Archives for category: Discontinuity

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).

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.



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

It has always been my impression that ‘Space’Time’ is discontinuous, or, to be more precise, that what we perceive is necessarily discontinuous — I am slowly coming round to the view that there may be an underlying reality which is, if you like, ‘continuous’, the perceived reality being a sort of froth on the surface of this deeper reality. Our Western scientific viewpoint, partly because of the influence of Newton and his ‘Theory of Fluxioms’, has always favoured continuity. But now some physicists are seriously re-considering the matter.
“We often speak of the fabric of space, as if it were continuous, but is it instead a kind of patchwork of jittering quantized bits?” writes Mariette DiChristina, the editor in chief of  Scientific American (in February 2012 issue).
The Director of Fermilab Particle Physics Centre, Craig Hogan, is planning an experiment which may “change what we currently think we know about the nature of space and time” (DiChristina).
“According to Hogan, in a bitlike world, space itself is quantum — it emerges from the discrete, quantized bits at the Planck scale (Note 1). (…) It does not sit still, a smooth backdrop to the cosmos. Instead, quantum fluctuations make space bristle and vibrate, shifting the world around with it. “Instead of the universe being this classical, transparent, crystaalline-type ether,” says Nicholas B. Suntzeff, an astronomer at Texas A&M University, “at a very, very small scale, there are these little foamlike fluctuations. It changes the texture of the universe tremendously.”  from “Is Space Digital?”  by Michael Moyer, (Scientific American, February 2012).

Notes :  (1)  The term ‘Planck scale’ can refer either to a space or time scale. Planck time is about 5.39 × 10 (exp –44) secs and Planck length is about 1.6 × 10 (exp –35) metres  (from Wikipedia)

I first conceived the idea of the new ‘science’ of ‘Eventrics’ some thirty years ago. At the time, I had just come back from a long period abroad and one of the main reasons I returned to civilisation was to study mathematics (via the OU) — even though mathematics was a subject for which I had shown no aptitude at school and had always heartily detested. My aim in following this surprising course of action was to better understand the adversary — rather in the manner of certain Syrian or Persian princes who travelled to Rome to acquire a military education before returning to their countries to start a revolt.
However, the reverse happened : I found myself seduced by the elegance and power of the axiomatic mathematical method and, so to speak, went over to the enemy.  A little later, when I began pondering about events and their interconnections, I automatically started off in the manner of Euclid  by formulating certain  basic axioms and postulates (see earlier post) and tried to draw some conclusions from them. I soon saw that a new symbolic system was required and I did manage to concoct a somewhat cumbersome method of classifying event-chains according to certain criteria. I got more and more involved, not to say obsessed, with these speculations and spent most nights endlessly discussing Eventrics and related topics with the only person I saw anything of at the time, Marion Rouse, a true kindred spirit unfortunately now long deceased.
But the system obstinately refused to ‘take off’.  With hindsight I can now see that certain computer ‘systems’  such as ‘cellular automata’, being developed at this precise moment in America, were the sort of tools I needed and was groping towards — but these developments were still little known in Europe and anyway all this was taking place at a far more exalted scholastic level than mine. So the new science of ‘Eventrics’ never got off the drawing board and, although the idea remained at the back of my mind, it is only very recently that, after browsing through suitcases full of mildewed exercise books and clamp files, that I have finally decided to put some of this strange stuff into the public domain. As an arch-Luddite (by temperament anyway) I originally viewed the Iinternet as a deadly threat to humanity, but once I started using it, I found that the ‘bitty’ format of blogs exactly suited my style.
So far, so good. But nonetheless I still carried on assuming that if Eventrics was ever to come to anything, it would have to be thrown into a rigorous axiomatic mould with appropriate mathematical symbolism and so on and so forth. Two days ago, though, I had a sort of Eureka moment. The material was still refusing to do as it was told and I found myself drifting into a more informal presentation — encouraged by coming across Taleb’s book The Black Swan where the author lauds the merits of working ‘bottom up’ rather than ‘top down’ (1). Now, the key idea of Ultimate Event Theory is discontinuity : the theory completely breaks with the entire mathematic-physical Western tradition of continuity and infinite divisibility which still casts a long shadow over science even in this quantum era.  Surely, I said to myself,  the theory, since it is the study of radical discontinuity, should by rights be developed in a discontinuous manner. So it should ! I resolved to make no further attempt, at this stage in the game anyway, to throw the rapidly accumulating material into a mould where it clearly did not want to go.
What’s the alternative ?  To allow, or rather encourage, a theory to develop ‘organically’ as things do in the natural world : this approach is especially appropriate in this century now that biology has clearly taken over from physics as the leading science. Nature does not bother too much with mathematics — far, far less than mathematicians imagine — it proceeds  by trial and error, fits and starts, threshes around in all directions until something that works turns up (a new species). As a matter of fact most important human developments started off like this as well  : even the mechanical/mathematical revolution which culminated in Newton’s Mechanics evolved painfully over a period of at least three centuries with all sorts of people contributing the odd block to the growing edifice — who, today, has heard of Oresme or Horrocks for example ? The fully fledged Mechanical view of the world, perhaps the most successful intellectual paradigm to date, had to wait for the genius of Newton to gather all these disparate strands together into a mighty synthesis.
It is clear to me, and seemingly to a growing number of other people, that Western society is undergoing a new paradigm shift at the moment : something is painfully emerging from the welter of discordant and scarcely intelligible ideas spawned by the twentieth century. I believe that progress in understanding the world and our place within it will come, not from making the current mathematical and conceptual apparatus even more abstruse, but rather from ‘going back to basics’ and re-examining the basic concepts of physical science. Hopefully, my ideas concerning events and event-chains, naive though they inevitably are at the moment, will bear fruit somewhere sometime in someone’s head. I intend to open up the field, starting with what is inside my head  : I shall no longer try to fit  my ideas into a formal strait-jacket but let them come out pell-mell, though maintaining a certain spasmodic surveillance noentheless.
My strategy at the moment, inasmuch as I have one, is to itemise various snippets that I sense could be important, trusting to Providence that somehow (changing the metaphor) these paths through the scrub and wilderness will eventually converge and an oasis will be there in front of us. One of the basic assumptions of Ultimate Event Theory is that, once certain collections of heterogeneous events have developed cohesion, they will attract other events to themselves, leading to yet larger conglomerations : this is an entirely ‘mechanical’ process, pretty much independent of the people concerned or the precise nature of the events. Eventually (sic) a fully fledged theory will ‘emerge’ without any one person having deliberately created it : the principle being to ‘give events enough rope’,  either to hang themselves, or tie themselves into an elegant seaman’s knot. We will see whether and how soon this happens and who will join me in this venture into the (not entirely) unknown.

