What is random? That which cannot be predicted with any confidence. But there is a weak and a strong sense to ‘unpredictable’. We might say that the motion of a leaf blown about by the wind is ‘random’ ― but then that may simply be because we don’t know the exact speed and direction of the wind or the aerodynamic properties of this particular leaf. In classical mechanics, there is no room for randomness since all physical phenomena are fully determined and so could in principle be predicted if one had sufficient data. Indeed, the French astronomer Laplace claimed that a super-mind, aware of the current positions and momenta of all particles currently in existence could predict the entire future of the universe from Newtonian principles.

In practice, of course, one never does know the initial conditions of any physical system perfectly. Whether this is going to make a substantial difference to the outcome hinges on how sensitively dependent on the initial conditions the system happens to be. Whether or not the flap of a butterfly’s wings in the bay of Tokyo could give rise to a hurricane in Barbados as chaos theory claims, systems that are acutely sensitive to initial conditions undoubtedly exist, and this is, of course, what makes accurate weather forecasting so difficult. Gaming houses retire dice after a few hundred throws because of inevitable imperfections creeping in and a certain Jagger made a good deal of money because he noted that certain numbers seemed to come up slightly more often than others on a particular roulette wheel and bet on them. Later on, he guessed that the cause was a slight scratch on this particular wheel and there seems to have been something in this for eventually the management thwarted him by changing the roulette wheels every night (Note 1). All sorts of other seemingly ‘random’ phenomena turn out, on close examination, to exhibit a definite bias or trend: for example, certain digits turn up in miscellaneous lists of data more often than others (Bensford’s Law) and this bias, or rather its absence, has been successfully used to detect tax fraud.

There is, however, something very unsatisfactory about the ‘unpredictable because of insufficient data’ definition of randomness: it certainly does not follow that there is an inherent randomness in Nature, nor does chaos theory imply that this is the case either. Curiously, quantum mechanics, that monstrous but hugely successful creation of modern science, does maintain that there is an underlying randomness at the quantum level. The radioactive decay of a particular nucleus is held to be not only unforeseeable but actually ‘random’ in the strong sense of the word ― though the bulk behaviour of a collection of atoms can be predicted with confidence. Likewise, genetic mutation, the pace setter of evolution, is regarded today as not just being unpredictable but, in certain cases at least, truly ‘random’. Randomness seems to have made a strong and unexpected come-back since it is now a key player in the game or business of living ― a bizarre volte-face given that science had previously been completely deterministic.

The ‘common sense’ meaning of randomness is the lack of any perceived regularity or repeating pattern in a sequence of events, and this will do for our present purposes (Note 2). Now, it is extremely difficult to generate a random sequence of events in the above sense and in the recent past there was big money involved in inventing a really good random number generator. Strangely, most random number generators are not based on the behaviour of actual physical systems but depend on algorithms deliberately concocted by mathematicians. Why is this? Because, to slightly misquote Moshe, “complete randomness is a kind of perfection”(Note 3).

The more one thinks about the idea of randomness, the weirder the concept appears since a truly ‘random’ event does not have a causal precursor (though it usually does have a consequence). So, how on earth can it occur at all and where does it come from? It arrives, as common language puts it very well, ‘out of the blue’.

Broadly speaking there are two large-scale tendencies in the observable universe: firstly the dissipation of order and decline towards thermal equilibrium and mediocrity because of the ‘random’ collision of molecules, secondly the spontaneous emergence of complex order from processes that appear to be, at least in part, ‘random’. The first principle is enshrined in the 2nd Law of Thermo-dynamics : the entropy (roughly extent of disorder) of a closed system always increases, or (just possibly) stays the same. Contemporary biologists have a big problem with the emergence of order and complexity in the universe since it smacks of creationism. But at this very moment the molecules of tenuous dispersed gases are clumping together to form stars and the trend of life forms on earth is, and has been for some time, a movement from relative structural simplicity (bacteria, archaea &c.) to the unbelievable complexity of plants and mammals. Textbooks invariably trot out the caveat that any local ‘reversal of entropy’ must always be paid for by increased entropy elsewhere. This is, however, not a claim that has been, or ever could be, comprehensively tested on a large scale, nor is it at all ‘self-evident’ (Note 4). What we do know for sure is that highly organized structures can and do emerge from very unpromising beginnings and this trend seems to be locally on the increase ― though it is conceivable that it might be reversed.

