Time and place

“Absolute, true and mathematical time, of itself, and from its own nature, flows equably without relation to anything external”  (Newton, Principia Scholium to Definition VIII)

Newton does not say whether there are any absolute units or measures to his absolute time, i.e. whether any exact meaning is to be given to the term ‘moment’. Rather he implies that there are no such units since time ‘flows’, i.e. is continuous. He does, however, contrast ‘absolute time’ with “relative, apparent, and common time” which is “some sensible and external (whether accurate or unequable) measure of duration by the means of motion…. such as an hour, a day, a month, a year”.

He also believes that each object has what he calls a ‘place’ which fixes it in absolute space and absolute time. “All things are placed in time as to order of succession; and in space as to order of situation. It is from their essence or nature that they are places; and that the primary places of things should be movable, is absurd. These are therefore the absolute places; and translations out of these places are the only absolute motions” (Ib.)

This view is to be contrasted with Leibnitz’s which sees the position and motion of bodies as essentially relative: a body’s ‘place’ merely indicates where it is in relation to other bodies at a given moment. This ‘relational’ approach has been adopted by several modern physicists beginning with Mach. As Lee Smolin puts it, “Space is nothing apart from the things that exist; it is only an aspect of the relationships that hold between things” (Smolin, Three Roads to Quantum Gravity p. 18). Much the same goes for time: “Time also has not absolute meaning…..Time is described only in terms of change in the network of relationships that describes space” (Smolin, Ib.)

Motion

What about ‘change of place with respect to time’ or motion? To determine a body’s motion we have to establish what a body’s ‘place’ was before motion began and the same body’s ‘place’ when motion has ceased. Newton concedes that the ‘parts of space’ cannot be seen and so we have to assume that there is a body which is immoveable and measure everything with respect to it. “From the positions and distances of things from any body considered as immovable, we define all places” (p. 8). But is there such a thing as an immovable body ? Newton is undecided about this though he would like to answer in the affirmative. He writes, “It may be that there is no body really at rest” but a few lines further on he adds that “in the remote regions of the fixed stars, or perhaps far beyond them, there may be some body absolutely at rest”. However, since such bodies are so far away, they are of little use as reference points practically speaking while “absolute rest cannot be determined from the position of bodies in our region”.

Newton concludes that we have to make do with ‘relative places’ though he is clearly bothered by this since it means that motion will also have to be treated as relative. This leads straight on to the Galilean ‘law’ that rest and constant straight line motion cannot be distinguished. Newton’s position is, however,  to be contrasted with the modern interpretation of Galileo’s claim. For Newton is not saying that ‘rest’ and ‘constant straight line motion’ are ‘equivalent’. Instinctively, he feels that there must be such a thing as ‘absolute place’ and ‘absolute rest’ and is chagrined that he cannot provide a reliable test to distinguish true rest from motion.

When discussing circular motion Newton invokes the backdrop of the ‘fixed stars’ which “ever remain unmoved and do thereby constitute immovable space”. Thus, it is, according to Newton, possible to distinguish between relative and absolute circular motion because in the latter case there is a force at work which makes a body “recede from the axis of circular motion”.  He gives the celebrated example of a bucket of water suspended by a chord which is twisted and then released so that the water climbs up the sides of the vessel. Today we would use the example of a merry-go-round which is, according to Newton’s test, a genuine case of absolute circular motion since we feel a definite force pushing us towards the outer edge.

Continuous and Discontinuous

When analysing the perceived motion of bodies, Newton treats motion as at once continuous and discontinuous: a projectile or a planet is never at rest as it pursues its path (unless interrupted by something in the way). But if a body is accelerating it does not have a constant velocity at that instant, and if it is in motion at any particular instant it cannot have a precise position. At the end of the day, provided we make the time interval small enough, it would appear that everything is at rest. However, the use of ‘infinitesimals’ allows one to decrease the time interval down to ‘almost nothing’ so that we can speak of  a body’s  ‘instantaneous velocity’ ― despite this being a contradiction in terms. In effect, the Infinitesimal Calculus which Newton co-invented, allows him to have his cake and eat it too as Bishop Berkeley pointed out to Newton and his supporters. This is probably the main reason why Newton avoids calculus methods as such in the Principia employing instead cumbersome geometrical constructions which in effect treat motion as an infinite succession of stills. Newton struggles to defend the logic behind his treatment in the beginning of Book I which treats the Motion of Bodies. But he cannot decide whether the ‘ultimate ratio’ of distance versus time ― what we call dy/dt ― is ever actually attained, a rather important point (Note 1).

