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Zeno's Paradox[1]

The "Arrow Paradox" of Zeno was treated by Newton in a practical fashion by assuming that there was an interval of time δt for which Zeno's problem had only theoretical interest. This is satisfactory for its original purpose - measuring ponderous motions of macroscopic bodies - and it can be carried surprisingly far down into microscopic motions. It is not a solution to Zeno's problem merely a statement that by a neat mathematical dodge (d/dt), it is not necessary to solve Zeno's problem for a wide range of dynamical problems. In particular it fails to say anything about what happens as δt0 if the process is continued beyond the degree called for by the particular dynamic problem. If δt could be identically equal to zero, then Zeno's problem would be solved, but although the mathematics avoids making δt=0 it gives no hint to indicate any particular range of δt Where the concept (d/dt) breaks down. This is because the laws of dynamics are formulated in terms of lumped parameters. The inertial attributes of a ponderous body are distributed, but the Mass of the body is a lumped parameter, and the use of the term Mass has already implied a Newtonian by-passing of Zeno's paradox. If the body is given a distributed inertia, and then the attempt is made to make δt0 before assuming that the parameter M may be lumped, it is found that the whole system must shrink with δt and tend to vanish with δt.

The effect may be seen to better advantage with an electrical example.

[An incomprehensible doodle was originally included here. B.E.N.]

Fig. 1. Short length of line, open ended, of delay δt.[2]

Develops its effective impedance at low frequencies, ie where 1/δt & f are discrepant.

Then note that any physical attempt to make δt0 makes the system shrink.

This result is not surprising when it is realised that Newtonian dynamics merely avoids Zeno, and gives a set of rules which apply where certain simplifying assumptions may be made.

These assumptions are in essence:

(1) That there is an identifiable entity.
(2) That it is in an identifiable place.
(3) That it is at an identifiable time.

These become, when extracted and idealised, M, L, and T, and can support a complete dynamic theory, indeed they can be extended to give an electro-dynamic theory.

This would be more than satisfactory, were it not that the theory is often asked to describe a complete Cosmos without adequate thought being given to the nature of its simplifying assumptions. If M, L, and T are to be attributed to all physical phenomena, then it is worth checking the necessary assumptions and finding whether they do in fact apply universally.

An examination[3] along these lines leads to the conclusion that the assumptions of dynamics are of limited validity, as expected, and that they fail completely when applied to phenomena which "travel with the velocity of light". This leads to the further conclusions:

a. That M, L, and T cannot be the universal descriptive form for a comprehensive scheme to describe objective phenomena.
b. That light cannot have a velocity and that the velocity of light is a complete misnomer.

These are negative conclusions, and open to criticism as such. If M, L, and T are rejected, what is there to put in its place? To which the answer is: Information, where the flow of information is elapsing time. This information, if there is enough of it, and if it is patterned or ordered in a way which is acceptable to the human observer, can be interpreted to form certain concepts. In their least ambiguous form these concepts are effectively time-invariant, and comprise a system of fixed objects or things. At the other extreme is unobservability. In between are the ambiguous regions where the variability is insufficient to destroy the power to recognise to a lesser accuracy.

Given enough information (δt must not be too small for this), it is possible to make a satisfactory intuitive compromise, as is done in language, example, the man walks; or the formalised compromise of Newtonian dynamics, example, the mass has a velocity; but these are indeed compromises. Man has an integrating time-constant for making these compromises, which may be between about 1/5 - 1/100 sec., depending on the sense organ concerned, and scientific instruments act as time-transformers of information which effectively extend these time-constants in both directions. But it remains a limited region of a larger domain, and strictly subject to the condition of the particular compromise. But above all, the compromise which allows the idealisation into M, L, and T is irrelevant to the problem of describing the raw information, as also are M, L, and T themselves. It is quite useless to ask what carries the information, or what it is made of or where and when it is, since these concepts are secondary and merely appear as part of the primary process.

