Archives for category: Newton

CALCULUS

“He who examines things in their growth and first origins, obtains the clearest view of them” Aristotle.

Calculus was developed mainly in order to deal with two seemingly intractable problems: (1) how to estimate accurately the areas and volumes of irregularly shaped figures and (2) how to predict physical behaviour once you know the initial conditions and the ‘rates of change’.
We humans have a strong penchant towards visualizing distances and areas in terms of straight lines, squares and rectangles ― I have sometimes wondered whether there might be an amoeba-type civilization which would do the reverse, visualizing straight lines as consisting of curves, and rectangles as extreme versions of ellipses. ‘Geo-metria’ (lit. ‘land measurement’) was, according to Herodotus, first developed by the Egyptians for taxation purposes. Now, once you have chosen a standard unit of distance for a straight line and a standard square as a unit of area, it becomes a relatively simple matter to evaluate the length of any straight line and any rectangle (provided they are not too large or too distant, of course). Taking things a giant step forward, various Greek mathematicians, notably Archimedes, wondered whether one could in like manner estimate accurately the ‘length’ of arbitrary curves and the areas of arbitrarily shaped expanses.

At first sight, this seems impossible. A curve such as the circumference of a circle is not a straight line and never will become one. However, by making your unit of length progressively smaller and smaller, you can ‘measure’ a given curve by seeing how many equal little straight lines are needed to ‘cover’ it as nearly as possible. Lacking power tools, I remember once deciding to reduce a piece of wood of square section to a cylinder using a hand plane and repeatedly running across the edges. This took me a very long time indeed but I did see the piece of wood becoming progressively more and more cylindrical before my eyes. One could view a circle as the ‘limiting case’ of a regular polygon with an absolutely enormous number of sides which is basically how Archimedes went about things with his ‘method of exhaustion’ (Note 1).

It is important to stop at this point and ask under what conditions this stratagem is likely to work. The most important requirement is the ability to make your original base unit progressively smaller at each successive trial measurement while keeping them proportionate to each other. Though there is no need to drag in the infinite which the Greeks avoided like the plague, we do need to suppose that we can reduce in a regular manner our original unit of length indefinitely, say by halving it at each trial. In practice, this is never possible and craftsmen and engineers have to call a halt at some stage, though, hopefully, only when an acceptable level of precision has been attained. This is the point historically where mathematics and technology part company since mathematics typically deals with the ‘ideal’ case, not with what is realizable or directly observable. With the Greeks, the gulf between observable physical reality and the mathematical model has started to widen.

What about (2), predicting physical behaviour when you know the initial conditions and the ‘rates of change’? This was the great achievement of the age of Leibnitz and Newton. Newton seems to have invented his version of the Calculus in order to show, amongst other things, that planetary orbits had to be ellipses, as Kepler had found was in fact the case for Mars. Knowing the orbit, one could predict where a given planet or comet would be at a given time. Now, a ‘rate of change’ is not an independently ‘real’ entity: it is a ratio of two more fundamental items. Velocity, our best known ‘rate of change’, does not have its own unit in the SI system ― but the metre (the unit of distance) and the second (the unit of time) are internationally agreed basic units. So we define speed in terms of metres per second.

Now, the distance covered in a given time by a body is easy enough to estimate if the body’s motion is in a straight line and does not increase or decrease; but what about the case where velocity is changing from one moment to the next? As long as we have a reliable correlation between distance and time, preferably in the form of an algebraic formula y = f(t), Newton and others showed that we can cope with this case in somewhat the same way as the Greeks coped with irregular shapes. The trick is to assume that the supposedly ever-changing velocity is constant (and thus representable by a straight line) over a very brief interval of time. Then we add up the distances covered in all the relevant time intervals. In effect, what the age of Newton did was to transfer the exhaustion procedure of Archimedes from the domain of statics to dynamics. Calculus does the impossible twice over: the Integral Calculus ‘squares the circle’, i.e. gives its area in terms of so many unit squares, while the Differential Calculus allows us to predict the exact whereabouts of something that is perpetually on the move (and thus never has a fixed position).

For this procedure to work, it must be possible, at least in principle, to reduce all spatial and temporal intervals indefinitely. Is physical reality actually like this? The post-Renaissance physicists and mathematicians seem to have assumed that it was, though such assumptions were rarely made explicit. Leibnitz got round the problem mathematically by positing ‘infinitesimals’ and ultimate ratios between them : his ‘Infinitesimal Calculus’ gloriously “has its cake and eats it too”. For, in practice, when dealing with an ‘infinitesimal’, we are (or were once) at liberty to regard it as entirely negligible in extent when this suits our purposes, while never permitting it to be strictly zero since division by zero is meaningless. Already in Newton’s own lifetime, Bishop Berkeley pointed out the illogicality of the procedure, as indeed of the very concept of ‘instantaneous velocity’.

The justification of the procedure was essentially that it seemed to work magnificently in most cases. Why did it work? Calculus typically deals with cases where there are two levels, a ‘micro’ scale’ and a ‘macro scale’ which is all that is directly observable to humans ― the world of seconds, metres, kilos and so on. If a macro-scale property or entity is believed to increase by micro-scale chunks, we can (sometimes) safely discard all terms involving δt (or δx) which appear on the Right Hand Side but still have a ‘micro/micro’ ratio on the Left Hand Side of the equation (Note 2). This ‘original sin’ of Calculus was only cleaned up in the late 19th century by the key concept of the mathematical limit. But there was a price to pay: the mathematical model had become even further away removed from observable physical reality.

The artful concept of a limit does away with the need for infinitesimals as such. An indefinitely extendable sequence or series is said to ‘converge to a limit’ if the gap between the suggested limit and any and every term after a certain point is less than any proposed non-negative quantity. For example, it would seem that the sequence ½; 1/3; ¼……1/n gets closer and closer to zero as n increases, since for any proposed gap, we can do better by making n twice as large and 1/n twice as small. This definition gets round problem of actual division by zero.

But what the mathematician does not address is whether in actual fact a given process ever actually attains the mathematical limit (Note 3), or how near it gets to it. In a working machine, for example, the input energy cannot be indefinitely reduced and still give an output, because there comes a point when the input is not capable of overcoming internal friction and the machine stalls. All energy exchange is now known to be ‘quantized’ ― but, oddly, ‘space’ and ‘time’ are to this day still treated as being ‘continuous’ (which I do not believe they are). In practice, there is almost always a gulf between how things ought to behave according to the mathematical treatment and the way things actually do or can behave. Today, because of computers, the trend is towards slogging it out numerically to a given level of precision rather than using fancy analytic techniques. Calculus is still used even in cases where the minimal value of the independent variable is actually known. In population studies and thermo-dynamics, for example, the increase δx or δn cannot be less than a single person, or a single molecule. But if we are dealing with hundreds of millions of people or molecules, Calculus treatment still gives satisfactory results. Over some three hundred years or so Calculus has evolved from being an ingenious but logically flawed branch of applied mathematics to being a logically impeccable branch of pure mathematics that is rarely if ever directly embodied in real world conditions.                                         SH

Note 1 It is still a subject of controversy whether Archimedes can really be said to have invented what we now call the Integral Calculus, but certainly he was very close.

Note 2 Suppose we have two variables, one of which depends on the other. The dependent variable is usually noted as y while the independent variable is, in the context of dynamics, usually t (for time). We believe, or suppose, that any change in t, no matter how tiny, will result in a corresponding increase (or decrease) in y the dependent variable. We then narrow down the temporal interval δt to get closer and closer to what happens at a particular ‘moment’, and take the ‘final’ ratio which we call dy/dt. The trouble is that we need to completely get rid of δt on the Right Hand Side but keep it non-zero on the Left Hand Side because dy/0 is meaningless ― it would correspond to the ‘velocity’ of a body when it is completely at rest.

Note 3   Contrary to what is generally believed, practically all the sequences we are interested in do not actually attain the limit to which they are said to converge. Mathematically, this does no9t matter — but logically and physically it often does.

Time and place

“Absolute, true and mathematical time, of itself, and from its own nature, flows equably without relation to anything external”  (Newton, Principia Scholium to Definition VIII)

Newton does not say whether there are any absolute units or measures to his absolute time, i.e. whether any exact meaning is to be given to the term ‘moment’. Rather he implies that there are no such units since time ‘flows’, i.e. is continuous. He does, however, contrast ‘absolute time’ with “relative, apparent, and common time” which is “some sensible and external (whether accurate or unequable) measure of duration by the means of motion…. such as an hour, a day, a month, a year”.

He also believes that each object has what he calls a ‘place’ which fixes it in absolute space and absolute time. “All things are placed in time as to order of succession; and in space as to order of situation. It is from their essence or nature that they are places; and that the primary places of things should be movable, is absurd. These are therefore the absolute places; and translations out of these places are the only absolute motions” (Ib.)

This view is to be contrasted with Leibnitz’s which sees the position and motion of bodies as essentially relative: a body’s ‘place’ merely indicates where it is in relation to other bodies at a given moment. This ‘relational’ approach has been adopted by several modern physicists beginning with Mach. As Lee Smolin puts it, “Space is nothing apart from the things that exist; it is only an aspect of the relationships that hold between things” (Smolin, Three Roads to Quantum Gravity p. 18). Much the same goes for time: “Time also has not absolute meaning…..Time is described only in terms of change in the network of relationships that describes space” (Smolin, Ib.)

Motion

What about ‘change of place with respect to time’ or motion? To determine a body’s motion we have to establish what a body’s ‘place’ was before motion began and the same body’s ‘place’ when motion has ceased. Newton concedes that the ‘parts of space’ cannot be seen and so we have to assume that there is a body which is immoveable and measure everything with respect to it. “From the positions and distances of things from any body considered as immovable, we define all places” (p. 8). But is there such a thing as an immovable body ? Newton is undecided about this though he would like to answer in the affirmative. He writes, “It may be that there is no body really at rest” but a few lines further on he adds that “in the remote regions of the fixed stars, or perhaps far beyond them, there may be some body absolutely at rest”. However, since such bodies are so far away, they are of little use as reference points practically speaking while “absolute rest cannot be determined from the position of bodies in our region”.

Newton concludes that we have to make do with ‘relative places’ though he is clearly bothered by this since it means that motion will also have to be treated as relative. This leads straight on to the Galilean ‘law’ that rest and constant straight line motion cannot be distinguished. Newton’s position is, however,  to be contrasted with the modern interpretation of Galileo’s claim. For Newton is not saying that ‘rest’ and ‘constant straight line motion’ are ‘equivalent’. Instinctively, he feels that there must be such a thing as ‘absolute place’ and ‘absolute rest’ and is chagrined that he cannot provide a reliable test to distinguish true rest from motion.

When discussing circular motion Newton invokes the backdrop of the ‘fixed stars’ which “ever remain unmoved and do thereby constitute immovable space”. Thus, it is, according to Newton, possible to distinguish between relative and absolute circular motion because in the latter case there is a force at work which makes a body “recede from the axis of circular motion”.  He gives the celebrated example of a bucket of water suspended by a chord which is twisted and then released so that the water climbs up the sides of the vessel. Today we would use the example of a merry-go-round which is, according to Newton’s test, a genuine case of absolute circular motion since we feel a definite force pushing us towards the outer edge.

Continuous and Discontinuous

When analysing the perceived motion of bodies, Newton treats motion as at once continuous and discontinuous: a projectile or a planet is never at rest as it pursues its path (unless interrupted by something in the way). But if a body is accelerating it does not have a constant velocity at that instant, and if it is in motion at any particular instant it cannot have a precise position. At the end of the day, provided we make the time interval small enough, it would appear that everything is at rest. However, the use of ‘infinitesimals’ allows one to decrease the time interval down to ‘almost nothing’ so that we can speak of  a body’s  ‘instantaneous velocity’ ― despite this being a contradiction in terms. In effect, the Infinitesimal Calculus which Newton co-invented, allows him to have his cake and eat it too as Bishop Berkeley pointed out to Newton and his supporters. This is probably the main reason why Newton avoids calculus methods as such in the Principia employing instead cumbersome geometrical constructions which in effect treat motion as an infinite succession of stills. Newton struggles to defend the logic behind his treatment in the beginning of Book I which treats the Motion of Bodies. But he cannot decide whether the ‘ultimate ratio’ of distance versus time ― what we call dy/dt ― is ever actually attained, a rather important point (Note 1).

