In an earlier post I suggested that the vast majority of ultimate events appear for a moment and then disappear for ever : only very exceptionally do ultimate events repeat identically and form an event-chain. It is these event-chains made up of (nearly) identically repeating clusters of ultimate events that correspond to what we call ‘objects’ — and by ‘object’ I include molecules, atoms and subatomic particles. There is, thus, according to the assumptions of this theory, ‘something’ much more rudimentary and fleeting than even the most short-lived particles known to cojntemporary physics. I furthermore conjectured that the ‘flicker of existence’ of ephemeral ultimate events might one day be picked up by instrumentation and that this would give experimental support to the theory.  It may be that my guess will be realized more rapidly than I imagined since, according to an article in the February edition of Scientific American an attempt is actually being made to detect such a ‘background hum’  (though those concerned interperet it somewhat differently).

“Craig Hogan, director of the Fermilab Particle Astrophysics Center near Batavia, Ill., thinks that if we were to peer down at the tiniest sub-divisions of space and time, we would find a universe filled with an intrinsic jitter, the busy hum of static. This hum comes not from particles bouncing in and out of being or other kinds of quantum froth……..Hogan’s noise arises if space is made of chunks. Blocks. Hogan’s noise would imply that the universe is digital.”     Michael Moyer, Scientific American, February 2012
Moreover, Hogan thinks “he has devised a way to detect the bitlike structure of space. His machine ― currently under construction ― will attempt to measure its grainy nature.(…) Hogan’s interferometer will search for a backdrop that is much like the ether ― an invisible (and possibly imaginary) substrate that permeates the universe”.
         Various other physicists are coming round to the idea that ‘Space-Time’ is ‘grainy’ though Hogan is the first to my knowledge to speak unequivocally of a ‘backdrop’ permeated with ‘noise’ that has nothing to do with atomic particles or the normal quantum fluctuations. However, the idea that the universe is a sort of giant digital computer with these fluctuations being the ‘bits’ does not appeal to me. As I see it, the ‘information’ only has meaning in the context of intelligent beings such as ourselves who require data to understand the world around them, make decisions and so forth. To view the universe as a vast machine running by itself and carrying out complicated calculations with bits for information (a category that includes ourselves) strikes me as fanciful though it may prove to be a productive way of viewing things.      S.H.  30/08/12

Newton’s First Law states, in effect, that the ‘natural’ movement of a particle ― Newton says ‘innate’ ― is to travel in a straight line at constant speed. In consequence, any deviation from this ‘natural’  state requires explanation and the conclusion to be drawn is that the particle has become subject to interference from an outside source (Newton’s Second Law).
After the square and the rectangle, the circle is probably the best known regular shape though there are precious few examples of true circles to be found in nature  ─ if we are so familiar with it, this is largely because of the wheel, one of the most ancient human inventions. Plato, enamoured of elegance and simplicity rather than actuality, considered that heavenly bodies must follow circular paths because the circle was the ‘most perfect’ shape (Note 1). Greek astronomy, as we know, dealt with departures from circular motion by introducing epicycles, circles within or on other circles. It would probably be possible, though hardly convenient, to trace out any regular curve in this manner and the Ptolemaic system proved to be a satisfactory way of calculating planetary orbits for nearly two thousand years.
However, for reasons that are probably physiological in origin, we do not feel at ease with anything other than straight lines, squares and rectangles and a good deal of mathematics is concerned with the problem of ‘squaring the circle’ or, more generally, converting the areas and volumes of irregular shapes into so many squares. Newton, following Galileo, broke down circular motion into two straight-line components at right angles to each other  : one tangential to a point on the circumference of a circle and the other along the radius from that point to the centre of the circle. He furthermore proposed that circular motion could be modelled in the following manner : the original tangential velocity did not change in magnitude but did change repeatedly in its direction and by the same amount.  We do not normally view a change in direction as an ‘acceleration’ and Newton, writing in Latin, did not use the term. However, in Newtonian mechanics we consider any deviation from constant straight-line motion  to be an ‘acceleration’ which, according to  Newton’s Second Law must be due to the action of an  outside force. He coined the term centripetal or ‘centre-seeking’ force (after Latin peto, I seek) to characterise this force. What is more,  since gravitational attraction was permanent and did not change in magnitude over relatively small distances, a particle subject to such a force would deviate repeatedly from its current straight line motion and deviate by the same amount each time. If the original velocity was ‘just  right’, the particle would keep more or less at the same distance from the attracting body while perpetually changing direction : the result if the attracting body was very much larger being motion in a circle around that body.
There can be little doubt that Newton based his mathematical treatment of circular motion on personal experience while generalising the principles involved, by a giant leap of thought, to model the motion of heavenly bodies. On the second page of his Principia he mentions in his Definition V how “a stone, whirled about in a sling, endeavours to recede from the hand that turns it; and  by that endeavour, distends the sling, and that with so much the greater force, as it is revolved with the greater velocity” (Principia, Vol. I Motte’s translation). It is a matter of experience (not theory) that a whirling conker or other projectile exerts a definite pressure on your finger, or on any other object that serves as an axis of rotation, and that we feel this pressure more or less on the opposite side to where the conker is ‘at that moment’. Secondly, as Newton says, the faster the whirling motion, the more the string cuts into your finger. Thirdly, if we use a rubber  connection we can actually observe the ‘string’ being extended beyond its normal length. And finally, if we cut the string or otherwise break the connection between particle at the circumference and the centre, the particle flies off sideways : it never flies off directly inwards nor, as far as we can judge, does it continue to follow a circular trajectory. 

