On considering the examples just given, we are forcibly reminded of a phenomenon which occurs, just occasionally, in thousands of homes. It occurs whenever we see on our television screens a picture of the commentator sitting beside a television set which shows a picture of the commentator sitting beside a television set which shows a picture of the… etc. etc. At once we can see in this a physical model of all self-contradictions of the type, “All generalisations are unsound.” Moreover, it is a “living” model, unlike my diagram which is but a “dead” imitation, a very poor approximation.
How do we modify this living model to obtain a similar one for the related, but different, type exemplified by “This statement is a lie”? The answer is akin to the one concerning the two facing mirrors: in the studio the television camera moves towards the set beside the commentator. Then, while the size of our own screen remains unchanged, the series of screens shown in our picture enlarges until they all coincide. In that process the commentator, his chair, the floor and all parts of the picture external to the series of screens, disappear at the screens’ edges. When the series of screens moves into coincidence with our own screen, our picture becomes completely blank! This vividly illustrates, even more clearly than the touching facing-mirrors, the flat, blank contradiction of the two “internal” or mutually included sides of “This statement is a lie.”
As we have seen, the Aristotelian view of self-contradiction, as per Aristotle’s Principle, is that it does not exist. Here, in the mutual inclusion of opposite reflections of our facing-mirrors model, and again in the mutual inclusions of opposite representations of the studio scene as between TV-camera and studio set, we have living proof of the existence of the interpenetration, the integration and even the identity of opposites! The mutual inclusions – note very carefully – refer to the reflections not to the mirrors, to the representations of the studio scene and not to the camera and set. If it now be objected, since these mutual inclusions are of images only, and not of the actual mirrors or television gadgets, that this principle of mutual inclusion of opposites does not apply to reality at all, the answer is simple. Reality is not exhausted by actual mirrors, actual television apparatus, etc. etc. The images, the reflections, the representations are also part of reality; that is, they are actual reflections, images, representations. Indeed, the very objection – should it be made – involves a similar type of representation. In short, whatever we say, we can only argue the toss about reality in terms of representations of reality. This is the crunch which makes nonsense of all objections, arbitrary bans, barriers and taboos, however abstruse, which seek to exclude the positive utilisation of selfcontradiction as the useful principle it is.
If it is asked how the principle applies usefully in the models, e.g. as an “internal standard,” we can start by considering the situation when our facing-mirrors are far apart, say at opposite ends of a large hall – or similarly, when the TV-camera and set are at opposite ends of a large studio. At these initial positions the portion of the reproduced scene which shows self-reproduction is very small indeed. If, then, one of the, opposed mirrors or instruments is removed to the horizon, we could say that the portion was so indefinitely small as to rank, for all practical purposes, as nil. If we now imagine the “reproducer” on the horizon moved nearer and nearer to its opposite “reproducer, we can see that, from nil, the portion of the reproduced scene which shows self-reproduction grows and grows until, when the facing-mirrors are touching (or the series of TV-screens coincide) that portion becomes all of the scene. It is at this very point our mirror or TV-screen becomes a complete blank! We characterise this result by saying that “all self-reproduction = none,” or that “total mutual inclusion = none.” Now, as “self-reproduction,” “self-reflection” etc. simply refer to the mutual inclusion of opposed reproductions or reflections, and therefore model the principle of mutual inclusion of opposites, we can infer from the foregoing that the principle applies to itself without upsetting or destroying the system of which it is part. It is thus self-consistent. As we have seen, the same cannot be said of Aristotle’s Principle.