More precisely, Bishop Berkeley made it clear that Newton was contradicting himself by first assuming that x has an increment and then, “in order to reach the result, allows the increment to be zero, i.e. assumes that there was no increment.” Modern mathematics gets round this problem by defining the ‘limit’ to an ‘infinite series’ in such a way that it is not required that this limiting value is actually attained ― indeed in practically all cases of interest it cannot be. The price we have to pay for this rationalization of the Calculus is loss of contact with physical reality. Even if Newton  had been capable of formulating the concept of a ‘limit’ in the precise modern sense I doubt if he would have employed it. Why not? Because Newton, like Leibnitz, and like practically every other ‘natural philosopher’ of the time, was a realist. In Newton’s time mathematics had not yet separated into ‘pure’ and ‘applied’ and the question as to whether infinitesimals ‘existed’ or not was the same sort of question as asking whether atoms existed. Pre-modern mathematics required infinitesimals to get tangible results which could be checked and usually turned out to be correct. But Newton was pragmatist enough to realize that, taken literally, Calculus methods made little sense.

Boyer naturally champions the modern view. “His [Berkeley’s] argument is of course absolutely valid as showing that instantaneous velocity has no physical reality, but this is no reason why, if properly defined or taken as an undefined notion, it should not be admitted as a mathematical abstraction” (Boyer, The History of the Calculus p. 227).
But  why should one allow mathematics to wag the tail of physics to this extent? The real world cannot be handwaved into irrelevance just because it hampers the style of pure mathematicians. It is in fact deeply shocking that contemporary physics, on the face of it  the most ’down to earth’ of the sciences, has been transformed into a piece of recondite pure mathematics. For mathematics, as a logico-deductive system, does not and cannot guarantee the existence of anything. Yet, for all that,  most of us would like to know what is ‘really real’ and what is imagination: science is not the same thing as science fiction.

Boyer claims that mathematics only deals in ‘relations’ not actualities which is all right up to a point ― but one has to ask, ‘relations between what sort of things?’ Since mathematics is a symbolic system, either its symbols ‘represent’  or stand in for realities of some  sort or they do not, in which case they are simply decorative in the same sort of way in which embroidery patterns are decorative. It is quite conceivable that a different intelligent species might use embroidery or textile design as a way of communicating truths about the cosmos but our species has not gone down this route and has restricted its scientific pattern-making to geometrical drawings and algebra.

Newton’s Approach and UET

How does Newton’s idea of ‘absolute time and place’ play out in terms of the basic assumptions of Ultimate Event Theory? In UET ‘ultimate events’ replace Newton’s ‘bodies’ and ‘time’ is the rate at which ultimate events succeed each other. Newton’s assumption of ‘absolute time’ is tantamount to suggesting that there is a fixed, universal and absolute rate at which certain events succeed each other and which is entirely regular. Most sequences of events, of course, probably only approximate to this measure in principle it is always there in the background. There is, then, a sort of metronome according to whose ticks all other sequences can be measured, and towards which all actual rates of actual events tend.

Is such an ‘absolute rate’ conceivable? (This is a different question to whether it actually exists.) If we assume, as I do, that all ultimate events have the ‘same extent’, i.e. occupy spatially  equivalent ‘places’ on the Locality, and last for exactly the same ‘length of time’, we obtain a basic regular rate if (and only if) the intervals between successive ultimate events are equal. And the simplest case would be when the interval between successive events is a minimum, i.e. just enough to keep the events separate, like a cell membrane that is one molecule thick.

As far as I am concerned, I believe that Newton was right in thinking that every object must have a ‘place’: this strikes me as even more necessary when dealing with events, which are by definition transitory, than with semi-permanent objects. I simply cannot conceive of ‘space’, or whatever is out there, as being simply composed of ‘relations’. But the UET conception is not the same as Newton’s concept of absolute space and time since the latter provide a fixed framework whether or not anything exists  inside this framework or not. One should conceive of an ‘ultimate’ rate for event chains as a constraint or asymptotic limit to which actual event sequences may tend, rather than something existing independently of all actual events. What it means, however, is that the time variable δt cannot be arbitrarily diminished, so there is always a final ratio of distance versus time for consecutive events. Moreover, one could state as a postulate that a ‘rest’ event sequence, the equivalent of a stationary object, proceeds at the minimal rate, i.e. one ksana at a time (sic) with the distance between any two consecutive events in an event-chain being  a minimum ― or, alternatively, a maximum.

SH  18/03/15

Note 1 Newton writes, “There is a limit which the velocity at the end of the motion may attain, but not exceed. This is the ultimate velocity”. But, a little further on, he changes tack and writes, “those ultimate ratios with which quantities vanish are not truly the ratios of ultimate quantities, but limits towards which the ratios of quantities decreasing without limit do always converge, and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to, till the quantities are diminished in infinitum”.

Note 2 And again, “Berkeley was unable to appreciate that mathematics was not concerned with a world of “real” sense impressions. In much the same manner today some philosophers criticize the mathematical conceptions of infinity and the continuum, failing to realize that since mathematics deals with relations rather than with physical existence, its criterion of truth is inner consistency rather than plausibility in the light of sense perception or intuition”.

Galileo’s Ship

It was Galileo who opened up the whole subject of ‘inertial frames’ and ‘relativity’, which has turned out to be of the utmost importance in physics. Nonetheless, he does not actually use the term ‘inertial frame’ or formulate a ‘Principle of Relativity’ as such.

Galileo wrote his Dialogue Concerning the Two World Systems, Ptolemaic and Copernican in 1616 to defend the revolutionary Copernican view that the Earth and the planets moved round the Sun. The Dialogue, modelled on Plato’s writings, takes the form of a three day long discussion where Salviati undertakes to explain and justify the heliocentric system to two friends, one of whom, Simplicius, advances various arguments against the heliocentric view. One of his strongest objections is, “If the Earth is moving, why do we not feel this movement?” Salviati’s reply is essentially this, “There are many other circumstances when we do not feel we are moving just so long as our motion is steady and in a straight line”.

Salviati asks his friends to conduct a ‘thought experiment’, ancestor of innumerable modern Gedanken Experimenten. They are to imagine themselves in “the main cabin below decks on some large ship” and this, given the construction at the time, meant there would have been no portholes so one would not be able to see out. The cabin serves as a floating laboratory and Galileo’s homespun apparatus includes “a large bowl of water with some fish in it”, “a bottle that empties drop by drop into a narrow-mouthed vessel beneath it”, a stick of incense, some flies and butterflies, a pair of scales and so on. The ship, presumably a galley, is moving steadily on a calm sea in a dead straight line. Galileo (via Salviati) claims that the occupants of the cabin would not be able to tell, without going up on deck to look, whether the ship was at rest or not. Objects will weigh just the same, drops of water from a tap will take the same time to fall to the ground, the flies and butterflies will fly around in much the same way, and so on — “You will discover not the least difference in all the effects named, nor could you tell from any of them whether the ship was moving or standing still” (Note 1).

Now, it should be said at once that this is not at all what one would expect, and not what Aristotle’s physics gave one to expect. One might well, for example, expect the flies and butterflies flying about to be impelled towards the back end of the cabin and even for human beings to feel a pull in this direction along with many other noticeable effects if the ship were in motion, effects that one would not perceive if the ship were safely in the dock.

What about if one conducted experiments on the open deck?  It is here that Galileo most nearly anticipates Newton’s treatment of motion and indeed Einstein himself. Salviati specifies that it is essential to decide whether a ‘body’ such as a fly or butterfly falls, or does not fall, within the confines of the system ‘ship + immediate environment’ ─ what we would call the ship’s ‘inertial frame’. Salviati concedes that flies and butterflies “separated from it [the ship] by a perceptible distance” would indeed be prevented from participating in the ship’s motion but this would simply be because of air resistance. “Keeping themselves near it, they would follow it without effort or hindrance, for the ship, being an unbroken structure, carries with it a part of the nearby air”. This mention of an ‘unbroken structure’ is the closest Galileo comes to the modern concept of an ‘inertial frame’ within which all bodies behave in the same way. As Salviati puts it, “The cause of all these correspondences of effects is the fact that the ship’s motion is common to all the things contained within it, and to the air also” (Dialogue p. 218 ).

Now, the claim that all bodies on and in the ship are and remain ‘in the same state of motion’ is, on the face of it, puzzling and counter-intuitive. For one might ask how an object ‘knows’ what ‘frame’ it belongs to and thus how to behave, especially since the limits of the frame are not necessarily, or even usually, physical barriers. Galileo does not seem to have conducted any actual experiments relating to moving ships himself, but other people at the time did conduct experiments on moving ships, dropping cannon balls, for example, from the top of a mast and noting where it hit the deck. According to Galileo’s line of argument, a heavy object should strike the deck very nearly at the foot of the mast if the ship continued moving forward at exactly the same speed in a straight line whereas the Aristotelians, on their side, expected the cannon ball to be shifted backwards from the foot of the mast by an appreciable distance. The issue  depended on which ‘structure’, to use Galileo’s term, a given object belonged to. For example, a cannonball dropped by a helicopter that happened to be flying over the ship at a particular moment, belongs to the helicopter ‘system’ and not to the system ‘ship’. In consequence, its trajectory would not be the same as that of a cannonball dropped from the top of a mast ─ unless the helicopter and ship were, by some fluke, travelling at an identical speed and in exactly the same direction.

By his observations and reflexions Galileo thus laid the foundations for the modern treatment of bodies in motion though this was not really his intention, or at any rate not at this stage in the argument. Newton was to capitalize on his predecessor’s observations by making a clearcut distinction between the velocity of a body which, other things being equal, a body retains indefinitely and a body’s acceleration which is always due to an outside force.

Families of Inertial Frames

In the literature, ‘inertial frame’ has come to mean a ‘force-free frame’, that is, a set-up where a body inside some sort of a, usually box-like, container remains at rest unless interfered with or, if considered to be already in straight line constant motion, retains this motion indefinitely. But neither Galileo nor Newton used the term ‘inertial reference frame’ (German: Inertialsystem) which seems to have been coined by Ludwig Lange in 1885.

The peculiarity of inertial frames is, then, that they are, physically speaking, interchangeable and cannot be distinguished from one another ‘from the inside’. Mathematically speaking, ‘being an inertial frame’ is a ‘transitive’ relation : if A is an inertial frame and B is at rest or moves at constant speed in a straight line relative to A, then B is also an inertial frame. We have, then, a vast family of ‘frames’ within which objects allegedly behave in exactly the same way and which, when one  is inside such a frame, ‘feel’ no different from one another.

It is important to be clear that the concept of ‘inertial frame’ implies (1) that it is not possible to tell, from the inside, whether the ‘frame’ (such as Galileo’s cabin or Einstein’s railway coach) is at rest or in straight line constant motion, and (2) that it is not possible to distinguish between two or more frames, neither of which are considered to be stationary, provided their motion remains constant and in a straight line. These two cases are distinct: we might, for example, be able to tell whether we were moving or not but be unable to decide with precision what sort of motion we were in ─ to distinguish, for example, between two different straight-line motions at constant speed. As it happens, Galileo was really only concerned with the distinction between being ‘at rest’ and in constant straight-line motion, or rather with the alleged inability to make such a distinction from inside such a ‘frame’, since it was this inability which was relevant to his argument. But the lumping together of a whole host of different straight-line motions is actually a more important step conceptually though Galileo himself did not perhaps realize this.