The Modern Derivation of the formula for centrifugal force

In the modern treatment of motion in a circle we have a particle which is at rest at time t = 0 at position P₀ on the circumference of a circle of radius r. It is propelled in a direction tangential to the circumference of the circle at an initial  velocity of v metres/sec.  At time t it has reached the point P₁ having travelled vt metres while the angle at O has turned through θ radians.

        Since it has travelled in a straight line along the tangent it has covered r tan θ metres while a particle travelling along the circumference has travelled in the same t seconds a distance of rθ metres if θ is in radians. If we have an angular velocity of ω radians per second this distance is rωt metres.  Now, v, the speed of the particle travelling along the tangent, is not the same as rθ/t = rω, the speed of a particle travelling along the circumference, since r tan θ > rθ.

v/rω = (r tan θ)/t   or  v/r  =  ω (tan θ)/θ)
                               rθ/t  

For very small angles v ≈ rω since the limit of tan θ/θ is 1 as θ → 0

Now, the particle supposedly always maintains its constant speed but each time it is in contact with the circumference changes direction and pursues a path along the tangent at that point. In the usual  modern treatment, we take two velocity vectors v₁ and v₂ corresponding to the velocities at two points on the circumference of a circle when the angle at the centre has turned through θ radians. These two velocities are, according to Newton’s treatment of circular motion, equal in magnitude but differ in direction and the angle between the two vectors is the same as the angle at the centre. We now evaluate the closing velocity vector which is considered “for very small angles” to be approximately equal to vθ as if this closing vector, instead of being a straight line, were part of the circumference of a circle with radius v metres subtended by the angle θ. We thus end up, after some manipulation, with an acceleration vector of rω² =  v²/r metres per second per second, setting ωr equal to v. Since F = mass × acceleration , the centrifugal force is mv²/r for a particle of mass m.
Thus according to textbooks aimed at engineers and technicians. In textbooks aimed at pure mathematicians it is  stressed that what we have here is a double ‘passage to a limit’, that involving v and rω and that involving 2v sin  θ/2 and vθ . Fpr, in reality the closing vector would have length 2v sin (θ/2) which goes to vθ because the ratio (2v sin (θ/2)) : vθ  = v sin (θ/2) : v (θ/2) tends to the limit v as θ/2 → 0. But what these textbooks do not make clear is that these limits are never attained since tan θ is never strictly equal to θ and sin θ/2 is never strictly equal to θ/2 for any 0 < θ < p/2.     

