Archives for category: Infinitesimal Calculus

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.

In its present state, Ultimate Event Theory falls squarely between two stools : too vague and ‘intuitive’ to even get a hearing from professional scientists, let alone be  taken seriously, it is too technical and mathematical to appeal to the ‘ordinary reader’. Hopefully, this double negative can be eventually turned into a double positive, i.e. a rigorous mathematical theory capable of making testable predictions that nonetheless is comprehensible and has strong intuitive appeal. I will personally not be able to take the theory to the desired state because of my insufficient mathematical and above all computing expertise : this will be the work of others. What I can do is, on the one hand, to strengthen the mathematical, logical side as much as I can while putting the theory in a form the non-mathematical reader can at least comprehend. One friend in particular who got put off by the mathematics asked me whether I could not write something that gives the gist of the theory without any mathematics at all. Thus this post which recounts the story of how and why I came to develop Ultimate Event Theory in the first place some thirty-five years ago.

Conflicting  beliefs

Although scientists and rationalists are loath to admit it, personal temperament and cultural factors play a considerable part in the development of theories of the universe. There are always individual and environmental factors at work although the accumulation of unwelcome but undeniable facts may eventually overpower them. Most people today are, intellectually speaking, opportunists with few if any deep personal convictions, and there are good reasons for this. As sociological and biological entities we are strongly impelled to accept what is ‘official doctrine’ (in whatever domain) simply because, as a French psycho-analyst whose name escapes me famously wrote, “It is always dangerous to think differently from the majority”.
At the same time, one is inclined, and in some cases compelled, to accept only those ideas about the world that make sense in terms of our own experience. The result is that most people spend their lives doing an intellectual balancing act between what they ‘believe’ because this is what they are told is the case, and what they ‘believe’ because this is what their experience tells them is (likely to be) the case. Such a predicament is perhaps inevitable if we decide to live in society and most of the time the compromise ‘works’; there are, however, moments in the history of nations and in the history of a single individual when the conflict becomes intolerable and something has to give.

The Belief Crisis : What is the basis of reality?

Human existence is a succession of crises interspersed with periods of relative stability (or boredom). First, there is the birth crisis (the most traumatic of all), the ‘toddler crisis’ when the infant starts to try to make sense of the world around him or her, the adolescent crisis, the ‘mid-life’ crisis which kicks in at about forty and the age/death crisis when one realizes the end is nigh. All these crises are sparked off by physical changes which are too obvious and powerful to be ignored with the possible exception of the mid-life crisis which is not so much biological as  social (‘Where am I going with my life?’ ‘Will I achieve what I wanted?’).
Apart from all these crises ─ as if that were not enough already ─  there is the ‘belief crisis’. By ‘crisis of belief’ I mean pondering the answer to the question ‘What is real?’ ‘What do I absolutely have to believe in?’. Such a crisis can, on the individual level, come at any moment, though it usually seems to hit one between the eyes midway between the adolescent ‘growing up’ crisis and the full-scale mid-life crisis. As a young person one couldn’t really care less what reality ‘really’ is, one simply wants to live as intensely as possible and ‘philosophic’ questions can just go hang. And in middle age, people usually find they want to find some ‘meaning’ in life before it’s all over. Now, although the ‘belief crisis’ may lead on to the ‘middle age meaning crisis’ it is essentially quite different. For the ‘belief crisis’ is not a search for fulfilment but simply a deep questioning about the very nature of reality, meaningful or not. It is not essentially an emotional crisis nor is it inevitable ─ many people and even entire societies by-pass it altogether without being any the worse off, rather the reverse (Note 1).
Various influential thinkers in history went through such a  ‘belief crisis’ and answered it in memorable ways : one thinks at once of the Buddha or Socrates. Of all peoples, the Greeks during the Vth and VIth centuries BC seem to have experienced a veritable epidemic of successive ‘belief crises’ which is what  makes them so important in the history of civilization  ─ and also what made the actual individuals and city-states so unstable and so quarrelsome. Several of the most celebrated answers to the ‘riddle of reality’ date back to this brilliant era. Democritus of Abdera answered the question, “What is really real?” with the staggering statement, “Nothing exists except atoms and void”. The Pythagoreans, for their part, concluded that the principle on which the universe was based was not so much physical as numerical, “All is Number”. Our entire contemporary scientific and technological ‘world-view’ (‘paradigm’) can  be traced back to the  two giant thinkers, Pythagoras and Democritus, even if we have ultimately ‘got beyond’  them since we have ‘split the atom’ and replaced numbers as such by mathematical formulae. In an equally turbulent era, Descartes, another major ‘intellectual crisis’ thinker, famously decided that he could disbelieve in just about everything but not that there was a ‘thinking being’ doing the disbelieving, cogito ergo sum (Note 2).
In due course, in my mid to late thirties, at about the time of life when Descartes decided to question the totality of received wisdom, I found myself with quite a lot of time on my hands and a certain amount of experience of the vicissitudes of life behind me to ponder upon. I too became afflicted by the ‘belief crisis’ and spent the greater part of my spare time (and working time as well) pondering what was ‘really real’ and discussing the issue interminably with the same person practically every evening (Note 3).

