So far, we have only-considered cases where the modulus is finite. In these systems, infinity appears as implied (therefore potential only) by the indefinite repetition of the finite modular system.
That is it is implied by the self-representation or self-reproduction of the system. The finite – modular system itself consists of the finite set of elements to which the internal standard (or, what amounts to the same, the modulus) sets the limits, and to which therefore it applies. I must now draw attention to the very important sense in which any individual what ever is symbolic of itself – or self-representative – and this again generates a potential infinity, since a symbol or representative is also an individual. Thus anything, including a symbol, is a symbol of itself, i.e. is self-symbolic, generating an infinite sequence of symbols all coincident and identical with the original. Hence the Law of Identity: A = A, and therefore A = A = A… etc. ad infinitum. Russell says somewhere that the relation of identity is reflexive without limit. The Law of Identity is the basis for the Laws of Tautology I mentioned earlier which, strictly speaking, are simply special ways (intensional and extensional) of expressing the Law of Identity and stating formally and abstractly what I have stated here.
This potential infinity implied by self-representation is clearly demonstrated in our physical models (mirrors and TV-system) where the whole of a set of elements is reproduced – apparently in endless succession – in a part of that whole. Naturally, the infinite succession does not actually appear, but it is clearly implied by what does appear. The part where the self-reproductions take place is the negative part, or element, of the set. It can be thought of, firstly, as an “empty” space, or a kind of womb, within the whole set of positive elements, filled with or pregnant with, the reproductions. Indeed, if we look at biological self reproduction sufficiently abstractly, we can see that it conforms to the essential structures we are describing. In biological terms, the potentially infinite succession of whole sets of elements is, of course, the implied infinity of succeeding generations of whole individuals or pair-sets (male and female) of opposited individuals, born one set out of others.
Holes, or spaces, or absences which can be regarded as functional parts of something – such as the bung-hole of a beer-barrel are negative elements in otherwise positive systems. However, looked at in such a context, these negative parts are not just merely negative. They are also definite and positive, as -x is also +(-x). The beer-barrel is not complete without its bung-hole, which plays an essential, active and positive part in the total functioning of the barrel. Now, we saw in our imaginary “horizon-to-facing-face” experiment that the negative part, in which the whole is reproduced in our physical models, can be varied from virtually “none” to virtually “all.” The latter case, we saw, resulted in a blank screen (or mirror). In the experiment, however, the extreme limits were only approximated. The limits are actually reached only in the case where the enlarging negative part becomes the “empty” negative whole, completely coincident with and containing the whole positive set of elements, which contains itself – and its negative “space” – over and over again ad infinitum. That is to say, the extreme limit is realised only in the complete “identity-of-a-thing-with-itself” as stated in the infinite tautology of the Law of Identity: A = A = A… This is the complete selfreproduction which we saw equates with zero to form our internal standard of self-consistency, namely A = 0. Thus every individual, in so far as it is an identity, a one, can be represented as a modular system: 1 = 1 = 1… or, what is the same, 1 = 0 (mod 1). It is “the identity modulus.” Let us see how we can “obtain this result as part of generalising our model, the modular circle.
The modulus may be e.g. the number 12, as in the case of the 12-hour clock where 12 midnight is also origin, the 0th hour, and 2 = 14, 3 = 15, 2 . 3 = 18, -5 = 7, 2(15 – 3) = 0, all modulo 12, etc. etc. Let us vary the modulus. What about 2 = 0 (mod 2)? We have already met this one in the form of the repeating electric switch: 1 + 1 = 0 (mod 2). It turns out to have the same structure as the modular form of Boolean algebra. It also exhibits the abstract structure of such paradoxes as “This statement is a lie,” as we have seen: 1 = -1 (mod 2). It represents, too, the abstract form of the two touching facing-mirrors (which modelled the paradox). How “low” may we go? With integers, of course, the lower limit is 1 = 0 (mod 1) which is, as we have just seen above, the modular form of the Law of Identity. Can we have fractional moduli? Yes. Instead of 12 for the twelve-hour clock, we could call it half-a-day: 1/2 = 0 (mod 1/2) in which case the hours must be written accordingly, e.g. one hour is l/12 of 1/2 = 1/24 day, so that 1/24 = 13/24 (mod 1/2). What is the lowest limit? The answer: 1/infinity = 0 (mod 1/infinity), which defines e. g. a geometrical point our modular circle has shrunk to zero! This is “the zero modulus.”
