We have, then, at any instant a three-dimensional grid extending in all directions composed of locations which can receive one and only one ultimate event. I mentioned in the last post (Co-ordinate Systems) that the best rough model of physical reality would be a three-dimensional framework traced out by lights within which pinpoints of coloured light, representing ultimate events, would occasionally make their appearance.  The entire three-dimensional framework with rectangular axes is set up to flash on and off rhythmically and when the framework of axes disappears, so do the coloured lights.  In the majority of cases the pinpricks of light never appear again. However, occasionally the pinpricks do reappear and, if the light machine is speeded up like a cine-camera, the coloured pinpricks eventually coalesce and form lines of coloured light, either straight or curved. This represents the case when ultimate events acquire persistence (to be discussed later) and form a repeating event-chain.
We consider  the simplest case of a single repeating ultimate event, one that reoccurs identically at each successive instant. Also, for simplicity, I shall reduce the three-dimensional grid to a single line. The trajectory is thus traced out as a sequence of occupied squares on a repeating array of lines stretching out indefinitely in both directions. The ‘line’ is not a material object, of course, it is simply a set of positions where ultimate event can have occurrence. When empty, a position on this line will be marked □ and when occupied by an ultimate event will be marked  ■.    We have :

…□□□□□■□□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□□……..
……………………………………….

Now suppose that this ultimate event creates a facsimile of itself in an immediately adjacent cubicle and that, at all subsequent instants, the ‘daughter event’, marked in red, is displaced to the right by a single cell. ‘Right’ and ‘left’ are, of course, purely conventional since the substratum on which the ultimate events have occurrence is, by hypothesis, homogeneous and isotropic, i.e. directionless, but  once a one direction has been selected we keep to it. We have :

…□□□□□■□□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□□□□□□□………..
…□□□□□■□□□□□□□□□□□□□□□□□□□……..
…………………………………………………………..

Joining up the coloured squares, we obtain in each case a straight line and we can easily imagine other straight lines representing different event-chains that differ only by the number of squares each diagonal  moves to the right at each successive instant. (The more complex case when an event-chain ‘jumps’ lines will be considered later.) Thus :

…□□□□□■□□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□□□□□□□□………..
…□□□□□■□□□□□□□□□□□□□□□□□□□□□……..
…………………………………………………………..

Each chain can be commenced and extended indefinitely if 1) we specify the starting position which we denote ☼ (in the above it is the same for all three chains), and 2) we specify the original increase in a chosen direction, either no increase at all or □, □□, □□□ and so on (Note 1).
This family of event-chains is to be distinguished from event-chains where either the increase is completely arbitrary or in some way depends on the current position of an event in a growing event-chain such as :

…□□□□□■□□□□□□□□□□□□□……..
…□□□□□□■□□□□□□□□□□□……..
…□□□□□□■□□□□□□□□□□□□……..
…□□□□□□□□□■□□□□□□□□□□□……..
…□□□□□□□□□□□■□□□□□□□□□……..
…□□□□□□□□□□□□□□■□□□□□□□□□□□………..
…□□□□□□□□□□□■□□□□□□□□□□□□□□□□……..
…………………………………………………………..

It is probable that the above event-chain, or rather its beginning stages, can be given by a formula, and probably by more than one, but, whether or not this is so, such an event-chain belongs to a completely different category from the ones shown above. Why is this? Because the evolution of such an event-chain cannot be gauged from two arbitrary successive positions only.
Now consider
…□□□□□■□□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□□□□□□………..
…□□□□□■□□□□□□□□□□□□□□□□□□………..
…□□□□□■□□□□□□□□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□□□……..
…□□□□□■□□□□□□□□□□□□□□………..
…□□□□□■□□□□□□□□□□□□□□□□□□□……..