So. Were Galileo in the cabin of a ship moving at a steady pace of, say, 10 knots, he would, so he claims, not be able to differentiate between what goes on inside such a cabin from what goes on in a similar cabin of a similar ship not moving at all or one moving at a speed of 2 or 20 or 200 or even 2,000 knots supposing this to be possible. Now, this is an extremely surprising fact (if it is indeed a fact) since Ship A and Ship B are not ‘in the same state of motion’ : one is travelling at a certain speed relative to dry land and the second at a quite different speed relative to the same land. One would, on the face of it, expect it to be possible to tell whether a ship were ‘in motion’ as opposed to being at rest, and, secondly, to be able to distinguish between two states of straight line constant motion with different speeds relative to the same fixed mass of land. Newton himself felt that it ought to be possible to distinguish between ‘absolute rest’ and ‘absolute motion’ but conceded that this seemed not to be possible in practice. He was obviously somewhat troubled by this point as well he might be.

Galileo’s Ship is not a true Inertial Frame

As a matter of fact, it would not only be possible but fairly easy today to tell whether we are at rest or in motion when, say, locked up without radio or TV communication in a windowless cabin of an ocean liner. All I would need to carry out the test successfully would be a heavy pendulum, a means to support it so that it can revolve freely, a good compass, and a certain amount of time. Foucault demonstrated that a heavy pendulum, suspended with the minimum possible friction from the bearings so that it can move freely in any direction, will appear to swing in a circle : the Science Museum in London and countless other places have working Foucault pendulums. The time taken to make a complete circuit depends on one’s latitude — or, more correctly, the time it takes the Earth to revolve around the pendulum depends on what we choose to call latitude. A Foucault pendulum suspended at the North Pole would, so we are assured, take 24 hours to make a full circuit and a similar one at the Equator would not change its direction of swing at all, within the margins of experimental error. By timing the swings carefully one could thus work out whether the ship was changing its latitude, i.e. moving ‘downwards’ in the direction of the South Pole, or ‘upwards’ in the direction of the North (geographical) pole. On the other hand, a ship at rest, whatever its latitude, would show no variation in the time of swing ─ again within the limits of scientific error.

However, suppose I noted no change in the period of the Foucault pendulum. I would now have to decide whether my ship, galley or ocean liner, was stationary relative to dry land or was moving at constant speed along a great circle of latitude. This is rather more difficult to determine but could be managed nonetheless even with home-made instruments. One could examine  the ‘dip’ of a compass needle which points downwards in regions above the Equator and upwards in regions south of the Equator ─ because the compass needle aligns itself according to the lines of force of the Earth’s magnetic field. Again, any change in the angle of dip would be noticeable and there would be changes as the ship moved nearer the magnetic south or north poles. Nor is this all. The magnetic ‘north pole’ differs appreciably from the geographical north pole and this discrepancy changes as we pursue a great circle path along a latitude : so-called isoclinics, lines drawn through places having the same angle of dip, are different from lines of latitude. There are also variations in g, the acceleration due to gravity at the Earth’s surface, because of the Earth’s slightly irregular shape, its ‘oblateness’ which makes the circumference of the Earth measured along the Equator markedly different from that measured along a great circle of longitude passing through the poles. And so, despite Galileo’s claim to the contrary, there would be slight differences in the weight of objects in the cabin at different moments if the ship were wandering about. Only if the Earth were a perfect sphere with the magnetic poles precisely aligned with the geographical poles, would such tests be inconclusive. But a perfect sphere does not exist in Nature and never will exist unless it is manufactured by humans or some other intelligent species.
Galileo’s claim is thus not strictly true : it is a typical case of an ‘ideal situation’ to which actual situations approximate but which they do not, and cannot, attain.

Einstein’s Generalizations

But, one might go on to argue, the discrepancies mentioned above only  arose because Galileo’s ship was constrained to move on a curved surface, that of the ocean : what about a spaceship in ‘empty space’?

The full Principle of Relativity, Galileian or early Einsteinian,  asserts that there is no way to distinguish from the inside between conditions inside a rocket stationary with respect to the Earth, and conditions inside one travelling at any permissible constant ‘speed’ in a straight line relative to the Earth. It is routinely asserted in textbooks on the Special Theory of Relativity that there would indeed be no way to distinguish the two cases provided one left gravity out of the picture.

Newton made Galileo’s idealized ship’s cabin into the arena where his laws of motion held sway. An object left to its own devices inside a recognizable container-like set-up (an inertial system) would either remain stationary or, if already moving relative to the real or imagined frame, would keep moving in a straight line at constant speed indefinitely. This is Newton’s First Law. Any deviation from this scenario would show that there was an outside force at work ─ and Newton, knowing nothing of interior chemical or nuclear forces, always assumed that any supposed force would necessarily come from the outside. Thus, Newton’s Second Law.

So, supposing I let go of a piece of wood I hold in my hand in this room, which I take as my inertial frame, what happens to it? Instead of remaining where it was when I had it in my hand, the piece of wood falls to the ground and its speed does not stay the same over the time of its trajectory but increases as it falls, i.e. is not constant. And if I throw a ball straight up into the air, not only does it not continue in a vertical line at constant speed but slows down and reverses direction while a shot fired in the air roughly northwards will be deflected markedly to the right because of the Earth’s rotation (if I am in the northern hemisphere). Neither this room nor the entire Earth are true inertial frames : if they were Newton’s laws would apply without any tinkering about. To make sense of the bizarre trajectories just mentioned it is necessary to introduce mysterious forces such as the gravitational pull of the Earth or the Coriolis ‘force’ produced by its rotation on its own axis.

As we know, Einstein’s theory of Special Relativity entirely neglects gravity, and introducing the latter eventually led on to the General Theory which is essentially a theory of Gravitation. Einstein’s aim, even in 1905, was quite different from Galileo’s. Whereas Galileo was principally concerned to establish the heliocentric theory and only introduced his ship thought-experiments to deal with objections, Einstein was concerned with identifying the places (‘frames’) where the ‘laws of physics’ would hold in their entirety, and by ‘laws’ he had in mind not only Newton’s laws of motion but also and above all Maxwell’s laws of electro-magnetism. Einstein’s thinking led him on to a search for a ‘true’ inertial frame as opposed to a merely stationary frame such as this room since the latter is certainly not a ‘force-free’ frame. Einstein, reputedly after speculating about what would happen to a construction worker falling from the scaffolding around a building, decided that a real or imaginary box falling freely under the influence of gravitation was a ‘true’ inertial   frame. Inside such a frame, not only would the ‘normal’ Newtonian laws governing mechanics hold good but the effects of gravity would be nullified and so could be legitimately left out of consideration. Such a ‘freely-falling frame’ would thus be the nearest thing to a spaceship marooned in the depths of space far away from the influence of any celestial body.

A freely falling frame is not a true inertial frame

So, would it in fact be impossible to distinguish from the inside between a box falling freely under the gravitational influence of the Earth and a spaceship marooned in empty space? The answer is, perhaps surprisingly, no. In a ‘freely falling’ lift dropping towards the Earth, or the centre of any other massive body, there would be so-called ‘tidal effects’ because the Earth’s gravitational field is not homogeneous (the same in all localities) and isotropic (the same in all directions). If one released a handful of ball-bearings or a basketful of apples in a freely falling lift, the ball-bearings or apples at the ‘horizontal’ extremities would curve slightly towards each other as they fell since their trajectories would be directed towards the centre of the Earth rather than straight downwards. Likewise, the top and bottom apples would not remain the same distance apart since the forces on them, dependent as they are on the distances of the two apples from the Earth’s centre of mass, would be different and this difference would increase as the falling lift accelerated.

It turns out, then, that, at the end of the day, Einstein’s freely falling lift is not a great deal better than Galileo’s ship ─ although both are good enough approximations to inertial frames, or rather are very good imitations of inertial frames. One can, of course, argue in Calculus manner that the strength of the Earth’s gravitational field will be the same over an ‘infinitesimally small region’ ─ though without going into further details about the actual size of such a region. Newton’s Laws in their purity and integrity are thus only strictly applicable to such ‘infinitesimal’ regions in which case there will inevitably be abrupt transitions, i.e. ‘accelerations’, as we move from one infinitesimal region to another. The trajectory of any free falling object will thus not be fluent and continuous but jerky at a small enough scale.

For that matter, it is by no means obvious that a spaceship marooned in the  middle of ‘empty’ space is a true ‘inertial frame’. According to Einstein’s General Theory of Relativity, Space-time is ‘warped’ or distorted by the presence of massive objects and this space-time curvature apparently extends over the whole of the universe ─ albeit with very different local effects. If the universe is to be considered a single entity, then strictly speaking there is nowhere inside it which is completely free of ‘curvature’, and so there is nowhere to situate a ‘true’ inertial frame.

What to Conclude?

So where does all this leave us? Or, more specifically, what bearing does all this discussion have on the theory I am attempting to develop ?

In Ultimate Event Theory, the basic entities are not bodies but point-like ultimate events which, if they are strongly bonded together and keep repeating more or less identically, constitute what we view as objects. In its most simplistic form, the equivalent of an ‘object’ is a single ultimate event that repeats indefinitely, i.e. an event-chain, while several ‘laterally connected’ event-chains make up an event cluster. There is no such thing as continuous motion in UET and, if this is what we understand by motion, there is no motion. There is, however, succession and also causal linkage between successive ultimate events which belong to the ‘same’ event-chain.

Although I did not realize this until quite recently, one could say that the equivalent of an ‘inertial frame’ in UET is the basic ‘event-capsule’, a flexible though always finite region of the event Locality within which every ultimate event has occurrence. There is no question of the basic ‘building block’ in Eventrics ‘moving’ anywhere : it has occurrence at a particular spot, then disappears and, in some cases, re-appears in a similar (but not identical) spot a ksana (moment) later. One can then pass on to imagining a ‘rest event-chain’ made up of successive ultimate events sufficiently far removed from the influence of massive event-clusters for the latter to have no influence on what occurs. This is the equivalent, if you like, of the imaginary spaceship marooned in the midst of empty space.

So, where does one go from here? One thing to have come out of the endless discussions about inertial frames and their alleged indistinguishability (at least from the inside), is that the concept of ‘motion’ has little if any meaning if we are speaking of a single object whether this object or body is a boat, a particle, ocean liner or spaceship. We thus need at least two ‘objects’, one of which is traditionally seen as ‘embedded’ in the other more or less like an object in a box. In effect, Galileo’s galley is related to the enclosing dry land of the Mediterranean or, at the limit, to the Earth itself including its atmosphere. The important point is being able to relate an object which ‘moves’ to a larger, distinctive object that remains still, or is perceived to remain so.

In effect, then, we need a system composed of at least two very different ‘objects’, and the simplest such system in UET is a ‘dual event-system’ made up of just two event-chains, each of which is composed of a single ultimate event that repeats at every ksana. Now, although any talk of such a system ‘moving’ is only façon de parler , we can quite properly talk of such a system expanding, contracting or doing neither. If our viewpoint is event-chain A , we conceive event-chain B to be, for example, the one that is ‘moving further away’ at each ksana, while if we take the viewpoint of event-chain B, it is the other way round. The important point, however, is that the dual system is expanding if this distance increases, and by distance increasing we mean that there is a specified, finite number of ultimate events that could be ‘fitted into’ the space between the two chains at each ksana.

This is the broad schema that will be investigated in subsequent posts. How much of Galileo’s, Newton’s and early Einstein’s assumptions and observations do I propose to carry over as physical/philosophic baggage into UET?