Motion in a Circle according to Ultimate Event Theory

As before we have a particle P is at r units of distance from O, the centre of an imaginary circle. It is projected along the tangent at P₀ with speed v units distance/units of time and at time t it is at P₁ having traversed vt units of distance. At time t the angle at O has gone through θ radians so that vt = r tan θ.
So far, so good : everything is as in the normal treatment. However, note that t is integral being ‘so many ksanas’ ― a ‘ksana’ is the minimal interval of time, equivalent of the vague ‘instant’. Also, v is a rational number since it is the ratio of an integral number of grid positions to t ksanas. If the ‘particle’ (repeating ultimate event) takes 4 ksanas to reappear 7 grid positions further along, the speed of reappearance is 7/4 spaces/ksana. It is not to be supposed that the particle covers two and a quarter spaces in one ksana since the spaces are indivisible, only that four ksanas later the particle reappears seven positions to the right (or left as the case may be) to where it was before. Moreover, these distances are ‘unstretched’, i.e. the grid positions are conceived to occupy ‘all’, or practically all, of the distance between two occupied positions. Just as in Special Relativity, we consider measurements made from inside an inertial system to be ‘standard’, in Ultimate Event Theory we start by evaluating distances from the standpoint of a regular repeating event not subject to influence from other event-chains. The repeating event E is such an event since it is (not yet) subject to an outside influence : it maintains a constant speed of s spaces per t ksanas and its speed if s/t spaces per ksana. r is likewise integral in Ultimate Event Theory since it is so many (unstretched) spaces from O.
When P reaches P₁ it is no longer at r spaces from O but a somewhat larger distance (and the same applies to event E when it repeats at this particular grid position). The distance P₀ O has thus been ‘stretched’ using the analogy of a string connecting two bodies. Stretched by how much?  By Pythagoras,
    (OP₁)²   = r² + r² tan² θ)  or OP₁ = r √(1² +  tan² θ) and the extension is this quantity less the radius, or  r √(1 + tan² θ) – r  =  r √(sec² θ) – r = r (1/cos θ  – 1)

        Now this spatial distance does not have to be integral, that is, correspond to a whole number of positions capable of receiving ultimate events. (Note 1).
The initial displacement of the particle, once it has been impelled along its path, is unaccelerated and thus, by definition, not subject to an outside force. However, this displacement does have an effect on a similar initially unaccelerated body situated at the centre O (since the distance between the two particles has increased). This ‘action’ at O calls forth an ‘equal and opposite reaction’ according to Newton’s Third Law and exactly four ksanas later the ‘particle’ appears on the circumferenceat P₂ with the same speed and proceeds along the tangent at P₂    ― this according to Newton’s treatment of motion in a circle.
In event terms we have an event E originally appearing at P₀ exactly r spaces from O, reappearing at P₁ outside the imaginary circle t ksanas later, and finally appearing at P₂ where it will travel along the tangent at P₂  at the same speed v = m spaces per t ksana ― in this particular case 7/4 spaces/ksana.
The first displacement calls forth an ‘equal and opposite reaction’ according to Newton’s Third Law and exactly four ksanas later the ‘same’ ultimate event repeats on the circumference of the imaginary circle at P₂. By hypothesis, it has  the same ‘reappearance speed’ of v spaces/ksana ― in this particular case 7/4 spaces/ksana. Although it has the same speed, event E has been subject to an outside influence since it has changed its straight-line event-direction : it has been, in Newtonian terms, accelerated and subject to an outside force. This force, or rather  influence, originates in  the reaction of the event-chain situated at the centre O .  The reaction will be, by Newton’s Laws, equal in magnitude to the original force to which the body (event-cluster) at O was subject.
It is essential to grasp the succession of events. When travelling from P₀ to P₁, the particle/event is not subject to any outside influence : this influence is in operation only for the second t ksanas during the passage from P₁ to P₂. Similarly, once at P₂, the particle/event is again free of outside influence for t ksanas and the entire cycle action/reaction repeats indefinitely. This is so in the Newtonian treatment as well but there is a tendency, even in authoritative text books, to assume that because action and reaction take place in such swift succession that they are strictly simultaneous.  
So how do we evaluate the strength of the restoring force which in Newtonian Mechanics is known as the centripetal or centre-seeking force (from Latin peto I seek) ? We can do this by comparing the distances along the radial direction and noting the changes in speed. The particle/event starts off with zero speed in this direction since it is situated on the circumference; it is at a distance r (1/cos θ  – 1) from the centre O t ksanas later and t ksanas later still it has once more zero radial speed having recovered its original configuration.