Temperamental Inclinations or Prejudices

My temperament (genes?) combined with my experience of life pushed me in certain well-defined philosophic directions. Although I only  started formulating Eventrics and Ultimate Event Theory (the ‘microscopic’ part of Eventrics) in the early nineteen-eighties and by then had long since retired from the ‘hippie scene’, the heady years of the late Sixties and early Seventies provided me with my  ‘field notes’ on the nature of reality (and unreality), especially the human part of it. The cultural climate of this era, at any rate in America and the West, may be summed up by saying that, during this time “a substantial number of people between the ages of fifteen and thirty decided that sensations were far more important than possessions and arranged their lives in consequence”. In practice this meant forsaking steady jobs, marriage, further education and so on and spending one’s time looking for physical thrills such as doing a ton up on the M1, hitch-hiking aimlessly around the world, blowing your mind with drugs, having casual but intense sexual encounters and so on. Not much philosophy here but when I and other shipwrecked survivors of the inevitable débâcle took stock of the situation, we retained a strong preference for a ‘philosophy’  that gave primary importance to sensation and personal experience.
The physical requirement ruled out traditional religion since most religions, at any rate Christianity in its later public  form, downgraded the body and the physical world altogether in favour of the ‘soul’ and a supposed future life beyond the grave. The only aspect of religion that deserved to be taken seriously, so I felt, was mysticism since mysticism is based not on hearsay or holy writ but on actual personal experience. The mystic’s claim that there was a domain ‘beyond the physical’ and that this deeper reality can to some degree actually be experienced within this life struck me as not only inspiring but even credible ─ “We are more than what we think we are and know more than what we think we know” as someone (myself) once put it.
At the same time, my somewhat precarious hand-to-mouth existence had given me a healthy respect for the ‘basic physical necessities’ and thus inclined to reject all theories which dismissed physical reality as ‘illusory’, tempting though this sometimes is (Note 4). So ‘Idealism’ as such was out. In effect I wanted a belief system that gave validity and significance to the impressions of the senses, sentio ergo sum to Descartes’ cogito ergo sum or, better, sentio ergo est :  ‘I feel therefore there is something’.

Why not physical science ?

Why not indeed. The main reason that I didn’t decide, like most people around me,  that “science has all the answers” was that, at the time, I knew practically no science. Incredible though this seems today, I had managed to get through school and university without going to a single chemistry or physics class and my knowledge of biology was limited to one period a week for one year and with no exam at the end of it.
But ignorance was not the only reason for my disqualifying science as a viable ‘theory of everything’. Apart from being vaguely threatening ─ this was the era of the Cold War and CND ─ science simply seemed monumentally irrelevant to every aspect of one’s personal daily life. Did knowing about neutrons and neurons make you  more capable of making more effective decisions on a day to day basis? Seemingly not. Scientists and mathematicians often seemed to be less (not more) astute in running their lives than ordinary practical people.
Apart from this, science was going through a difficult period when even the physicists themselves were bewildered by their own discoveries. Newton’s billiard ball universe had collapsed into a tangled mess of probabilities and  uncertainty principles : when even Einstein, the most famous modern scientist, could not manage to swallow Quantum Theory, there seemed little hope for Joe Bloggs. The solid observable atom was out and unobservable quarks were in, but Murray Gell-Mann, the co-originator of the quark theory, stated on several occasions that he did not ‘really  believe in quarks’ but merely used them as ‘mathematical aids to sorting out the data’. Well, if even he didn’t believe in them, why the hell should anyone else? Newton’s clockwork universe was bleak and soulless but was at least credible and tactile : modern science seemed nothing more than a farrago of  abstruse nonsense that for some reason ‘worked’ often to the amazement of the scientists themselves.
There was another, deeper, reason why physical science appeared antipathetic to me at the time : science totally devalues personal experience. Only repeatable observations in laboratory conditions count as fact : everything else is dismissed as ‘anecdotal’. But the whole point of personal experience is that (1) it is essentially unrepeatable and (2) it must be spontaneous if it is to be worthwhile. The famous ‘scientific method’ might have a certain value if we are studying lifeless atoms but seemed unlikely to uncover anything of interest in the human domain — . the best ‘psychologists’ such as  conmen and dictators are sublimely ignorant of psychology. Science essentially treats everything as if it were dead, which is why it struggles to come up with any strong predictions in the social, economic and political spheres. Rather than treat living things as essentially dead, I was more inclined to treat ‘dead things’ (including the universe itself) as if they were in some sense alive.

Descartes’ Thought Experiment

Although I don’t think I had actually read Descartes’ Discours sur la méthode at the time, I had heard about it and the general idea was presumably lurking at the back of my mind. Supposedly, Descartes who, incredibly, was an Army officer at the time, spent a day in what is described in history books as a poêle (‘stove’) pondering the nature of reality. (The ‘stove’ must have been a small chamber close to a source of heat.) Descartes came to the conclusion that it was possible to disbelieve in just about everything except that there was a ‘thinking  being’, cogito ergo sum. To anyone who has done meditation, even in a casual way, Descartes’ conclusion appears by no means self-evident. The notion of individuality drops away quite rapidly when one is meditating and all one is left with is a flux of mental/physical impressions. It is not only possible but even ‘natural’ to temporarily disbelieve in the reality of the ‘I’ (Note 5)─ but one cannot and does not disbelieve in the reality of the various sensations/impressions that are succeeding each other as ‘one’ sits (or stands).