For all that, it seems that there really are such things as truly random events and they keep on occurring. What can one conclude from this? That, seemingly, there is a powerful mechanism for spewing forth random, uncaused events, and that this procedure is, as it were, ‘hard-wired’ into the universe at a very deep level. But this makes the continued production of randomness just as mysterious, or perhaps even more so, than the capacity of whatever was out there in the beginning to give rise to complex life!

The generation of random micro-events may in fact turn out to be just about the most basic and important physical process there is. For what do we need to actually produce a ‘world’? As far as I am concerned, there must be something going on, in other words we need ‘events’ and these events require a source of some sort. But this source is remote and we don’t need to attribute to it any properties except that of being a permanent store of proto-events. The existence of a source is not enough though. Nothing would happen without a mechanism to translate the potential into actuality, and the simplest and, in the long run, most efficient mechanism is to have streams of proto-events projected outwards from the source at random. Such a mechanism will, however, by itself not produce anything of much interest. To get order emerging from the primeval turmoil we require a second mechanism, contained within the first, which enables ephemeral random events to, at least occasionally, clump together, and eventually build up, simply by spatial proximity and repetition, coherent and quasi-permanent event structures (Note 5). One could argue that this possibility, namely the emergence of ‘order from chaos’, however remote, will eventually come up ― precisely because randomness in principle covers all realizable possibilities. A complex persistent event conglomeration may be termed a ‘universe’, and even though an incoherent or contradictory would-be ‘universe’ will presumably rapidly collapse into disorder, others may persist and maybe even spawn progeny.

So, which tendency is going to win out, the tendency towards increasing order or reversion to primeval chaos? It certainly looks as if a recurrent injection of randomness is necessary for the ‘health’ of the universe and especially for ourselves ― this is one of the messages of natural selection and it explains, up to a point, the extraordinarily tortuous process of meiosis (roughly sexual reproduction) as against mitosis when a cell simply duplicates its DNA and splits in two (Note 6). But there is also the “nothing succeeds like success” syndrome. And, interestingly, the evolutionary biologist John Bonner argues that microorganisms “are more affected by randomness than large complex organisms” (Note 7). This and related phenomena might tip the balance in favour of order and complexity ― though specialization also makes the larger organisms more vulnerable to sudden environmental changes.                                                                 SH

 

Note 1 This anecdote is recounted and carefully analysed in The Drunkard’s Walk by Mlodinow.

 Note 2 Alternative definitions of randomness abound. There is the frequency definition whereby, “If a procedure is repeated over and over again indefinitely and one particular outcome crops up as many times as any other possible outcome, the sequence is considered to be random” (adapted from Peirce). And Stephen Wolfram writes: “Define randomness so that something is considered random only if no short description whatsoever exists of it” (Stephen Wolfram).

 Note 3 Moshe actually wrote “Complete chaos is a kind of perfection”.

Note 4 “The vast majority of current physics textbooks imply that the Second Law is well established, though with surprising regularity they say that detailed arguments for it are beyond their scope. More specialized articles tend to admit that the origins of the Second Law remain mysterious” (Stephen Wolfram, A New Kind of Science p. 1020

 Note 5 This is essentially the principle of ‘morphic resonance’ advanced by Rupert Sheldrake. Very roughly, the idea is that if a certain event, or cluster of events, has occurred once, it is slightly more likely to occur again, and so on and so on. Habit thus eventually becomes physical law, or can do. At bottom the ‘Gambler’s Fallacy’ contains a grain of truth: I suspect that current events are never completely independent of previous similar occurrences despite what statisticians say. Clearly, for the theory to work, there must be a very slow build-up and a tipping point after which a trend really takes off. We require in effect the equivalent of the Schrodinger equation to show how initial randomness evolves inexorably towards regularity and order.

Note 6. In meiosis not only does the offspring get genes from two individuals rather than one, but there is a great deal of ‘crossing over’ of segments of chromosomes and this reinforces the mixing process.

Note 7 The reason given for this claim is that there are many more developmental steps in the construction of a complex organism and so “if an earlier step fails through a deleterious mutation, the result is simple: the death of the embryo”. On the other hand “being small means very few developmental steps, with little or no internal selection” and hence a far greater number of species, witness radiolaria (50,000) and diatoms (10,000). See article Evolution, by chance? in the New Scientist 20 July 2013 and Randomness in Evolution by John Bonner.

 

 

 

 

 

 

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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.