The uppermost limit? I call this “the infinity modulus.” This leads us straight to the Paradox Principle itself: infinity = 0 (mod infinity). In both of the limiting cases the modular suffix is logically redundant. Thus we write 1/infinity = 0 for the lower limit and 00 = 0 for the upper limit. What has now happened to our modular circle? Just as the lower limiting case defines (on the model) a general point, so the upper limiting case defines a general line (a curve with zero curvature) i.e. a straight line (Fig. VI):
What are the consequences of generalising the modulus to the infinite case? They are many. The first is that infinity = 0 not only represents the infinite case but, at the same time, it includes all the other cases as well! This seeming impossibility is because we have now broken right through the Aristotelian barrier which prevented us from passing inferentially from extensional to intensional forms of the same thing, or the same sort of thing, and vice versa. It prevented us from thinking logically of something like the universe – i.e. as actually (extensionally) containing or including all elements but which at the same time, since it is one of those elements, mentally (intensionally) distinguishable from all the others, necessarily excludes them! Thus we have to think of the universe as having mutually contradictory properties: of including and excluding everything at the same time. As such, the universe was never a proper Arisotelian entity, that is, logically acceptable! The problem was solved in Aristotelian terms, as we saw, by arbitrarily and permanently isolating extension from intension. And the easiest way of doing just that, a way followed by most authors of modern textbooks, is not to think about the problem at all or to ignore one of the duality entirely, usually “intension.” We have now, with the Paradox Principle, solved the problem anew – positively this time – and linked the two together again, so that we may pass logically, i.e. according to rule, from one to the other. The great dichotomy has been bridged!
The outcome is a new universe, the conception of a modular universe which includes the old or non-modular as a special case, and a new infinite to represent it: an infinite modulus, or modular infinite in perpetual oscillation from extension to intension and back again. Whereas the lower limit, 1/infinity = 0, of our generalisation involved the familiar “idempotent” or extensional form of the infinite, i.e. infinity squared = infinity, the upper limit yields the new modular (extensional/intensional) infinite: infinity = 0 implies, by the usual rules, but ignoring the barrier to division by zero, infinity/0 = 1, and therefore infinity squared = 1 and generally, infinity squared times n = 1, infinity squared times n plus one = 1. (In Whitehead and Russell’s Principia, p. 62, a similar oscillation appears but between “false” and “truth” values instead of intension and extension; the infinite regress involved is not mentioned explicitly, though the oscillation which has the effect of eliminating intension is said to be implied by the Liar Paradox, and is identifiable, i.e. isomorphic, with the oscillation I have noted, namely, 1 = -1, or 1 + 1 = 0, modulo 2.)
The oscillatory values for the modular infinite become interpretable in terms of complex “relations of relations” in extension and intension when we begin to construct a logic based on the new principle. Such a logic is identical with a new general algebra (an algebra of extensive or concrete generality, as distinct from intensive or abstract generality) which I have called “dialectic or general algebra.” It is obtained simply by adding the Paradox Principle to the postulates of common algebra and developing the system accordingly. The first result is to convert Aristotelian opposites (positives and negatives) into complements:
If infinity = 0 (Paradox Principle),
and x + (-x) = 0 (Aristotle’s Principle),
then x + (-x) = infinity (Law of Complements).
This result is the familiar logical principle which states that any class, x, and its complement (-x) – now read “non-x” – are together identical with the universe class. We are now enabled to pass freely from negatives to complements or vice versa. The freedom to pass within the system over what, in highly abstract systems, are rigid, barriers to such passage, extends further and is the great characteristic of dialectic algebra. Moreover, it is this very freedom which is responsible for the distinctive quality of our thinking and behaviour, namely: self-adaptive versatility. Dialectic or general algebra thus duplicates the concrete generality of human thought. Its importance for the design of computers and sophisticated problem solving machines with high self adaptive versatility is obvious. Apart from the development of a concrete general algebra the Paradox Principle links self-contradiction with undetectability problems in physics (e.g. the ether, the Fitzgerald-Lorenz contraction, the expanding universe, etc.) and also links questions of undetectability with questions of “undecidability” in logic and mathematics, these latter being special cases of undetectability (of proofs, principles of consistency, existence, etc.).
More generally: changes within a system which are uniform for all elements of the system cannot, for obvious-reasons, be measured by specific standards which are themselves elements of the system. This general undetectability “within the system” is expressed, as we have seen, by our general “internal standard,” and recalls the discussion in mathematical logic about “functions which are arguments to themselves.” The conditions under which specific external standards can detect and measure such changes, and under which such standards are available or non-available, is an important subject connected with problems of dimensionality and infinity, on the one side, and with cybernetics and feedback systems, on the other. The Paradox Principle links these separated topics together to produce new results generalising the notion of “feedback” – a modular structure in the process. I propose to deal with all these and other applications in future papers.
Flexibility and self-adaptive versatility go hand in hand. In their most extreme forms they are more characteristic of art than of abstract science. But it is these characteristics which are of the very essence of the Paradox Principle. Thus the latter is a unifying principle. It is a builder of bridges, a maker of roads through barred territories, an opener of communications where none exists. Where Aristotle’s Principle “irreconcilably” divides with an impassable fence, the Paradox Principle reunites with an openable gate, even if it be also an oscillating or closable gate.
Grasmere, Westmoreland, England.
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