Descartes’ thought experiment nonetheless seemed  suggestive and required, I thought, more precise evaluation. Whether the ‘impressions/sensations’ are considered to be mental, physical or a mixture of the two, they are nonetheless always events and as such have the following features:

(1) they are, or appear to be, ‘entire’, ‘all of a piece’, there is no such thing as a ‘partial’ event/impression;

(2) they follow each other very rapidly;

(3) the events do not constitute a continuous stream, on the contrary there are palpable gaps between the events (Note 6);

(4) there is usually a connection between successive events, one thought ‘leads on’ to another and we can, if we are alert enough, work backwards from one ‘thought/impression’ to its predecessor and so on back to the start of the sequence;

(5) occasionally ‘thought-events’ crop up that seem to be  completely disconnected from all previous ‘thought-events’, arriving as it were ‘out of the blue.’.

Now, with these five qualities, I already have a number of features which I believe must be part of reality, at any rate individual ‘thought/sensation’  reality. Firstly, whether my thoughts/sensations are ‘wrong’, misguided, deluded or what have you, they happen, they take place, cannot be waved away. Secondly, there is always sequence : thought ‘moves from one thing to another’ by specific stages. Thirdly, there are noticeable gaps between the thought-events. Fourthly, there is  causality : one thought/sensation gives rise to another in a broadly predictable and comprehensible manner. Finally, there is an irreducible random element in the unfolding of thought-events — so not everything is deterministic apparently.
These are properties I repeatedly observe and feel I have to believe in. There are also a number of conclusions to be drawn from the above; like all deductions these ‘derived truths’ are somewhat less certain than the direct impressions, are ‘second-order’ truths as it were, but they are nonetheless compelling, at least to me. What conclusions? (1) Since there are events, there  must seemingly be a ‘place’ where these events can and do occur, an Event Locality. (2) Since there are, and continue to be, events, there  must be an ultimate source of events, an Origin, something distinct from the events themselves and also (perhaps) distinct from the Locality.
A further and more radical conclusion is that this broad schema can legitimately be generalized to ‘everything’, at any rate to everything in the entire known and knowable universe. Why make any hard and fast distinction between mental events and their features and ‘objective’ physical events and their features? Succession, discontinuity and causality are properties of the ‘outside’ world as well, not just that of the private world of an isolated thinking individual.
What about other things we normally assume exist such as trees and tables and ourselves? According to the event model, all these things must either be (1) illusory or irrelevant (same thing essentially) (2) composite and secondary and/or (3) ‘emergent’.
Objects are bundles of events that keep repeating more or less in the same form. And though I do indeed believe that ‘I’ am in some sense a distinct entity and thus ‘exist’, this entity is not fundamental, not basic, not entirely reducible to a collection of events. If the personality exists at all ─ some persons  have doubts on this score ─ it is a complex, emergent entity. This is an example of a ‘valid’ but not  fundamental item of reality.
Ideas, if they take place in the ‘mind’, are events whether true, false or meaningless. They are ‘true’ to the extent that they can ultimately be grounded in occurrences of actual events and their interactions, or interpretations thereof. I suppose this is my version of the ‘Verification Principle’ : whatever is not grounded in actual sensations is to be regarded with suspicion.  This does not necessarily invalidate abstract or metaphysical entities but it does draw a line in the sand. For example, contrary to most contemporary rationalists and scientists, I do not entirely reject the notion of a reality beyond the physical because the feeling that there is something ‘immeasurable’ and ‘transcendent’ from which we and the world emerge is a matter of experience to many people, is a part of the world of sensation though somewhat at the limits of it. This reality, if it exists, is ‘beyond name and form’ (as Buddhism puts it) is ‘non-computable’, ‘transfinite’. But I entirely reject the notion of the ‘infinitely large’ and the ‘infinitely small’ which has bedevilled science and mathematics since these (pseudo)entities are completely  outside  personal experience and always will be. With the exception of the Origin (which is a source of events but not itself an event), my standpoint is that  everything, absolutely everything, is made up of a finite number of ultimate events and an ultimate event is an event  that cannot be further decomposed. This principle is not, perhaps, quite so obvious as some of the other principles. Nonetheless, when considering ‘macro’ events ─ events which clearly can be decomposed into smaller events ─ we have two and only two choices : either the process comes to an end with an ‘ultimate’ event or it carries on interminably (while yet eventually coming to an end). I believe the first option is by far the more reasonable one.
With this, I feel I have the bare bones of not just a philosophy but a ‘view of the world’, a schema into which pretty well everything can be fitted ─ the contemporary buzzword is ‘paradigm’. Like Descartes emerging from his ‘stove’, I considered  I had a blueprint for reality or at least that part of it amenable to direct experience. To sum up, I could disbelieve, at least momentarily,  in just about everything but not that (1) there were events ; (2) that events occurred successively; (3) were subject to some sort of omnipresent causal force with  occasional lapses into lawlessness. Also, (4) these events happened somewhere (5) emerged from something or somewhere and (6) were decomposable into ‘ultimate’ events that could not be further decomposed.  This would do for a beginning, other essential features would be added to the mix as and when required.                                                                             SH