 

What is time? Time is succession. Succession of what? Of events, occurrences, states. As someone put it, time is Nature’s way of stopping everything happening at once.

In a famous thought experiment, Descartes asked himself what it was not possible to disbelieve in. He imagined himself alone in a quiet room cut off from the bustle of the world and decided he could, momentarily at least, disbelieve in the existence of France, the Earth, even other people. But one thing he absolutely could not disbelieve in was that there was a thinking person, cogito ergo sum (‘I think, therefore I am’).
Those of us who have practiced meditation, and many who have not, know that it is quite possible to momentarily disbelieve in the existence of a thinking/feeling person. But what one absolutely cannot disbelieve in is that thoughts and bodily sensations of some sort are occurring and, not only that, that these sensations (most of them anyway) occur one after the other. One outbreath follows an inbreath, one thought leads on to another and so on and so on until death or nirvana intervenes. Thus the grand conclusion: There are sensations, and there is succession.  Can anyone seriously doubt this?

 Succession and the Block Universe

That we, as humans, have a very vivid, and more often than not  acutely painful, sense of the ‘passage of time’ is obvious. A considerable body of the world’s literature  is devoted to  bewailing the transience of life, while one of the world’s four or five major religions, Buddhism, has been well described as an extended meditation on the subject. Cathedrals, temples, marble statues and so on are attempts to defy the passage of time, aars long vita brevis.
However, contemporary scientific doctrine, as manifested in the so-called ‘Block Universe’ theory of General Relativity, tells us that everything that occurs happens in an ‘eternal present’, the universe ‘just is’. In his latter years, Einstein took the idea seriously enough to mention it in a letter of consolation to the son of his lifelong friend, Besso, on the occasion of the latter’s death. “In quitting this strange world he [Michel Besso] has once again preceded me by a little. That doesn’t mean anything. For those of us who believe in physics, this separation between past, present and future is an illusion, however tenacious.”
Never mind the mathematics, such a theory does not make sense. For, even supposing that everything that can happen during what is left of my life has in some sense already happened, this is not how I perceive things. I live my life day to day, moment to moment, not ‘all at once’. Just possibly, I am quite mistaken about the real state of affairs but it would seem nonetheless that there is something not covered by the ‘eternal present’ theory, namely my successive perception of, and participation in, these supposedly already existent moments (Note 1). Perhaps, in a universe completely devoid of consciousness,  ‘eternalism’ might be true but not otherwise.

Barbour, the author of The End of Time, argues that we do not ever actually experience ‘time passing’. Maybe not, but this is only because the intervals between different moments, and the duration of the moments themselves, are so brief that we run everything together like movie stills. According to Barbour, there exists just a huge stack of moments, some of which are interconnected, some not, but this stack has no inherent temporal order. But even if it were true that all that can happen is already ‘out there’ in Barbour’s Platonia (his term), picking a pathway through this dense undergrowth of discrete ‘nows’ would still be a successive procedure.

I do not think time can be disposed of so easily. Our impressions of the world, and conclusions drawn by the brain, can be factually incorrect ― we see the sun moving around the Earth for example ― but to deny either that there are sense impressions and that they appear successively, not simultaneously, strikes me as going one step too far. As I see it, succession is an absolutely essential component  of lived reality and either there is succession or there is just an eternal now, I see no third possibility.

What Einstein’s Special Relativity does, however, demonstrate is that there is seemingly no absolute ‘present moment’ applicable right across the universe (because of the speed of light barrier). But in Special Relativity at least succession and causality still very much exist within any particular local section, i.e. inside a particular event’s light cone. One can only surmise that the universe as a whole must have a complicated mosaic successiveness made up of interlocking pieces (tesserae).

Irreversibility
In various areas of physics, especially thermo-dynamics, there is much discussion of whether certain sequences of events are reversible or not, i.e. could take place other than in the usual observed order. This is an important issue but is a quite different question from whether time (in the sense of succession) exists. Were it possible for pieces of broken glass to spontaneously reform themselves into a wine glass, this process would still occur successively and that is the point at issue.

Time as duration

‘Duration’ is a measure of how long something lasts. If time “is what the clock says” as Einstein is reported to have once said, duration is measured by what the clock says at two successive moments (‘times’). The trick is to have, or rather construct, a set of successive events that we take as our standard set and relate all other sets to this one. The events of the standard set need to be punctual and brief, the briefer the better, and the interval between successive events must be both noticeable and regular. The tick-tock of a pendulum clock provided such a standard set for centuries though today we have the much more regular expansion and contraction of quartz crystals or the changing magnetic moments of electrons around a caesium nucleus.

