Archives for category: Descartes

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.

 

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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.

 

In daily life we do not use co-ordinate systems unless we are engineers or scientists and even they do not use them outside the laboratory or factory. If we wish to be passed a certain book or utensil, we do not say it has x, y and z co-ordinates of (3, 5, 7)  metres relative to the left hand bottom corner of the room ― anyone who behaved in such a way would be considered half-mad. We specify the position of an object by saying it is “on the table”, “below the sink”, “near the Church”, “to the right of the Post Office” and so on. As Bohm pointed out in an interview, these are, mathematically speaking, topological concepts since they do not involve distance or angles. In practice, in our daily life, we define an object’s position by referring it to some prominent object or objects whose position(s) we do know. Aborigines and other roving peoples start off by referring their position to a well-known landmark visible for miles around and refer subsequent focal points to it, in effect using a movable origin or set of origins. In this way one advances  step by step from the known to the unknown instead of plunging immediately into the unknown as we do when we refer everything to a ‘point’ like the centre of the Earth, something of which we have no experience and never will have. We do much the same when directing someone to an object in a room : we relate a hidden or not easily visible object by referring to large objects whose localization is well-known, is imprinted permanently on our mental map, such as a particular table, chair, sink and so on. Even when we do not know the exact localization of the object, a general indication will at least tell us where to look ― “It is on the floor”. Such a simple and informative (but inexact) statement would be nearly impossible to put into mathematical/scientific language precisely because the latter is exact, too exact for everyday use.
I have gone into this at some length because it is important to bear in mind how unnatural scientific and mathematical co-ordinate systems are. Such systems, like so much else in an ‘advanced’ culture, are patterns that we impose on natural phenomena for our convenience and which have no  independent existence whatsoever (though scientists are rather loath to admit this). So why bother with them ? Well, for a long time humanity did not bother with such things, getting along perfectly well with more rough and ready but also more user-friendly systems like the local reference point directional system, or the ‘person who looks like so-and-so’ reference system. It is only when society became urban and started manufacturing its own goods rather than taking them directly from nature that such things as  geometrical systems and co-ordinate systems became necessary. The great advantage of the GPS or rectangular  three-dimensional co-ordinate system is that such systems are universal, not local, though this is also their drawback. Such artifices give us a way of fixing the position of  any object anywhere,  by using three, and only three, numbers. Using topological concepts such as ‘on’, ‘under’, ‘behind’ and so on, we commonly need more than three directional terms and the specifications tend to differ markedly depending on the object we are looking for, or the person we are talking to. But the ‘scientific’ co-ordinate system works everywhere ― though it is useless for practical purposes if we do not know, cannot see or remember the point to which everything is related. When out walking, the scientific system is only necessary when you are lost, i.e. when the normal local reference point system has broken down. Anyone who went hiking and looked at their computer every ten minutes to check on their position would be a fool and, if ever deprived of electronic devices, would never be able to find his or her way in the wilderness because he would not be able to pick up the natural cues and clues.
Why rectangular axes and co-ordinates? As a matter of fact, we  sometimes do use curved lines instead of straight ones since this is what the lines of latitude and longitude are, but human beings, when they do think quantitatively, almost always tend to think in terms of straight lines, squares, cubes and rectangles, shapes that do not exist in Nature (Note 1). The ‘Method of Exhaustion’, ancestor of the Integral Calculus, was essentially a means of reducing the areas and volumes of irregular figures to so many squares (Note 2). I have indeed sometimes wondered whether there might be an intelligent species for whom circles were much more natural shapes than straight lines and who would evaluate the area of a square laboriously in terms of epicycles whereas we evaluate the area of a circle by turning it into so many half rectangles, i.e. triangles. Be that as it may, it seems that human beings cannot take too much curved reality and I doubt if even a student of General Relativity ever thinks in curvilinear Gaussian co-ordinates.
