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A Riddle

September 21, 2011

Had things gone differently, I might have made the study of paradoxes into my life’s work.

First, because they are beautiful.

But second, because they are useful. A paradox moves through a natural lifecycle: from fundamental antinomy, to mathematical or conceptual paradox, to mere ‘linguistic’ paradox, to interesting party trick, to lame party trick. These moves deceive us into believing that issues have been resolved, insight gained, thinking clarified. Would that it were so! But of course nothing has been resolved. We are the ones who have changed. The course of our thinking subtly modified to flow around and over and past the obstruction. The paradox remains, lurking like Jörmungandr, gnawing away at the heart of the world, poisoning everything. A testament to the vileness of man, to the fact that his mind is so monstrous that it will take logic, which is perfect, and make it abominable.

The child’s instinct is correct. The child says: “that cannot be!” The child realizes that the world is perfect, and so the paradox must not reside in the world. The child is wiser than the philosopher, who sees the cracked and twisted image, and would rather take it to be real than inspect the glass. The child, on hearing about the barber who shaves all men who do not shave themselves, immediately grasps the simple truth: there is no such barber, there can be no such barber, the fact that such a non-barber merrily lives out his non-existence within our imaginations is nothing more than crushing evidence of how profoundly warped our imaginations really are.

And so paradoxes are useful in two ways: they engender a spirit of humility, and they reveal to us those things which do not exist. The second utility is not to be discounted. Non-existence is a funny property, and objects exhibiting it can be tricky to find. But once the non-existence of a thing has been established, we are usually moved to reexamine any rules or principles which would imply the existence of the non-existent thing. This is the least pathological manifestation of the universal reaction to paradoxes: avoidance. Sometimes, it even brings us closer to the truth.

One of the more unseemly avoidance reactions of the recent past was that which the mathematical world performed in response to the paradox that Russell discovered. The story is all the more heartbreaking because it started out on such a promising note. Russell discovered a thing which, like the non-existent non-barber, did not exist: in particular, he discovered the non-existent set of all sets that do not contain themselves. His discovery of this non-existent thing was complicated by the fact that if it had existed, it would have existed non-physically. Some people felt that even perfectly well-behaved non-physical things had at best a rather tenuous grasp on existence. Civilized and progressive folks such as myself don’t look too kindly on that sort of physicalist chauvinism, but this was a less enlightened time.

Happily, the non-existence of Russell’s non-set was eventually recognized, but the hysterical overreaction that followed nearly overwhelmed all the good that came from the discovery. “Axiomatize!” was the order of the day, and everybody momentarily forgot what a set was. Perfectly decent and well-behaved sets such as the set of all sets, or the set of all sets that contain themselves, were thrown onto the pyre. Sacrifices to appease the fearsome paradox. Collateral damage. Regrettable casualties.

And so we come to the riddle: What is the actual reason for the non-existence of Russell’s non-set? Whence does the paradox really spring? What can be done to our conception of a set such that Russell’s non-set is no longer implied, but the good and useful sets are saved from the abyss?

I don’t have the answers to these questions, but over the coming months I’ll be sharing a couple of non-answers with you. Perhaps, by seeing where they fail, we can grope our way towards the crux of the problem, the pinch in our conceptual fabric that distorts our image of the world.

This I know: the time for fear and denial is past. A set, like this blog, is just a collection of related things.

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