Continuous or discontinuous?

 A pendulum clock records and measures time in a discontinuous fashion: you can actually see, or hear, the minute or second hand flicking from one position to another. And if we have an oscillating mechanism such as a quartz crystal, we take the extreme positions of the cycle which comes to the same thing.
However, this schema is not so evident if we consider ‘natural’ clocks such as sundials which are based on the apparent continuous movement of the sun. Hence the familiar image of time as a river which never stops flowing. Newton viewed time in this way which is why he analysed motion in terms of ‘fluxions’, or ‘flowings’. Because of Calculus, which Newton invented, it is the continuous approach which has overwhelmingly prevailed in the West. But a perpetually moving object, or one perceived as such, is useless for timekeeping: we always have to home in on specific recurring configurations such as the longest or shortest shadow cast. We have to freeze time, as it were, if we wish to measure temporal intervals.

Event time

The view of time as something flowing and indivisible is at odds with our intuition that our lives consist of a succession  of moments with a unique orientation, past to future, actual to hypothetical. Science disapproves of the latter common sense schema but is powerless to erase it from our thoughts and feelings: clearly the past/present/future schema is hard-wired and will not go away.
If we dispense with continuity, we can also get rid of  ‘infinite divisibility’ and so we arrive at the notion, found in certain early Buddhist thinkers, that there is a minimum temporal interval, the ksana. It is only recently that physicists have even considered the possibility that time  is ‘grainy’, that there might be ‘atoms of time’, sometimes called chronons. Now, within a minimal temporal interval, there would be no possible change of state and, on this view, physical reality decomposes into a succession of ‘ultimate events’ occupying  minimal locations in space/time with gaps between these locations. In effect, the world becomes a large (but not infinite) collection of interconnected cinema shows proceeding at different rates.

Joining forces with time 

The so-called ‘arrow of time’ is simply the replacement of one localized moment by another and the procedure is one-way because, once a given event has occurred, there is no way that it can be ‘de-occurred’. Awareness of this gives rise to anxiety ― “the moving finger writes, and having writ/ Moves on, nor all thy piety or wit/Can lure it back to cancel half a line….”  Most religious, philosophic and even scientific systems attempt to allay this anxiety by proposing a domain that is not subject to succession, is ‘beyond time’. Thus Plato and Christianity, the West’s favoured religion. And even if we leave aside General Relativity, practically all contemporary scientists have a fervent belief in the “laws of physics” which are changeless and in effect wholly transcendent.
Eastern systems of thought tend to take a different approach. Instead of trying desperately to hold on to things such as this moment, this person, this self, Buddhism invites us to  ‘let go’ and cease to cling to anything. Taoism goes even further, encouraging us to find fulfilment and happiness by identifying completely with the flux of time-bound existence and its inherent aimlessness. The problem with this approach is, however, that it is not clear how to avoid simply becoming a helpless victim of circumstance. The essentially passive approach to life seemingly needs to be combined with close attention and discrimination ― in Taoist terms, Not-Doing must be combined with Doing.

Note 1 And if we start playing with the idea that  not only the events but my perception of them as successive is already ‘out there’, we soon get involved in infinite regress.

 

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”.

 

Space-Time

Minkowski, Einstein’s old teacher of mathematics, inaugurated  the hybrid ‘Space-Time’ which is now on everyone’s lips. In an address delivered not long before his death in 1908 he said the now famous lines,

“Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”

         But why should Minkowski, and whole generations of scientists, have ever thought that ‘space’ and ‘time’ could be completely separate in the first place? Certain consequences of a belief in ‘Space-Time’ in General Relativity do turn out to be  scarcely credible, but there is nothing weird or paradoxical per se about the idea of ‘time’ being a so-called fourth dimension. To specify an event accurately it is convenient to give three spatial coordinates which tell you how far the occurrence of this event is, or will be, along three different directions relative to an agreed  fixed point.  If I want to meet someone in a city laid out like a grid as New York is (more or less), I need to specify the street, say Fifth Avenue, the number of the building and the floor (how high above the ground it is). But this by itself will not be enough for a successful meet-up : I also need to give the time of the proposed rendez-vous, say, three o-clock in the afternoon. The wonder is, not that science has been obliged to bring time into the picture, but that it was possible for so long to avoid mentioning it (Note 1)

