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But there are different kinds of contradictions. For our purposes we can narrow them down to two, the paradox and the antinomy. Sometimes a contradiction is only apparent; it’s not really there. That’s a paradox. Other times it is really, solidly there. That is an antinomy. (The difference is often ignored.) Presented in the right way at the right time, both can teach us much.

A paradox arises when two opposing truths are set up against each other as in “The more you learn, the less you know”. The contradiction is not really there. It is only apparent because the more you know, the more you become aware of how much you don’t know. That is a paradox. You can think of lots of them. They are interesting and fun.

But there is no playing around with antinomies. They are extremely interesting, but they are not fun, for the contradiction is really there, and it can be a torment. If the one part is true, the other part cannot be true—but it is. A famous antinomy, often mislabeled a paradox, is the race between Achilles and the turtle. If the turtle gets any kind of a head start, Achilles will never catch him because before Achilles reaches the spot where the turtle was when Achilles started, the turtle will have moved on.  Achilles, then, will have a new starting spot but the goal will have moved on again. And on and on forever. The contradiction is between logic which says Achilles will never close the gap and experience which says it is only a matter of a few steps. That is a real contradiction.  Another antinomy, one that is a little more serious, is between determinism and free will. The obviously sensible view is that in life one thing leads to another, the chain of causation. But a free will choice is not part of that chain, for if the choice is free it is not caused, but if it is caused it is not free. You may have one or the other but you cannot have both. Free will and fixed fate.  An antinomy.