Note 1  Many extremely successful societies seem to have been perfectly happy in  avoiding the ‘intellectual crisis’ altogether : Rome did not produce a single original thinker and the official Chinese Confucian world-view changed little over a period of more than two thousand years. This was doubtless  one of the main reasons why these societies lasted so long while extremely volatile societies such as VIth century Athens or the city states of Renaissance Italy blazed with the light of a thousand suns for a few moments and then were seen and heard no more.

Note 2 Je pris garde que, pendant que je voulais ainsi penser que tout était faux, il fallait nécessairement que moi, qui le pensais, fusse quelquechose. Et remarquant que cette vérité : je pense, donc je suis, était si ferme et si assure, que toutes les autres extravagantes suppositions des sceptiques n’étaient capables de l’ébranler, je jugeai que je pouvais le reçevoir, sans scrupule, pour le premier principe de la philosophie que je cherchais.”
René Descartes, Discours sur la Méthode Quatrième Partie
“I noted, however, that even while engaged in thinking that everything was false, it was nonetheless a fact that I, who was engaged in thought, was ‘something’. And observing that this truth, I think, therefore I am, was so strong and so incontrovertible, that the most extravagant proposals of sceptics could not shake it, I concluded that I could justifiably take it on  board, without misgiving, as the basic proposition of philosophy that I was looking for.”  [loose translation]

Note 3  The person in question was, for the record, a primary school teacher by the name of Marion Rowse, unfortunately now long deceased. She was the only person to whom I spoke about the ideas that eventually became Eventrics and Ultimate Event Theory and deserves to be remembered for this reason.

Note 4   As someone at the other end of the social spectrum, but who seemingly also went through a crisis of belief at around the same time, put it, “I have gained a healthy respect for the objective aspect of reality by having lived under Nazi and Communist regimes and by speculating in the financial markets” (Soros, The Crash of 2008 p. 40).
According to Boswell, Dr. Johnson refuted Bishop Berkeley, who argued that matter was essentially unreal, by kicking a wall. In a sense this was a good answer but perhaps not entirely in the way Dr. Johnson intended.  Why do I believe in the reality of the wall? Because if I kick it hard enough I feel pain and there is no doubt in my mind that pain is real — it is a sensation. The wall must be accorded some degree of reality because, seemingly, it was the cause of the pain. But the reality of the wall, is, as it were, a ‘derived’ or ‘secondary’  reality : the primary reality is the  sensation, in this case the pain in my foot. I could, I argued to myself, at a pinch, disbelieve in the existence of the wall, or at any rate accept that it is not perhaps so ‘real’ as we like to think it is, but I could not disbelieve in the reality of my sensation. And it was not even important whether my sensations were, or were not, corroborated by other people, were entirely ‘subjective’ if you like, since, subjective or not, they remained sensations and thus real.

Note 5 In the Chuang-tzu Book, Yen Ch’eng, a disciple of the philosopher Ch’i  is alarmed because his master, when meditating, appeared to be “like a log of wood, quite unlike the person who was sitting there before”. Ch’I replies, “You have put it very well; when you saw me just now my ‘I’ had lost its ‘me’” (Chaung-tzu Book II. 1)

Note 6 The practitioner of meditation is encouraged to ‘widen’ these gaps as much as possible (without falling asleep) since it is by way of the gaps that we can eventually become familiar with the ‘Emptiness’ that is the origin and end of everything.

‘Speed’ is not a primary concept in the Système Internationale d’Unités  : it is defined by means of two quantities that are primary, the unit of length, the metre,  and the unit of time, the second. ‘Speed’ is the ratio distance/time and its unit is metres/second.
It is, I think, possible to disbelieve in the reality of motion but not to disbelieve in the reality of distance and time, at least in some sense.
The difficulty with the concept of motion and the associated notions of speed and velocity, is that we have somehow to combine place (exact position) and change of place for  if there is no change in a body’s position, it is motionless. The concepts of ‘exact position’ and movement are in fact irreconcilable (Note 1)  : at the end of the day we have to decide which of the two we consider to be more fundamental. For this reason there are really only two consistent theories of motion, the continuous process theory and the cinematographic theory.
The former can be traced at least as far back as Heraclitus, the Ionian philosopher for whom “all things were a-flowing” and who likened the universe to “a never ending fire rhythmically rising and falling”. Barrow, Newton’s mathematics teacher, was also a proponent of the theory and some contemporary physicists, notably Lee Smolin, seem to belong to this camp.
Bergson goes so far as to seriousoly assert that, when a ‘moving object’ is in motion, it does not occupy any precise location whatsoever (and he is not thinking of Quantum Wave Theory which did not yet exist). He writes,
“… supposons que la flèche puisse jamais être en un point de son trajet. Oui, si la flèche, qui est en mouvement, coincidait jamais avec une position, qui est de l’immobilité. Mais la flèche n’est jamais a aucun point de son trajet”.
(“Suppose that the arrow actually could be at a particular point along its trajectory. This is possible if the arrow, which is on the move, ever were to coincide with a particular position, i.e. with an immobility. But the arrow never is at any point on its trajectory”.)
So how does he explain the apparent fact that, if we arrest a ‘moving’ object we always find it at a particular point ? His answer is that  in such a case we ‘cut’ the trajectory and it falls, as it were, into two parts. But this is like the corpse compared to the living thing ― c’est justement cette continuité indivisible de changement qui constitue la durée vraie” (“It is precisely the indivisible continuity of change that constitutes true durastion”) .