Succession

 Now, if you start off with ‘events’, which are by definition ‘punctual’ and impermanent, rather than things or ‘matter’ you cannot avoid bringing time into the picture from the start: indeed one  might be inclined to say that ‘time’ is a good deal more important than space. Events happen ‘before’ or ‘after’ each other; what happened yesterday preceded what happened this morning, and you read the previous sentence before you started on the current one. The very idea of ‘simultaneous’ events, events that have occurrence ‘at the same time’, is a tricky concept even without bringing Special Relativity into the picture. But  the idea of succession is both clearcut and basic and one could, as a first bash, even define ‘simultaneous’ events negatively as bona fide  occurrences that are not temporally ordered.

So, when I started trying to elaborate an ‘event-orientated’  world-view, I felt I absolutely had to have succession as a primary ingredient : if anything it came higher up the list than ‘space’. Originally I tried to kick off with a small number of basic assumptions (axioms or postulates) which seemed absolutely unavoidable. One such assumption was that most events are ‘ordered temporally’, that they have occurrence successively, ‘one after the other’ ─ with the small exception of so-called ‘simultaneous events’. Causality also seemed to be something I could not possibly do without and causality is very much tied up with succession since it is usually the prior event that is seen as ‘causing’ the other event in a causal pair. Again, one might tentatively  defined ‘simultaneous events’ as events which cannot have a direct causal bond, i.e. function as cause and effect (Note 2). And,  in an era innocent of Special Relativity and light cones, one might well define space as  the totality of all distinct events that are not temporally ordered.

From an ‘event-based’ viewpoint,  chopping up reality into ‘space’ and ‘time’ is not fundamental : all we require is a ‘place’ where events can and do have occurrence, an  Event Locality. Such a Locality starts off empty of events but has the capacity to receive them, indeed I have come to regard ultimate events as in some sense concretisations or condensations of an underlying substratum.

 Difference between Space and Time

 There is, however, a problem with having a single indivisible entity whether we call it ‘Space-Time’ or simply ‘the Locality’. The two parts or aspects of this creature are not at all equivalent. Although I believe, as some physicists have suggested, that, at a certain level, ‘space’ is ‘grainy’, it certainly appears to be continuous : we do not notice any dividing line, let alone a gap, between the different spatial ‘dimensions’ or between different spatial regions. We don’t have to ‘add’ the dimension height to pre-existing dimensions of length and width for example : experience always provides us with a three-dimensional physical block of reality (Note 3). And the fact that the choice of directions, up/down, left/right and so on, is more often than not completely arbitrary suggests that physical reality does not have inbuilt directions, is ‘all-of-a-piece’.

Another point worth mentioning is that we seem to have a strong sense of being ‘at rest’ spatially : not only are we ‘where we are’, and not where we are not, but we actually feel this to be the case. Indeed we tend to consider ourselves to be at rest even when we know we are moving : when in a train we consider that it is the other things, the countryside, that are in motion, not us. It is indeed this that gives Galileo’s seminal concept of inertia its force and plausibility; in practice all we notice is a flagrant disturbance of the ‘rest’ sensation, i.e. an ‘acceleration’.

What about time? Now it is true that time is often said to ‘flow’ and we do not notice any clearcut temporal demarcation lines any more than we notice spatial ones. Nonetheless, I would argue that it is much less natural and plausible to consider ‘time’ as a continuum because we have such a strong sense of sequence. We continually break up time into ‘moments’ which occur ‘one before the other’ even though the extent of the moment varies or is left vague. Sense of sequence is part of our world and since our impressions are themselves bona fide events even if only subjective ones, it would appear that sequence is a real feature of the physical world. There is in practice always an arrow of time, an arrow which points from the non-actual to the actual. Moreover, the process of ‘actualization’ is not reversible : an event that has occurrence cannot be ‘de-occurred’ as it were (Note 4).

