Harold Walsby: Infinity and Self-Contradiction

Soon after the discovery of the calculus by Newton and Leibniz, problems of consistency in mathematics arose which centred around the concept of “infinity,” i.e. “infinitesimals” or “infinitely small quantities.” The inconsistencies, together with ensuing disputes among mathematicians and philosophers, were not allayed until the middle of the last century. Whether or not Weierstrass, Dedekind and others really laid them to rest still remains to be seen. The fact is that whenever we find self-contradictions, somewhere around in the offing, so to speak, there is usually to be found some particular form of the infinite. Thus it would appear that the problem of self-contradiction is intimately connected with infinity. And as that problem is itself still attempts to resolve it, new doubts raised about problem.

Indications of this intimate connection are legion if only one looks for them. I will limit myself to a few. Analysis, the mathematics of continuously varying quantities, connects motion (which appears continuous) and infinity. The first well-known instance of this connection in history is, of course, in the famous Paradoxes of Zeno. But their connection in the Paradoxes depended on Zeno’s paramount object to show that self-contradictions were involved in motion, that therefore motion and change were illusory. The concept of infinity was essential to that object. This early partnership of self-contradiction and infinity was to be maintained throughout the history of mathematics and its disputes, right up to the present. Since ancient times it has been the habit of mathematicians to regard continuity in terms of its opposite, division, with the aid of the concept of infinity – providing us with yet another example of our now-familiar pattern of self-contradiction, namely: “infinite division = non-division.” Thus continuity is said to be “infinitely divisible.” (But only “unofficially,” “off the record,” of course!)

The connection between infinity and self-contradiction was well known to the Greek geometers, who early devised their straightedge-and-compass constructions especially to avoid problems of infinity and inconsistency. This is probably the earliest instance of a major “avoidance” technique in mathematics. I have already mentioned avoidance attitudes with respect to self-contradiction and classic paradoxes. Avoidance techniques are simply ritualistic attitudes or practices (often hallowed by authority and/or custom) the function of which is to continue to prohibit the crossing of a mental barrier. They are thus like real taboos. And like real taboos, they function as undercover rules which cannot be put into a system in an open, workable, explicit form. Other instances of such avoidance devices are hallowed verbal phrases like “tending to infinity” and “becomes indefinitely large.” Just as self-contradiction has to be avoided at all costs – by various means, including wrecking the argument – so the corresponding avoidance rituals associated with infinity are centred in the emotionally-tense ban upon considering it “actual” or “complete.” The alleged dangers of so treating it are, indeed, undermined by the self-contradiction involved in the orthodox view which, once again, we find is founded on the fixed distinction, the alleged impassable barrier, between “actuality” and “signification” – in a system where all such entities are, in either case, signified. Furthermore, we see that an unending process may be incompletable, conditionally (that is to say, is simply uncompleted) or unconditionally so… i.e. is completely incompletable. In short, the infinite is not merely unending. It is unendable, unconditionally, absolutely, completely. However, these taboos vary considerably in their intensity and emotive content as we pass from one branch of mathematics to another. Compare, e.g., analysis with projective geometry, or the theory of sets.

In this connection it is interesting to note the emotive name given to self contradiction by both Poincaré and Russell in their correspondence (of the early 1900’s) on the paradoxes raised by Cantor’s theory of sets. They averred the paradoxes were due to what they called “vicious circles,” and Russell erected a principle for “the avoidance of vicious circles.” Unfortunately for the author, however, the cure turned out to be worse than the disease: it spawned more paradoxes than ever. His remedy was that whatever involved all of a collection must not be part of the collection. That is to say: whatever, i.e. any one of a first collection of entities, involves all of a second collection of entities, must never be one or part of this second collection. The two collections, in short, must always be kept mutually exclusive; they must never be allowed to intersect or interpenetrate. This would be sound but for the claim to unqualified universality implied by “must never,” “must always,” etc. Wherever “never,” “always,” “universality,” are unconditional and applicable without exception, the concept of infinity is lurking, since there is implied an unending series of applications. Needless to say, terms like “always,” “never” etc. can be implicit, as well as explicit, in statements.

We saw that it was this very universality-without-limit (of applicability) which turned Aristotle’s Principle into a self-contradiction. Although we now suspect that we shall have to live with and use self-contradictions, we must remember that their meaningfulness depends upon their context. What is meaningless in terms of Aristotle’s Principle is not, any longer, necessarily so for us. Hence, in the light of the foregoing sections, we can see that Russell’s “Vicious-circle Principle” is really no more than an arbitrary insistence that Aristotle’s Principle shall apply as between two collections or classes related in a very special way: i.e. where the two classes can actually be regarded as mutually interpenetrating, and where the are of the type of conventional opposites called “contraries” – as e.g. “content and form,” “intenstion and extension,” “meaning and symbol,” “concept and thing,” “subject and object,” “mind and matter,” signification and actuality,” etc. etc.

