Archives for category: Newton

Newton on Velocity

August 5, 2012 //

Archimedes gave us the fundamental principles of Statics and Hydrostatics but somehow managed to avoid founding the science of dynamics though, as a practising civil and military engineer, he must have had to deal with the mechanics of moving bodeis. The Greek world-view, that which has been passed down to us anyway, was essentially geometrical and the truths of geometry are, or were conceived to be, ‘timeless’ : they referred to an ideal world where spherical bodies really were perfectly spherical and straight lines perfectly straight.
Given the choice between exact positionaing and movement (which is change of position) you are bound to lose out on one or the other. But science and technology must somehow encompass both exact position and ‘continuous’ movement. So how did Newton cope with the slippery idea of velocity ? From a pragmatic point of view, supremely well since he put dynamics on a firm footing and went so far as to invent a new form of mathematics tailor-made to deal with the apparently erratic motions of heavenly bodies — his ‘Method of Fluxions’ which eventually became the Differential Calculus. Strangely, however, Newton completely avoided Calculus methods in his Principia and relied entirely on rational argument supplemented by cumbersome, essentially static, geometrical demonstrations. Why did he do this? Probably, because he felt himself to be on uncertain ground mathematically and philosophically when dealing with velocity.
If you are confronted with steady straight line motion you don’t need Calculus — ordinary arithmetic such as even the ancient Babylonians and Egyptians employed is quite adequate. But, precisely, Newton was interested in the displacements of objects subject to a force, thus, by definition, not in constant straight line motion. And when the force was permanent, as was the case when dealing with gravitational attraction, the consequent motions of the boldies were never going to be constant (if change of direction was taken into account).
Mathematically speaking, speed is simply the first derivative of displacement with respect to time, and velocity, a vector quantity, is ‘directed speed’, speed with a direction attached to it. The modern mathematical concept of a ‘limit’ artfully avoids the question of whether a ‘moving’ particle actually attains a particular speed at a particular moment : it is sufficient that the difference between the ratio distance covered/time elapsed and the proposed limit can be made “smaller than any finite quantity” as the time intervals are progressively reduced. This is a solution only to the extent that it removes the problem from the domain of reality where it originated. For the world of mathematics is an ideal, not real world though in some cases there is a certain overlap.
Newton was not basically a pure mathematician, he was a mathemartical realist and a hard-nosed materialist (at least in his physics). He was obviously bothered by the question that today you are not allowed to ask, namely “Did the particle attain this limit or did it only get very close to it?”
It is often said that Newton did not have the modern mathematical concept of limit, but he came as close to it as was possible for a consistent realist. He speaks of “ultimate ratios” “evanescent quantities” and, unlike Leibnitz, tends to avoid infinitesimals if he can possibly manage to. He sees that there is indeed a serious logical problem about these diminishing ratios somehow carrying on ad infinitum and yet bringing the particle to a standstill.

“Perhaps it may be objected that there is no ultimate proportion of evanescent quantities; because the proportion, before the quantities have vanished, is not the ultimate, and when they are vanished, is none. But by the same argument, it may be alleged that a body arriving at a certain place, is not its ultimate velocity: because the velocity, before the body comes to the place, is not its ultimate velocity; when it has arrived, is none.”
Newton, Principia Translation Andrew Motte

Note that Newton speaks of ‘at a certain place’ and ‘its place’, making it clear that he believes there really are specific positions that a moving particle occupies. He continues :

“But the answer is easy; for by ultimate velocity is meant that with which the body is moved, neither before it arrives at its last place and the motion ceases, nor after, but at the very instant it arrives; that is, that velocity with which the body arrives at its last place and with which the motion ceases. And in like manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish.”
Newton, Principia Translation Andrew Motte

But this implies that there is a definite final velocity :

“There is a limit which the velocity at the end of the motion may attain, but not exceed. This is the ultimate velocity.”

Well and good, but Newton now has to meet the objection that, if the ‘ultimate ratios’ are specific, so also, seemingly, are the ‘ultimate magnitudes’ (since a ratio is a comparison between two quantities). This would seem to imply that nothing can properly be compared with anything else or, as Newton puts it, that “all quantities consist of incommensurables, which is contrary to what Euclid has demonstrated”.

