Archives for category: Bishop Berkeley

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”.

 

One can trace contemporary physics back to the suggestion, or intuition, of certain ancient Greeks, especially Democritus and Epicurus, that at bottom reality is composed of atoms, minute indestructible ‘bodies’ that combine with each other to form  objects. The most important addition to this ultra-reductionist picture of reality was Newton’s idea of particular forces operating between bodies composed of atoms, both short-range contact forces and long-range non-contact forces. And forces could only operate because of ‘mass’.  So what exactly is ‘mass’? Newton originally defined it as the “quantity of matter within an object”, an intuitively clear definition but one that physics has largely discarded today. And Newton himself obviously envisaged  mass as something much more elusive and more metaphysical than a mere question of numbers of atoms and how densely packed together they were. Inertial mass was a property that objects possessed which could be measured by their capacity to resist forces that attempted to change their state of motion. And gravitational mass measured a body’s capacity to respond to a particular kind of force operating at a distance.

Now, the starting point of Ultimate Event Theory is the notion that ‘objects’, which are relatively stable and permanent things, consist, not of smaller relatively stable and long-lasting objects such as atoms, but of what I call ‘ultimate events’. And ‘ultimate events’  are inherently unstable in the sense that they appear and disappear almost as soon as they have appeared; also, while some ultimate events occur again and again at successive instants, i.e. repeat, most do not. Instead of being a collection of solid objects, physical reality, according to this view, is rather a sort of cosmic kaleidoscope or cinema show where the successive stills are run through so fast that we can’t keep up and perceive them as continuous movement. In effect, instead of basing our notion of physical ‘reality’ on our perception of solidity and permanence around and inside us, Ultimate Event Theory appeals rather to our  sense of the transience of everything and everyone. Time thus becomes the basic dimension rather than space. This mode of perception seems to be more ‘Eastern’ than ‘Western’ since two of the chief religions/philosophies of the East, Buddhism and Taoism, emphasize transience, indeed make it the cornerstone of the entire conceptual system.

Now, it is my contention, or ‘intuition’ if you like, that a system of physical science could have been developed on such premises especially by certain Indian Buddhists during the first few centuries of our era. That it was not can be ascribed on the one hand to the greater difficulty of experimenting usefully with transient items such as ultimate events (the dharmas of Hinayana Buddhism) rather than relatively permanent solid objects. But there was also a cultural reason:  Buddhist thinkers did develop a sophisticated kind of psychology (Abhidharma) and a form of logic but this was mainly because both these disciplines were useful in making converts and in making sense of their own meditative experiences. But these people had very little interest in the physical world per se, tending to view it, if not as a complete delusion, as at any rate a barrier to ‘deliverance’ and ‘enlightenment’. There was thus insufficient motivation within this particular intellectual milieu for developing a branch of knowledge devoted exclusively to physical matters as happened in the West during the fifteenth and sixteenth centuries.

Mass  

In any case, all such speculation about what ‘might have happened if…’ is irrelevant. The question I ask myself  is, “Can any sort of a coherent physical system be developed from the premise that the basic elements of existence are not ‘things’ but ephemeral ‘ultimate events’?” And one of the very first sub-questions that arises is: “What is the equivalent of ‘mass’ in this system?”
Returning to the ‘classical’ Newtonian concept of inertial mass, we see that it tends to ‘keep things as they are’, hence the term ‘inertia’ with its largely negative overtones. In Ultimate Event Theory (UET) there can, of course, be no question of ‘keeping things as they are’ in the usual sense, since the innate tendency of ultimate events, is, by hypothesis, to disappear at once, not to remain. But there must, seemingly, be some similar or equivalent ‘property’ for there to be a ‘physical world’ at all, or indeed anything observable and perceptible whether ‘real’ or ‘delusory’.