 Notes :  

(1) The terms ‘bottom up’ and ‘top down’ are Stockmarket trader jargon — Taleb, the author of The Black Swan was (and possibly still is) an options trader. Economists tend to work ‘top down’, i.e. they start with the theories and try to fit the facts to the theory; traders tend to use whatever methods they find work for them and  any ‘theory’ there may be is just a generalisation from actual experience. Western science, stemming as it does from the Greeks and given a strong philosophic impetus by Plato, started off as a largely ‘top down’ affair and, despite the emphasis on experiment and observation, this legacy is still very much with us, particularly in physics which has today become little more than a branch of (very abstruse) applied mathematics.



The test of a model depends on what it can predict, though this is not the only consideration : models which stimulate the mind because they are ‘intuitively clear’ have proved to be extremely helpful in the development of science even if they were eventually discarded.
Anyone wishing to lay the foundations for a new science on the basis of preliminary assumptions must steer a narrow course between two extremes. On the one hand, he must beware of wrenching unjustifiable conclusions from the premises because he ‘knows exactly where he wants to land up’. On the other hand, there is no point in threshing around in the dark and hoping for the best : once it is clear that one line of argument is leading nowhere, it must be abandoned. How do we know it is leading nowhere ?  Often we don’t, but we can appeal to our own or others’ experience to judge how we are progressing. For example, de Sitter’s model derived from Einstein’s Equations of General Relativity was clearly wrong (or rather not applicable to the case that concerned us) since it predicted a universe completely empty of matter. In other cases, early scientists were eventually proved ‘right’ (though not necessarily for the reasons they believed at the time), for example Huyghens’ wave theory of light.

        I shall attempt to avoid these two extremes. My sketchy knowledge of advanced physics and current experimentation (HLC and so on) could actually be an advantage in the sense that I am by no means sure ‘where I want to land up’, so I am less likely to fudge things. As for the second danger, a manifestly absurd conclusion will (hopefully) prompt me to re-examine my original assumptions and add to them, and, if this does not work, simply admit that something has gone wrong. But at this stage in the game it would be unfair to  expect, and even foolish to desire, anything but the broadest qualitative predictions : being too specific in one’s forecasts too early can all too easily block off diverging avenues worth exploring.

Before drawing any conclusions, I will briefly review in an informal manner, the preliminary assumptions on which the whole of Ultimate Event Theory is based. Broadly speaking, In a nutshell, I consider that “the universe is composed of events rather than things”. Although I have listed some properties of ‘events’ as I see them, at this stage I have to assume that the notion of an ‘event’, or at any rate the distinction between an event and a ‘thing’,  is ‘intuitively clear’. Ultimate events associate together to form ‘ordinary’ events but cannot themselves be further decomposed — which is why thet are called ‘ultimate’. They occupy ‘spots’  on the ‘Locality’ — the latter beihng, for the moment, nothing more than a  sufficiently large expanse able to accommodate as many ultimate events as we are likely to need. Ultimate events are ‘brief’ : they flash into and out of existence lasting for the space of a single ‘chronon’, the smallest temporal interval that can exist, in this ‘universe’ at any rate. A definite ‘gap’ exists between successive appearance of ultimate events : physical reality is discontinuous through and through (Note 1). Some ultimate events acquire  what I call ‘dominance’ : this enables them to repeat identically, perhaps  associate with other stabilized ultimate event and influence event clusters. ‘Objects’, a category that includes molecules and some elementary particles (but perhaps not quarks) — are viewed as relatively persistent, dense event-clusters. Dominance is not conserved on the grand scale : there will always be some ultimate events that  pass out of existence for ever, while there are also ultimate events which come into existence otherwise than by a causal process (random events) (Note 1).