To start with, what we can say in advance is that the actual distance (in terms of possible positions for ultimate events) between two event-chains does not seem to matter very much. Although Galileo, or Salviati, does not see fit to mention the point ─ he doubtless thinks it too ’obvious’ ─ it is notable that, whether the ship is in motion or not, the objects inside Galileo’s cabin do not change wherever the ship is, neglecting the effects of sun and wind, i.e. that position as such does not bring about changes in physical behaviour. This is not a trivial matter. It amounts to a ‘law’ or ‘principle’ that carries over into UET, namely that the Event Locality does not by itself seem to affect what goes on there, i.e. we have the equivalent of the principle of the ‘homogeneity’ and ‘isotropy’ of Space-time. As a contemporary author puts it : “The homogeneity of space means that all points in space are physically equivalent, i.e. a transportation of any object in space does not affect in any way the processes taking place in this object. The homogeneity of time must be understood as the physical indistinguishability of all instants of time for free objects. (By a free object we mean an object which is far from all surrounding objects so that their interaction can be neglected.)”  Saxena, Principles of Modern Physics  2.2)

What about the equivalent of velocity? Everything we know about so-called ‘inertial systems’ in the Galileian sense suggests that, barring rather recondite magnetic and gravitational effects, the velocity of a system does not seem to matter very much, provided it is constant and in a straight line. Now, what this means in UET terms is that if successive members of two event-chains get increasingly separated along one spatial direction, this does not affect what goes on in each chain or cluster so long as this increase remains the same. What does affect what goes on in each chain is when the rate of increase or decrease changes : this not only means the system as a whole has changed, but that this change is reflected in each of the two members of the dual system. When travelling in a car or train we often have little idea of our speed but our bodies register immediately any abrupt substantial change of speed or direction, i.e. an acceleration.  This is, then, a feature to be carried over into UET since it is absolutely central to traditional physics.

Finally, that there is the question of there being a limit to the possible increase of distance between two event-chains. This principle is built into the basic assumptions of UET since everything in UET, except the extent of the Locality itself, has an upper and lower limit. Although there is apparently nothing to stop two event-chains which were once adjacent from becoming arbitrarily far apart at some subsequent ksana provided they do this by stages, there is a limit to how much a dual system can expand within the ‘space’ of a single ksana. This is the (now) well-known concept of there being an upper limit to the speed of all particles. Newton may have thought there had to be such a limit but if so he does not seem to have said so specifically : in Newtonian mechanics a body’s speed can, in principle, be increased without limit. In UET, although there is no continuous movement, there is a (discontinuous) ‘lateral space/time displacement rate’ and this, like everything else is limited. In contrast to orthodox Relativity theory, I originally attempted to make a distinction between such an unattainable upper limit, calling it c, and the highest attainable rate which would be one space less per ksana. This means one does not have the paradox of light actually attaining the limit and thus being massless (which it is in contemporary physics). However, this finicky separation between c s0/t0 and c* = (c – 1) s0/t0 (where s0 and t0 are ‘absolute’ spatial and temporal units) may well prove to be too much of a nuisance to be worth maintaining.  SH 21/11/14

NOTES

Note 1  This extract and following ones are taken from Drake’s translation of Dialogue concerning two world systems by Galileo Galilei (The Modern Library)

Note : Recent posts have focused on ‘macroscopic’ events and event-clusters, especially those relevant to personal ‘success’ and ‘failure’. I shall be returning to such themes eventually, but the point has now come to review the basic ‘concepts’ of ‘micro’ (‘ultimate’) events. The theory ─ or rather paradigm ─ seems to  know where it wants to go, and, after much trepidation, I have decided to give it its head, indeed I don’t seem to have any choice in the matter.  An informal ─ but nonetheless tolerably stringent ─ treatment now seems more appropriate than my original attempted semi-axiomatic presentation. SH   26/6/14

Beginnings

It is always necessary to start somewhere and assume certain things, otherwise you can never get going. Contemporary  physics may be traced back to Democritus’ atomism, that is to the idea that ‘everything’ is composed of small ‘bodies’ that cannot be further divided and which are indestructible ─ “Nothing exists except atoms and void” as Democritus put it succinctly. What Newton did was essentially to add in the concept of a ‘force’ acting between atoms and which affects the motions of the atoms and the bodies they form. ‘Classical’, i.e. post-Renaissance  but pre twentieth-century physics, is based on the conceptual complex atom/body/force/motion.

Ultimate Event Theory (UET), starts with the concept of the ‘event’. An event is precisely located : it happens at a particular spot and at a particular time, and there is nothing ‘fuzzy’ about this place and time. In contrast to a solid object an ‘event’ does not last long, its ‘nature’ is to appear, disappear and never come back again. Above all, an event does not ‘evolve’ : it is either not at all or ‘in one piece’. Last but not least, an ultimate event is always absolutely still : it cannot ‘move’ or change, only appear and disappear. However, in certain rare cases it can give rise to other ultimate events, either similar or dissimilar.

Rejection of Infinity

The spurious notion of ‘infinity’ is completely excluded from UET: this clears the air considerably and allows one to deduce at once certain basic properties about events. To start with, macroscopic events, the only ones we are directly aware of, are not (in UET) made up of an ‘infinite’ number of ‘infinitely small’ micro-events: they are composed of a particular, i.e. finite, number of ‘ultimate events’ ─ ultimate because such micro-events cannot be further broken down (Note 1).

Size and shape of Ultimate Events

Ultimate events may well  vary in size and shape and other characteristics but as a preliminary simplifying assumption, I assume that they are of the same shape and size, (supposing these terms are even meaningful at such a basic level). All ultimate events thus have exactly the same ‘spatio/temporal extent’ and this extent is an exact match for the ‘grid-spots’ or  ‘event-pits’ that ultimate events occupy on the Event Locality. The occupied region may be envisaged as a cuboid of dimensions su × su × su , or maybe a sphere of radius su ,  or indeed any shape of fixed volume which includes three dimensions at right angles to each other.
Every ultimate event occupies such a ‘space’ or ‘place’ for the duration of a single ksana of identical ‘length’ t0. Since everything that happens is reducible to a certain number of  ultimate events occupying fixed positions on the Locality, ‘nothing can happen’ within a spatial region smaller than su3 or within a ‘lapse of time’ smaller than t0. Though there may conceivably be smaller spatial and temporal intervals, they are irrelevant since Ultimate Event Theory is a theory about ‘events’ and their interactions, not about the Locality itself.

Event Kernels and Event Capsules

The region  su3 t0  corresponds to the precise region occupied by an individual ultimate event. As soon as I started playing around with this simple model of precisely located ultimate events, I saw that it would be necessary to introduce the concept of the ‘Event Capsule’. The latter normally has a much greater spatial extent than that occupied by the ultimate event itself : it is only the small central region known as the ‘kernel’ that is of spatial extent su3, the relation between the kernel and the capsule as a whole being somewhat analogous to that between the nucleus and the enclosing atom. Although each ‘emplacement’ on the Locality can only receive a single ultimate event, the vast spatial region surrounding the ‘event-pit’ itself is, as it were, ‘flexible’. The essential point is that the Event Capsule, which completely fills the available ‘space’, is able to expand and contract when subject to external (or possibly also internal) forces.
There are, however, fixed limits to the size of an Event Capsule ─ everything except the Event Locality itself has limits in UET (because of the Anti-Infinity Axiom). The Event Capsule varies in spatial extent from the ‘default’, maximal size of s03 to the  absolute minimum size of u3which it attains when the Event Capsule has shrunk to the dimensions of the ‘kernel’ housing a single ultimate event.

Length of a ksana

The ‘length’ of a ksana, the duration or ‘temporal dimension’ of an ultimate event, likewise of an Event Capsule, does not expand or contract but, by hypothesis, always stays the same. Why so? One could in principle make the temporal interval flexible as well but this seems both unnecessary and, to me, unnatural. The size of the enveloping capsule should not, by rights, have anything to do what actually occurs inside it, i.e. with the ultimate event itself, and, in particular, should not affect how long an ultimate event lasts. A gunshot is the same gunshot whether it is located within an area of a few square feet, within a square kilometre or a whole county, and it lasts the same length of time whether we record it as simply having taken place in such and such a year, or between one and one thirty p.m. of a particular day within this year.

Formation of Event-Chains and Event-clusters

In contrast to objects, a fortiori organisms, it is in the nature of an ultimate event to appear and then disappear for ever : transience and ephemerality are of the very essence of Ultimate Event Theory. However, for reasons that we need not enquire into at present, certain ultimate events acquire the ability to repeat more or less identically during (or ‘at’) subsequent ksanas, thus forming event-chains. If this were not so, there would be no universe, no life, nothing stable or persistent, just a “big, buzzing confusion” of ephemeral ultimate events firing off at random and immediately subsiding into darkness once again.
Large repeating clusters of events that give the illusion of permanence are commonly known as ‘objects’ or ‘bodies’ but before examining these, it is better to start with less complex entities. The most rudimentary  type of event-chain is that composed of a single ultimate event that repeats identically at every ksana.

‘Rest Chains’

Classical physics kicks off with Galileo’s seminal concept of inertia which Newton later developed and incorporated into his Principia (Note 2). In effect, according to Galileo and Newton,  the ‘natural’ or ‘default’ state of a body is to be “at rest or in constant straight-line motion”. Any perceived deviation from this state is to be attributed to the action of an external force, whether this force be a contact force like friction or a force which acts from a distance like gravity.
As we know, Newton also laid it down as a basic assumption that all bodies in the universe attract all others. This means that, strictly speaking, there cannot be such a thing as a body that is exactly at rest (or moving exactly at a constant speed in a straight line) because the influence of other massive bodies would inevitably make such a body deviate from a state of perfect rest or constant straight-line motion. And for Newton there was only one universe and it was not empty.
However, if we  consider a body all alone in the depths of space, it is reasonable to dismiss the influence of all other bodies as entirely negligible ─ though the combined effect of all such influences is never exactly zero in Newtonian Mechanics. Our ideal isolated body will then remain at rest for ever, or if conceived as being in motion, this ‘motion’ will be constant and in a straight line. Thus Newtonian Mechanics. Einstein replaced the classical idea of an ‘inertial frame’ with the concept of a ‘free fall frame’, a region of Space/Time where no external forces could trouble an object’s state of rest ─ but also small enough for there to be no variation in the local gravitational field.
In a similar spirit, I imagine an isolated event-chain completely removed from any possible interference from other event-chains. In the simplest possible case, we thus have a single ultimate event which will carry on repeating indefinitely (though not for ‘ever and ever’) and each time it re-appears, this event will occupy an exactly similar spatial region on the Locality of size s03 and exist for one ksana, that is for a ‘time-length’ of to.  Moreover, the interval between successive appearances, supposing there is one, will remain the same. The trajectory of such a repeating event, the ‘event-line’ of the chain, may, very crudely, be modelled as a series of dots within surrounding boxes all of the same size and each ‘underneath’ the other.

True rest?

Such an event-chain may be considered to be ‘truly’ at rest ─ inasmuch as a succession of events can be so considered. In such a context, ‘rest’ means a minimum of interference from other event-chains and the Locality itself.
Newton thought that there was such a thing as ‘absolute rest’ though he conceded that it was apparently not possible to distinguish a body in this state from a similar body in an apparently identical state that was ‘in steady straight-line motion’ (Note 3). He reluctantly conceded that there were no ‘preferential’ states of motion and/or rest.
But Newton dealt in bodies, that is with collections of  atoms which were eternal and did not change ever. In Ultimate Event Theory, ‘everything’ is at rest for the space of a single ksana but ‘everything’ is also ceaselessly being replaced by other ‘things’ (or by nothing at all) over the ‘space’ of two or more ksanas. In the next post I will investigate what meaning, if any, is to be given to ‘velocity’ ‘acceleration’ and ‘inertia’ in Ultimate Event Theory.       SH  26/6/14

Note 1  One could envisage the rejection of infinity as a postulate, one of the two or three most important postulates of Ultimate Event Theory, but I simply regard the concept of infinity as completely meaningless, as ‘not even wrong’.         I do, however,  admit the possibility of the ‘para-finite’ which is a completely different and far more reasonable concept. The ‘para-finite’ is a domain/state where all notions of measurement and quantity are meaningless and irrelevant : it is essentially a mystical concept (though none the worse for that) rather than a mathematical or physical one and so should be excluded from natural science.
The Greeks kept the idea of actual infinity firmly at arm’s length. This was both a blessing and, most people would claim, also a curse. A blessing because their cosmological and mathematical models of reality made sense, a curse because it stopped them developing the ‘sciences of motion’, kinematics and dynamics. But it is possible to have a science of dynamics without bringing in infinity and indeed this is one of the chief aims of Ultimate Event Theory.