In UET it is supposed that some ultimate events, by processes at present unknown but probably involving chance repetition, form themselves into event-chains and repeating event clusters. It is the latter, i.e. identically repeating event clusters, that we perceive as objects. The equivalent of ‘mass’ would seem to be persistence since ‘persistence’ is not so strongly associated with continuous existence as mass (Note 1). Note that, in contrast to mass which in the Newtonian system is everywhere, persistence is not a property possessed by all ultimate events but only those of  a certain class ─ though these are the ones we are normally interested in. However, once an ultimate event acquires persistence, it seems to retain it, if not indefinitely at least for a considerable length of time, thus giving rise to an impression of solidity and permanence. And repeating event clusters’ (‘objects’) usually remain ‘where and as they are’ unless interfered with in some way, i.e. made subject to an external ‘force’. Following Newton, one is thus tempted to define ‘force’ as something that stops a persistent  event or event cluster from carrying on repeating identically in the same manner.

Note, however, that in UET, everything is, as it were, pushed one stage further back : it is thus necessary to assume some sort of ‘existence-force’ for event-chains without which there would be nothing but a chaos of momentarily existing ‘ultimate events’.

Elementary or ‘Static’ Persistence

 An ultimate event, then, for reasons unknown ─ but which may have something to do with the pre-occurrence of identical or similar events within a neighbouring region of the Locality ─ repeats identically once and keeps on doing so thus forming an event-chain. Now, since the ‘repeat event’ is not, strictly speaking, the ‘same thing’ as the original ultimate event, there is an extra variable which comes into play in UET and which does not appear in the classical concept of an object, namely the re-appearance rate of an event-chain.

Suppose an ultimate event that has occurrence at one ksana and repeats at the very next ksana. It does not necessarily keep repeating at the same rate which in this case is 1/1 or one appearance per ksana. It might shift to a rate of one appearance every three ksanas, one appearance every five ksanas and so forth, or it might have an irregular repeat rate but for all that still keep repeating. Since ksanas, the ‘ultimate’ temporal intervals, are so small compared to ‘macroscopic’ time intervals, a different repeat rate on the ‘ultimate’ level would not be distinguishable to our senses, or perhaps even to our most accurate current instruments. Nonetheless, in order to keep things simple, I shall start by assuming that an event chain has a 1/1 reappearance rate even though there are all sorts of other possibilities. So the basic ‘persistence’ of an event within an event-chain is set at one occurrence per ksana unless stated otherwise.

In matter-based physics, an object can ‘move’ relative to some other stable easily recognizable object, or relative to a recognizable spot considered the ‘origin’ if we are dealing with a co-ordinate system. But an object cannot meaningfully be said to move ‘relative to itself’ : it is always where it is when it is. However, since an event-chain is by definition discontinuous, being composed of a succession of discrete ultimate events, perhaps with appreciable gaps between such appearances, the situation is rather different. Can we meaningfully consider that a particular ultimate event’s reappearance is shifted to the right or left of its previous position? One’s first inclination is to say, yes, but this immediately gets one into difficulties. An ultimate event is conceived as ‘appearing on’ a backdrop, the Event Locality, or perhaps is better viewed as a ‘localized concretisation’ of this backdrop. Since, by hypothesis, this backdrop (the Event Locality) is ‘neutral’ with respect to what occurs in or on it and is not itself endowed with directions, it makes little sense to speak of the trajectory of an isolated event-chain being ‘straight’ or ‘crooked’ or ‘curved’ with respect to this backdrop ─ although it does still make sense to speak of an event-chain having a certain re-appearance rate. If each ultimate event in an event-chain were conscious, it would consider itself and the entire chain to be ‘at rest’, to be stationary, just as we conceive of ourselves as being stationary and the countryside drifting by when in a train (if it is smooth running). So, if we are to speak of ‘lateral drift’ to right or left at successive ksanas, i.e. to introduce the notion of ‘speed’ into UET, this ‘’lateral  shift’ must be related to some real or hypothetical event-chain which is itself regarded as ‘stationary’, i.e. as composed of ultimate events repeating identically at an equivalent spot at successive ksanas. This issue is by the way not specific to UET since it comes up in classical physics : even in Newton’s own time there was considerable, and often heated, discussion about whether one could meaningfully talk of a body isolated in the middle of space as being ‘at rest’ or ‘in motion’ (Note 2).