The predictions are as follows:

(1)  There will always be an irreducible background ‘flicker’   because of the discontinuity of physical reality. This ‘rate of existence’ varies : essentially it depends on how many positions on the Locality are ‘missed out’ in a particular event-chain. The rate of most event chains is so rapid that it is virtually imperceptible — though, judging by certain passages in the writings of Plato and J-J Rousseau, some people seem to have thought they perceived it.  But there should be some ‘extended’ event chains whose flicker can be recognized by the instruments we now have, or will shortly develop (Note 2).

(2)  The current search for ‘elementary particles’ will turn up a very large quantity of heterogeneous ‘traces’ which are too brief and too rare to be dignified with the title of ‘elementary particle’.  The reason for this is the vast majority of ultimate events do not repeat at all : they flash into existence and disappear for ever.

(3)  The number of ‘elementary particles’ detected by colliders and other instruments will increase though some will never be detected again : this is so because ultimate events are perpetually forming themselves into clusters but also  ‘breaking up’ into their component parts, in some cases dematerializing completely.

(4) Certain ‘elementary particles’ will pass clean through solid matter without leaving a trace : this will tend to occur every time the (relative) speed of a very small event cluster is very large while  and the ‘thickness’ of the lumped cluster is small in the direction of travel (Note 3).

(5)  There will always be completely new event-clusters and  macroscopic events, so the future of the universe is not completely determinate. This is so because not all ultimate events are brought into existence by previously existing ones : some ultimate  events originate not in K1 (roughly what is known as the physical universe) but in K01 , the source of all events. If these ‘uncaused events’ persist, i.e. acquire self-dominance, or come to dominate existing event clusters, something completely new will have come into existence —  though whether it persists depends on how well it can co-exist with already well-established event-clusters. In brief, there is an irreducible random element built into the universe which stops it being fully determinate.

Notes :

(1) Since putting up this post on January 18th, I have come across what might be confirmation (od a sort) of this prediction. The February 2012 edition of Scientific American includes a mind-blowing article, Is Space Digital? by Michael Moyer. “Craig Hogan, director of the Fermilab Center ….thinks that if we were to peer down at the tiniest subdivisions of space and time, we would find a universe filled with an intrinsic jitter, the busy hum of statuc. This hum comes not from particles bouncing in and out of being or other kinds of quantum froth that physicists have argued about in the past. Rather Hogan’s noise would come about if space was not, as we have long assumed, smooth and continuous, a glassy backdrop to the dance of fields and particles. Hogan’s noise arises if space is made of chunks. Blocks. Bits.”  This is not just a passing thought, for Hogan “has devised an experiment to explore the buzzing at the universe’s most fundamental scales.”
I originally thought that what I call the ‘flicker of existence’ would remain forever beyond the reach of our instrumentation and said as much in the original draft of this post. However, after thinking about the amazing advances made already, I added, perhaps prophetically, “There should be some ‘extended’ event chains whose flicker can be recognized by the instruments we now have, or will shortly develop.”  Maybe Hogan’s is one of them.
However, I do not ‘buy’ the current trend of envisaging the universe as a super computer  — for Hogan my ‘flicker of exietence’ is  ‘digital noise’. The analogy universe/computer strikes me as being too obviously rooted in what is becoming the dominant human activity — computing. I wouold have thought the ‘universe’  had better things to do than just process information. Like what for example ?  Like bringing something new into existence from out of itself, actualizing what is potential. In a nutshell : the ‘universe’ (not a term I would choose) is creative not computational. But I suppose one cannot expect trained scientists to see things in this light.    S.H. (7/2/12)

(2) Heidegger put it more poetically, “Being is shot through with nothingness”.

(3)   This happens because a rapid event cluster ‘misses out’ more event locations on its path, so the chance of the two clusters ‘colliding’, i.e. ‘competing’ for the same spots on the Locality, is drastically reduced.

(4) This is so because not all ultimate events are brought into existence by previously existing ones : some ultimate  events originate not in K1 (roughly what is known as the physical universe) but in K01 , the source of all events. If these ‘uncaused events’ persist, i.e. acquire self-dominance, or come to dominate existing event clusters, something completely new will have come into existence —  though whether it persists depends on how well it can co-exist with already well-established event-clusters. In brief, there is an irreducible random element built into the universe. (This is quite apart from Quantum Indeterminacy which in any case would disappear if a coherent ‘hidden variables’ theory replaces the orthodox one.)

I will discuss in a subsequent post whether modern experiment and observation gives any support to these predictions.

(Note: This is not a review of the best-selling book, The Black Swan, The Impact of the Highly Improbable, by Nassim Nicholas Taleb : I shall merely discuss its relevance to the scientific study of events and their interactions, what I call ‘Eventrics’. S.H.)