The cinematographic theory of movement finds its clearest expression in certain Indian thinkers of the first few centuries AD —:
“Movement is like a row of lamps sending flashes one after the other and thus producing the illusion of a moving light. Motion consists in a series of immobilities. (…) ‘Momentary things,’ says Kamalasila, ‘cannot displace themselves, ‘because they bdisappear at that very place at which they have appeared’.” Stcherbatsky, Buddhist Logic vol. I pp.98-99

For almost as long as I can remember, I have always had a strong sense that ‘everything is discontinuous’, that there are always breaks, interludes, gaps. By this I do not just mean breaks between lives, generations, peoples and so on but that there are perceptible gaps between one moment and the next. Now, western science, partly  because of the overwhelming influence of Newton and the Infinitesimal Calculus he invented, has definitely leaned strongly towards the process theory of motion, as is obvious from the colossal importance of the notion of continuityin the mathematical sciences.
But the development of physical science requires both the notion of ‘continuous movement’ and precise positioning. Traditional calculus is, at the end of the day, a highly ingenious, brilliantly successful but hopelessly incoherent procedure as Bishop Berkeley pointed out in Newton’s own time. Essentially Calculus has its cake and eats it too since it represents projectiles in continuous motion that yet occupy precise positions at every interval, however brief (Note 2).
In Ultimate Event Theory exact position is paramount and continuous motion goes  by the board. Each ultimate event is indivisible,  ‘all of a piece’, and so, in this rather trivial sense, we can say that every ultimate event is ‘continuous’ while it lasts (but it does not last long). Also, K0 , the underlying substratum or event Locality may be considered to be ‘continuous’ in a rather special sense, but this need not bother us anyway since K0 is not amenable to direct observation and does not interact with the events that constitute the world we experience. With these two exceptions, “Everything is discontinuous”. This applies to ‘matter’, ‘mind’, ‘life’, movement, anything you like to think of.    Furthermore, in the UET model, ultimate events have occurrence in or on three-dimensional grid-points on the Locality, but these grid-points are not pressed right up against one another (as in certain other  models such as that of Lee Smolin). No, there are (by hypothesis) real, and in principle measurable, breaks between one grid-position and the next and consequently between one ultimate event and its neighbours if there are any, or between each of its its consecutive reappearances.
Furthermore, in the UET model, ultimate events have occurrence in (or on) three-dimensional grid-points on the Locality, but these grid-points are not pressed right up against one another as they are in certain other discontinuous physical  models (Note 3). In Ultimate Event Theory there are real, and, in principle, measurable gaps breaks between one grid-position and the next and consequently between one ultimate event and its neighbours if there are any, or between each of its consecutive reappearances.
What we call a ‘body’ or ‘particle’ is a (nearly) identically repeating event cluster which, in the simplest case, consists of a single endlessly repeating ultimate event. The trajectory of the repeating event as it ‘moves’ (appears/reappears) from one three-dimensional frame to the next may be presented in the normal way as a line — but it is a broken, not a continuous line.
It is a matter of common experience that certain ‘objects’ (persisting event-clusters) change their position relative to other repeating event-clusters.  For illustrative purposes, we consider three event-chains composed of single events that repeat identically at every ksana (roughly ‘instant’). One of these three event-chains, the black one Z is considered to be ‘regular’ in its reappearances, i.e. to occupy the equivalent grid-point at each ksana. Its trajectory or eventway will be represented by a column on black squares where each row is a one-dimensional representation of what in reality is a three-dimensional region of the Locality. The red and green event-chains, X  and  Y  are displaced to the right laterally by one and three grid-positions relative to at each ksana (Note 4).

X   Y                              Z
…□□□□□□□□□□□□□□□■□……..
…□□□□□□□□□□□□□□□□……..
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…□□□□□□□□□□□□□□□□■□□□□……..
…□□□□□□□□□□□□□□□□■□□□□……..
…□□□□□□□□□□□□□□□□□■□□□□□□□□□………..
…□□□□□□□□□□□□□□□□□■□□□□□□□□□□……..
…………………………………………………………………..