And it is noteworthy that one very seldom feels oneself to be ‘at rest’ temporally, i.e. completely unaware of succession and variation. The sensation is so rare that it is often classed as  ‘mystical’, the feeling of being ‘out of time’ of which T.S. Eliot writes so eloquently in The Four Quartets. Heroin and certain other drugs,  by restricting one’s attention to the present moment and the recent past, likewise ‘abolish time’, hence their appeal. In the normal way,  even when deprived of all external physical stimuli, one still retains the sensation of there being a momentum and direction to one’s own thoughts and inner processes : one idea or internal sensation follows another and one never has any trouble assigning order (in the sense of sequence) to one’s inner feelings and thoughts. It is now thought that the brain uses parallel processing on a big scale but, if so, we are largely unaware of this incessant multi-tasking. Descartes in his thought experiment of being entirely cut off from the outside world and considering what he simply could not doubt, might well have concluded that sequence, rather than the (intemporal) thinking ego, was the one item that could not be dispensed with. For one can temporarily disbelieve in one’s existence as a particular person but not in the endless succession of thoughts and subjective sensations that stream through one’s mind/brain.

All this will be dismissed as belonging to psychology rather than physics. But our sense impressions and thoughts are rooted in our physiology and should not be waved aside for that very reason : in a sense they are the most important and inescapable ‘things’ we have for without them we would be zombies. Physical theories that deny sequence, that consider the laws of physics to be ‘perfectly reversible’, are both implausible and seemingly  unliveable, so great is our sense of ‘before and after’. Einstein towards the end of his life decided that it followed from General Relativity that everything happened in an ‘eternal present’. He took this idea seriously enough to mention it in a letter to the son of his college friend, Besso, on receiving news of the latter’s death, writing “For those of us who believe in physics, this separation between past, present and future is only an illusion, however tenacious”.

Breaks in Time

 If, then, we accept succession as an unavoidable feature of lived reality, are we to suppose that one moment shifts seamlessly into the next without any noticeable demarcation lines, let alone gaps? Practically all physicists, even those who toy with the idea that Space-time is in some sense ‘grainy’, seem to be stuck with the concept of a continuum. “There is time, but there is not really any notion of a moment in time. There are only processes that follow one another by causal necessity” as Lee Smolin puts it in Three Roads to Quantum Gravity..

But I cannot see how this can possibly be the case, and this is precisely why the ‘time dimension’ of the Event Locality is so different from the spatial one. If I shift my attention from two items in a landscape, from a rock and its immediate neighbourhood to a tree, there is no sense that the tree displaces the rock : the two items can peaceably co-exist and do not interfere with each other. But if one moment follows another, it displaces it, pushes it out of the way, as it were, since past and present moments, prior and subsequent events, cannot by definition co-exist ─ except perhaps in the inert way they might be seen to co-exist in an Einsteinian perpetual now. And all the attributes and particular features of a given moment must themselves disappear to make way for what follows. We do not usually see this happening, of course, because most of the time the very same objects are recreated and our senses do not register the transition. We only notice change when a completely different physical feature replaces another one, but the same principle must apply even if the same feature is recreated identically. Since a single moment is, in its physical manifestation, three-dimensional, all these three dimensions must go if a new moment comes into being.

Whether there is an appreciable gap between moments apart from there being a definite change is an open question. In the first sketch of Ultimate Event Theory I attribute a fixed extent to the minimal temporal interval, the ksana, and I allow for the possibility of flexible gaps between ksanas. The phenomenon of  time dilation is interpreted as the widening of the gap between ksanas rather than as an extension of the ‘length’ of a ksana itself. This feature, however, is not absolutely essential to the general theory.

What we actually perceive and consider to constitute  a ‘moment’ is, of course, a block containing millions of ksanas since the length of a ksana must be extremely small (around the Planck scale). However, it would seem that ksanas do form blocks and that there are transitions between blocks and that sometimes, if only subliminally, we are aware of these gaps. Instead of being a flowing river, ‘time’ is more like beads on a string though the best image would be a three-dimensional shape pricked out in coloured lights that is switched on and off incessantly.

Mosaic Time

Temporal succession is either a real feature of the world or it is not, I cannot see that there is a possible third position. In Einstein’s universe “everything that can have occurrence already has occurrence” to put things in event terms. “In the ‘block universe’ conception of general relativity….the present moment has no meaning ─ all that exists is the whole history of the universe at once, timelessly. When laws of physics are represented mathematically, causal processes which are the activity of time are represented by timeless logical implications…. Mathematical objects, being timeless, don’t mhave present moments, futures or pasts”  (Lee Smolin, It’s Time to Rewrite time in New Scientist 20 April 2014)  

This means that there is no free will since what has occurrence cannot be changed, cannot be ‘de-occurred’. It also makes causality redundant as Lee Smolin states. One could indeed focus on certain pairs of events and baptise them ‘cause and effect’ but, since they both have occurrence, neither of them has brought the other about, nor has a third ‘previous’ event brought both of them about simultaneously. Causality becomes of no account since it is not needed.