Now, if we wish to accurately pinpoint the position of an object, we can do so, as stated, using only three distances plus the specification of the origin. (In the case of an object on the surface of the Earth we use latitude and longitude with the assumed origin being the centre of the Earth, the height above sea level being the third ‘co-ordinate’.) However, this is manifestly inadequate if we wish to specify the position, not of an object, but of an event. It would be senseless to specify an occurrence such as a tap on the window or a knife thrust to the heart by giving the distance of the occurrence from the right hand corner of the room in which it took place. It shows what a space-orientated culture we live in that it is only relatively recently that it has been found necessary to tack on a ‘fourth’ dimension to the other three and a lot of people still find this somewhat bizarre. For certain cultures, Indian especially, time seems to have been more significant than space (inasmuch as the two can be separated) and, had modern science developed there rather than in the West, it would doubtless have been very different. For a long time the leading science and branch of mathematics in the West was Mechanics, which studies the motions of rigid bodies that change little over brief periods of time. But from the point of view of Eventrics, what we familiarly call an ‘object’ is simply a relatively persistent event-cluster and the only reason we do not need to specify a time co-ordinate is that this object is assumed to be unchanging at least over ‘small’ intervals of time. Even the most stable objects are always changing, or rather they flash into existence, disappear and (sometimes) reoccur in a more or less identical shape and position with respect to nearby ‘objects’.
Instead of somehow tacking on a mysterious ‘fourth dimension’ to the familiar three spatial dimensions, Ultimate Event Theory posits discrete ‘globules’ or three-dimensional grids spreading out in all possible directions, each of which can receive one, and only one, ultimate event. The totality of possible positions for ultimate events constitutes the enduring  base-entity which I shall refer to as K0, or rather the only part of K0 with which we need to concern ourselves at the moment. It is misleading, if not meaningless, to refer to  this backdrop or substratum as ‘Space-Time’. Although I believe that ‘succession’ and ‘co-existence’ really do exist ― since events can and do occur ‘in succession’ and can also exist ‘at the same moment’  ― ‘Space’ and ‘Time’ have  no objective existence though one understands (sometimes) what people have in mind when they use the terms. Forf me ‘Space’ and ‘Time’ are basically mental constructs but I believe that the ultimate events themselves really do exist and likewise I believe that there really is an ‘entity’ on whose ‘surface’ ultimate events have occurrence. Newton fervently believed in the ‘absolute’ nature of Space and Time but his contemporary Leibnitz viewed  ‘Space’ as nothing but the sum-total of instantaneous relations between objects and some  contemporary physicists such as Lee Smolin (Note 3) take a similar line. For me, however, if there are events there must be a ‘somewhere’ on or in which these events can and do occur. Indeed, I take the view that the backdrop is more fundamental than the ultimate events since they emerge from it and are  essentially just momentary surface disturbances on it, froth on the ocean of K0.
For the present purposes it is, however, not so very important how one views this underlying entity, and what one calls it, it is sufficient to assume that it exists and that ultimate events are localized on or within it. K0 is assumed to be featureless and homogeneous, stretching indefinitely in all possible directions. For most of the time its existence can be neglected since all that we can observe and experiment with are the events themselves and their inter-relations. In particular, Kdoes not exert any ‘pressure’ on event-clusters or offer any  noticeable resistance to their apparent movements although it does seem to restrict them  to specific trajectories. As Einstein put it, referring to the ether, “It [the ether] has no physical effects, only geometrical ones”. (Note 4) In the terms of Ultimate Event Theory, what this means is that there are, or at least might be, ‘preferred pathways’ on the surface of K0 and, other things being equal, persisting event-clusters will pursue these pathways rather than others. Such  pathways and their inter-connections are inherent to K0  but are not fixed for all time because the landscape and its topology is itself affected by the event-clusters that have occurrence on and within it.
