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A Review of The Ways of Paradox

June 1st, 2006

Throughout the history of mathematics, paradoxes have been used to challenge its pillars, those assumptions and inference methods that seem to support all that we know through mathematical reasoning. When a blow from a paradox is struck, the impact can be devastating, paralyzing branches of mathematics for generations. But as the dust settles, mathematics adjusts and moves on, until the next paradox is found. In this way, paradoxes can be seen as losing their vigor over time – a particularly pernicious paradox of one century can be a harmless brain teaser of another. W.V. Quine, in the The Ways of Paradox, takes the reader on a tour of some of these more notable paradoxes of mathematical reasoning, and along the way forms both a general definition for a paradox and a classification of paradoxes according to their impact.

In general, Quine claims, a paradox is “just any conclusion that at first sounds absurd but that has an argument to sustain it”. The difference between paradoxes, however, is in the contents of the argument. In some cases, a paradox can reveal a true, but completely unintuitive or startling, conclusion from true premises. Paradoxes of this sort are called veridical. Paradoxes can also lead to false conclusions from false, but deceptive, premises. These Quine names falsidical. Lastly, paradoxes can lead to self-contradictions, but via air-tight (at least by contemporary standards) reasoning. These, the most destructive of all paradoxes, are termed antinomies.

As mentioned, a paradox’s classification can change as the method of mathematical reasoning around it adjusts and evolves. Paradoxes that we now consider veridical were at one time quite powerful antinomies. One example of a contemporarily viewed veridical paradox is the paradox of the Barber, proposed by Russell in 1918. The story goes: there exists a barber in a village that shaves all and only all men that do not shave themselves. By simple examination, it’s obvious to see the contradiction. If the barber shaves himself, then he shaves someone who shaves himself, a contradiction. However, if the barber does not shave himself, then he doesn’t shave all men who do not shave themselves, namely himself! This paradox seems as though it could be harmful for first order logic, however, the way out is just to deny the possibility of such a barber – and in this way the paradox can be considered veridical, and rather innocuous. Veridical paradoxes, it seems, do not alter mathematical reasoning, but only demand better clarification of it.

Falsidical paradoxes, however, can force an adjustment in mathematical thought. For example, Zeno’s paradox of the Achilles and Tortoise, where the fast rabbit can only cut the distance between the slow hare but never surpass him, reveals a fallacy in reasoning about infinite time (namely that infinite time series must add to all of eternity). Another paradox that reaches a false conclusion is misleading proof of 2 = 1: x = 1, so x^2 = x, so x^2 – 1 = x – 1, so x + 1 = 1, and therefore 2 = 1! What’s interesting is that anyone can find a false proof to “prove” a false conclusion; these are called fallacies. What makes a fasidicus paradox different, however, is in the absurdity of the conclusion.

Antinomies are quite different from the aforementioned paradoxes. These yield contradictions, but from accepted methods of reasoning. Quine explains, they “establish that some tacit and trusted pattern of reasoning must be made explicit and henceforward be avoided or revised”. In some cases, these revisions are slight or trivial. For example, consider the following paradox: ‘Yields a falsehood when appended to its own quotation’ yields a falsehood when appended to its own quotation.

This self-referential sentence yields an obvious contradiction. The sentence is true, since it is appended to its own quotation, but cannot be true since it says it yields a falsehood. The escape, devised by Tarski, is to consider a hierarchy of truth locutions, such that if a sentence T refers to another sentence S, then the truth of T will be given a higher subscript than that of S (for example true1 over true0). In this way, the grammar is essentially redefined, and renders the paradox meaningless.

Antinomies are not always so easily avoided; Russell’s paradox of set theory has proved especially recalcitrant since put forth in 1901. Using the basic definitions of sets, Russell constructed a set, call it S, that contains all the sets that are not members of themselves. The paradox is quite evident. If a S is a member of itself, then there is a contradiction, since it contains only sets that are not members of itself. On the other hand, if S is not a member of itself, then by definition it is a member of itself – another contradiction. Russell’s paradox struck hard the foundations of mathematics. It seemed as though to eliminate the paradox would be to give up the principle of class-existence when defining sets, for instance B, where B = { a | a A }. This notion is obviously fundamental to set construction. To remedy, a compromise of sorts was reached: “happily we can indeed withhold the principle of class existence from cases where the membership condition mentions membership, without unsettling those brances of mathematics that make only incidental use of classes.”

I am currently living in Pittsburgh, PA, working as a Senior Technical Consultant for Summa, and studying as a part-time graduate student in Philosophy at Carnegie Mellon University.