More precisely, Bishop Berkeley made it clear that Newton was contradicting himself by first assuming that x has an increment and then, “in order to reach the result, allows the increment to be zero, i.e. assumes that there was no increment.” Modern mathematics gets round this problem by defining the ‘limit’ to an ‘infinite series’ in such a way that it is not required that this limiting value is actually attained ― indeed in practically all cases of interest it cannot be. The price we have to pay for this rationalization of the Calculus is loss of contact with physical reality. Even if Newton  had been capable of formulating the concept of a ‘limit’ in the precise modern sense I doubt if he would have employed it. Why not? Because Newton, like Leibnitz, and like practically every other ‘natural philosopher’ of the time, was a realist. In Newton’s time mathematics had not yet separated into ‘pure’ and ‘applied’ and the question as to whether infinitesimals ‘existed’ or not was the same sort of question as asking whether atoms existed. Pre-modern mathematics required infinitesimals to get tangible results which could be checked and usually turned out to be correct. But Newton was pragmatist enough to realize that, taken literally, Calculus methods made little sense.

Boyer naturally champions the modern view. “His [Berkeley’s] argument is of course absolutely valid as showing that instantaneous velocity has no physical reality, but this is no reason why, if properly defined or taken as an undefined notion, it should not be admitted as a mathematical abstraction” (Boyer, The History of the Calculus p. 227).
But  why should one allow mathematics to wag the tail of physics to this extent? The real world cannot be handwaved into irrelevance just because it hampers the style of pure mathematicians. It is in fact deeply shocking that contemporary physics, on the face of it  the most ’down to earth’ of the sciences, has been transformed into a piece of recondite pure mathematics. For mathematics, as a logico-deductive system, does not and cannot guarantee the existence of anything. Yet, for all that,  most of us would like to know what is ‘really real’ and what is imagination: science is not the same thing as science fiction.

Boyer claims that mathematics only deals in ‘relations’ not actualities which is all right up to a point ― but one has to ask, ‘relations between what sort of things?’ Since mathematics is a symbolic system, either its symbols ‘represent’  or stand in for realities of some  sort or they do not, in which case they are simply decorative in the same sort of way in which embroidery patterns are decorative. It is quite conceivable that a different intelligent species might use embroidery or textile design as a way of communicating truths about the cosmos but our species has not gone down this route and has restricted its scientific pattern-making to geometrical drawings and algebra.

Newton’s Approach and UET

How does Newton’s idea of ‘absolute time and place’ play out in terms of the basic assumptions of Ultimate Event Theory? In UET ‘ultimate events’ replace Newton’s ‘bodies’ and ‘time’ is the rate at which ultimate events succeed each other. Newton’s assumption of ‘absolute time’ is tantamount to suggesting that there is a fixed, universal and absolute rate at which certain events succeed each other and which is entirely regular. Most sequences of events, of course, probably only approximate to this measure in principle it is always there in the background. There is, then, a sort of metronome according to whose ticks all other sequences can be measured, and towards which all actual rates of actual events tend.

Is such an ‘absolute rate’ conceivable? (This is a different question to whether it actually exists.) If we assume, as I do, that all ultimate events have the ‘same extent’, i.e. occupy spatially  equivalent ‘places’ on the Locality, and last for exactly the same ‘length of time’, we obtain a basic regular rate if (and only if) the intervals between successive ultimate events are equal. And the simplest case would be when the interval between successive events is a minimum, i.e. just enough to keep the events separate, like a cell membrane that is one molecule thick.

As far as I am concerned, I believe that Newton was right in thinking that every object must have a ‘place’: this strikes me as even more necessary when dealing with events, which are by definition transitory, than with semi-permanent objects. I simply cannot conceive of ‘space’, or whatever is out there, as being simply composed of ‘relations’. But the UET conception is not the same as Newton’s concept of absolute space and time since the latter provide a fixed framework whether or not anything exists  inside this framework or not. One should conceive of an ‘ultimate’ rate for event chains as a constraint or asymptotic limit to which actual event sequences may tend, rather than something existing independently of all actual events. What it means, however, is that the time variable δt cannot be arbitrarily diminished, so there is always a final ratio of distance versus time for consecutive events. Moreover, one could state as a postulate that a ‘rest’ event sequence, the equivalent of a stationary object, proceeds at the minimal rate, i.e. one ksana at a time (sic) with the distance between any two consecutive events in an event-chain being  a minimum ― or, alternatively, a maximum.

SH  18/03/15

Note 1 Newton writes, “There is a limit which the velocity at the end of the motion may attain, but not exceed. This is the ultimate velocity”. But, a little further on, he changes tack and writes, “those ultimate ratios with which quantities vanish are not truly the ratios of 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”.

Note 2 And again, “Berkeley was unable to appreciate that mathematics was not concerned with a world of “real” sense impressions. In much the same manner today some philosophers criticize the mathematical conceptions of infinity and the continuum, failing to realize that since mathematics deals with relations rather than with physical existence, its criterion of truth is inner consistency rather than plausibility in the light of sense perception or intuition”.

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