In normal parlance, Y is a ‘faster’ event-chain (relative to Z) than X and its speed relative to Z is three grid-positions (I shall henceforth say ‘places’) per ksana . The speed of  X  relative to Z is one place/ksana. (It is to be remarked that Y reappears on the other side of  Z without ‘colliding’ with it).
Of course, this is a simplified picture : in reality event-chains will be more spread out, i.e. will consist of many more than a single element per ksana; also,  there is no reason a priori why they should be made up of events that reappear during every ksana. But the point is that ‘velocity’ in Ultimate Event Theory is a straight numerical ratio (number of grid-positions)/(number of  ksana)  relative to a regular repeating event-chain whose trajectory is considered to be vertical.  Note that Y reappears on the other side of  Z without ‘colliding’ with it.      S.H.  27/7/12

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Note 1 :     “A particle may have a position or it may have velocity but it cannot in any exact sense be said to have both” (Eddingon).

Note 2 :  Barrow, Newton’s geometry teacher, wrote, “To every instant of time, I say, there corresponds some degree of velocity, which the moving body is considered to possess at that instant”. Newton gave mathematical body to this notion in his ‘Theory of Fluxions’, his version of what came to be known as the Infinitesimal Calculus.

Note 3      According the Principle of Relativity, there is no absolute direction for a straight event-line, and any one of a family of straight lines can be considered to be vertical. Other things being equal, we consider ourselves to be at rest if we do not experience any jolts or other disturbances and thus our ‘movement’ with that of Z, a vertical line.  However, if we were ‘moving’, i.e. appearing and reappearing at regular intervals, alongside or within (straight) event-chains or  Y, we would quite legitimately consider ourselves to be at rest and would expect our event-lines to be represented as vertical.
Z
…□□□□□□□□□□□□□□□□□■□……..
…□□□□□□□□□□□□□□□□□……..
…□□□□□□□□□□□□□□□■□□……..
…□□□□□□□□□□□□□□■□□□□……..
…□□□□□□□□□□□□□■□□□□……..
…□□□□□□□□□□□□■□□□□□□□□□□………..
…□□□□□□□□□□■□□□□□□□□□□□……..
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The point is that in classical physics up to and including Special Relativity the important distinction is not between rest and constant straitght-line motion but between accelerated and unaccelerated motion, and both rest and constant straight-line motion count as unaccelerated motion. This capital distinction was first made by Galileo and incorporated into Newton’s Principia.
The distinction between ‘absolute’ rest and constant straight-line motion thus became a purely academic question of no practical consequence. However. by the end of the nineteenth century, certain physicists argued that it should be possible after all to distinguish between ‘absolute rest’ and constant straight-line motion by an optical experiment, essentially because the supposed background ether ought to offer a resistance to the passage of light and this resistance ought to vary at different times of the year because of the Earth’s orbit. The Michelsen-Morley experiment failed to detect any discrepancies and Einstein subsequently introduced as an Axiominto his Theory of Special Relativity the total equivalence of all inertial systems with respect to the laws of physics. He later came to wonder whether there really was such a thing as a true inertial system and this led to the generalisation of the Relativity principle to take in any kind of motion whatsoever, inertial systems being simply a limiting case.
What I conclude from all this is that (in my terms) the Locality does not interact physically with the events that have occurrence in and on it; however, it seems that there are certain privileged pathways into which event-chains tend to fall. I currently envisage ultimate events, not as completely separate entities, but as disturbances of the substratum, K , disturbances that will, one day, disappear without a trace. The Hinayana Buddhist schema is of an original ‘something’ existing in a state of complete quiescence (nirvana) that has, for reasons unknown, become disturbed (samsara) but which will eventually subside into quiescence once again. The time has come to turn this philosophic schema into a precise physical theory with its own form of mathematics, or rather symbolic system, and my aim is to contribute to this development as much as is possible. Others will take things much, much further but the initial impulse has at least been given.

Note 4  Of course, this is a simplified picture : in reality event-chains will be more spread out, i.e. will consist of many more than a single element per ksana; also,  there is no reason a priori why they should be made up of events that reappear during every ksana.