Even though I have argued that co-ordinate systems are entirely man-made and have no independent reality, in practiced I have found it impossible to proceed without an image at the back of my mind of a sort of fluid rectangular co-ordinate system consisting of an indefinite number of positions where ultimate events can and sometimes do occur. Ideally, instead of using two dimensional diagrams for a four-dimensional reality, we ought to have a three-dimensional framework, traced out by lights for example, and which appears and reappears at intervals ― possibly something like this is already in use. The trajectory of an object (i.e. repeating event-chain or event-cluster) would then be traced out, frame  by frame,  on this repeating three-dimensional co-ordinate backdrop. This would be a far more truthful image than the more convenient two dimensional representation.
One point should be made at once and cannot be too strongly stressed. Whereas the three spatial dimensions co-exist and, as it were, run into each other ― in the sense that a position (x, y, z) co-exists alongside a position (x1, y1, z1) ― ‘moments of time’ do not co-exist. This may seem obvious enough since if ‘moments of time’ really did co-exist they would be simultaneous, in effect the ‘same’ moment. And if all moments co-existed there would be nothing but an eternal present and no ‘time’ at all (Note 4).  But there is an unexpected and drastic consequence : it means that for the next ‘moment in time’ to come about, the previous one must disappear and along with it everything that existed at that moment. If we had an accurate three–dimensional optical model, when the lights defining the axes were turned off, everything framed by the optical co-ordinate system, pinpoints of coloured light for example, would by rights also disappear.
Rather few Western thinkers and scientists have ever realized that there is a problem here, let alone resolved it. (And there is no problem if we assume that existence and ‘Space-Time’ and everything else is ‘continuous’ but I do not see how this can possibly be the case and Ultimate Event Theory is based on the hypothesis that it is not the case.) Most scientists and philosophers in the West have assumed that it is somehow inherent in the make-up of objects and, above all, human beings to carry on existing, at least for a certain ‘time’. Descartes was the great exception : he concluded that it required an effort that could only come from God Himself to stop the whole universe disintegrating at every single instant. To Indian Buddhists, of course, the ephemeral nature of reality was taken for granted, and they ascribed the re-appearance and apparent continuity of ‘objects’, not to a supernatural Being,  but to the operation of a causal Force, that of ‘Dependent Origination’ (Note 4). Similarly, in Ultimate Event Theory, it is not the appearance or disappearance of ultimate events that requires explanation ― it is their ‘nature’, if you like,  to flash into and out of existence ― but rather it is the apparent solidity and continuous existence of ‘things’ that requires explanation. (Note 5) This is taking the Newtonian schema one step back : instead of ascribing the altered motion of a particle to an external force, it is the continuing existence of a ‘particle’ that requires a ‘force’, in this case a self-generated one.
Although Relativity and other modern theories have done away with all sorts of things that classical physicists thought  practically self-evident, the idea of a physical/temporal continuum is not one of them. Einstein, no less than Newton, believed that Space and Time were continuous. “The surface of a marble table is spread out in front of me. I can get from any point on this table to any othe point by passing continuously from one point to a ‘neighbouring’ one and, repeating this process a (large) number of times, or, in other words, by going from point to point without executing ‘jumps’. (…) We express this property of the surface by describing the latter as a continuum” (Einstein, Relativity p. 83).  To me, however, it is not possible to go from one point to another without a ‘jump’ as Einstein he puts it — quite the reverse, physical reality is made up of ‘jumps’. Also, the idea of a neighbourhood is quite different in Ultimate Event Theory : there are not an ‘infinite’ number of positions between a point on where an ultimate event has occurrence and another point where a different ultimate event has occurrence (or will have, has had, occurrence) but only a finite number. This number is not relative but absolute (though the perceived or inferred ‘distances’ may differ according to one’s standpoint and state of motion). And, of course, the three dimensional co-ordinate system we find appropriate need not necessarily be rectangular but might be curvilinear as in General Relativity.   S.H.   8 July 2012

