Speed is not one of the seven basic SI units (the metre, kilogram, second, ampere, kelvin, candela & mole). It is a ‘derived unit’ defined in terms of the basic standard of length, or distance, the metre, and the basic standard of time, the second. Being entirely secondary, dependent on other entities for its very existence, one might very reasonably concluide that it is fictitious — though a useful concept for all that. Curiously, very few Western thinkers have taken this view though this is perhaps fortunate since otherwise we would not have the science of dynamics. On the other hand, amongst Indian Buddhist thinkers, disbelief in the reality of motion (and by implication speed in the normal sense of the word) is the norm rather than the rare exception. Vasubandhu writes, *“There is no motion….Things do not move, they have no time to do so, they disappear as soon as they appear”* (quoted Stcherbarsky).

Speed is “the rate of change of distance against time”, metres/sec or, more precisely, so many metres *per* second (there is no such thing as a metre-second). The concept of motion does not specifically involve time, but if a body is not in motion, it has no speed, or, if you like, has zero speed. ‘Intuitively’, as the mathematicians write (with a certain sneer since intuition is not reliable), we feel that there is an ‘absolute’ difference between being in a state of rest and being in a state of motion. Already in ancient times, Zeno had pointed out in his Paradox of the Chariot, much less well know than his Achilles and the Tortoise Paradox, that the motion of a chariot in a chariot race was different, depending on whether we were judging it with reference to the spectators, or to other competitors. He concluded that a body could at one and the same time have “all kinds of completely different speeds” — a remarkably perceptive observation.

Subsequently, Galileo came to the conclusion that the really important distinction to be made was not between ‘rest’ and ‘motion’ as such but between ‘constant’ and ‘accelerated’ motion. To some extent Galileo had been anticipated by the important and neglected medieval thinker, Oresme, (**Note 1) **but Galileo seems to have been the first to introduce the notion of an ‘inertial frame’, as it was eventually called, one of the the most fruitful ideas in the whole of physics. In one of his dialogues, Galileo observed that, if one were in a cabin inside a ship on a calm sea and had no window, it would not be possible to tell by conducting physical experiments whether the ship were in a state of constant motion or at rest. As Einstein eventually put it, *The laws of physics are the same in all inertial frames. *

Newton viewed both Space and Time as absolutes in the sense that they were not dependent on subjective human viewpoints or states of motion, though he conceded that it was in practice often impossible to locate an object precisely **(Note 2**)**.** This viewpoint is not unrelated to Newton’s profound, if unorthodox, religious belief : God would know the ‘real’ position of an object even if we couldn’t always precisely define it. What of motion? Since Space and Time were absolutes, Newton found himself propelled to believe in absolute motion as well *: “IV. Absolute motion, is the translation of a body from one absolute place into another*” **(Principles, I, 6** **ff**). However, he quickly realised that if we want to determine a body’s ‘absolute motion’, as opposed to its apparent motion, we need some repeating event-sequence that we know to be absolutely regular, and no such sequence appeared to be available. Clocks needed resetting and even the Sun’s course and that of the planets changed slightly. *“It may be, that there is no such thing as an equable motion, whereby time may be accurately measured. (…) **It may be that there is no body at rest, to which the places and motions of others may be referred.”
*But the problem with the ‘notion of motion’ runs much deeper still. Not only is it in practice impossible to determine a body’s ‘real’ motion, as opposed to its motion in relation to some other body which is itself in some sort of motion, but it is impossible to attribute at one and the same time an exact position to a body and motion. As Sextus Empiricus realized long ago

*. “If a thing moves, it moves either in the place where it is or in that where it is not; but it moves neither in the place where it is (for it remains therein) nor in that which it is not (for it does not exist therein); therefore nothing moves*.”

So one has, seemingly, to give up one of two things : either exact position or motion. Mathematically, of course, one has one’s cake and eats it : one gets round the difficulty by the (contradictory) concept of ‘instantaneous velocity’ and other artifices of Calculus, also by the unproven assumption that “space and time are infinitely divisible” supposing even the last sentence is meaningful. Calculus works (more or less) but a mathematical wand cannot bring an impossibility into existence : an object simply cannot be at once in a particular spot

*and*in motion.

So what has Ultimate Event Theory to say on the matter ? In a nutshell : If by motion we are to understand

*continuous*motion (and this is the usual unstated assumption), then there is no motion. But if by motion we simply understand “being in different spots at different times” (as Bertrand Russell put it), then, certainly there is ‘motion’ — though what we like to think is motion is in reality a succession of stills like the cinema

**(Note 3**)

**.**

Ultimate Event Theory assumes ‘absolute’ position :

*“Every ultimate event occupies one, and only one, spot on the Locality”*(Axiom of Locality). It also assumes Succession though, on re-reading my Axioms, I find that this is not specifically stated — this will be remedied in due course. An ultimate event, like a Buddhist

*dharma*, is fixed, static and evanescent.

*“Momentary things cannot displace themselves because they disappear at that very place at which they have appeared*” (Kamalasila). And since everything is, by assumption, made up of distinct ultimate events, no-thing moves (though there can be change of circumstance, i.e. event environment).

A so-called ‘object’ is, in Ultimate Event Theory, a dense cluster of distinct ultimate events which repeat more or less identically. A spot on the Locality is, if you like, two dimensional where one dimension, the spatial, is itself subdivided into three. An ultimate event occupies this spot, the entire spatial dimension with the event itself disappears to be replaced by an identical one and the event repeats. We will assume, for simplicity’s sake, that there is only one ultimate event and that it repeats at every ‘temporal layer’, i.e. chronon after chronon. A connected repeating chain of events can be conceived as a sequence of dots or pixels that are so close that they appear to form a continuous line. This ‘line’ is either straight or not straight. In the first case we have the equivalent of the ‘motion’ of a body which is “either at rest or in constant straight line motion”, in the second we have the case of ‘accelerated’, or simply irregular.