S.H.  22/7/12

Speed is not one of the seven basic SI units (the metre, kilogram, second, ampere, kelvin, candela & mole). It is a ‘derived unit’ defined in terms of the basic standard of length, or distance, the metre, and the basic standard of time, the second. Being entirely secondary, dependent on other entities for its very existence, one might very reasonably concluide that it is fictitious — though a useful concept for all that. Curiously, very few Western thinkers have taken this view though this is perhaps fortunate since otherwise we would not have the science of dynamics. On the other hand, amongst Indian Buddhist thinkers, disbelief in the reality of motion (and by implication speed in the normal sense of the word) is the norm rather than the rare exception. Vasubandhu writes, “There is no motion….Things do not move, they have no time to do so, they disappear as soon as they appear” (quoted Stcherbarsky).
Speed is “the rate of change of distance against time”, metres/sec or, more precisely, so many metres per second  (there is no such thing as a metre-second). The concept of motion does not specifically involve time, but if a body is not in motion, it has no speed, or, if you like, has zero speed. ‘Intuitively’, as the mathematicians write (with a certain sneer since intuition is not reliable), we feel that there is an ‘absolute’ difference between being in a state of rest and being in a state of motion. Already in ancient times, Zeno had pointed out in his Paradox of the Chariot, much less well know than his Achilles and the Tortoise Paradox, that the motion of  a chariot in a chariot race was different, depending on whether we were judging it with reference to the spectators, or to other competitors. He concluded that a body could at one and the same time have “all kinds of completely different speeds” — a remarkably perceptive observation.
Subsequently, Galileo came to the conclusion that the really important distinction to be made was not between ‘rest’ and ‘motion’ as such but between ‘constant’ and ‘accelerated’ motion. To some extent Galileo had been anticipated by the important and neglected medieval thinker, Oresme, (Note 1) but Galileo seems to have been the first to introduce the notion of an ‘inertial frame’, as it was eventually called, one of the the most fruitful ideas in the whole of physics.  In one of his dialogues, Galileo observed that, if one were in a cabin inside a ship on a calm sea and had no window, it would not be possible to tell by conducting physical experiments whether the ship were in a state of constant motion or at rest. As Einstein eventually put it, The laws of physics are the same in all inertial frames.
Newton viewed both Space and Time as absolutes in the sense that they were not dependent on subjective human viewpoints or states of motion, though he conceded that it was in practice often impossible to locate an object precisely (Note 2). This viewpoint is not unrelated to Newton’s profound, if unorthodox, religious belief  : God would know the ‘real’ position of an object  even if we couldn’t always precisely define it. What of motion?  Since Space and Time were absolutes, Newton found himself propelled to believe in absolute motion as well : “IV. Absolute motion, is the translation of a body from one absolute place into another(Principles, I, 6 ff). However, he quickly realised that if we want to determine a body’s ‘absolute motion’, as opposed to its apparent motion, we need some repeating event-sequence that we know to be absolutely regular, and no such sequence appeared to be available.  Clocks needed resetting and even the Sun’s course and that of the planets changed slightly. “It may be, that there is no such thing as an equable motion, whereby time may be accurately measured. (…) It may be that there is no body at rest, to which the places and motions of others may be referred.”
But the problem with the ‘notion of motion’ runs much deeper still. Not only is it in practice impossible to determine a body’s ‘real’ motion, as opposed to its motion in relation to some other body which is itself in some sort of motion, but it is impossible to attribute at one and the same time an exact position to a body and motion. As Sextus Empiricus realized long ago. “If a thing moves, it moves either in the place where it is or in that where it is not; but it moves neither in the place where it is (for it remains therein) nor in that which it is not (for it does not exist therein); therefore nothing moves.”
So one has, seemingly, to give up one of two things : either exact position or motion. Mathematically, of course, one has one’s cake and eats it : one gets round the difficulty by the (contradictory) concept of ‘instantaneous velocity’ and other artifices of Calculus, also by the unproven assumption that “space and time are infinitely divisible” supposing even the last sentence is meaningful. Calculus works (more or less) but a mathematical wand cannot bring an impossibility into existence : an object simply cannot be at once in a particular spot and in motion.
So what has Ultimate Event Theory to say on the matter ?  In a nutshell : If by motion we are to understand continuous motion (and this is the usual unstated assumption), then there is no motion. But if by motion we simply understand “being in different spots at different times” (as Bertrand Russell put it), then, certainly there is ‘motion’ — though what we like to think is  motion is in reality a succession of stills like the cinema (Note 3).
Ultimate Event Theory assumes ‘absolute’ position : “Every ultimate event occupies one, and only one, spot on the Locality” (Axiom of Locality). It also assumes Succession though, on re-reading my Axioms, I find that this is not specifically stated — this will be remedied in due course. An ultimate event, like a Buddhist dharma, is  fixed, static and evanescent. “Momentary things cannot displace themselves because they disappear at that very place at which they have appeared” (Kamalasila). And since everything is, by assumption, made up of distinct ultimate events, no-thing moves (though there can be change of circumstance, i.e. event environment).
A so-called ‘object’ is, in Ultimate Event Theory, a dense cluster of distinct ultimate events which repeat more or less identically. A spot on the Locality is, if you like,  two dimensional where one dimension, the spatial, is itself subdivided into three. An ultimate event occupies this spot, the entire spatial dimension with the event itself disappears to be replaced by an identical one and the event repeats. We will assume, for simplicity’s sake, that there is only one ultimate event and that it repeats at every ‘temporal layer’, i.e. chronon after chronon. A connected repeating chain of events can be conceived as a sequence of dots or pixels that are so close that they appear to form a continuous line. This ‘line’ is either straight or not straight. In the first case we have the equivalent of the ‘motion’ of a body which is “either at rest or in constant straight line motion”, in the second we have the case of ‘accelerated’, or simply irregular.
Is there any distinction to be made between a strictly vertical straight line of dots (representing events) and a slanting line?  Seemingly not : what is for one ‘observer’ a vertical line is slanting for someone else. Normally, we consider ourselves to be at rest even if we are (in the normal sense of the word) in motion : looking out of the window of a a train we perceive the countryside flashing by whereas in reality it is more we who are flashing by the static countryside. The important distinction is between a regularly spaced sequence of dots, whether ‘straight’ or slanting and one which is not, which curves round or changes its direction from chronon to chronon.
The ‘equivalence’ of different ‘inertial’ sequences is a somewhat surprising fact of experience, it is not at all what one would expect. What it means in terms of my world-picture is that the successive layers of the Locality can not only ‘slide past each other’, as it were, but that they have no discernable fixed orientation one to another. This is difficult to believe but, like Newton, we can conclude that if there is a true ‘absolute’ orientation, we do not have the means to identify it so it can be discounted.
However, there are some things that we can deduce right away. For a start, there must be some limit on the spatial separation between successive ultimate events if they are to remain connected. This limit is a pure number : it is given by the maximum possible difference between two spatial spots, i.e. spots belonging to the same ‘layer’, compatible with the bonding of events that constitute an ‘object’. (This is Einstein’s brilliant insight that there is —or ought by rights to be — a physical limit to thenoperation of causality, namely the speed of light in a vacuum.) It is irrelevant which of two straight lines we consider ‘vertical’ and which ‘slanting’ : all that matters is the number of spaces in between. Since we are speaking of a single chronon, the limiting event ratio is  No. spatial spots/1 chronon, i.e a pure number.  Should anything exceed this limit, the subsequent event-chains will immediately ‘dissociate’ to use a term from chemistry.
Of course, we are not able to determine this limit from one chronon to the next — though possibly we will be able to one day. But we can approximate to this limit by making a simple ratio of spations/chronons  where a spation is simply the lateral distance between two successive events which are recongizably part of the same event-chain. The point is that, once again, this limit is a pure number, and a rational number at that, n/m where n, m are integers.
So far, it has been assumed that the events of this event-chain  do not miss out any layers. But, supposing they do, there must likewise  be a limit to the number of layers, i.e. chronons, that they are allowed to miss out without dissociation. And for any sort of event-chain, the ‘speed’ is given by a straight ratio. A rapidly moving object (read dense event-cluster) will cover more spaces laterally than a comparable slower moving object. And so it goes on.  The limitations of the vocabulary of this site do not enable me to give suitable pictures but this will come later.
What is the advantage of this schema? Principally this : it enables one to believe in the succession and change without falling into contradiction or evoking fantastic entities such as instantaneous velocities or infinitesiamlly small or large intervals. A whole post will be devoted to the Pardoxes of Zeno, but it can be said now that this schema resolves all of them. A closing door does not have to pass through an ‘infinite’ number of positions before it is shut but only a (large) finite number of spots on the Locality. Achilles is able to overtake the tortoise because ‘time’ is constituted by successive layers and there are not an infinite number of them between any two. (It may well be that Achilles gets in front of the tortoise without ever being actually exactly abreast of it.) And finally, the arrow reaches its goal because it occupies a finite number of positions between the bowstring and the target. The arrow, a dense cluster of ultimate events, is always at rest (is where it is) but it occupies spots on successive temporal layers.
And the price of this coherent theory ?  There is always a price. and here this would seem to be that the door, the arrow, the tortoise and even Achilles are not continuously existing but ‘exist’ at successive moments (chronons) only — “Being is shot through with nothingness” (Heidegger).