“But,” Newton continues, “this objection is founded on a false supposition. For those ultimate ratios with which quantities vanish are not truly the ratios of the ultimate quantities, but limits towards which the ratios of quantities decreasing without limit do always converge; and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to, till the quantities are diminished in infinitum.” (Newton, Principia Translation Andrew Motte)

The last phrase (’till the quantities are diminished in infinitum’) seems to be tacked on. I was expecting as a grand climax, “to which they approach nearer than by any given difference, but never go beyond, nor in effect attain” full stop. This would make the ‘ultimate ratio’ something akin to an asymptote, a quantity or position at once unattained and unattainable. But this won’t do either because, after all, the particle does pass through such an ultimate value (‘limit’) since, were this not the case, it would not reach the place in question, ‘its place’. Bringing in infinity at the last moment (‘diminished in infinitum’ ) looks like a sign of desperation.
A little later, Newton is even more equivocal
“Therefore if in what follows, for the sake of being more easily understood, I should happen to mention quantities as least, or evanescent, or ultimate, you are not to suppose that quantities of any determinate magnitude are meant, but such as are conceived to be always diminished without end.”

But what meaning can one give to “quantities….diminished without end” ? None to my mind, except that we need such quantities to do Calculus, but this does not make such concepts any more reasonable or well founded. The issue, as I said before, has ceased to worry mathematicians because they have lost interest in physical reality, but it obviously did worry Newton and still worries generations of schoolboys and schoolgirls until they are cowed into acquiescence by their elders and betters. The fact of the matter is that to get to ‘its place’ (Newton’s phrase) a particle must have a velocity that is the ‘ultimate ratio’ between two quantities (distance and time) which are being ‘endlessly diminished’ and yet remain non-zero.
In Ultimate Event Theory there is no problem since there is always an ultimate ratio between the number of grid positions displaced in a certain direction relative to the number of ksana required to get there. When doing mathematics, we are not going to specify this ratio, supposing even we knew it : it is for the physicist and engineer to give a value to this ratio, if need be, to the level of precision needed in a particular case. But we know (or I do) that δf(t)/δt has a limiting value (Note 1)which we may call df(t)/dt if we so wish. Note, however, that the actual ‘ultimate ratio’ will almost always be more (or less) than the derivative since there will be non-zero terms that need to be taken into account. Also, the actual limiting value will vary according to the processes being studied since manifestly some event-chains are ‘faster’ than others (require less ksanas to reach a specified point). Nonetheless, the normal derivative will usually be ‘good enough’ for practical purposes, which is why Calculus is employed when dealing with processes that we know to be strictly finite such as population growth or radio-active decay. S.H.

Note 1 : Why must there always be a limiting value? Because δt can never be smaller than a single ksana — one of the basic assumptions of Ultimate Event Theory.

(The opening image is from a painting by Jane Maitland) S.H. 5/08/12

Categories Limits, Newton, Uncategorized

Motion

July 28, 2012 //

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

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

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

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

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

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

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

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

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

S.H. 22/7/12

Categories Buddhism, Hinayana, Infinitesimal Calculus, Newton, Uncategorized

Co-ordinate Systems

July 8, 2012 //

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, K₀does 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 K₀and, 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 K₀and, 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

Categories Buddhism, Descartes, Discontinuity, Einstein, Newton, Uncategorized

Laws of Motion in Ultimate Event Theory

July 2, 2012 //

Anyone who presents a radically new scientific theory must expect hostility, ridicule and stupefaction. Up to a point (up to a point) this is even healthy, since a society where new ways of viewing reality hoved on the horizon every two years or so would be bewildering in the extreme. What generally happens is that the would-be innovator is told that everything that is true in the new theory is already contained in the current theory, while everything that differs from the existing theory is almost certainly wrong. The new theory is thus either redundant or misguided or both.
And yet we need new theories, by which I do not mean extensions of the current paradigm, or patched up versions, but something that really does start with substantially different first principles. Viable new ways of viewing the world are not easy to come by, and inventing a symbolic system appropriate to the new view is even more difficult.

Now, it is quite legitimate to keep in full view features of the official theory that are solidly based, provided one rephrases them in terms of the competing theory. Ideally, one would like to see the assumptions of the new theory leading to something similar but, clearly, it is all too easy to fudge things up when one knows where one would like to end up. Such an attempt is, however, instructive since it focuses attention on what extra assumptions apart from the basic postulates are necessary if one wants to find oneself in a certain place. But if predictions of the new theory don’t differ from the existing one, there is little justification for it, although the new theory may still have a certain explanatory power, intuitive or otherwise, which the prevailing theory lacks.
Now, at first sight, Ultimate Event Theory, may appear to be nothing more than an eccentric and pretentious way of presenting the same stuff. Instead of talking of molecules and solid objects, Eventrics and Ultimate Event Theory speak of ‘event-clusters’, ‘event-chains’ and the like. But since the ‘laws’ governing these new entities must, so the argument goes, be the very same laws governing solid bodies and atoms, the whole enterprise seems pointless. Certainly, I am quite happy to do mechanics without continually reinterpreting ‘body’ as ‘relatively persistent event-cluster’ — I would be crazy to behave otherwise. However, as I examine the bases of modern science and re-interpret them in terms of the principles of Eventrics, I find that there are marked differences not only in the basic concepts but, occasionally, in what can be predicted. There are, for example, Newtonian concepts for which I cannot find any precise equivalent and the modern concept of Energy, not in fact employed by Newton, which has become the cornerstone of modern physics, is conspicuously absent (Note 1). There are also predictions that can be made on the basis of UET that completely conflict with experiment amd observation (Note 2) but at least such discrepancies focus my attention on this particular area as a problematic one.
I start by examining Newton’s Laws of Motion, perhaps the most significant three sentences ever to have been penned by anyone anywhere.
They are :
1. Every body continues in its state of rest or uniform straight-line motion unless compelled to change this state by external imposed forces.
2. Change of a body’s state of motion is proportional to the appled force and takes place in the direction of the straight line in which the force acts.
3. To every action there is an equal and oppositely directed action.