Is there any distinction to be made between a strictly vertical straight line of dots (representing events) and a slanting line? Seemingly not : what is for one ‘observer’ a vertical line is slanting for someone else. Normally, we consider ourselves to be at rest even if we are (in the normal sense of the word) in motion : looking out of the window of a a train we perceive the countryside flashing by whereas in reality it is more we who are flashing by the static countryside. The important distinction is between a regularly spaced sequence of dots, whether ‘straight’ or slanting and one which is not, which curves round or changes its direction from chronon to chronon.

The ‘equivalence’ of different ‘inertial’ sequences is a somewhat surprising fact of experience, it is not at all what one would expect. What it means in terms of my world-picture is that the successive layers of the Locality can not only ‘slide past each other’, as it were, but that they have no discernable fixed orientation one to another. This is difficult to believe but, like Newton, we can conclude that if there is a true ‘absolute’ orientation, we do not have the means to identify it so it can be discounted.

However, there are some things that we can deduce right away. For a start, there must be some limit on the spatial separation between successive ultimate events if they are to remain connected. This limit is a pure number : it is given by the maximum possible difference between two spatial spots, i.e. spots belonging to the same ‘layer’, compatible with the bonding of events that constitute an ‘object’. (This is Einstein’s brilliant insight that there is —or ought by rights to be — a physical limit to thenoperation of causality, namely the speed of light in a vacuum.) It is irrelevant which of two straight lines we consider ‘vertical’ and which ‘slanting’ : all that matters is the number of spaces in between. Since we are speaking of a single chronon, the limiting event ratio is

**No. spatial spots/1 chronon**, i.e a pure number. Should anything exceed this limit, the subsequent event-chains will immediately ‘dissociate’ to use a term from chemistry.

Of course, we are not able to determine this limit from one chronon to the next — though possibly we will be able to one day. But we can approximate to this limit by making a simple ratio of

**spations/chronons**where a spation is simply the lateral distance between two successive events which are recongizably part of the same event-chain. The point is that, once again, this limit is a pure number, and a rational number at that,

**n/m**where

**n, m**are integers.

So far, it has been assumed that the events of this event-chain do not miss out any layers. But, supposing they do, there must likewise be a limit to the number of layers, i.e. chronons, that they are allowed to miss out without dissociation. And for any sort of event-chain, the ‘speed’ is given by a straight ratio. A rapidly moving object (read dense event-cluster) will cover more spaces laterally than a comparable slower moving object. And so it goes on. The limitations of the vocabulary of this site do not enable me to give suitable pictures but this will come later.

What is the advantage of this schema? Principally this : it enables one to believe in the succession and change without falling into contradiction or evoking fantastic entities such as instantaneous velocities or infinitesiamlly small or large intervals. A whole post will be devoted to the Pardoxes of Zeno, but it can be said now that this schema resolves all of them. A closing door does not have to pass through an ‘infinite’ number of positions before it is shut but only a (large) finite number of spots on the Locality. Achilles is able to overtake the tortoise because ‘time’ is constituted by successive layers and there are not an infinite number of them between any two. (It may well be that Achilles gets in front of the tortoise without ever being actually exactly abreast of it.) And finally, the arrow reaches its goal because it occupies a finite number of positions between the bowstring and the target. The arrow, a dense cluster of ultimate events, is always at rest (is where it is) but it occupies spots on successive temporal layers.

And the price of this coherent theory ? There is always a price. and here this would seem to be that the door, the arrow, the tortoise and even Achilles are not

*continuously*

*existing*but ‘exist’ at successive moments (chronons) only

*— “Being is shot through with nothingness*” (Heidegger).

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

**(2 )** “1.

*Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without regard to anything external…..*

*2. Absolute space, in its own nature, without regard to anything external, remains always similar and immovable…..”*

Newton,

*Principles I. 6*

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

“It is to be observed that, in consequence of the denial of the infinitesimal and in consequence of the allied purely technical view of the derivative of a function, we must entirely reject the notion of a *state *of motion. Motion consists *merely* in the occupation of different places at different times, subject to continuity as explained in Part V. There is no transition from place to place, no consecutive moment or consecutive position, no such thing as velocity except in the sense of a real number which is the limit of a certain set of quotients. The rejection of velocity and acceleration as physical facts (i.e. as properties belonging *at each instant* to a moving point, and not merely real numbers expressing limits of certain ratios) involves, as we shall see, some difficulties in the statement of the laws of motion; but the reform introduced by Weierstrass in the infinite calculus has rendered this rejection imperative.”

Russell, *Principles of Mathematics, *i, p. 473*
*What is remarkable (and ridiculous) is that Russell rejects velocity and acceleration “as physical facts”, but apparently believes in both as “numbers expressing limits of certain ratios”. Motion is reduced to something strictly mathematical, but for all that real ! God knows what Bishop Berkeley, who flummoxed Newton with his shrewd criticisms of the latter’s early versions of the Infinitesimal Calculus, would have made of this. It is true that the postulates of Ultimate Event Theory also require some “re-statement of the Laws of Motion” but at least I believe that something is going on : my ultimate events are perfectly real, not mathematical constructions.

*S.H.*