Note 1 :  Extremely few natural objects have the appearance of our standard geometrical shapes, and the only ones that do are microscopic like rock crystals and radiolaria.

Note 2 : Geometry means literally ‘land measurement’ and was first developed for practical reasons —“According to most accounts, geometry was first discovered in Egypt, having had its origin in the measurement of areas. For this was a necessity for the Egyptians owing to the rising of the Nile which effaced the proper boundaries of everyone’s lands” (Proclus, Summary). Herodotus says something similar, claiming that the Pharaoh Ramses II distributed land in equal rectangular plots and levied an annual tax on them but that, subsequently, owners applied for tax reductions when their land got swept away by the overflowing Nile. To settle such disputes surveyors toured the country and had to work out accurately how much land had been lost. See Heath, A History of Greek Mathematics Vol. 1 pp. 119-22 from which these quotations were taken.

Note 3: “Space is  nothing apart from the things that exist; it is only an aspect of the relationships that hold between things” (Lee Smolin, Three Roads to Quantum Gravity, p. 18)

Note 4 : In the terms of Ultimate Event Theory, what this means is that there are, or at least might be, ‘preferred pathways’ on the surface of K0 and, other things being equal, persisting event-clusters will pursue these pathways rather than others. Such  pathways and their inter-connections are inherent to K0  but are not fixed for all time because the landscape and its topology is itself affected by the event-clusters that have occurrence on and within it.

  Note 5 : This is the same force that operates within a single existence, or causal chain of individual existences, in which case it is named Karma (literally ‘activity’). The entire aim of meditation and related practices is to eliminate, or rather to still, this force which drives the cycle of death and rebirth. The arhat (Saint?) succeeds in doing this and is thus able to enter the state of complete quiescence that is nirvana ― a state to which, eventually, the entire universe will return. The image of something completely still, like the surface of a mountain lake, being disturbed and these disturbances perpetuating themselves could prove to be a useful schema for a future physics. It is a very different paradigm from that of indestructible atoms moving about in the void which we inherit from the Greeks. In the  new paradigm, it is the underlying and invisible ‘substance’ that endures while everything we think of as material is a passing eruption on the surface of this something. The enorm ous event-cluster we currently call the ‘universe’ will thus not expand for ever, nor contract back again into a singularity : it will simply evaporate, return to the nothingness (that is also everything) from which it once emerged. In my unfinished SF novel The Web of Aoullnnia, the future mystical sect the Yther make this idea the cornerstone of their cosmology and activities ― Yther  is a Lenwhil Katylin term which signifies ‘ebbing away’. Interested readers are referred to my personal site www.sebastianhayes.com

Our civilization is, and always has been, object based, tactile, much more concerned with space than with time.  The main reason would seem to be that we inherit our science and mathematics from the Greeks and, for historic and perhaps also temperamental reasons, they hankered after the stability and permanence which, for most of their era, they did not possess (note 1). The Greeks excelled above all in two areas where change is not immediately apparent or was deliberately eliminated : namely astronomy and geometry. Archimedes formulated the basic laws of statics and hydrostatics but the world had to wait nearly another two thousand years before Galileo and Kepler gave us the laws of motion. This preoccupation with solid matter and with changelessness is by no means a universal attitude : Buddhism, for example, has been described as “essentially a long meditation on impermanence” while Taoism positively exults in the ebb and flow of natural processes and counsels us, if we want to achieve happiness, to ‘abandon ourselves to the flow’.
Even when movement did make its belated appearance in ‘classical’ (i.e. Renaissance to mid 19th century) physics the early scientists dealt primarily with ‘extended bodies’, ‘inertial systems’, ‘ideal’ gases, and viewed the whole of Nature as subject to unchanging mathematical laws laid down by the Creator. When thermodynamics in the nineteenth century gave physics the notion of ‘entropy’ there was general consternation since it seemed to show that the whole marvellous machine was running down and that the universe would ‘in time’ become uninhabitable as all temperatures became uniform. Then came Einstein and Relativity and the world of science was never the same again (note 2).
The difficulty that so many people, even some physicists, have in dealing with four dimensions instead of three only demonstrates the materialistic bias of our whole civilization. Obviously, if we are dealing with relatively persistent and unchanging entities such as rocks or lumps of metal, we can afford to neglect the time dimension since such substances  remain the same, or very nearly the same, from one day to the next, often from one century to another. But if we are talking about events, this is quite another matter.
Suppose a staccato event such as a pistol shot or a sudden flash of lightning. To locate this event spatially for the benefit of someone who was not present, it is sufficient to specify three, and no more than three, distances from a fixed point, say the corner of the room for the pistol shot. These three directions do not need to be at right angles to each other but in general this is the most convenient way of proceeding : an imaginary rectangle, extendable in all directions, with one corner baptised the ‘origin’ gives us a coordinate system. Alternatively, we can imagine a perfect sphere, make its centre the origin and precisely identify any spot by giving its distance from the centre and two angles. Latitude and longitude suffice to locate Waterloo if we mean the village in Belgium (or what is left of it), but this is not much use if we are referring to the battle. Human history is much more concerned with what happened than where such and such an event took place — what does it matter where Napoleon was when he decided to invade Russia?
The time dimension is, in reality, more fundamental than the spatial one, as Pearse correctly observed. One can (just about) imagine a world without extendable bodies, but not a world where nothing happens — one would not call it a world. All subjective events, wishes, fancies and so forth, though technically speaking localized ‘in our head’, have nothing to do with space : this is precisely why dreams and fantasies are attractive — they liberate us from the tyranny of spatial boundaries. This is a rather important point : it shows how the ‘time dimension’ is relatively independent of the spatial one, is dislocated from it, as it were, at least in our consciousness, if not in reality.