Stepping, now, from consideration of physical models to mathematical ones, we pause to think of the situation of our TV-model just before the series of TV-screens moved into coincidence. Let us imagine that the electron beam scanning the screen, instead of passing from side to side, traverses a spiral path which goes first around the perimeter of our screen delineating the image of the perimeter of the first or largest screen shown in our picture. Then, on its next journey round, the beam moves in just a fraction, delineating the edge of the second or next largest screen in our picture, and so on. Let us imagine, further, that we have arranged for the traveling light-spot to confine its traverses to the perimeters of the images; so that, as these perimeters all move into coincidence, the light-spot path, following the expanding perimeters, moves outwards with them and comes to rest on the very extreme perimeter of our own screen, with the light spot continuing to circle this path over and over again. We can think of these latter “endlessly-repeated” cycles of the light-spot at the extreme edge of our screen as actually delineating, one after another, the endless series of, perimeters which now coincide with the perimeter of our own screen. If we now take our imaginative picture, and erase all the non-essentials, we are left with a circular path of an endlessly moving point of light. This path we can represent on paper, simplified in form, as a circle. The circle, then, is my first mathematical model, considered as the path of a moving point.
The vast structure of disciplines we call “mathematics” (falsely, because strictly, mathematics is only a particular type or class of auxiliary languages) has been erected, according to the orthodox tradition, upon the basis of the postulates of logic. Which logic? From our viewpoint, any logic which contains Aristotle’s Principle unlimited by another, explicitly stated and more general, is Aristotelian. In view of the contents of the foregoing pages, we may now begin to suspect that the mathematical edifice contains, unrecognised and unstated, and therefore implicit, the very principle towards a statement of which we are now rapidly moving. To define this principle let us first approach it from an Aristotelian standpoint.
Let us first consider the so-called “number line,” the axis of real numbers, sometimes called “the algebraic scale,” realised in the horizontal X axis (below) imagined as extended “indefinitely,” i.e. infinitely, in both directions. “Opposites” are defined on this line as pairs of points, representing numbers, which are equidistant from an arbitrary point, 0, the origin, measured in arbitrary units but in opposite directions. All numbers to the right of 0 are called “positive,” those to the left, “negative.” The coincidence, or mutual inclusion, of opposites always equates to 0. This “coincidence” is imagined either (a) as the, “to-and-fro” journey of a moving point from 0 and back again, or (b) as the telescoping together of two segments of line of equal length, measured oppositely from 0. Both cases are represented algebraically by adding the opposites, x and -x. However, the orthodox interpratation of any number, +x, on the axis is subject to an ambiguity, one which I have just indicated in (a) and (b). The germ of our new principle lies hidden here, and we shall, now state the ambiguity.
According to the Aristotelian tradition any number plus/minus X lying “on the axis” X, may be interpreted in two different ways: (1) as representing a discrete point, namely the xth point counting positively or negatively from 0 in arbitrary units; and (2) as representing a continuous distance or segment of line intervening between the discrete points 0 and x. I will call (1) “the point interpretation,” and (2) “the line interpretation.” Normally, no confusion arises between these two interpretations because the laws of ordinary arithmetic, algebra, etc. apply in the same way to both.
Periodicity, or regular recurrence, is found almost universally in nature, and is a very familiar notion in science and mathematics. The conventional model for periodic functions is a circle drawn in a Cartesian frame with centre at origin, 0, and an orbiting radius-line, yielding continuously varying angle-ratios, in terms of which, when plotted on a time-graph, the periodicity is defined. The mathematics of periodicity is highly developed and has been so since Napoleonic times. But we have seen that there are certain aspects of recurrence models which lie entirely outside the scope of Aristotle’s Principle, and therefore outside of all Aristotelian systems. It is these non-Aristotelian aspects of recurrence that I propose to define in terms of our new model – which, although nearly complete, still needs to be taken one stage further in its development. As with the conventional model for periodicity, our model is also a circle, but seen as a locus of an endlessly-circulating point. (Note carefully the two models (1) circle with orbiting point; (2) circle with orbiting radius-line.) Of the three features we wish to bring together in our model, “recurrence” and “infinity” are well represented; we need now to bring out the third and main feature of self-contradiction the “positive coincidence” of opposites. By “positive coincidence,” I mean to stress the difference of our coincidence, the non-Aristotelian which seeks to preserve the paradox – from the Aristotelian coincidence which equates to zero, not to preserve, but to get rid of, to expel, self-contradiction.