Before Westerners colonised Australia, it was assumed that all swans were white. However, it is now known that a few (very few) are black. The term ‘Black Swan’, though a brilliant choice of name for an important phenonenon, is somewhat misleading. For Taleb is not concerned with unusual birds but exclusively with unusual events. Moreover, although the term could, by extension, be applied to any kind of event, Taleb seems to be entirely concerned with human events, either general historical ones like wars and financial crashes, or, on an individual level, serendipitous circumstances leading to an important discovery like penicillin or a chance encounter in a bar leading to  a career change.
Taleb lists three characteristics of Black Swan events : “rarity, extreme impact, retrospective (though not prospective) predictability” (Note 1). These are best broken down into four : rarity, unpredictability, extreme impact, subsequent plausibility. Taleb rightly emphasizes how readily commentators and historians are to list the causes of, say, the outbreak of WWI the 1929 Wall Street Crash after the event, but how slow people living at the time were to even realise that anything special was about to happen. Human blindness and inveterate bias is an important issue but it is not my main concern : I wish to determine whether there are any discernable ‘general principles’ controlling the occurrence of events. Nonetheless, the fact that a reasonable attempt at showing the steps leading up to a devastating crisis like the outbreak of WWI or the 1929 and 2008 Wall Street crashes does show that a Black Swan event is not only possible — because it took place — but can be made to appear plausible, even inevitable (after the event).  Had the event under scrutiny been totally fantastic, historians would have had a much harder job trying to trace such an event’s antecedents, and, unless it was a very well verified and important event, they would have  been very happy to ignore it completely. ( Scientists and rationalists do this all the time when confronted with aberrant apparently ‘psychic’ phenomena on the principle that what they cannot explain casually cannot exist and, if they do, are best left well alone.)
The restrospective ‘plausibility’ of a Black Swan event is something to be born in mind for future reference, but the really important features of a Black Swan event are (1) unpredictability and (2) extreme impact with rarity  coming very much in third place (since Black Swan events turn out not to be so rare as all that) (Note 2). Nassim Taleb has in effect divides (human and hisotrical) events into two classes, one comprising ‘ordinary events’, those taking place in Mediocristan as he puts it, and one containing Black Swan events — those taking place in Extremistan.  The difference between these two classes is not so much the rarity of  Black Swan events as (1) their apparent lack of causal antecedents combined with (2) their sudden appearance and (3) their colossal consequences.
‘Normal’ human event-chains proceed by small increments that can be roughly figured out in advance : a nation becomes steadily wealthier as new markets open up or alternatively poorer as sources of valuable raw materials become exhausted, an individual gets promoted every five years until he retires and so on. Each step is defined by the previous one and such progress, if it is modelled mathematically, gives a so-called ‘continuous’ function. In the terms of Eventrics, there is a steady flow of what I call ‘dominance’ from one macroscopic event to the next — dominance, to be explained in due course, is roughly  causality viewed as a perfectly real ‘force’.
Black Swan events are not like this at all : they seem to appear from nowhere,  strike like a ‘bolt out of the blue’ (a significant expression) and have consequences that may last centuries in the case of historical turning-points  or a lifetime in the case of personal reversals of fortune. As an example of historical Black Swans take the rise of Islam, a movement that swept through half the known world, emerging from an insignificant, arid, scarcely civilized region no one bothered about, or, even more incredibly, the largest land Empire the world had yet seen emerging like a volcano from the steppes of Outer Mongolia. More recently, we have had 9/11,  the financial crisis of 2008, the sudden appearance of China as the twenty-first century’s super-power when not so long ago we had people in laundromat suits reading the Little Red Book and collecting scrap iron for “backyard steel foundries” (Mao’s own prhase). The examples are endless : it is actucally more difficult to find examples of ‘ordinary’, predictable events that have been turned out to be historical turning points, though there are one or two.
As for personal life :”Look into your own existence. Count the significant events, the technological changes, and the inventions that have taken place in our environment since you were born and compare them to what was expected before their advent. How many of them came on schedule ? Look into your own personal life….. How often did these things [important changes] occur according to plan?” (The Black Swan p. xix)
I would certainly concur with Taleb when he says that history is discontinuous (“History and societies do not crawl. They make jumps” (p. 11)), that history is dominated by Black Swan events and that such events seem to be on the increase (“The modern world being Extremistan, is dominated by rare — very rare — events“(p. 61) (Note 3).
Two questions immediately spring to mind : (1) Are Black Swan events truly random, or only unpredictable because of insufficient data? (2) Is there a ‘hidden pattern’ to Black Swan events, a driving force concealed from our vision ?
On the first point, Taleb is adamant that Black Swan events are truly random,  unpredictable through and through, not just because our instruments are too coarse or our information inadequate. He makes this an article of faith, not to say an obsession, and, in a sense, he ought to know what he is talking about since he has spent much of his adult life as a trader.  He is careful to distinguish true indeterminism even from the sort of randomness we associate with Chaos Theory since the latter somehow combines strict determinism with unpredictability (amongst other reasons, because the systems studied in Chaos Theory are very sensitive to the initial conditions). What do I think? I remain open minded  for the moment : though there is no doubt in my mind that Black Swan ‘causality’, if it exists at all, is totally different from the usual “Event A brings about event B” sort of causality. Black Swan events do not seem to originate in the everyday world, as if there were some sort of underground well of events which occasionally forces itself through to the surface, spewing out molten lava.
So, what of ‘hidden patterns’, ‘hidden historical sources’? Taleb, as a systematic  Sceptical Rationalist, gives any such ideas short shrift :  those who think they can read the writing on the wall, people like Plato, Hegel, Marx, Spengler et al. are deluding themselves — and doubtless he would say the same about my own humble attempts in the same direction. Few people living after the 2008 financial crash would quarrel with Taleb’s scathing attacks on economic experts, all of whom despite their sophisticated mathematical and computer telescopes, somehow managed not to see the hurricane that was blowing their way. Indeed, I would go further than Taleb (who sensibly suggests that the Nobel Prize for Economics be abolished) by also suggesting that Economics Faculties be closed down world-wide, especially the London School of Economics. However, market analysis, like weather forecasting, is a notoriously complex area and the ‘experts’ are not so much to blame for getting it wrong as for arrogantly refusing to admit that they did get it wrong. Weather forecasters have in fact been challenged and have mended their ways : after getting long-term (month or more) weather forecasting palpably wrong they have (in the UK) decided to restrict themselves to short-term forecasting where they do a good job.
But there is certainly no need to abandon the whole principle of drawing inferences from previous instances — which Taleb tends to rubbish as ‘tunneling’. The adage, “Where there is pattern, there is significance”, though it can at times mislead, has underpinned almost all of mankind’s greatest intellectual and cultural developments, whereas nothing great has ever been achieved by doubt — certainly not the great scientific discoveries such as gravitation and evolution which were wild generalisations from very inadequate evidence. The human mind and psyche does not work well with negatives : scepticism is a useful restraining force, a counter weight, but no more than that.