‘Time’ has in common parlance two very different meanings  which are frequently confused : it sometimes means Succession (before and after) and sometimes Duration (how long?). Of these two meanings, there can be no doubt that the first, succession, is the more important since the only way we can evaluate duration meaningfully is by measuring the interval between two prominent events, the first and last in a chain of events. On the other hand, succession has no connection to duration : event B succeeds event A whether the interim is scarcely perceptible or centuries long. A person’s ‘life’ spans the inerval between a first event, the moment of birth (or sometimes the moment of conception) and a last event, the moment of death. Today, the ‘times’ of these two events can be specified with great precision and there can be no doubt about which of the two events comes before the other and that they really are the first and the last (of this particular sequence anyway). The extent of a lifetime or other interval between two successive events is always measured by referring it to some repeated event or cycle which we have reason to believe is regular, for example the periodic reappearance of the sun (so many days), or its return to the same spot as judged by the shadow cast on a stone (so many years). Otherwise, we have man-made devices on the same principle, swinging pendulums, vibrating atoms and so on. One could in principle measure a time interval  with tolerable precision by recording one’s own heartbeats. In all such cases we have (1) a first and last event in a specific event chain, and (2) a (finite) number of events taking place in an adjacent event chain, or, in the case of heart beats, ‘within’ the very event chain we are measuring.
The Axioms of Ultimate Event Theory make this even more decisive. In a causally connected event chain — for the moment we assume that it is possible to recognize such a chain — there are always single events which we can take as ‘first’ and ‘last’. Every macroscopic event is made up of a finite number of ultimate events (Axiom of Finitude) and every ultimate event is precisely localized (Axiom of Locality). Moreover, in much the same way as we can, or could, measure a life-span in terms of heartbeats, we can measure the duration of any connected event chain by the number of (actual or possible) intermediate events. This number will be large but will not be infinite (Axiom of Finitude). Of course, in practice there may be serious difficulties in deciding whether such and such macroscopic events are causally connected or not, i.e. belong to the same chain, and the number of intermediary ultimate events will not, with our current technology at any rate, be ascertainable. But in principle this could be done. Such an event chain would be entirely determined both as to its constituent ultimate events and their number. (More will be said about this in a subsequent post.) We know athat, because of Relativity, there will be serious problems about relating this particular event chain to all the other event chains going on ‘at the same time’ but if we consider only a particular connected chain, and other chains occurring in its immediate vicinity, these problems do not arise.
Four, and seemingly no more than  four, specifications are required to localize an event exactly. However, this does not mean that the four ‘dimensions’ enter the arena ‘on the same footing’ : manifestly they do not. The three spatial dimensions are inextricably intertwined and how we label them, i.e. which we call length, breadth, depth, x, y or z is obviously neither here nor there — provided we keep to the same labels we have assigned them to a particular case. I conceive of ‘space’ as being ‘continuous’ in the sense of there not being any obvious breaks or barriers between specific spots which can receive ultimate events, even though (by the Axiom of Locality combined with the Axiom of Finitude) there is not an ‘infinite’ number of possible locations for events between any two given spots. The three spatial dimenions -dimensions are thus (1) arbitrarily labelled with respect to the three different directions; (2) are at right angles to each other (in the normal coordinate representation); and (3) are continuous in that they ‘run into each other’ without any apparent breaks. But none of this is true of the time dimension (which renders all this talk about the fourth dimension being compared to ‘adding a third dimension to a flat surface’ completely vacuous). Why is this not true? Because the time dimension has only one possible direction and this direction is (I believe) imposed on us — the so-called Arrow of Time. Secondly, the time dimension does not  ‘run into’ the three spatial dimensions but is very much out on its own, which is precisely why it was possible for so long to neglect it. Thirdly and most important of all, the time dimension is not continuous.
Suppose a set of objects in a particular neighbourhood at a particular moment, the contents of this room for example. A moment later, I see the same set of objects to all intents and purposes as they were before — though I  know there have been some slight changes at an invisible level. I cannot conceive for the life of me how this room can get from what it is at one moment to what it is at another, later moment, except by disappearing and reappearing. Strangely enough, few Western thinkers have addressed this problem since nearly all of them assume that it is ‘natural’ for things to keep on existing from one moment to the next and, moreover, to make the transition without any kind of an interim. Descartes is about the only Western philosopher I know who was worried by the question, and he resolved it, to his own satisfaction at least, by invoking the perpetual intervention of God who, at each and every moment, prevented the entire universe from disappearing without trace by recreating it anew — the so-called ‘Theory of Continuous Creation’ (Note 3).
Now, I believe that the movement from present to future is discontinuous, that “Being is shot through with nothingess” as Heidegger put it in a memorable phrase. Not only that, I believe that we have a certain instinctive awareness of this breakage in ‘the flow of time’. For Newton and countless others, ‘time’ is imaged as a stream, a fluid at any rate that ‘flows’ in a particular direction — an idea that goes back to Heraclitus. The classic Indian Buddhist simile is quite other : it is that of a lamp flashing on and off repeatedy (note 4) . I find this simile much more plausible, and indeed, I have never felt at ease with the whole concept of ‘continuity’, which is not a physical but a mathematical concept. We now know that all transfers of energy are not continuous, as they were once thought to be, but are subject to quantum laws (come in chunks). Fluids appear to be  continuous but in reality they are made up of molecules which we can even ‘see’, if only via an electron microscope. The last bastions of the continuous are ‘space’ and ‘time’ and, even here, some physicists are already suggesting that the ‘fabric of Space-Time’  might be ‘grainy’, as they put it.         S.H.