If now Zeno's paradox is re-examined it is seen to be quite valid. An object, completely defined, cannot move. However if the information is not completely time-invariant it can be used to define a near-object which is indistinguishable for all practical purposes from one completely defined, together with a description of its motion. This is the familiar everyday world. The small qualification that everyday moving objects lack by an infinitesimal amount the complete qualifications of a fully-defined object is necessary in order to be prepared for the occasion when the motion is increased infinitely. If no infinitesimal term is included there is nothing in the formal structure of the theory to stop the infinite motion being a fixed mass moving with an infinite velocity. It has been left to experiment to indicate the invalidity of this; but there is still no hint in the dynamically derived theory as to a preferred adjustment, and in default it has been usual to consider a limited velocity and an infinite mass. Logically, this is just as unsatisfactory, but if it is recognised, following Zeno, that however small the velocity there is some loss of conformity to the nature of an object, it follows naturally that taken to a limit of infinite motion there is a likelihood of vanishing objectivity. This would form a logically satisfactory theoretical framework in which with zero motion the abstraction "object" is well defined, but with infinite motion the necessary time-invariance has vanished completely and there is no basis whatever for defining an object, and naturally following that, the concept velocity vanishes too. This may be restated in terms of what could happen in a narrower example of Information, a simple communication channel. This will be defined by its band-width, within which may appear a signal of regular pattern, in its simplest form a time-invariant sinusoidal oscillation. This oscillation can be described simply in conventional terms with no theoretical limits to accuracy. This reformulation is in essence a recoding to make manifest the narrow bandwidth and lack of information in its time-invariant structure. The act of recognition of its structure made possible by its lack of variability is analogous to the complex act of recognition made possible by the lack of motion of an object (which thereby becomes time-invariant and recognisable as an object). The act in one case is to say this is in the form Aejwt, and in the other to say this is in the form Dog or Cat or some other object where the act of recognition is extremely complex, and the communication channels are many and various.

Continuing the parallel, the simple sinusoid may be found to have a slight variation - almost imperceptible, but just enough to say over a period of time that A or w is changing with time. It is still sensible to say that there is a recognisable sinusoidal oscillation of which the periodicity say, or perhaps its amplitude or its phase is changing; but it is to an extent an arbitrary choice of method of description. If for example the phase changed, imperceptibly at first, but increasingly as time went on, it would be possible to reassess the frequency at a new value with zero phase change. In contradistinction, if there is a firm conviction that the signal must be a sinusoid and a change with time, then almost any waveform can be squeezed to fit the mould. In the limit when variability has reached the limit allowed by the communication channel it would still be possible to describe the received signal in terms of a time-invariant function subject to variations. The logical contradiction is easy to spot when considering a communications channel, because when the time-invariant function becomes variable up to the limit set by the channel, it is obvious that there is a complete lack of coherent structure, and the form of description in terms of a coherent structure becomes wholly unsuitable. Limits are set to the validity of a descriptive method by the rate of flow of the communication channel.

It is less obvious, but equally logical to expect the rate of flow of information in any circumstance to set a limit. As this limit is approached the idea of a "variable invariant" changes from a useful working compromise to a plainly unsupportable contradiction in terms.

The terms "object" and "motion" are just such a pair of contradictions, allowable where the variable content, the motion, is small compared with the rate of flow of the information source from which it is derived. At this point the parallel must be abandoned because in the larger region of general theories there is no larger-yet framework against which the flow can be measured. Perhaps all that can be said is that the flow of information cannot be assumed infinite, and if it is not there will be secondary evidence of this when increasing difficulty is found in making high-variability phenomena fit the low-variability forms of description. The particular example where this has occurred above all others is the one treated in this article - the conceptual apparatus of an object and its velocity. If an attempt is made to discover increased velocity as defined in ordinary terms (where movement as discussed above is infinitesimal), it is found that there is a limiting velocity "c" and an increase in mass. This can be taken either as evidence for a state of affairs along conventional lines but subject to the correction terms; or as evidence that the flow of information is limited and that the limits are being approached. If the latter explanation is preferred, then the so-called limiting velocity "c" is not a velocity at all but an indirect statement about the flow of information.

The logic has here been developed by treating[4] one part of a general problem of suitable descriptive forms. It is capable of being applied over the range of experimental phenomena. In its general form the choice seems to be between describing the world in terms of combinations of elemental objects - minute and subatomic perhaps, but still essentially differently coloured billiard balls whose combinations define information; or of hoping to reverse the order of derivation and start with information as basic, and expect to find that certain kinds of information can be codified and classified as "objects", but also finding that information of high variability is of low codability but still has a place in the descriptive scheme.

The latter system seems to be the more powerful and the more elegant, and as such worth consideration to replace the existing Cartesian/Newtonian framework.

[1] There are several paradoxes associated with Zeno. This one refers to the "Arrow Paradox", a good explanation of which may be found here.

[2] I do not understand what this diagram is intended to represent, therefore no attempt has been made to redraw it, and it has been reproduced from the original manuscript. [B.E.N.]

[3] The original manuscript has an asterisk and the word "insert" written at this point above the text. [B.E.N.]

[4] I interpreted this word as "treating", but in fact I could not read it. [B.E.N.]