Note 2  Galileo only introduced the concept of an ‘inertial frame’ to meet the obvious objection to the heliocentric theory, namely that we never feel the motion of the Earth around the Sun. Galileo’s reply was that neither do we necessarily detect the regular motion of a ship on a calm sea ─ the ship is presumably being rowed by well-trained galley-slaves. In his Dialogue Concerning the Two World Systems, (pp. 217-8 translator Drake) Galileo’s spokesman, Salviati, invites his friends to imagine themselves in a makeshift laboratory, a cabin below deck (and without windows) furnished with various homespun pieces of equipment such as a bottle hung upside down with water dripping out, a bowl of water with goldfish in it, some flies and butterflies, weighing apparatus and so on. Salviati claims that it would be impossible to know, simply by observing the behaviour of the drips from the bottle, the flight of insects, the weight of objects and so on, whether one was safely moored at a harbour or moving in a straight line at a steady pace on a calm sea.
Galileo does not seem to have realized the colossal importance of this thought-experiment. Newton, for his part, does realize its significance but is troubled by it since he believes ─ or at least would like  to believe ─ that there is such a thing as ‘absolute motion’ and thus also ’absolute rest’. The question of whether Galileo’s principle did, or did not, cover optical (as opposed to mechanical) experiments eventually gave rise to the theory of Special Relativity. The famous Michelsen-Morley experiment was, to everyone’s surprise at the time, unable to detect any movement of the Earth relative to the surrounding ‘ether’. The Earth itself had in effect become Galileo’s ship moving in an approximately straight line at a steady pace through the surrounding fluid.
Einstein made it a postulate (assumption)  of his Special Theory that “the laws of physics are the same in all inertial frames”. This implied that the observed behaviour of objects, and even living things, would be essentially the same in any ‘frame’ considered to be ‘inertial’. The simple ‘mind-picture’ of a box-like container with objects inside it that are free to move, has had tremendous importance in Western science. The strange thing is that in Galileo’s time vehicles  ─ even his ship ─ were very far from being ‘inertial’, but his idea has, along with other physical ideas, made it possible to construct very tolerable ‘inertial frames’ such as high-speed trains, ocean liners, aeroplanes and space-craft.

Note 3  Newton is obviously ill at ease when discussing the possibility of ‘absolute motion’ and ‘absolute rest’. It would seem that he believed in both for philosophical (and perhaps also religious) reasons but he conceded that it would, practically speaking, be impossible to find out whether a particular state was to be classed as ‘rest’ or ‘straight-line motion’. In effect, his convictions clashed with his scientific conscience.

“Absolute motion, is the translation of a body from one absolute place into another. Thus, in a ship under sail, the relative place of a body is that part of the ship which the body possesses, or that part of its cavity which the body fills, and which therefore moves together with the ship, or its cavity. But real, absolute rest, is the continuance of the body in the same part of that immovable space in which the ship itself, its cavity and all that it contains, is moved. (…) It may be, that there is no such thing as an equable motion, whereby time may be accurately measured. (…) Instead of absolute places and motions we use relative ones; and that without any inconvenience in common affairs: but in philosophical disquisitions, we ought to abstract from our senses and consider things themselves, distinct from what are only sensible measures of them. For it may be that there is no body really at rest, to which the places and motions of others may be referred.”
Newton, Principia, I, 6 ff.

Cromwell and Providence

In a previous post I suggested that the three most powerful men in Western history who acquired their position other than by inheritance, were Napoleon, Cromwell and Hitler. Of the three, if we judge them over the whole of their careers, Cromwell was undoubtedly the most successful. He started his adult life as an impoverished country squire with no connections at court but ended up as Lord Protector of a country that was already overtaking France and Spain and would soon vie with Holland for world-wide naval and mercantile supremacy. Once put in command of an army, unlike Napoleon and Hitler, Cromwell never lost a battle and he died quietly in his bed not especially bothered by his latter-day unpopularity. It is true that he did not aspire to world domination like Napoleon and Hitler, but this is part of the reason that he succeeded more completely than they did : he cut his coat according to the cloth available while they did not.
But why exactly was he so successful?

The sense of mission

From the standpoint of Eventrics, individuals succeed because they ‘go with the flow of events’. Either consciously or unconsciously, they identify dominant event patterns, or rather patterns that are about to become dominant, and they ‘put their faith in them’, allowing the momentum of the events to do most (but not all) of the work. They must also be quick to change course and dissociate themselves from the onward rush of events when this is no longer to their advantage.
Machiavelli was perhaps the first to state openly (but not the first to believe) that the pursuit and retention of power is a technique that can be learned just like horsemanship or perspective drawing. However, although Machiavelli gives some useful general principles, the technique cannot be taught but must be learned in situ, in the thick of events. In what does it consist? Essentially, in achieving the right mix of self-confidence and reliance on one’s own judgment combined with scrupulous attention to detail, to the day to day drift of unpredictable and uncontrollable event currents. Very few individuals have what it takes to do both. Messianic individuals neglect practical considerations, believing it to be beneath them, while sound technicians lack the breadth of vision and occasional recklessness without which nothing great can be achieved.
Clearly, an individual who believes him or herself to be divinely inspired has a vast advantage over ‘ordinary people’, provided, of course, that he possesses a minimum of technical ability and worldly knowledge. Most people do not know what they want or even what they believe : someone who clearly knows both commands respect, devotion and admiration (also fear). However, most ‘men of mission’ become blinded by their own success as Hitler, Napoleon and even Julius Caesar did alongside a host of lesser figures.

The Puritan mentality that Cromwell embodied was admirably suited indeed to the challenges and opportunities of the English civil war.       The concept of Providence ─ useful precisely because it was vague ─ was an enormous advance on pagan ideas of ‘fate’ which encourage defeatism, or the rock-hard belief in ‘Divine Right’ such as prevailed in the Royalist camp which encouraged recklessness. Instead of looking for ‘signs’ in the sense of omens and presages, the Puritans tried to discern an underlying ‘logic of events’ which they interpreted as the workings of Providence.
Moreover, the moral earnestness of the Puritan forced him to act, and if his acts were successful this was proof that God was on his side ─ “Duties are ours, events are the Lord’s” as Samuel Rutherford put it concisely. But, on the other hand, since every good Puritan was convinced that he was fallible, failure did not reflect on God, only on man’s faulty reading of the writing on the wall. This provided a useful check on the megalomania to which messianic individuals and movements are prone.
The Puritans in effect had it both ways, hence their success both in England and in the Netherlands (where they victoriously opposed the might of Spain). They picked up on the psychological advantages that come from believing oneself to be guided by God, but were protected by the peculiarities of their belief system from the dual dangers of inactivity (Quietism) and cocksureness (messianism).

Cromwell as ‘God’s Englishman’

Cromwell definitely believed that he was guided by God ─ “A divine presence hath gone along with us in the late great transactions of the nation”, he wrote in a letter. Again, on receiving the Scottish command, he wrote to Richard Major, “Truly I have been called unto them [these functions] by the Lord.” And many other people at the time bore witness to the same feeling. John Desborough thought that “It is [God] himself that hath raised you up amongst men, and hath called you to high enjoyments”. Fairfax himself, originally Cromwell’s superior, speaks of “The constant presence and blessing of God that have accompanied him”.
Moreover, this belief in ‘election’ was very widespread : there was nothing at all unusual in the idea that one could be, and indeed should be, guided by the Divine Will. And the Divine Will was for the Puritan, but not the Catholic or even High Church Anglican, directly knowable without the intermediary of priest or monarch. Cromwell, though extraordinary in the degree of his success, could (and did) believe himself to be ‘typical’, to be nothing out of the ordinary except inasmuch as he had, for some inscrutable reason, been given a larger task to fulfil (Note 1).
And the peculiarity of the Protestant, and more specifically Puritan, mentality safeguarded Cromwell from the self-intoxication of, say, the 19th century Mahdi (the victor at Khartoum) who enjoyed as much initial success as Cromwell and was, for a while, believed to be invincible. Almost alone of all dictators and military commanders, Cromwell demonstrates a certain humility and, incredible though it sounds, a seemingly genuine lack of personal ambition ! He really did have’greatness thrust upon him’. A moment after writing that he sees the hand of the Lord in his (Cromwell’s) sudden advancement, he goes on to say that he is “not without assurance that he [God] will enable this poor worn and weak servant to do his will”. This conception of being a ‘servant’ who is without merit in himself is rooted in the peculiarly Puritan (originally Calvinistic) idea that the ‘elect’ are not chosen for their merit but simply because God so pleases.
The permanent feeling of inadequacy which is central to the Puritan mentality, also meant that no one, however high and mighty he might be, would be tempted to ‘rest on his laurels’. And this in turn meant that a military commander or ruler always had to pay as much attention to detail as a village carpenter — as Cromwell indeed did. Cromwell, who trained and personally led the Parliamentarian cavalry was not a dashing glamorous figure like Prince Rupert on the Royalist side. Cromwell trained his cavalry to advance at the trot, not at the gallop, and always to reform in good order whether they broke through the enemy ranks or not. Cromwell also held out for a strictly no nonsense approach to enlistment and advancement ─ “The State in choosing men to serve it, takes no notice of their opinions”. This may seem commonsensical enough today but was well-nigh unheard of in the bigoted seventeenth century and unusual even as late as the latter nineteenth century.