Now, although there is no such thing as velocity in the continuous sense usually implied in normal physics and everyday speech, there is in UET a perceptible ‘lateral drift’ of successive ultimate events relative to an actual or hypothetical ‘landmark event-chain’ whose constituent ultimate events form a ‘straight line’, or are supposed to do so. If the successive constituents of an event-chain are randomly situated to right and left relative to this landmark event-chain, the event-chain in question does not have a proper displacement rate ─ though nonetheless the ultimate events are somehow bonded together, one bringing into existence the next. But if there is a clearcut displacement pattern, the event-chain can be said to have a ‘velocity’, or the UET equivalent. And if the pattern of successive appearances resembles a straight line, we have an event-chain with a constant lateral displacement rate. In such a case, we can say that not only does the event-chain have ‘persistence’ but that its displacement rate also has persistence. An event-chain can thus have two kinds of ‘persistence’ : ‘existence persistence’ and ‘lateral displacement persistence’,  roughly the equivalents of the ‘rest mass’ and ‘kinetic energy’ of a particle in Newtonian mechanics. A single ultimate event cannot, of course, have ‘displacement persistence’ if it does not already have ‘existence persistence’ ─  though it can have the latter without the former, i.e. be the equivalent of stationary.

And if we take over Newton’s Laws of Motion and re-state them in ‘event’ rather than ‘object’ terms, we can define ‘force’ as that which affects an event-chain’s persistence, either its ‘displacement persistence’ alone or its ‘existence persistence’. In the latter case, an event-chain is replaced by another event-chain or simply annihilated. The doctrine of the ‘conservation of mass-energy’, if taken over into UET, would forbid complete annihilation without replacement, i.e. the simple disappearance of an event-chain without any sequels. It may, however, turn out not to be the case that event-chains must always be replaced by other ones : at any rate the question is left open. If we do introduce the equivalent of a conservation principle, this would mean that as soon as even a single event-chain formed, there would be an endless succession of events since each time one chain terminated it would give rise to another. Such an ‘event-universe’ would thus be endless, ‘infinite’ if you like. However, my feeling is that nothing physical is endless and that not only can event-chains terminate without giving rise to other ones, but that this must happen eventually for all event-chains, i.e. the ‘event-universe’ will simply disappear and, as it were, return to what gave rise to it in the first place. Indeed, according to the ‘Anti-Infinity Principle’, nothing can continue for ever except the background or origin.

Energy?

 There does not seem to be any obvious equivalent of ‘energy’ in UET ─ though one must remember that the notion only really entered classical physics during the middle of the 19th century. ‘Energy’ cannot be perceived directly anyway, only inferred, and is, in the framework of Newtonian physics, simply the  “capacity to do work”. One could perhaps view an ultimate event’s  capacity to repeat (or give rise to a different event) as the UET equivalent of ‘energy’, and the reality of repetition as the equivalent of mass ─ though I am not sure this is a meaningful distinction. Clearly, if an ultimate event does not possess the capacity to repeat, it cannot give rise to an  event-chain, while if it does in point of fact repeat, clearly it had the prior capacity to do so. One might also envisage a more general ‘existence capacity’ which covers the two cases of an ultimate event repeating exactly and alternatively  giving rise to a different event (an eventuality we have not treated yet). This would correspond to the generalized notion of ‘energy’ in normal physics where ‘energy’ always exists and passes through different forms. But basically it would seem that there are no obvious exact parallels to the dual concepts of mass and energy in matter-based physics. Had an ‘event-based’ physics ever been developed, or were one to evolve now, it would require  its own concepts and categories, not all of which would necessarily correspond to the familiar ones we have and which themselves required centuries to evolve.          SH   1/1/15

 Note 1  Spinoza apparently believed that the most essential feature of anything real is its ‘striving’ to remain what it is. “Each thing, as far as it can by its own power, strives to persevere in its being….The striving by which each thing strives to persevere in its own being is nothing but the actual essence of the thing” Spinoiza, Ethics Part III quoted by Sheldrake, The Science Delusion.
This is interesting because a ‘striving to persevere’ is not the same thing as a capacity to persevere.

 Note 2  Bishop Berkeley, in criticising Newton, wrote that

Up, down, right, left, all directions and places are based on some relation and it is necessary to suppose another body distant from the moving one…..so that motion is relative in its nature, it cannot be understood until the bodies are given in relation to which it exists, or generally there cannot be any relation if there are no terms to be related. Therefore, if we imagine everything is annihilated except one globe, it would be impossible to imagine any movement in this globe” (quoted in Rosser, Introductory Relativity p. 276)