Taleb resuscitates the ‘Turkey Paradox’ (orginally Chicken Paradox) invented by Bertrand Russell, a man who has a deeply destructive influence on modern logic and mathematics. The ‘paradox’ goes something like this. A turkey, well-fed and well-protected by its owner for a good period of time, say forty days, concludes (or would conclude if it could reason) that, on the basis of the past record, it is going to continue to be well-fed and well-protected indefinitely. On Day 41 the turkey is slaughtered since it is Thanksgiving Day. “”Jumping to conclusions!” Unsound inference!” “Incomplete Induction!” scream Hume and Russell and all the rest of the rationalist posse. Not really. On Day 41, unbeknown to the turkey, the conditions of the problem have dramatically changed and that is why the forecast, based on outmoded conditions, turns out to be wrong. The lesson to draw is not that ‘incomplete induction is risky’ (we know that already) but rather that one should make sure that the conditions have not suddenly changed in a way that is not at first sight evident. Neither Hume nor Russell nor Popper was a practising scientist, businessman or military commander — and they would have failed in all three areas. The adroit general or businessman knows that there will never be enough data relating to economic conditions or the movements of enemy troops, he thus learns to make rapid conclusions on the basis of thoroughly inadequate data : a skill that can only really be learned by practise in the market place or on the  battlefield. In terms of the Turkey Problem, one tries to guess whether the conditions have remained the same or not and one develops a certain flair or scent for this — Taleb himself, who is dismissive of  the ability of most millionnaires, recognizes that Soros had this.
Science is a somewhat different case.  The very existence of science rests on the unprovable (and conceivably mistaken)  assumption that identical conditions produce identical results (Note 3). If we did not believe this, we would not trouble ourselves to repeat a scientific test : the very same experiment carried out in a different laboratory on a different day might well give completely different results. The ancient Greeks, likewise the Chinese, did not have make this assumption which is one reason why neither quite managed to develop natural science as we understand it today though they came very close. The Greeks, under the influence of Plato, considered that the sublunary world was subject to random influences, ‘chaotic’ in our terms, and thus could not be reduced to behaviour governed by a handful of ‘physical laws’. Heavenly bodies were different : they were regular in their movements. The result was a very advanced astronomy and mathematics but a somewhat defective mechanics despite Archimedes (no science of movement, dynamics). We have extended the application of ‘natural law’ to the entire universe, “One universe, one set of laws”. This enormous intellectual gamble seems to have paid off by and large, though it has been found necessary to exclude the inflationary period of the early universe, and there are some indications that the basic physical ‘constants’ have changed over time.  The “one universe. one set of laws” assumption is by no means self-evident: it is justified, not by its inherent plausibility, since it is not plausible, but uniquely only by its results — “One judges a tree by its fruits”. Similarly, the adage “Where there is pattern, there is significance” — which Taleb catigates as the ‘narrative fallacy’ —  has, by and large, served humanity very well and will continue to do so despite all the dreadful cautionary tales of poitivists and sceptics. You learn to distinguish between the important and unimportant patterns, false prophecies and true, on the job, not in the study, and in some cases, errors of judgement can be as fruitful as genuine discoveries, witness the wild guesses of intuitive mathematicians such as Fermat and Ramanujan (Note 4).
So, can any reasonable conclusions be drawn from Taleb’s discussion of Black Swan events, other than “Take care how you go” ? I think, yes, and Taleb, even though he categorically declined to provide specific forecasts on repeated occasions when interviewed, does as a matter of fact sneak in one or two predictions  — remarkably (since the book was written before the 2008 Wall Street crash) he says that “the government-sponsored institution Fanny Mae, seems to be sitting on a barrel of dynamite” (Note 5).
If we consider large historical Black Swans like the two world wars of the 20th century, we see that there is a family resemblance — the same big European powers opposed each other on much the same terrain — but the second Black Swan was not an identical repeat, indeed with respect to tactics it was almost exactly the reverse. WWI was a trench war that lasted nearly four years; WWII, as far as France was concerned, was a Blitzkrieg that lasted a few weeks. Likewise, the 2008 financial meltdown has a family resemblance to 1929 in America but is not an identical repeat. One can, from this and other evidence, hazard a good guess that successive large-scale Black Swans are never identical repeats : the preparations made for the ‘next’ Black Swan (the Maginot Line) tend to be quite useless, indeed counter-productive, since they draw attention away from the coming danger, and thus increase its eventual effects. This, combined with the very plausible guess that Black Swan events are becoming more and more frequent, allows one to hazard certain guesses about the 21st century.
Taleb remarks shrewdly that the supposedly ‘safe bets’ in the case of both banks and regimes may well be the most dangerous of all : not so long ago Iceland was considered a pretty safe country to invest in and standard economic wisdom says that government bonds are the safest of all investments, even though it is now whole countries, including Spain and Italy, that are on the verge of defaulting. Taleb says at one point that apparently ‘safe’ (but oppressive) regimes may be the very ones to fall, on the principle that the taller you are, the harder you fall. He mentions as examples Syria and Saudi Arabia and, since his book was written long before the Arab Spring, this is pretty good going.
I will myself stick my neck out and hazard a few guesses. It may well be that one of the affluent Arab Middle Eastern kingdoms, which no one pays much attention to at the moment, will precipitate the next financial crisis — as nearly happened with Abu Dhabi. I also predict that the natural resource that will most likely precipitate war and economic turmoil is not oil that everyone is bothered about but water (Note 6) .
What about science and technology? It seems clear that the leading science of the 21st century will be biology, but biology, though something of a Black Swan science when DNA was discovered, is hardly an outlier at the moment, is on the contrary advancing steadily year by year, the difference being that the steps it takes are getting larger. Traditional mathematics — or rather modern traditional mathematics — will decline in importance and (hopefully) be largely replaced by more flexible modelling such as computer simulations, cellular automata, ‘evolutionary computing’ and so forth.  The trouble with current mathematics is that it is fixed, inert, capable of m0delling change and motion (up to a point) but, by definition, incapable of growing from within itself. Everything unexpected is kept outside the hallowed domain which is both mathematics’s strength and elegance, but also its hopeless limitation. In the turbulent environment of today, we need symbolic systems that do not exclude the random and the uncontrollable, the source of evolutionary innovation but, on the contrary, welcome it into the system while nonetheless keeping it under control — “The price of freedom is endless vigilance”. We want computer programmes that advance by trial and error and take initiatives themselves : strangely enough, since writing these lines yesterday I have come across two articles in back issues of the New Scientist dealing with this very issue (Note 7).  As for theoretical physics, it looks like it will continue cascading into total unintelligibility until the basic concepts are rigorously re-examined and a radically new outlook emerges. Enough of all that, the rest is history.
Notes : (1) “What we call here a Black Swan… is an event with the following three attributes :
First, it is an outlier, as it lies outside the realm of regular expectations, because nothing in the past can con vincingly point to its possibility. Second, it carries an extreme impact. Third, in spite of its outlier status, human nature makes us concoct explanations for its occurrence after the fact, making it explainable and predictable. I stop and summarize the triplet : rarity, extreme impact, and retrospective (though not prospective) predictability.”