Ignoring for the moment the  problem of the reversal of direction for the red event-chain, what we have here is two event-chains, each with the same first and last event, one of which (the black one)  has apparently kept to the ‘same’ position on the line while the members of the other event chain have gone to the right and then back to the left. Now, one of the original suppositions was that all the ‘grids’ on Kwere identical in size, and that every ultimate event fills a site completely. Thus the total occurrence time (where the ‘time of transition’ from one occurrence to the next is not taken into account), is the same for both event-chains : fifteen instants. And so the red event-chain is seemingly no ‘longer’ than the black !       This is contrary to all observation and I am tempted to add, to all reason. Clearly, by any normal reckoning, the red trajectory is lomnger than the black : it is at any rate not identical in length. But if we keep to the original assumption about the ‘size’ of the grid positions, there is no way the two trajectories can be different in length unless we conclude that the transition times, the ‘breaks’ in the flow of events, are different for different event-chains. We thus have the concept of the backdrop, whatever it is, being somehow ‘elastic’ or, more precisely, not being measurable in the normal way that solid objects are measurable. This is, up to a point, acceptable since the backdrop is not part of the ‘normal’ physical world.
What makes things worse, however,  is that, according to the Principle of Special Relativity, we could just as well have made the red event-chain a vertical line and the black line one that first of all slants to the left and then to the right to rejoin the red chain. Each description is, according to SR, equally valid — though I think it misleading to say they are ‘equivalent’ which manifestly they are not. If the black event-chain were part of an event-cluster with an observing agent on board, ‘he’ or ‘she’ would, in the case presented, consider himself to be motionless and the red spacecraft to be in constant motion, and the red observer would think the same. Of course, Special Relativity might be mistaken but there a large number of observations which suggest that the two different representations, each consistent within its own terms, really are equally legitimate and that there is no way of deciding which is ‘true’ — they both are. Any alternative theory — and there have been several proposed — has to explain why, for example, it does not seem to be possible to determine from inside an inertial system whether it is ‘really’ at rest or in straight line constant motion.
Note, however, that there is a big difference between the schema of Eventrics and that of classical and modern science, including Relativity theory. In Eventrics, by hypothesis, every event-chain is composed of a finite number of ultimate events (see original assumptions) and so has associated with it an ‘event-number’ which is not relative, does not depend on one’s actual or hypothetical standpoint or state of motion, but which is absolute. In classical and modern science the trajectories of ‘continuously moving objects’ are conceived as being composed of an infinite number of  instantaneous locations. So, since twice or three times infinity equals infinity, a ‘longer’ trajectory has not passed through any more actual or possible point-locations  than a shorter one  –  which is hard to believe, to say the least.    (To be continued)

Note 1 :  Mathematically, this is definition by recursion or  f(n+1) = f(n) + r     r = 0, 1, 2. 3…..    f(0) = Ο. In each case, the increase f(n+1) – f(n) = r and does not change however far the event-chain is extended. In particular, the coming increase does not depend on the current position of an event relative to a fixed (repeating) point. Ideally, all mathematical functions that describe actual behaviour should be defined by recursion since this would seem to be much closer to what actually goes on in nature. The analytic formula y = f(x) provides a ‘God’s eye’ view of the world : all possible values of the dependent variable are given ‘in one fell swoop and, as Ullmo put it rather well, “it is our fault if we have to discover piecemeal all the features of the curve”. This way of proceeding suited the world-view of the early scientists perfectly for they all, to a man, were firm believers, Descartes, Kepler, Leibnitx, Newton, Boyle…   However, we know that in biology trial and error (subject to certain overriding physical constraints) is the rule and both species and organisms proceed step by step from a given departure point. With recursion you only need to know the starting point and how to get from one position to the next.
I do not know whether all anaytic functions can be presented recrusively and vice-versa — I think someone has proved they can’t be — but the two presentations are certainly not ‘equivalent’. Even such a simple function as y = x²  is quite tricky to define recursively while one of the simplest and most important recursive functions f(n+ 1) = f(n) + f(n – 1)  gives a very complicated analytic formula.

Note 2 :  It was essentially the qualitative distinction between these two classes of event-chains that, perhaps more than anything else, gave rise to the formidable development of physical science in the West. Though anticipated to some extent by Oresme, Galileo was the first to grasp the signal importance of the distinction. The straight lines represent constant change, and the simplest case of constant change is no change at all (rest), whereas all other curves show acceleration, changing change as Oresme put it. Newton, following on from Galileo, classed straight line motion or rest as the ‘natural’ state which required no explanation : any deviation from this equilibrium state denoted the presence of a force which, since Newton did not know modern chemistry, was assumed to be an external force.
It is remarkable that Galileo hit upon the idea of what we today call an ‘inertial system’ since, in his day, there were no smooth running means of transportation like our trains and aircraft. Galileo, in his thought experiment, supposed that he was in a cabin without a window in a boat on a perfectly calm sea and he asked himself if it was possible, by performing various tests inside the cabin only to decide if he and the ship were at rest or moving at constant speed in a straight line. He decided that it was not possible. Einstein took up Galileo’s ‘Principle ofn Relativity’ and made it the cornerstone, along with the constancy of the speed of light, of the Theory of Special Relativity, viz. The laws of physics take the same form in all inertial systems as he put it.  Subsequently, Einstein cast doubt on the validity of the concept of an ‘inertial system’ and extended his theory to cover all forms of motion.
In the terms of Eventrics, the trajectories noted as the three straight lines in the first diagram “are equivalent” (though I would not quite put it like this). ‘Relativity’ comes about because if an observer were ‘moving’ at the same rate as the black squares, he or she would judge himself to be at rest and objects synchronised with the red or green squares to be ‘moving to his right at a steady pace. However, an observer ‘moving’ at the rate of the red or green squares would judge himself to be at rest and an object appearing and disappearing at the same rate as the black squares to be moving steadily towards his left (or vice-versa).

(To be continued 11/7/12)

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