Now, the particle (event) has constant speed v that and so, were it not for the restraining influence from the centre, would have, during the second t ksanas,  travelled another vt spaces along the tangent line from P₀ and the angle at the centre would have gone through another θ radians. The particle’s distance from P₂ on  the circumference O would thus be

        √ (r² tan² 2θ – r²) – r  = r √sec² θ  – r  = r(1 – cos 2θ)/cos 2θ

        We wish to have this distance and the eventual radial acceleration in terms of r and v rather than θ, since r and v are ‘macroscopic’ values that can be accurately measured whereas θ is microscopic ― what used to be called ‘infinitesimal’.
To bring the particle to P₂ the above distance has had to be reduced to zero in the space of t (not 2t) seconds since the reaction did not begin until the particle was at P₁. We thus evaluate        r (1 – cos 2θ) /t²   where, as before, t = r tan θ/v
                                                                cos 2θ

                =   (v²/r)  (1 – cos 2θ)    = (v²/r)  (2 cos² θ)(cos² θ)
                              cos 2θ tan² θ             (1 – 2cos² θ)(sin² θ)   

Since cos 2θ  =  2 cos² θ –1
        1 –  cos 2θ  =  1 – (2 cos² θ – 1) = 2(1 – cos² θ)   =  2 sin² θ

Consequently         (v²/r)  (2 sin² θ)(cos² θ)     =   2(v²/r) (cos² θ)
                                             (2cos² θ – 1)(sin² θ)              (2cos² θ – 1)

         Since θ is not a macroscopic value and is  certainly extremely small, we may take the limit as θ → 0  (Note 2).

Lim  as θ → 0     (cos² θ)       =           1     .       =   1
                        (2cos² θ – 1)      2 –1/cos² θ            

        The acceleration is thus 2 v²/r spaces per ksana per ksana  and the  centrifugal ‘force’ on a particle of mass m is 2m v²/r (Note 3).

This is a most significant result since it differs from that derived by traditional methods which is just v²/r .  The latter is the average value over 2t ksanas since, during the first t ksanas, the centrifugal force was not active. It will be objected that the particle ‘in effect’ follows a circular path around the circumference at all moments but, if this really were the case, the particle would never obey Newton’s First Law but would be permanently accelerated (since it would never follow a straight line trajectory). That the deviation is for normal practical purposes entirely negligible is irrelevant : the aim of an exact science  should be to deduce what actually goes on and anyway it is by no means certain that the discrepancy might, in some cases, be significant.  Whereas in the conventional treatment, the centrifugal force is permanent, in Ultimate Event Theory, it is, like all reactions, intermittent : it is only operative over t moments out of every 2t.  It is by no means inconceivable that accurate experiments could detect this intermittence, just as they might one day detect the difference between v and rω .    S.H. ­­­­­­­­­­­­­­­

_____________________________________ 

Note 1. In reality, all orbiting bodies eventually move away from the centre of attraction or fall towards it as even the Moon does — but so slowly that the difference from one year to the next is only a few centimetres.   