(2) An actual Black Swan (the bird) has Taleb’s first attribute, ‘rarity’ and his third ‘unpredictability’ (since it was not even known to exist) but it completely lacks his second attribute, ‘extreme impact’. Who, apart from a few professionals, cares whether the bird is classed as a swan or not ? Viruses and  archaea are closer to being true ‘Black Swans’ since their belated and very unexpected discovery has provoked a mini-revolution in biology.

(3) Of course, it is not true in Quantum Mechanics that “the same conditions produce identical results” but this is precisely why QM, in the orthodox Copenhagen interpretation, is so worrisome.

(4) Fermat was an amateur mathematician — he was a jurist by profession  — and, though regarded as the founder of an important branch of mathematics, Number Theory, gave few proofs. He claimed to have the proof of his famous Last Theorem, but famously noted that “it was too long to go into the margin” of the book where he noted it down, Diophantus’s Arithmetica. Modern mathematicians think his proof was almost certainly spurious.
The  strange largely self-taught Hindu mathematician, Ramanujan, produced a whole lot of stunning mathematical theorems while working with chalk and slate-board (because he couldn’t afford ink and paper) on the verandah of his parents’ house near Madras in the early twentieth century. Hardy, the leading British pure mathematician of the day and himself a model of mathematical ‘rigour’, nonetheless had the breadth of vision to recognize Ramanujan’s genius and invited him to come to Cambridge. This was perhaps a mixed blessing for mathematics since it seems that Ramanujan’s really creative work, some of which turned out to be wrong, wilted in the aridity of modern ‘prove it or be damned’ mathematics (Ramanujan didn’t bother with proofs) and only flowered again briefly when he returned to India to die at the age of 32. Hardy said that “Ramanujan’s mistakes were as remarkable as his correct theorems” — or something to that effect.  Today, Ramanujan would stand even less chance of being recognized (except perhaps via the Internet) as few universities, let alone the Royal Society, would welcome such a maverick into their ranks and I doubt if any contemporary Hardy would have the vision  and above all the generosity to aid such a person.