The Extraordinary vs. the Impossible

Given all this, the wonder is, not that Cromwell and his followers, succeeded so well but that Cromwell has had so few imitators. What, then, were the weak points of this apparently unbeatable mixture of self-confidence and humility, grand strategy and careful attention to detail?
The Cardinal de Retz, himself one of the leaders of a movement not unlike the ‘Great Rebellion’, the Fronde, says that it is essential that “resolution should run parallel with judgment” and especially with what he calls ‘heroic judgment’ which “is able to discern the extraordinary from the impossible”. Now, Cromwell was able to make this all-important distinction. Overcoming a Royalist army staffed by experienced officers and backed by powerful foreign powers was certainly extraordinary but, as it appears, not impossible. Even more extraordinary on the psychological level is that a group of men should have dared to try and execute a monarch and have felt no guilt about doing so. And Cromwell’s endless stream of military successes against all comers is certainly out of the ordinary.
Cromwell was, however, able to discern what was, for historical reasons, impossible, namely the creation of a modern egalitarian democratic state. Movements like the Levellers and the Diggers were too much ahead of their time since they called, on the one hand, for a wholesale redistribution of land to all who were prepared to work it and, incredibly, even suggested the creation of a ‘Welfare State’, an idea that did not become a reality anywhere in the world until two and a half centuries later.
The trouble with believing that you are guided by God and that you know God’s Will, is that God is, by definition, all-powerful. He can thus, if He so wishes, bring about not only the extraordinary but also the impossible. This is a very dangerous doctrine for anyone who exercises power to adhere to. One of the most important tenets of the Enlightenment was that even God (in whom Voltaire and the Encyclopaedists believed) was limited by the physical laws of the universe He had created and did not intervene directly in the day to day running of the universe, nor was his intervention needed ─ Newton’s Laws sufficed.
But even the toned-down Protestant vision of a Calvin or a Cromwell was, at bottom, a shaky intellectual edifice: it could explain success and temporary setbacks but could make no sense of disaster. Cromwell, fortunately for him, did not live to see the Restoration. The surviving Puritans were faced not so much with a political and social crisis as a profound psychological one. If success showed that God was on your side, why had the Royalists triumphed? Even given that the Puritans themselves, like the Jews in a similar situation, were fallible and sinful, it remained difficult, if not impossible, to explain why God should seemingly have ‘changed sides’.
From the Restoration on most Non-conformists eschewed politics, and they certainly eschewed the use of force. Many retired, like the Quakers, into Quietism on the one hand while directing their burning need for activity into industrial and commercial channels. The Puritans had failed to bring about a complete social revolution, but they did more or less single-handed bring about the Industrial Revolution since practically all the inventors and forward-looking industrialists from Newcomen to Watt to Darby could be broadly described as ‘Puritans’ in belief and mentality (Note 2).
Since Cromwell, there has been no one who has so successfully combined the two opposing features of a sense of mission and intense practicality (Note 3). Bismarck had the Realpolitik but lacked the messianism, Hitler the messianism but not the rationality and attention to detail at any rate in the military sphere.      SH

Note 1    “There was nothing unusual in the belief which henceforth governed Cromwell’s actions ─ that he was directly guided by the Divine Will. He did not, of course, regard himself as the infallible interpreter of God’s wishes, but he tested his actions no longer by the criticism of his own reason, but by their effectiveness. If he did God’s will, he must succeed; failure meant that the divine contact had somewhere broken down ─ that there had been sin. This it was which gave him his buoyant confidence when things went well, and drove him to an agony of prayer when things went wrong.”
C.V. Wedgwood, Oliver Cromwell

Note 2 Darlington, in his monumental work, The Evolution of Man and Society, singles out 16 individuals as ‘founders of the Scientific Revolution in Britain born between 1620 and 1800’, and 17 individuals as ‘founders of the Industrial Revolution born between 1650 and 1810’.
Of the scientists, only five are described as coming from the ‘gentry’ and only two, Halley and Malthus, gained entrance to an English university after 1662 when a Bill excluded religious non-conformists from places of higher learning. The Quakers weigh in strongly on the science side with Dalton, Young and Davy and field Darby as one of the prime movers of the Industrial Revolution. The steam engine is exclusively a Non-conformist achievement: Newcomen was a Baptist, Savery of Huguenot origin, Watt a Scottish Presbyterian and Trevithick a Methodist.
In conclusion, it would seem that neither the Anglican Church nor Oxford and Cambridge had much to do with either the industrial or scientific revolution in Britain, at any rate prior to the mid nineteenth century. Newton himself was a Unitarian but stifled his doubts though it is said that his suspected non-conformism stopped him from becoming Master of Trinity College, Cambridge. Halley was denied the chair of astronomy at Oxford on religious grounds. Mercifully the Royal Institution after 1800 dropped the religious requirement, thus allowing Davy and Faraday to lecture there.
Interestingly, there is  only one self-avowed atheist in either list, the ironmonger Wilkinson, and I am not quite sure that Darlington was right there.

Note 3 Stalin comes nearest perhaps. Stalin died in his bed after a long reign marked by complete victory over his principal enemy, Trostsky, and an enormous gain of territory for the USSR. But the cost of Russian victory in WWII was very high and Stalin was partly responsible for this because he allowed himself to be outwitted by Hitler. He also made the disastrous error of purging his own army of many of his most experienced and ablest officers for ideological reasons ─ something Cromwell would never have done.

Although, in modern physics,  many elementary particles are extremely short-lived, others such as protons are virtually immortal. But either way, a particle, while it does exist, is assumed to be continuously existing. And solid objects such as we see all around us like rocks and trees are also assumed to carry on being rocks and trees from start to finish even though they do undergo considerable changes in physical and chemical composition. What is out there is  always there when it’s out there, so to speak.
However, in Ultimate Event Theory (UET) the ‘natural’ tendency is for everything to flash in and out of existence and most ultimate events, the ‘atoms’ or elementary particles of  Eventrics,  disappear for ever leaving no trace and even with more precise instruments than we have at present, wouldshow up as a sort of faint permanent background ‘noise’, a ‘flicker of existence’. Certain ultimate events, those that have acquired persistence ─ we shall not for the moment ask how and why they acquire this property ─ are able to bring about, i.e. cause, their own re-appearance and eventually to constitute a repeating event-chain or causally bonded sequence. And some event-chains also have the capacity to bond to other event-chains, eventually  forming relatively persistent clusters that we know as matter.  All apparently solid objects are, according to the UET paradigm, conglomerates of repeating ultimate events that are bonded together ‘laterally’, i.e. within  the same ksana, and ‘vertically’, i.e. from one ksana to the next. And the cosmic glue is not gravity or any other of the four basic forces of contemporary physics but causality.

The Principle of Spatio/Temporal Continuity

Newtonian physics, likewise 18th and 19th century rationalism generally, assumes what I have referred to elsewhere as the Postulate of Spatio-temporal Continuity. This postulate or principle, though rarely explicitly  stated in philosophic or scientific works,  is actually one of the most important of the ideas associated with the Enlightenment and thus with the entire subsequent intellectual development of Western society (Note 1). In its simplest form, the principle says that an event occurring here, at a particular spot in Space-Time (to use the traditional term), cannot have an effect there, at a spot some distance away without having effects at all (or at least most or some) intermediate spots. The original event, as it were, sets up a chain reaction and a frequent image used is that of a whole row of upright dominoes falling over one after the other once the first has been pushed over. This is essentially how Newtonian physics views the action of a force on a body or system of bodies, whether the force in question is a contact force (push/pull) or a force acting at a distance like gravity ─ though in the latter case Newton was unable to provide a mechanical model of how such a force could be transmitted across apparently empty space.
As we envisage things today, a blow affects a solid object by making the intermolecular distances of the surface atoms contract a little and they pass on this effect to neighbouring atoms which in turn affect nearby objects they are in contact with or exert an increased pressure on the atmosphere, and so on. Moreover, although this aspect of the question is glossed over in Newtonian (and even modern) physics, each transmission of the original impulse  ‘takes time’ : the re-action is never instantaneous (except possibly in the case of gravity) but comes ‘a moment later’, more precisely at least one ksana later. This whole issue will be discussed in more detail later, but, within the context of the present discussion, the point to bear in mind is that,  according to Newtonian physics and rationalistic thought generally, there can be no leap-frogging with space and time. Indeed, it was because of the Principle of Spatio-temporal Continuity that most European scientists rejected out of hand Newton’s theory of universal attraction since, as Newton admitted, there seemed to be no way that a solid body such as   the Earth could affect another solid body such as the Moon thousands  of kilometres without affecting the empty space between. Even as late as the mid 19th century, Maxwell valiantly attempted to give a mechanical explanation of his own theory of electro-magnetism, and he did this essentially because of the widespread rock-hard belief in the principle of spatio-temporal continuity.

So, do I propose to take the principle over into UET? No, except possibly in special situations. If I did take over the principle, it would mean that certain regions of the Locality would soon get hopelessly clogged up with colliding event-chains. Indeed, if all the possible positions in between two spots where ultimate events belonging to the same chain had occurrence were occupied, event-chains would behave as if they were solid objects and one might as well just stick to normal physics. A further, and more serious, problem is that, if all event-chains were composed of events that repeated at every successive ksana, one would expect event-chains with the same ‘speed’ (space/time ratio with respect to some ‘stationary’ event-chain) to behave in the same way when confronted with an obstacle. Manifestly, this does not happen since, for example, photon event-chains behave very differently from neutrino event-chains even though both propagate at the same, or very similar, speeds.
One of the main reasons for elaborating a theory of events in the first place was my deep-rooted conviction ─ intuition if you like ─ that physical reality is discontinuous. I do not believe there is, or can be, such a thing as continuous motion, though there is and probably always will be succession and thus change since, even if nothing else is happening, one ksana is perpetually being replaced by another, different, one ─ “the moving finger writes, and, having writ, moves on” (Rubaiyat of Omar Khayyam). Moreover, this movement is far from smooth : ‘time’ is not a river that flows at a steady rate as (Newton envisaged it) but a succession of ‘moments’, beads of different sizes threaded together to make a chain and with minute gaps between the beads which allow the thread that holds them together to become momentarily visible.
If, then, one abandons the postulate of Spatio-temporal Continuity, it becomes perfectly feasible for members of an event-chain to ‘miss out’ intermediate positions and so there most definitely can be ‘leap-frogging’ with space and time. Not only are apparently continuous phenomena discontinuous but one suspects that they have very different staccato rhythms.

‘Atomic’ Event Capsule model

At this point it is appropriate to review the basic model.
I envisage each ultimate event as having occurrence at a particular spot on the Locality, a spot of negligible but not zero extent. Such spots, which receive (or can receive) ultimate events are the ‘kernels’ of much larger ‘event-capsules’ which are themselves stacked together in a three-dimensional lattice. I do not conceive of there being any appreciable gaps between neighbouring co-existing event-capsules : at any rate, if there are gaps they would seem to be very small and of no significance, essentially just demarcation lines. According to the present theory these spatial ‘event-capsules’ within which all ultimate events have occurrence cannot be extended or enlarged  ─ but they can be compressed. There is, nonetheless,  a limit to how far they can be squeezed because the kernels, the spots where ultimate events can and do occur, are incompressible.
I believe that time, that is to say succession, definitely exists; in consequence, not only ultimate events but the space capsules themselves, or rather the spots on the Locality where there could be ultimate events, appear and disappear just like everything else. The lattice framework, as it were, flicks on and off and it is ‘on’ for the duration of a ksana, the ultimate time interval (Note 2). When we have a ‘rest event-chain’ ─ and every event-chain is ‘at rest’ with respect to itself and an imaginary observer moving on or with it ─ the ksanas follow each other in close succession, i.e. are as nearly continuous as an intrinsically  discontinuous process can be.
According to the theory, the ‘size’ or ‘extent’ of a ksana cannot be reduced  ─ otherwise there would be little point in introducing the concept of a minimal temporal interval and we would be involved in infinite regress, the very thing which I intend to avoid at all costs. However, the distance between ksanas can, so it is suggested, be extended, or, more precisely, the distance between the successive kernels of the event capsules, where the ultimate events occur, can be extended. That is, there are gaps between events. As is explained in other posts, in UET the ‘Space/Time region’ occupied by the successive members of an event-chain remains the same irrespective of ‘states of motion’ or other distinguishing features. But the dimensions themselves can and do change. If the space-capsules contract, the time dimension must expand and this can only mean that the gaps between ksanas widen (since the extent of an ‘occupied’ ksana is cnstant. The more the space capsules contract, the more the gaps must increase (Note 3).  But, as with everything else in UET, there is a limiting value since the space capsules cannot contract beyond the spatial limits of the incompressible kernels. Note that this ‘Constant Region Principle’ only applies to causally related regions of space ─ roughly what students of SR view as ‘light cones’.