Even a little acquaintance with Special Relativity leads one to conclude that it is impossible to establish a universally valid ‘now’. Instead we have the two light cones, one leading back to the past and one to the future (the observer’s future), and a large region classed as ‘elsewhere’. It is notorious that the order of events in ‘elsewhere’, viewed from inside a particular light cone, is not fixed for all observers : for one observer it can be said that event A precedes event B and for another that event B precedes A. This indeterminacy if of little or no practical consequence since there is (within SR) no possibility of interaction between the two regions. However, it does mean that it is on the face of it impossible to speak of a universally valid ‘now’ ─ although physicists do use expressions like the “present state of the universe”.

I personally cannot conceive of a ‘universe’ or a life or indeed anything at all without succession being built into it : the timeless world of mathematics is not reality but a ‘take’ on reality. The only way to conceptually save succession while accepting some of the more secure aspects of Relativity would seem to be to have some sort of ‘mosaic time’, physical reality split up into zones. How exactly these zones, which are themselves subjective in that they depend on a real or imagined ‘observer’, fit together is not a question I can answer though certain areas of research into general relativity can presumably be taken over into UET.  One could perhaps define the next best thing to a universal ‘now’ by taking a weighted average of all possible time zones : Eddington suggested something along these lines though he neglected to give any details. Note that if physical reality is a mosaic rather than a continuum, it would in principle be possible to shift the arrangement of particular tesserae in a small way, exchange one with another and so on.                     SH 23/01/15

 

 Note 1 Time was left out of the picture for so long, or at any rate neglected, because the first ‘science’ to be developed to a high degree of precision in the West was geometry. And the truths of (Euclidian) geometry, if they are truths at all, are ‘timeless’ which is why Plato prized geometry above all other branches of knowledge except philosophy. Inscribe a triangle in a circle with the diameter as base line and you will always find that it is right-angled. And if you don’t, this is to be attributed to careless drawing and measurement : in an ‘ideal’ Platonic world such an angle has to be a right angle. How do we know? Because the theorem has been proved.

This concentration on space rather than time meant that although the Greeks set out the basic principles  of statics, the West had to wait another 1,600 years or so before Galileo more or less invented the science of dynamics from scratch. And the prestige of Euclid and the associated static view of phenomena remained so great that Newton, perversely to our eyes, cast his Principia into a cumbrous geometrical mould using copious geometrical diagrams even though he had already invented a ‘mathematics of motion’, the Calculus.

 Note 2   Kant did in point of fact defend the idea of ‘simultaneous causation’ where each of two ‘simultaneous’ events affects the other ‘at the same time’. He gave the example of a ball resting on a cushion arguing that the ball presses down on the cushion for the same amount of time as the cushion is deformed by the presence of the ball. And if we take Newton’s Third Law as operating exactly at the same time on or between two different objects, we have to accept the possibility of simultaneous causation.

Within Ultimate Event Theory, what would normally be  called ‘causality’ is (sometimes) referred to as ‘Dominance’. I chose this term precisely because it signifies an unequal relation between two events, one event, referred to as the ‘cause’, as it were ‘dominating’ the other, the ‘effect’. In most, though perhaps not all, cases of causal relations I believe there really is priority and succession despite Newton’s Third Law. I would conceive of the ball pressing on the cushion as occurring at least a brief moment before its effect ─ though this is admittedly debatable. One could introduce the category of ‘Equal Dominance’ to cover cases of  Kant’s ‘simultaneous causality’ between two events.

Note 3  I have always found the idea of Flatland, which is routinely trotted out in popular books on Relativity, completely unconvincing. I can more readily conceive of there being more than three spatial dimensions as there being a world with less than three : a line, any line, always has some width and height.

 Note 4. If it is possible for an event in the future to have an effect ‘now’, this can only be because the ‘future’ event has already somehow already occurred, whereas intermediate events between ‘now’ and ‘then’ have not. I cannot conceive of a ‘non-event’ having any kind of causal repercussion — except, of course, in the trivial sense that current wishes or hopes about the future might affect our behaviour. Such wishes and desires belong to the present or recent past, not to the future.

 

 

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) 

 

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)