Note 2 The ‘stretched distance’ between two occupied spaces  does not (by hypothesis) contain any more grid positions than an unstretched and the extra ‘distance’ is to be attributed to the widening of the gaps between grid-points, i.e. to the underlying (non-material) ‘substance’ that fills the Locality. This stretched distance is essentially a mathematical convenience that has no true existence. All that actually happens and that is, or could be,  observable is the occurrence of a particular ultimate event at a certain spot on the Locality at a particular ksana and its reappearance at a subsequent ksana at a spot that is not exactly equivalent to the previous position (relative to some standard repeating event-chain).

Note 2. It must be stressed that this limiting value, like practically all those which occur in Calculus, is not actually attained in practice though one assumes that in such a case as that considered the approximation is very good. The value of θ in any actual occurrence will certainly be small but not ‘infinitesimal’, a term which has no meaning in Ultimate Event Theory (and should have none in traditional mechanics either).

Note 3  It remains to give a satisfactory definition of ‘mass’ in terms of Ultimate Event Theory. Newton calculated mass with respect to area and density and spoke of it as “a measure of the quantity of matter in a body”. In traditional physics mass is not only measured by, but equated with, the ability of a particle to withstand any attempt to change its constant straight-line motion and in contemporary physics mass has virtually lost any ‘material’ meaning by being viewed as interchangeable with energy.

 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 K₀ were 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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The forces operating in a market are both exogenous, coming from the outside, and endogenous, coming from inside. Broadly speaking, those from the outside are studied in fundamental analysis and those from the inside in technical analysis. Fundamental analysis concentrates on such things as a company’s assets, balance sheet, style of management, state of the market and so forth. Technical analysis ultimately bases itself on human behaviour since the trends and counter trends originate in the minds, or rather in the emotional states, of the primates we are. But scientists are wrong to see the market as simply another complex system. Herding animal behaviour can certainly be observed (Note 1)but humans are at least partially aware of what is going on and can feed their views back into the system, thus modifying it. Also, the peculiarity of the market is that predator and prey are mixed together and each player takes on both roles at different moments.
What do we see when we examine charts and graphs of the Stock Market? We practically never see straight lines and smooth curves : market activity proceeds by short, sharp bursts which in the short term tend to wipe each other out while  nonetheless, allegedly, dancing to a hidden deeper music. The typical zig-zag
shape of the Dow and the FTSE demon strates how a burst of trading activity in one direction almost immediately gives rise to a burst in the opposing direction. However, since this correcting burst, a fortiori trend, is reactive not  pro-active, it is always one step behind. Like Alice in Wonderland, the counter-trend has a job even keeping up, let alone overtaking the main trend.
But eventually this is precisely what happens.  A tipping point occurs when the counter trend  ousts the main trend and itself becomes dominant. A stock is overvalued, certain influential traders realize this, sell their shares, the value of the stock depreciates and so on. Alternatively, a company starts recovering from a bad time, one or two sharp traders note this, buy when the price is temptingly low, and initiate a rally.
Need there be a reversal at all? Yes, because otherwise the market  would cease to function : there would be a credit bubble so big that it would swallow up everything else, or a crash so devastating that it would bring down the entire economy. If there is danger of this, those in charge immediately end the game by closing down the Stock Market and, if necessary, forbidding further trading until they change thew rules. In the tulip bubble of the 17^th century, the price of a single tulip allegedly rose so high that it equalled the price of a house : it could clearly not go up as far as the value of the entire country.
“The trend is your friend”  — but is it?  The people who win out in the Stock Market are precisely those who buck the trend, sell stock during a rising trend that is, according to them, about to peak (thus contributing to this  very levelling off), or buy stock when the  market value, according to them, is about to level off and subsequently rise. The art is in judging the relative strengths of trend and counter-trend and throwing one’s own weight into the scales.
There are, in the Stock Market, always  two opposed forces at work since all traders, inasmuch as they are traders, rather than buy-to-hold investors, are both buyers and sellers. Traders in shares do not generally wish to keep the shares they acquire and buyers of futures certainly do not want the vast quantities of corn or rice they buy in anticipation. They have a single purpose,  to make money, but to achieve this aim at least two completely opposed actions are required, buying and selling the very same item, often many times over. This is the main reason why the Stock Market is such a peculiar place. To make a profit, a trader must successively live on both sides of the Great Divide, like the alleged Afghan fighters in the Civil War who drew two salaries by fighting for one side during the day and the other side during the night. On top of that, each trader is, or aspires to be, an ‘objective’ analyst who has a God’s eye view of the battlefield from up above.
The market is a unique place in that volatility is the norm rather than the exception : unlike most ‘curves’ that are studied in mathematics, the ‘curves’ in market graphs are always jagged and nearly always go up or down, rarely sideways.  The market is in a permanent state of disequilibrium : indeed, if it were otherwise no money could be made, or very little. Can any very general ‘laws’ be formulated which might apply to other event environments?
First, a definition. In terms of Eventrics a (market) trend is an event-chain that has persistence (repeats itself), direction(goes up or down)and dominance (has the power to attract other events and make them copy itself). In this context, the direction’ is price movement while it is the increasing dominance of an event-chain that provokes herding behaviour, sometimes producing hysteria. Momentum, a vector quantity, may be defined as  volume × dominance, i.e. the ‘spread’ of an event-chain multiplied by its power to influence other events.