The third parameter of motion

In traditional physics, when considering an object or body ‘in motion’, we essentially only need to specify two variables : spatial position and time. Considerations of momentum and so forth is only required because it affects future positions at future moments, and aids prediction. To specify an object’s ‘position in space’, it is customary in scientific work to relate the object’s position to an imaginary spot called the Origin where three mutually perpendicular axes meet. To specify the object’s position ‘in time’ we must show or deduce how many ‘units of time’ have elapsed since a chosen start position when t = 0. Essentially, there are only two parameters required, ‘space’ and ‘time’ : the fact that the first parameter requires (at least) three values is not, in the present context, significant.
Now, in UET we likewis need to specify an event’s position with regard to ‘space’ and ‘time’. I envisage the Event Locality at any ‘given moment’ as being composed of an indefinitely extendable set of ‘grid-positions’. Each ‘moment’ has the same duration and, if we label a particular ksana 0 (or 1) we can attach a (whole) number to an event subsequent to what happened when t = 0 (or rather k = 0). As anyone who has a little familiarity with the ideas of Special Relativity knows, the concept of an  ‘absolute present’ valid right across the universe is problematical to say the least. Nonetheless, we can talk of events occurring ‘at the same time’ locally, i.e. during or at the same ksana. (The question of how these different  ‘time zones’ interlock will be left aside for the moment.)
Just as in normal physics we can represent the trajectory of an ‘object’ by using three axes with the y axis representing time and, due to lack of space and dimension, we often squash the three spatial dimensions down to two, or, more simply still, use a single ‘space’ axis, x (Note 4). In normal physics the trajectory of an object moving with constant speed will be represented by a continuous vertical straight line and an object moving at constant non-zero speed relative to an object considered to be stationary will be represented by a slanting but nonetheless still straight line. Accelerated motion produces a ‘curve’ that is not straight. All this essentially carries over into UET except that, strictly, there should be no continuous lines at all but only dots that, if joined up, would form lines. Nonetheless, because the size of a ksana is so small relative to our very crude senses, it is usually acceptable to represent an ‘object’s’ trajectory as a continuous line. What is straight in normal physics will be straight in UET. But there is a third variable of motion in UET which has no equivalent in normal physics, namely an event’s re-appearance rhythm.
Fairly early on, I came up against what seemed to be an insuperable difficulty with my nascent model of physical reality. In UET I make a distinction between an attainable ‘speed limit’ for an event-chain and an upper unatttainable limit, noting the first c * and the second c. This allows me to attribute a small mass ─ mass has not yet been defined in UET but this will come ─  to such ‘objects’ as photons. However, this distinction is not significant in the context of the present discussion and I shall  use the usual symbol c for either case. Now, it is notorious that different elementary particles (ultimate event chains) which apparently have the same (or very nearly identical) speeds do not behave in the same way when confronted with obstacles (large dense event clusters) that lie on their path. Whereas it is comparatively easy to block visible light and not all that difficult to block or at least muffle much more energetic gamma rays, it is almost impossible to stop a neutrino in its path, so much so that they are virtually undetectable. Incredible though it sounds, “about 400 billion neutrinos from the Sun pass through us every second” (Close, Particle Physics) but even state of the art detectors deep in the earth have a hard  job  detecting a single passing neutrino. Yet neutrinos travel at or close to the speed of light. So how is it that photons are so easy to block and neutrinos almost impossible to detect?
The answer, according to matter-based physics, is that the neutrino is not only very small and very fast moving but “does not feel any of the four physical forces except to some extent the weak force”. But I want to see if I can derive an explanation without departing from the basic principles and concepts of Ultimate Event Theory. The problem in UET is not why the repeating event-pattern we label a neutrino passes through matter so easily ─ this is exactly what I would expect ─ but rather how and why it behaves so  differently from certain other elementary event-chains. Any ‘particle’, provided it is small enough and moves rapidly, is likely, according to the basic ideas of UET, to ‘pass through’ an obstacle just so long as the obstacle is not too large and not too dense. In UET, intervening spatial positions are simply skipped and anything that happens to be occupying these intermediate spatial positions will not in any way ‘notice’ the passing of the more rapidly moving ‘object’. On this count, however, two ‘particles’ moving at roughly the same speed (relative to the obstacle) should either both pass through an  obstacle or both collide with it.
But, as I eventually realized, this argument is only valid if the re-appearance rates of the two ‘particles’ are assumed to be the same. ‘Speed’ is nothing but a space/time ratio, so many spatial positions against so many ksanas. A particular event-chain has, say, a ‘space/time ratio’ of 8 grid-points per ksana. This means that the next event in the chain will have occurrence at the very next ksana exactly eight grid-spaces along relative to some regularly repeating event-chain considered to be stationary. On this count, it would seem impossible to have fractional rates and every ‘re-appearance rate’ would be a whole number : there would be no equivalent in UET of a speed of, say, 4/7 metres per second since grid-spaces are indivisible.
However, I eventually realized that it was not one of my original assumptions that an event in a chain must repeat (or give rise to a different event) at each and every ksana. This at once made fractional rates possible even though the basic units of space and time are, in UET, indivisible. A ‘particle’ with a rate of 4/7 s0 /t0 could, for example, make a re-appearance four times out of every seven ksanas ─ and there are any number of ways that a ‘particle’ could have the same flat rate while not having the same re-appearance rhythm.

Limit to unitary re-appearance rate

It is by no means obvious that it is legitimate to treat ‘space’ and ‘time’ equivalently as dimensions of a single entity known as ‘Space/Time’. A ‘distance’ in time is not just a distance in space transferred to a different axis and much of the confusion in contemporary physics comes from a failure to accept, or at the very least confront, this fact. One reason why the dimensions are not equivalent is that, although a spatial dimension such as length remains the same if we now add on width, the entire spatial complex must disappear if it is to give rise to a similar one at the succeeding moment in time ─ you cannot simply ‘add’ on another dimension to what is already there.
However, for the the time being I will follow accepted wisdom in treating a time distance on the same footing as a space distance. If this is so, it would seem that, in the case of an event-chain held together by causality, the causal influence emanating from the ‘kernel’ of one event capsule, and which brings about the selfsame event (or a different one) a ksana later in an equivalent spatial position, must traverse at least the ‘width’ or diameter of a space capsule, noted s0 (if the capsule is at rest). Why? Because if it does not at least get to the extremity of the first spatial capsule, a distance of ½ s0  and then get to the ‘kernel’ of the following one, nothing at all will happen and the event-chain will terminate abruptly.
This means that the ‘reappearance rate’ of an event in an event-chain must at least be 1/1 in absolute units, i.e. 1 s0 /t0 , one grid-space per ksana. Can it be greater than this? Could it, for example, be  2, 3 or 5 grid-spacesper ksana? Seemingly not. For if and when the ultimate event re-appears, say  5 ksanas later, the original causal impulse will have covered a distance of 5 s0   ( s0 being the diameter or spatial dimension of each capsule) and would have taken 5 ksanas to do  this. And so the space/time displacement rate would be the same (but not in this case the actual inter-event distances).
It is only the unitary rate, the distance/time ratio taken over a single ksana, that cannot be less (or more) than one grid-space per ksana : any fractional (but not irrational) re-appearance rate is perfectly conceivable provided it is spread out over several ksanas.  A re-appearance rate of m/n s0/t0  simply means that the ultimate event in question re-appears in an equivalent spatial position on the Locality m times every n ksanas where m/n ≤ 1. And there are all sorts of different ways in which this rate be achieved. For example, a re-appearance rate of 3/5 s0/t0 could be a repeating pattern such as



and one pattern could change over into the other either randomly or, alternatively, according to a particular rule.
As one increases the difference between the numerator and the denominator, there are obviously going to be many more possible variations : all this could easily be worked out mathematically using combinatorial analysis. But note that it is the distribution of  and  that matters since, once a re-appearance rhythm has begun, there is no real difference between a ‘vertical’ rate of   and  ─ it all depends on where you start counting. Patterns only count as different if this difference is recognizable no matter where you start examining the sequence.
Why does all this matter? Because, each time there is a blank line, this means that the ultimate event in question does not make an appearance at all during this ksana, and, if we are dealing with large denominators, this could mean very large gaps indeed in an event chain. Suppose, for example, an event-chain had a re-appearance rate of 4/786. There would only be four appearances (black dots) in a period of 786 ksanas, and there would inevitably be very large blank sections of the Locality when the ultimate event made no appearance.

Lower Limit of re-creation rate

Since, by definition, everything in UET is finite, there must be a maximum number of possible consecutive non-reappearances. For example, if we set the limit at, say, 20 blank lines, or 200 03 2000, this would mean that, each time this was observed, we could conclude that the event-chain had terminated. This is the UET equivalent  of the Principle of Spatio-Temporal Continuity and effectively excludes phenomena such as an ultimate event in an event-chain making its re-appearance a century later than its first appearance. This limit would have to be estimated on the  basis of experiments since I do not see how a specific value can be derived from theoretical considerations alone. It is tempting to estimate that this value would involve c* or a multiple of c* but this is only a wild guess ─ Nature does not always favour elegance and simplicity.
Such a rule would limit how ‘stretched out’ an event-chain can be temporally and, in reality , there may not after all be a hard and fast general rule  : the maximal extent of the gap could decline exponentially or in accordance with some other function. That is, an abnormally long gap followed by the re-appearance of an event, would decrease the possible upper limit slightly in much the same way as chance associations increase the likelihood of an event-chain forming in the first place. If, say, there was an original limit of a  gap of 20 ksanas, whenever the re-appearance rate had a gap of 19, the limit would be reduced to 19 and so on.
It is important to be clear that we are not talking about the phenomenon of ‘time dilation’ which concerns only the interval between one ksana and the next according to a particular viewpoint. Here, we simply have an event-chain ‘at rest’ and which is not displacing itself laterally at all, at any rate not from the viewpoint we have adopted.

Re-appearance Rate as an intrinsic property of an event-chain

Since Galileo, and subsequently Einstein, it has become customary in physics to distinguish, not between rest and motion, but rather between unaccelerated motion and  accelerated motion. And the category of ‘unaccelerated motion’ includes all possible constant straight-line speeds including zero (rest). It seems, then,  that there is no true distinction to be made between ‘rest’ and motion just so long as the latter is motion in a straight line at a constant displacement rate. This ‘relativisation’ of  motion in effect means that an ‘inertial system’ or a particle at rest within an inertial system does not really have a specific velocity at all, since any estimated velocity is as ‘true’ as any other. So, seemingly, ‘velocity’ is not a property of a single body but only of a system of at least two bodies. This is, in a sense, rather odd since there can be no doubt that a ‘change of velocity’, an acceleration, really is a feature of a single body (or is it?).
So what to conclude? One could say that ‘acceleration’ has ‘higher reality status’ than simple velocity since it does not depend on a reference point outside the system. ‘Velocity’ is a ‘reality of second order’ whereas acceleration is a ‘reality of first order’. But once again there is a difference between normal physics and UET physics in this respect. Although the distinction between unaccelerated and accelerated motion is taken over into UET (re-baptised ‘regular’ and ‘irregular’ motion), there is in Ultimate Event Theory a new kind of ‘velocity’ that has nothing to do with any other body whatsoever, namely the event-chain’s re-appearance rate.
When one has spent some time studying Relativity one ends up wondering whether after all “everything is relative” and that  the universe is evaporating away even as we look it leaving nothing but a trail of unintelligible mathematical formulae. In Quantum Mechanics (as Heisenberg envisaged it anyway) the properties of a particular ‘body’ involve the properties of all the other bodies in the universe, so that there remain very few, if any, intrinsic properties that a body or system can possess. However, in UET, there is a reality safety net. For there are at least two  things that are not relative, since they pertain to the event-chain or event-conglomerate itself whether it is alone in the universe or embedded in the dense network of intersecting event-chains we view as matter. These two things are (1) the number of ultimate events in a given portion of an event-chain and (2) the re-appearance rate of events in the chain. These two features are intrinsic to every chain and have nothing to do with velocity or varying viewpoints or anything else.  To be continued SH

Note 2  Ideally, we would have a lighted three-dimensional framework flashing on and off and mark the successive appearances of the ‘object’ as, say, a red point of light comes on periodically when the lighted framework comes on.

Note 3 In principle, in the case of extremely high speed event-chains, these gaps should be detectable even today though the fact that such high speeds are involved makes direct observation difficult.