Notes :  (1) : During Plato’s lifetime Athens lost a long drawn out war with Sparta and was decimated by a plague. It is hardly surprising that Plato, and other men like him at the time, recoiled with horror from the human spectacle, taking refuge in a transcendent reality.

(2)  Paradoxically, the man who did more than anyone else to bring time to the centre of the stage, ended up by dispensing with it completely. During his last years, Einstein apparently came to the conclusion that everything takes place “in an eternal present”. He seems to have been serious about this, since he mentions this idea in a letter of condolence he wrote after hearing about the death of his undergraduate friend, Besso.

(3) Bergson “The world the mathematician deals with is a world that dies and is reborn at every instant, the world which Descartes was thinking about when he spoke of continuous creation” (Bergson, Creative Evolution p. 23-4). Stcherbatsky, who quotes this statement, remarks that “His [Descartes’] idea is quite Buddhistic and the Sanscrit translation sounds like a quotation from an Indian text” (Stcherbatsky, Buddhist Logic Vol. 1 Footnote p. 108).  The Buddhist theory is rather one of ‘Instantaneous Being’ since there is no Creator : the appearance and disappearance of dharma is a purely mechanical process, a sort of cosmic karma.

(4)    “The Buddhist theory of Universal Momentariness (ksanikatva’), converting the universe into a kind of cinema, meaintains that there is no other cause of destruction than origination, entities disappear as soon as they appear, the moment when the jar is broken a stroke of a hammer does noit differ in this respect from all preceding moments, since every moment a new or ‘other’ jar appears, constant desrtruction or renovation is inherent in every existence which is really a compact series of ever new moments.” Stcherbatsky, Buddhist Logic Vol. 2 Page 93 Footnote)