Notes :  (1)   In his excellent (albeit from my point of view largely misguided) book, The History of the Calculus, Boyer rescues Oresme from his undeserved oblivion. He writes, “Oresme had a clear conception not only of acceleration in general but also of uniform acceleration in particular. (…) He went farther and applied his idea of uniform rate of change and of graphical representation to the proposition that the distance travelled by a body at rest amd moving with uniform acceleration is the same as that which a body would traverse if it were to move for the same interval of time with a uniform velocity which is one-half the final velocity” (p. 83, op. cit.). This is remarkable indeed.

(2)  “1. Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without regard to anything external…..
2. Absolute space, in its own nature, without regard to anything external, remains always similar and immovable…..”
Newton, Principles I. 6

(3) Russell has a curious passage in Principles of Mathematics :

“It is to be observed that, in consequence of the denial of the infinitesimal and in consequence of the allied purely technical view of the derivative of a function, we must entirely reject the notion of a state of motion. Motion consists merely in the occupation of different places at different times, subject to continuity as explained in Part V. There is no transition from place to place, no consecutive moment or consecutive position, no such thing as velocity except in the sense of a real number which is the limit of a certain set of quotients. The rejection of velocity and acceleration as physical facts (i.e. as properties belonging at each instant to a moving point, and not merely real numbers expressing limits of certain ratios) involves, as we shall see, some difficulties in the statement of the laws of motion; but the reform introduced by Weierstrass in the infinite calculus has rendered this rejection imperative.”
Russell, Principles of Mathematics, i, p. 473
What is remarkable (and ridiculous) is that Russell rejects velocity and acceleration “as physical facts”,  but apparently believes in both  as “numbers expressing limits of certain ratios”.  Motion is reduced to something strictly mathematical, but for all that real ! God knows what Bishop Berkeley, who flummoxed Newton with his shrewd criticisms of the latter’s early versions of the Infinitesimal Calculus, would have made of this. It is true that the postulates of Ultimate Event Theory also require some “re-statement of the Laws of Motion” but at least I believe that something is going on : my ultimate events are perfectly real, not mathematical constructions.   S.H.