Note 4 This is not how we specify an object’s position in ordinary conversation. As Bohm pertinently pointed out, we in effect speak in the language of topology rather than the language of co-ordinate geometry. We say such and such an object is ‘under’, ‘over’, ‘near’, ‘to the right of’ &c. some other well-known  prominent object, a Church or mountain when outside, a bookcase or fireplace when in a room.
Not only do coordinates not exist in Nature, they do not come at all naturally to us, even today. Why is this? Chiefly, I suspect because they are not only cumbersome but practically useless to a nomadic, hunting/food gathering life style and we humans spent at least 96% of our existence as hunter/gatherers. Exact measurement only becomes essential when human beings start to manufacture complicated objects and even then many craftsmen and engineers used ‘rules of thumb’ and ‘rough estimates’ well into the 19th century.

A radically new theory, model or manner of viewing the world is inevitably going to be attacked as (1) absurd and (2) not new after all. Eventrics, which views physical reality as composed of ‘ultimate events’ (rather than atoms), can more justifiably be dismissed as crazy than being just the same old stuff in a different wrapping.
During the last few centuries the leading paradigm (consensus view of reality) has been a mixture of atomism which goes back to the Greeks, usefully combined with a belief in unalterable ‘laws of Nature’ which owes a lot to Judaeo-Christianity. The classical synthesis of Newton introduced the un-Greek (but actually very ancient) concept of force. A very satisfactory world-view emerged consisting of atoms moving around subject to contact or remote forces, the whole regulated by immutable laws laid down by God. 19th century science removed God from the picture and 20th century scientific discoveries have made matter a very tenuous substance indeed and have introduced forces unknown to Newton, nuclear, electro-magnetic and so on.
At the end of the day, however, we still work with a matter based world-view and in some cases atoms and molecules can actually be imaged, if not literally ‘seen’ with a microscope. So what is the difference between such a view and the viewpoint of Eventrics that I am proposing? Essentially this : that atoms are in some sense solid and they are extremely long-lasting if not quite eternal as Democritus and Newton imagined. Eventrics is the study of events and their interactions and Ultimate Event Theory is that part of the theory that deals with the ultimate constituents of all events, so-called ultimate events. UET thus bears roughly the same relation to Eventrics as nuclear physics does to physics. Now, ultimate events are quite definitely not substantial in any usual sense of the word and, more significantly, they only last for an incredibly short time span, that of the ksana (or chronon). The actual interval involved is yet to be determined but it would most likely be at least 10–36 seconds (the Planck scale).
This, however, is not all. It is true that certain  contemporary physicists have come round to the idea that there is a minimal temporal interval and the sort of ‘reality constituents’ they deal, in ‘causal nodes’ for example, do seem to be rather similar to my ultimate events. However, where my theory departs from current theories (apart from the lack of mathematical sophistication which will come later) is that I believe that ‘physical reality’ or ‘Space/Time’ or whatever else you want to call it, is radically discontinuous. There are, quite literally, gaps between ultimate events and, because ‘macroscopic events are conglomerates of ultimate events much as chemical compounds are conglomerates of atoms, there are gaps between everything we see and hear and, for that matter, are. This is certainly not a popular view in Western thought, scientific or not, and would be regarded by most people as lunatic. It was, however, a very widespread viewpoint amongst Hinayana Buddhists in India in the first few centuries AD. This is not, I hasten to add, the reason why I am adopting it : on  the contrary, I formed my rough view before I had even heard of Buddhism though, naturally, when I came across the views of these people via Stcherbatsky’s books on Buddhism, such views at once had a favourable audience. I must also insist that, although I do practise my own brand of meditation, I would not call myself a Buddhist and my interest in Buddhism is largely in the underlying physical theory, not the moral theory which is what these monks were essentially interested in (and still are).
The assumption that ‘physical reality’ is made up of ultimate events has two especially important consequences with regard to current science and mathematics. As far as mathematics is concerned, it means that Calculus and similar branches are all based on a false (though extremely useful) assumption, namely that everything, matter, energy, time, space and so on is “infinitely divisible’. According to UET this cannot be so since there is always a final ratio between input and output, dependent and independent variable. We know as a matter of fact that there is such a limit when dealing with energy exchanges (because of Planck’s quanta) but this principle is in UET extended to absolutely everything you can think of (perhaps with a single exception). There are ultimate time/space blocks, for example, and physical reality, instead of being modelled as a continuous substance should be conceived as a sort of mosaic with the ‘pieces’ being molecules of ultimate events. This, incidentally, raises the question of whether these pieces are separable in certain circumstances, whether, the building blocks can dissolve like certain chemical substances in solution.
The second consequence of the theory is that everything in physics is, as it were, pushed one stage backwards. We are familiar with the notion of inertia which may be described as the tendency of a body to maintain its current state of unaccelerated motion. Newton and Galileo also took for granted the idea that a solid object, or at any rate an atom, could not simply disappear : in other words that it was self-perpetuating. We now know that this is not entirely true since atoms can decay, though this remains a very rare occurrence in the portion of reality we can readily observe. But the idea of an object “remaining what it is” still seems pretty reasonable, at least as a first approximation.
Now, in UET, the ‘natural’ tendency of ultimate events is to disappear as soon as they come into existence and never to appear again. What we sense as solid objects must, then, be exceptions to the general rule and UET must provide some sort of reason for this state of affairs. ’Objects’ in UET are repeating event-chains which have acquired what I term ‘self-dominance’ ─ without this property they would not repeat ever. This is, incidentally, part of the preliminary assumptions of the theory, not something derived from experience. And the plausibility of this seemingly mad suggestion depends on what can be deduced from it.
As to the ‘gaps’ between ultimate events, although as humans there is little chance of us experiencing them directly, it may well be that we can experience gaps between blocks of ultimate events and many people have indeed had such experiences, or had some sort of experience that they interpreted in this fashion. As Heidegger put it, “Being is shot through with nothingness”.
This is a very radical assumption. It means that ‘reality’ has two ─ and essentially only two ─ states : On or Off, Existence or Non-existence. The ‘event-trajectory’ of any repeating event-chain (including ourselves) is gapped. Interestingly, this very basic conception fits in well with the current digital revolution and the victory of the digital two-state computer over the many-state analogue computer.
Is there anything that persists between ultimate events, anything that ‘fills the gap’? It is difficult to avoid the conclusion that there must be ‘something’ since otherwise it is hard to see why everything doesn’t just switch off permanently. This ‘something’ must, however, be so unphysical, so vacuous, so insubstantial, that it would be more pertinent to describe it as ‘Emptiness’ or ‘Void’ ─ and this is precisely how it is described in innumerable Hinayana Buddhist texts. Also, one notes the idea floating around contemporary physics ─ but I hasten to add without any ‘spiritual’ or religious meaning ─ that ‘nothingness’ is at the origin of everything and in particular at the origin of our universe which may be one of many. ‘Nothing’ is in fact making something of a come-back and even becoming scientifically respectable.
In a nutshell, the paradigm, or world-model of Ultimate Event Theory is that reality is made up of three, and only three, elements : (1)the ultimate events themselves, (2) the interconnections event-chains have with each other and (3) the underlying invisible immaterial ‘substance’ of which they are, as it were, concretisations or droplets of foam. As it happens, these features remind one of the three main constituents of the Buddhist world-view : dharma, karma, nirvana. The dharma correspond to the ultimate events, karma to the ‘causal force’ propelling events, nirvana to the underlying reality. I must once again add that the way I am using these terms is slightly different from the way Buddhists did and do use them since Buddhists  are concerned almost exclusively with the ‘moral law’ and with ways of hastening the extinction of the world-process and returning everything to the quiescent state of nirvana. The image of a calm pool of water (nirvana) being somehow for unknown reasons ‘stirred up’ or set into commotion seems, nonetheless, extremely apt.
It will be objected (and has been already vehemently) that all this is just speculation and metaphysics and essentially mumbo-jumbo, not science. To this I answer that science evolved from so-called ‘natural philosophy’ and that contemporary science is not just a prediction system but contains a (relatively) coherent world-view or world-system that is not regarded as philosophy only because the main tenets are not seriously questioned. It seems to be not only psychologically necessary but extremely useful to have some sort of a model of reality or picture in the mind, though naturally there are dangers in taking it too literally. On the whole science and mathematics have erred in taking their assumptions too seriously rather than in not having any : mathematicians, for example, apparently the least metaphysical of beasts, would be hard put to expel the idea of the ‘number line’ with all that that entails from their minds. And the crude Bohr model of the atom is still holding its ground against quantum mechanics and, in a sense, with good reason.
So what’s the point of having a different world-view or paradigm from the present one, especially since the current one  has led to such amazing results? To this I would say that one may be temperamentally inclined in a different direction and there is nothing wrong with that, though certainly that is no argument for the ‘truth’ or usefulness of the view. One judges a tree by its fruits. Firstly, the current world-view seems to be getting into serious conceptual difficulties and it may be that there is no way of patching it up : perhaps better to go back and take a different turning and see where it leads. It is to be hoped that this different direction will lead to some or most of  the undeniable benefits of the current method, but if that’s all it did there would be little point in my offering it to the public or even believing it myself. There are, however, certain assumptions that I do not feel obliged to take on board, or not quite to the same extent as contemporary scientists, and so I may be able to accommodate certain data that they cannot, or explain what is going on in a more comprehensible fashion. Science,or rather scientism, currently assumes that it holds the whole truth, at any rate in principle, and anything that doesn’t fit is either ignored or dismissed as coincidence. This is a rather unsatisfactory way of proceeding and plenty of scientists in private will admit that they have doubts about certain matters or are inclined to views that would get them into seriously trouble if expressed in public.
So what assumptions do I have in mind?  Well, one is the notion of there being immutable ‘laws of nature’ from which there can be no deviation, except in the sense of quantum uncertainties which are of no use to us practically speaking. Although there must seemingly be some underlying principles, much of the behaviour we see around us seems to be the result of habit, or blind repetition. Physicists readily agree to this when speaking of customs and human opinions but the same may apply to physical laws as well (as Sheldrake has dared to suggest). ‘Natural laws’ are, in terms of Eventrics, event-schemas which have been built up by countless repetitions and have become so strong they appear to be absolute. But perhaps they are not absolute after all.
For example, though not denying for a moment its utility, I do not have to subscribe to the absolute nature of the conservation of mass/energy. There is, as a matter of fact, no obvious equivalent of ‘energy’ in Eventrics, the nearest being the ‘force’ between blocks of events that I call Dominance. Energy is an entirely hypothetical concept in the sense that one cannot ee or hear or touch energy but only deduce its presence from certain effects. It is the work that could be done, not the work that actually is being done (work having the physical sense of force × distance). The idea that the universe has a specific amount of energy delivered at the Big Bang and which has been conserved ever since would mean, translated into the terms of Eventrics, that the total amount of ‘dominance’ is given and remains constant. This means, for one thing, that it cannot decrease. However, I am inclined to believe that this quantity can and indeed must decrease eventually since the entire universe (or what we call a universe) will eventually disappear and everything will return to the nothingness from which it came. There may also be local or temporary fluctuations in the level of dominance. Why will the universe disappear if the quantity of dominance (karma if you like) declines? Because according to my preliminary assumptions, it is in the nature of ultimate events not to persist but to disappear for ever, the persistent ones being the exceptions. Events that have acquired dominance can themselves persist and can interact with other chains producing new chains but eventually the entire machine will ‘run out of steam’. Also, there is a finite amount of dominance emanating from the ‘sink’, at least within the limits of ‘our’ universe. Events consist either of combinations and repeats of existing events + completely new events emerging from nothingness. Eventually, no new events will emerge and the existing event-chains will die away : this at any rate is how I view things.
This overview is necessarily general and subsequent posts will y derive from the schema conclusions which can be tested, or will give an alternative and perhaps preferable explanation of well-attested physical phenomena.     SH  5/2/13