Rule 1   A market event-chain’s momentum is rarely static but tends to increase or decrease, and to increase or decrease irregularly (not linearly). As the saying goes, “Nothing succeeds like success” and one ought to add, “Nothing fails so successfully as failure”.

But :

Rule 2   Every burst, a fortiori trend, ‘normally’ gives rise to a correcting burst/trend whose direction is opposite to the main trend but which always  lags slightly behind the main trend.

Rule 3  At a certain point (tipping point) the relation main trend/correcting trend reverses.

Rule 4  In certain circumstances the correcting trend  disappears altogether with devastating consequences for the entire event-environment : examples are a runaway credit bubble or, conversely, a run on a bank.

These rules are merely extrapolations from the data and will not enable you to actually make money on the Stock Exchange. To do this you need either to be lucky or be at once highly intuitive and self-disciplined (a rare combination); you also need to have a good understanding of what is going on in the outside world politically and economically and be able to derive rather more precise rules governing the fluctuations of share prices than the above (to say the least).
So what are these more precise rules? Elliott Wave Theory tells us and I refer the reader to thw writings of Elliott himself or his most lucid and successful follower, Prechter. There is, on the face of it, no special reason why ‘trends’ should be made up of three dominant and two counter-dominant ‘waves’ as Elliott believed. However, if this turns out actually to be the case, it can only be that there is some deep-rooted physical reason.  Elliott himself writes

“The forces that cause market trends have their origin in nature and human behaviour and can be measured in various ways. (…) Forces travel in waves….  [they] can be forecast with considerable accuracy by comparing the structure and extent of the waves”  (Elliott, Nature’s Law p. 81)

From my point of view, it is misleading to speak of market trends as composed of waves, since, mathematically speaking, a ‘wave’ is continuous whereas market activity is discrete, is composed of specific trades and these trades proceed by bursts, not continuously (Note 2). Also, Elliott does not say what causes these waves in the first place : he seems to think that they are universal and omnipresent — but are they ? If they are, why don’t we find them in other contexts?          S. H.      (30/06/12)

Note 1  “Shared mood trends [amongst traders] appear to derive from a herding impulse governed by the phylogenetically ancient, pre-reasoning portions of the brain. This emotionally charged mental drive developed through evolution to help animals survive, but it is maladaptive to frming successful expectations cocnerning future financial valuation” (Prechter, Conquer the Crash p. 25) As E.O. Wilson said in an interview for “We have Palaeolithic emotions, medieval institutions and godlike technology. That’s dangerous” (New Scientist 21 April 2012)

Note 2   “Whenever a market ‘gaps’ up or down on an opening, it simply registers a new value onm the first trade, which can be conducted by as few as two people” (Prechter, Conquer the Crash p. 93)     S.H.

         The image is  Perpetual Motion by  June Mitchell   (All rights reserved) .