(5) “The Black Swan” footnote p. 225 Penguin edition. Taleb also writes (p. 226), “We have moved from a diversified ecology of small banks, with varied lending policies, to a more homogeneous framework of firms that all resemble one another. True, we now have fewer failures, but when they occur… I shiver at the thought. I rephrase here : we will have fewer but mroe severe crises”.

(6)  Since writing this I have come across the following :  “A shortage of water is a more serious peril than any of the others mentioned in this report [concerning Pakistan]. Combined with fast growth of population, it is the true existential threat to Pakistan. (…) The study forecasts that by 2025 Pakistan’s annual water supply will fall short of demand by around 100 billion cubic metres, about half of the entire present flow of the Indus {!!]. In parts of the country the shortage is already acute.” (Going with the Flow in The Economist, February 11th 2012).    
Also, “We are facing a planet without enough water and with a rapidly warming atmosphere….” (Princess Sumaya of Jordan, Interview in New Scientist 18 February 2012)

(7)  This  development has already taken place though the mathematical fraternity, which even now does not accept an innovator such as Mandelbrot into its ranks, has not yet woken up to the fact :
“Evolutionary computing allows computers to do things they haven’t been programmed to do and is already being used to solve problems as diverse  as creating train timetables to designing aircraft” (“Move Over Einstein” by Justin Mullins, New Scientist, 19 March 2011)
        “Genetic algorithms mimic natural selection by describing a design as if it were a genome conbstructed from segments. Each segment describes a parameter of the invention, varying from its shape, say, to much finer grained aspects, such as electrical resistance or a chemical’s molecular affinities. By randomly changing some segments — or ‘mutating them’ — the algorithm improves the design. The best results are then bred together to improve things further (“The Next Wave” by Paul Marks, New Scientist, 14 May 2011).


