, Mj, Mk, G’, Gi, Gj) is:

 

Mi               G’                Mj

 

                  

            Gi              Gj

 

 

Mk

 

 

t (tp)	ω	ωtot
0	ωG	ωG
1	ωG’ + ωGi + ωGj	ωG + ωG’ + ωGi + ωGj
2	ωG’ + ωGi + ωGj	ωG + 2ωG’ +2ωGi + 2ωGj
N	ωG’’ + ωGi + ωGj	ωG + n(ωG’ + ωGi  + ωGj)

Silhouette, genetically engineered by thermonuclear fusion.

 

Always park on the bright si-ide of Garth.

 

 “But, the sign said nothing about electric wheel-stakes.”

 

To companion: “9 times six is, what?”

Companion: “half past seven”

 

Not dreamscape, nightmare nor wakefulness

But an image in central Holwick

Soon after sunrise

Winter solstice, or thereabouts

Twenty-nought-nought

Twenty hundred

The year that followed 1999

 

 

Acknowledgements

This work was part-funded by UK Education, EU scientific research and UK & US Military establishment contracts e.g. NNR2/2044/02 and DAJA45-86-C-0001.

 

I would like to thank the following individuals for their encouragement, support, criticism, help, care and/or assistance - J Anderson, EL Andreas, PG Austin, R Botteley, AM Blyth, S Boyadjiev, K Brown, C Burdett, T Cambridge, TW Choularton, N Cleminson, IS Connell, IG Cook, KL Davidson, JB Edson, HJ Exton, J Fenton, MW Gallagher, BA Gardiner, SG Gathman, MJ Gay, H Gerber, AM Harris, MK Hill, ML Hill, M Hopps, SG Jennings, J Latham, G de Leeuw, M Lynch, M Lynch Jr., BJ Mason, CS Mill, JC Nainby-Luxmore, CD O’Dowd, PM Park, K Rees, MH Smith, L Strevens, IM Stromberg, I Wriglesworth and N Yorkston.

 

Others in the families – Allen, Bainbridge, Bell, Boldans, Brewer, Byrom, Crossley, Consterdine, Drewe, Foster, Harris, Hill, Jackson, Laidlaw, Lee, Liddle, Lynch, May, Park, Robertshaw, Robson, Robinson, Smith, Sparkes, Stevens, Strathmore and Kinghorn, Thompson, Vallack, Weet and Wesson I thank, also.

 

Each has contributed.

 

Yours predictably, I guess,

Ian

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Ian E Consterdine

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