Thus the Vicious-circle Principle, just like a legal enactment, presupposes the actual possibility of that which it is arbitrarily banning! This is quite unlike Aristotle’s Principle, which explicitly asserts the actual impossibility of the interpenetration of opposites, and which therefore, as I have said, arbitrarily legislates against the possibility of that which it declares to be a universal impossibility. Hence the mess and mass of unresolved paradoxes involved in both the Vicious-circle Principle and in the theory to which it gave birth: the Theory of Logical Types. The latter, developed mainly by Russell (first published in 1908) appeared in 1910 under the joint authorship of Whitehead and Russell in their first volume of Principia Mathematica. The theory arbitrarily sets up, on the basis of the Vicious-circle Principle, certain “impassable” barriers such as those we have already met with in these pages, to render certain types of proposition “meaningless” and certain totalities “illegitimate.” This is supposed to result in a hierarchy of “logical” types that is, of functions and propositions – rather like a hierarchy of officials in which those of one level are arbitrarily barred from doing certain things with those of other levels. The authors, however, scuttle themselves from the very start of Chapter II of the Introduction (especially designed to explain the theory) with the following gem: “We shall, therefore, have to say that statements about ‘all propositions’ are meaningless.” Which, of course, is simply another way of asserting: “All propositions about ‘all propositions’ are meaningless.” There are numerous other assertions of the same kind. Such assertions are, indeed, unavoidable in propounding the theory, which circumstance renders it self-refuting from the start. Thus, when ushered into the world by its authors, it was already still-born. To found a theory designed to avoid self-contradiction upon such assertions as “You must never say ‘You must never say…'” surely touches the low-water mark in the history of logic? I shall examine the theory in detail elsewhere, but it seems clear that the reputation for sound profundity garnered by the Principia over the years is quite undeserved, and that the authors of the theory never thoroughly understood it themselves. Unfortunately, it still has considerable influence upon the academic world, especially upon the philosophically naiive, including writers of college textbooks.

The important thing to notice about the theory is that it leads, not only to more self-contradictions, but to an infinity of them. For example, the authors’ assertion quoted above leads to another: “All propositions about ‘all propositions about “all propositions” are meaningless’ are meaningless.”.. and this leads to another, and so on, ad infinitum. In sum, a statement which says, “All statements about ‘all statements’ are meaningless,” is not only self contradictory but, in its self-application, generates further selfcontradictory statements without end.

The irony in all this is that, earlier, when Poincaré had suggested that the paradoxes of set theory were due to treating the infinite as “complete” and “actual” (referred to above) Russell answered that Poincaré was mistaken and did not really appreciate the problems of the Vicious-circle Principle! In passing it is of interest to note that Poincaré – one of the greatest mathematical thinkers of all time, the last of the giants to take practically all mathematics as his province – (a) regarded the intuition of infinite recurrence as the foundation of mathematical and geometrical induction, two basic types of reasoning; (b) connected mathematical periodicity (a type of recurrence) with infinity in a special way, generalising periodicity in automorphic functions; (c) seems therefore, with Jevons, to have, believed it highly probable that all reasoning reduces to a single type: namely, what is true of one thing will be true of another with a certainty proportional to the degree of resemblance between them in all relevant respects; (d) thus, characterised mathematics as “the art of giving the same name to different things.”

For further evidence of the link between infinity and self-contradictions, let us look again at the example given above, apparently innocent of any suck link, “All generalizations are unsound.” As this example is actually a generalisation, that is, it takes the form of a generalisation, i.e. it is extensionally one, is a thing as such a symbol, apart from its concept or significance, etc. etc. The Theory of Types would bar us from discussing it “in the same, breath” as it’s meaning. We ignore the ban. Let us put the example in a more general form, one which is exceedingly common. For clarity I put a diagram beside each general proposition:

“All S is P”
“All S is P” is S
Therefore, “All S is P” is a P.

It will now be clear that if “All S is P” is a member of S and therefore of P, a reproduction of “All S is P” will appear in the original S, which reproduction in its turn, in order to be a true reproduction, must necessarily contain another reproduction inside the first, and so on and on in an infinite regress or sequence. I cannot indicate in the diagram more than the first reproduction or so. Another example I have given above is “This statement is a lie,” and it may not seem amenable to a similar demonstration. Let us see:

“This statement is a lie” (“All S is P“)

“This statement is a lie” is this statement (“All S is P” is S)

Therefore “This statement is a lie” is a lie (Therefore “All S is P” is P)

It may not be obvious where the infinite regress is in this example. In the previous example it was obvious because “All S is P” was reproduced in part of S, i.e. “All S is P” was rightly presumed to be one only of a larger number of members of S. The present case is slightly different S contains only one member and that one is “this statement.” Thus the “reproduction” of S is such that it is coincident with itself. Where, then, is the rest of P reproduced? The answer is that if S is coincident with itself, and P is reproduced to the same scale, then there will be no “room” for the rest of P inside S. But as the diagram is only a diagram, we may imagine that the reproductions of the rest of P have been squeezed up so that they occupy only the rim or border of S. This would be like taking two mirrors facing each other across a room (where each reproduces an indefinite sequence of images of itself inside the other) and bringing them together so that their faces were touching.

George Walford International Essay Prize

Harold Walsby: Infinity and Self-Contradiction

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