How does all this shape up in terms of Ultimate Event Theory?
It is first necessary to make clear what ‘motion’ means in the context of Ultimate Event Theory (UET). Roughly speaking motion is “being at different places at different times” (Bertrand Russell). Yes, but what is it that appears at the different places and what and where are these ‘places’? The answer in UET is : the ‘what‘ are bundles of ultimate events, or, in the simplest case, a single ultimate event, while the ‘places’ are three-dimensional grid-positions on the Locality, K₀ , where all ultimate events are motionless. Each constituent of physical reality is, thus, always ‘at rest’ and it is only meaningful to speak of ‘motion’ with respect to event-chains (sequences of ultimate events). But these event-chains do not themselves ‘move’ : the constituent events flash in and out of existence while remaining somehow bonded together (Note 3). It is all like a rhythmically flashing lamp that we carry around from room to room — except that there is no lamp, only a connected sequence of flashes. As Heraclitus put it, “No man ever steps into the same river twice” .
To clear the ground, we might thus take as the

Zeroth Law of Motion : There is no such thing as continuous motion.

We now introduce the idea of the successive appearance and disappearance of events which replaces the naïve concept of continuous motion.

First Law. The ‘natural tendency’ of every ultimate event is to appear once on the Locality at a single spot and never reoccur.

(Remark. When this does not happen, we have to suppose that something equivalent to Newton’s ‘Force’ is at work, i.e. something that is not itself composed of ultimate events but which can affect them, as for example displace them a position where they would be expected or simply enable them to re-occur (repeat more or less identically).

Second Law. When an event or event-cluster acquires ‘Dominance’ it is capable of influencing other ultimate events, but it must first of all acquire ‘Self-Dominance’, the power to repeat (nearly) identically.

From here on, the Laws are rephrasings of Newton though perhaps with an added twist:

Third Law. An ultimate event, or event-cluster, that has acquired self-dominance continues to repeat (nearly) identically in a straight line from instant to instant except when subject to the dominance of other event-chains.

(Remark: It is an open question whether an event or event-cluster that has acquired ‘Self-Dominance’, will carry on repeating indefinitely in this way, but for the moment we assume that it does.)

Fourth Law. The dominance of one event-chain over another is measured by the extent of the deviation from a straight line multiplied by the ‘event-momentum’ of the constituent events of the event-cluster.

(Remark. I am still searching for the exact equivalent of Newton’s excellent, and by no means obvious, concept of ‘momentum’ which gives us the ‘quantity’ of ‘matter-in-motion’ so to speak. Event-clusters obviously differ in their spread (number of grid-positions occupied), their density (closeness of the occupied places) and the manner of their reappearance at successive instants, but there are other considerations also, such as ‘intensity’ which need exploration.)

Fifth Law. In all interactions between event-clusters the dominance of one event-cluster over another is met by an equal and oppositely directed subsequent reverse dominance.

(Remark. Note that Newton’s Third Law (the Fifth in this list) is the only one of his laws that refers to events only (action/reaction) without mentioning bodies.)

Note 1. Newton did not use the term energy and even as late as the mid nineteenth-century physicists like Mayer and Helmholtz who did so much to develop the energy concept still talked of ‘Force’. J.J. Thomson (Lord Kelvin) seems to have been the first physicist to introduce the term into physics.

Note 2. For example, I find I am unable to explain why what we call light does not pass right through every possible obstacle as neutrinos almost always do — clearly this will require some new assumption.

Note 3. No event is ever exactly the same as any other, since, even if two ultimate events are alike in all other respects, they do not occupy the same position on the Locality.

SH 23/7/12

Categories Laws of Motion, Newton, Newton's Laws, Uncategorized

Is Space-Time Discontinuous?

January 27, 2012 //

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)

Categories Discontinuity, Newton, Scientific American

Stop-Start

January 25, 2012 //

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.

Categories Discontinuity, Euclid, Evolution, Newton