Although we cannot know for sure because his works have been lost, it seems that Democritus, the founder of atomic theory, believed that atoms, the ultimate constituents of the universe, were eternal. Newton certainly believed this was the case. But few classical scientists were prepared to envisage the universe as a plurality, a mosaic rather than a witewashed wall : notions of ‘matter’ and ‘continuity’ held sway even during the nineteenth-century when the molecular theory eventually emerged. Today, of course, matter has all but disintegrated into ghostly entities such as quarks.
Yet the notion of, if not matter, at least unchanging ‘material elements’ clings on and we are told that the atoms in our bodies were forged in exploding stars millions of years ago and will most likely carry on practically unchanged for millions of years yet. It is ‘continuity’ which has kept ‘matter’ (just about) alive and the idea of continuity, though clearly it has a strong psychological hold on the Western mind, is essentially nothing but a mathematical asssumption required by the Calculus. Incredibly, people are prepared to ‘believe’ in infinite universes and in quantum entanglement whereby cats (or humans) can be simultaneously alive and dead, or both or neither, but dismiss the idea of radical discontinuity without even giving it serious consideration.
From the standpoint which I am attempting to develop in these articles, matter is an illusion though a persuasive and convenient one : not only are there no rocks or stones but even electrons and protons and so forth are no more (but no less) than strongly persistent identically repeating event-clusters. It is not that I cannot bring myself to believe that the atoms of carbon in my body have been in existence for thousands of years : for me, they have not even existed continuously for a single second. Heraclitus wrote that “No man steps into the same river twice” which indeed he does not. A follower of his went further and said that this was wrong because a man “did not even step into the same river once” : a statement that few historians make much sense of. I assume what the follower meant was that the ‘man’ who steps into the river was not the ‘same man’ as the (pseudo) person who approached the riverbank or who would subsequently dry himself on the side — certainly, this would be the  Buddhist interpretation.
Even when thinkers stress the insubstantiality of matter, they baulk at discontinuity. Lee Smolin writes, “The illusion that the world consists of objects is behind many of the constructs of classical science” (“Three Roads to Quantum Gravity”, p. 52). My eyes lit up when I read this, likewise, a little further on, “From this new point of view [that brought about by quantum theory and relativity], the universe consists of a large number of events“. However, on a closer look, Lee Smolin is at the opposite end of the philosophic spectrum to myself : he believes that the ‘world’ consists of ‘processes’, that everything is always evolving. “Motion and change are primary. Nothing ‘is’ except in a very approximate and temporary sense” (p. 53). I find Lee Smolin’s ‘relational universe’ both unappealing and scarcely intelligible : on the contrary, in my universe ‘everything is’ but only for the space of a chronon, then it is gone. There are relations between event-clusters, but these relations are not themselves as fundamental as the events themselves and these ultimate events are ‘static’ — they may be replaced by similar or other events but in no way continuously evolve into them.
Smolin also rejects completely the idea of there being any gaps whatsoever in his ‘world processes’ and uses the (to me) entirely unconvincing analogy of beads on a chain or molecules of water touching each other without a void between them. My world is not only ‘not continuous’ but is ‘gapped’ : I believe we have a faint percpetion of these minute intervals.
All this is, perhaps, partly a matter of temperament. But it is certainly not the case that all people in all eras have believed that life and ‘matter’ and everything we see around us is continuous : the main evidence to the contrary is the success of Buddhism, the religion that tells us that everything is ephemeral. It is not quite clear whether Heraclitus held to the ‘process’ view or the ‘spasmodic’ view : Newton himself, a devotee of continuity, seems to have got the kernel of his theory of fluxions from Heraclitus. Plato, though he is best known for his theory of eternal Forms, has a curious passage quoted by Stcherbatsky :
“This suddenly is a halt, or break, in the flow of time, an extra-temporal condition, in which the subject has no existence, no attributes, though it revives again forthwith clothed in its new attri butes : a point of total negation or annihilation, during which the subject with all its attributes disappears.”
Whether because of the crushing weight of Newton’s genius, or for other reasons such as the Christian influence, not a single Western  scientific thinker has, to my knowledge, ever advanced a theory of radical physical discontinuity. Maybe the reason is that any such theory is ridiculous, untenable? Maybe. But to date, observation and experiment do not clinch the matter one way or the other and, certainly, in the last hundred years, discontinuity has had by far the better of it. Although Planck himself was initially frightened by his own discovery, it is now known for a fact that no energy transfer is ‘continuous’ — even physicists had previously modelled heat and light and magnetism and gravity as ‘continuous’. The ‘quantum’ is here to stay, like it or not. Even atoms have broken up into discrete elementary particles, themselves decomposed into quarks;  a few people are even suggesting that Space-Time itself is ‘grainy’. .
Would there be any differences between a physical theory positing radical discontinuity and what we have at the moment? Certainly, there are grave conceptual differences which would lead to very different interpretations of phenomena and, better still, to new experiments that would confirm predictions made from a discontinuous perspective. Let us start right at the beginning. As far as I can make out, present day orthodoxy assumes that, barring rather rare events such as neutrons decaying into protons and so forth, elementary particles stay the same for very long periods of time; also, the total amount of energy in the universe is held to be constant (1st Law of Thermo-dynamics). Now, from the standpoint of Ultimate Event Theory, the ‘natural’, the expected, ‘thing’ is for an ultimate event to appear and then disappear for ever. The apparently stable elements that make up conventional matter are event clusters that have acquired persistence via what I call ‘self-dominance’. My assumptions thus do not make me surprised when I read that all sorts of previously unknown ‘elementary particles’ are cropping up all the time and vanish within fractions of a second : this is precisely what I would expect. The difficulty for my theory is to explain why anything persists at all — though clearly some ‘things’ do.
The notion that ‘material objects’ (in which category I include molecules and elementary particles) keep on occurring simply because of a property I call ‘persistence’ suggests at once that if one wanted to get rid of an object, all one would need to do would be to remove this property. What would happen? In current physics, any disappearance ‘here’ is followed by reappearance elsewhere (perhaps in another form) but this is not a necessary assumption in Ultimate Event Theory. It is, in principle, possible for a repeating event-cluster (an object) to disappear completely without leaving any sort of trace at all : in my conception ultimate events are in the last resort only disturbances of an invisible and intangible eternal ‘something’ and even long-lasting ‘stable’ disturbances can, in special circumstances, simply evaporate. This is rank heresy since it is a denial of the Principle of Energy on which Thermo-dynamics is based. However, sweeping principles such as the Conservation of Energy cannot ever be proved in all possible cases, and indeed one or two daring people such as Prigogine and Wolfram have actually cast some doubt on the universality of the principle already. Clearly, energy conservation is widespread and basic, but it need not be fundamental and strictly universal; if it is violated occasionally, I would not be unduly bothered. Exactly what sort of procedures would have to be gone through to prevent a dense event-cluster from repeating, I cannot at this moment say, but if there is anything in this theory, someone will one day perform such an experiment successfully just as all sorts of other supposedly ‘impossible’ experiments have been accomplished successfully.
What other conclusions follow from my assumptions ? Since everything that exists is, according to me, perpetually appearing and disappearing presumably in a rhythmic manner, it should be possible to detect this permanent oscillation : most likely it has been detected already but has been given another explanation. As to how my assumptions fit, or rather do not fit, with other concepts and principles of contemporary physics, I am currently pondering : in a large number of cases because we are dealing with such a fine scale there would be little difference. But, from my point of view, the long-standing wave/particle controversy has a different sense : everything is, in the last resort, discontinuous so, if it is useful to speak of waves (which it is), we should not regard these waves as strictly continuous, ‘all of a piece’ — as many physicists do seem to view them even today. I cannot claim to resolve the double slit experiment and other conundrums but, if other people one day take up these ideas, it is not impossible that some of these paradoxes will be dealt with without involving us in infinite dimensions and all the extravagant paraphernalia of contemporary advanced physics. Even, I would guess that the actual dimensions of an ultimate event will be determined during the next two centuries : we have determined the values of physical objects that most people in the past thought would remain forever unknown so why not this?     S.H.