Tag: paradox

The Overthrow of Logical Arithmetic

The story of the clean-shaven barber is an example of the Russell Paradox. Throughout the late 19th century, logicians and analytical philosophers had been trying to establish a series of axioms that would provide a logical foundation for mathematical operations. Gottlob Fege was one of the great leaders of this enterprise. Over the course of his career Frege sought to establish a series of logical principles of inference which could support mathematical proofs, thus asserting that arithmetic could be treated logically. Of course, the problem with logic is that it can create contradictions which maths cannot support. One such contradiction was brought to light by Bertrand Russell in his famous letter to Frege refuting his theory of sets. Frege’s Basic Law V states that sets can be understood as equivalent if the constituents of set 1 are identical to the constituents of set 2. The law is important not so much because that it establishes equivalency, but rather that it sets up conditions for difference. If the contents of set 1 are not identical to set 2 then you can classify them differently and claim that there is logical justification for doing so. It seems perfectly air tight, but Russell discovered a case of illogic in which the law could be broken. The paradox is traditionally understood with the following formulation:

Let R be the set of all sets that are not members of themselves. If R qualifies as a member of itself, it would contradict its own definition as a set containing all sets that are not members of themselves. On the other hand, if such a set is not a member of itself, it would qualify as a member of itself by the same definition.

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The barbershop problem I spoke of in the last post is a more illustrative version of the contradiction. For the barber to belong to the set of men who shave themselves he would also belong to the set that is shaved by the barber and so would be disqualified from the set of men who shave themselves. Frege acknowledged the validity of the Russell’s challenge almost immediately and withdrew Basic Law V in the same volume in which in he introduced it.

Russell’s paradox does not explode set theory and the rationale for classification, but it does prove that sets cannot be treated as axiomatic and unfailingly valid. Before Russell, every well-defined collection was treated as though it could be understood as a logical set. After acceptance of the Russell paradox, it became necessary for one to prove that a collection is a set so that the principles of set theory could be applied to it. A lot of mathematicians ignore Russell’s paradox. Most math people are not attempting to form a logical excursus—math need not be founded in logic to be functional. The practical implications of Russell’s paradox kind of murky also. We still make classifications and treat like objects as logical sets regardless of whether our groupings create internal contradictions. What cannot be avoided, however, is that there is no logical foundation for classification. Classes must always be treated as potentially spurious; always vulnerable to dispute, always prone to failure.

Vicious Regress

While supposedly infinite, the regress produced by the why litany that I discussed in the last post is not vicious—at least it does not appear to be. There is an orderly chain of causality which branches outward from the initial premise and does not loop back to its original conditions. In a vicious regress, the predicate yields an outcome that fails to satisfy the predicate’s causal requirements and thus resorts back to the predicate. No progress is made. The initial problem is never resolved and is instead re-introduced in the solution.

The most famous examples of vicious regress is the Barbershop paradox, which goes as follows:

Imagine a town in which all the men are clean-shaven. There is one barber in the town, and he shaves all of the men who do not shave themselves. As a citizen of the town, the barber, like all the other men, is clean-shaven. Who then shaves the barber?

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The question is impossible to answer because the two parameters that determine clean-shavedness—that either you were shaved by the barber or you shaved yourself—contradict one another in the case of the barber himself. If the barber shaves himself, it violates the rule stipulating that the men who shave themselves are not shaved by the barber. Any answer to the question, who shaves the barber, initiates a vicious cycle where no solution can be provided without creating the problem all over again.

Often, when I share the barbershop paradox at parties or wherever, people want to try to solve it like it’s a riddle. The rules are constructed in such a way that there can be no answer, but this is logically unacceptable to people. If the barber is clean-shaven, there must be an explanation. Most often, people want to call the rules into question and say that one or both of the parameters must be permeable in some way. “Oh, the barber must be going to a different town for his shave” or “Maybe the barber doesn’t grow facial hair.” It’s a familiar position to adopt when confronted with a confusing outcome. I think we make these kinds of intuitive judgments all the time in our day to day lives, whenever we encounter an unexplained event really. The practical world will not abide antinomy like the barbershop paradox, so naturally, there always has to be something else at play. Something we’re not seeing, or that was left out of the report of the problem. It always has to come down to problem of comprehension: there are additional factors at work here of which we are not aware. It is practical wisdom to be suspicious of vicious regress because nature does not seem to support it. I don’t believe the paradox above is to be understood as demonstrable in any way. It is purely a game of logic and mathematics. It is an artificial construct conjured up by philosophers to help them conceive of the limits of abstract logic. I’m not sure I know what it means to say that paradox is imaginable but not practicable. It is said that man cannot dream a thing that does not have some analogy in the world. There is a good deal in mathematics that probably repudiates that claim.