We now develop our model in terms of the ambiguity I noted above respecting the line and point interpretations of any number, x, on the X axis. Taking any segment of the X axis, with 0 as the middle point, we bend it round forming a circle to coincide exactly with our model circle. We have merely imposed an arbitrary segment – which can be any amount we choose of the algebraic scale upon our circular locus. Subject to extensions and variations, our model is virtually complete (Fig. III):
Let us assume we control the moving point, x, starting and stopping it at will. Starting with x at 0, the point moves clockwise once round the circle and “recurs” at 0, where it stops. If we adopt the “point” interpretation of x (i.e. x represents the name of a discrete point) then the discrete point, x, and the discrete point, 0, are identical. If, however, we adopt the “line” interpretation, then x (representing the intervening line of its path round the circle) and 0, being distances, are not identical, they are opposites in a new sense. Let x move around once again (Fig. IV). As the distance, x, measured clockwise in arbitrary units from 0, increases, so x “grows away from” the distance 0, and increasingly approximates the circumference, c, of our locus. When x = c (in the line interpretation) then x = 0 (in the point interpretation). This situation mimics exactly those situations we met earlier, where “actuality” and “signification” (or “extension” and “intension,” “thing” and “concept,” “object” and “subject,” “reality” and “representation” etc. etc.) were barriers to the equation of entities, simply because Aristotle’s Principle – and its natural offspring, the Theory of Logical Types by asserting a fixed distinction of “incompatibility of type” between them, arbitrarily declared their equation to be impossible, meaningless or non-existent! No such barrier exists, however, for the line and point interpretations of x on the conventional axis. Nor shall we introduce or invent one now for our model, where x = c (line interpretation) and x = 0 (point interpretation).
In short, c = 0, where 0 is origin, is conditional upon the expression being the internal standard of the system of points and/or lines of our model locus. We ensure this condition by calling c “the modulus” of the system and indicate both this condition and the distinction of model from the conventional model for periodicity by the expression “a modular circle” (or “modular locus,” since the circle, as circle, is not essential).
We now see that our model is clearly related to various other mathematical subjects: arithmetic congruences, modular algebraic fields, infinite series, group theory, etc. Accordingly, we shall adopt or adapt such of their symbolic conventions as may be convenient for our purposes. We therefore write c = 0 to the modulus c, or c = 0 modulo c, or c = 0 (mod c). We do not us a causal congruence or equivalence symbol here since from our viewpoint so far it is logically redundant. The expressions “mod c,” “mod m,” etc. will therefore stand as equivalents, respectively, of the internal standards c = 0, m = 0, etc. since all moduli may be equated to zero. Turning back now to our TV-model, if there are two elements, x, y, in the self-repeating picture which represent the same thing, such as two representations of the commentator, or any other thing (of the indefinite sequence of, representations) then it will follow that x = y (mod c), where c is the finite positive set of elements which are indefinitely repeated. This is like saying “One o’clock p.m. = thirteen hours’-” or more shortly, 1 = 13 (mod 12). Indeed, our system of timekeeping is a modular system, and two “similar” modular systems, – e.g. the 12-hour clock and the 24-hour clock, can be put into a special kind of correspondence involving an appropriate function – which generates an infinite sequence of modular systems, thus producing a modular system of modular systems. The “internal standards” of the two timekeeping systems, the British and the Continental, are of course, respectively, 12 = 0 and 24 = 0.
continue reading The Paradox Principle by Harold Walsby (1967):
Dedication | Aristotle’s Principle | The Role of Logic | Do Self-Contradictions Exist? | Three Types of Contradictions | Meaningful Self-Contradictions | Infinity and Self-Contradictions | Models for Self-Contradiction | The Paradox Principle and Applications | Appendix