Puzzles that point out the limits of logical thinking, help improve it and otherwise deal with contradictions in life.
ILLUSTRATION: TOMASZ WALENTA
13 COMMENTS
If you wear glasses you’ve probably experienced the problem of being unable to find your glasses because you’re not wearing your glasses. There’s also the conundrum of needing to drink coffee before you’re capable of making coffee in the morning. More seriously, there’s the question of whether tolerant people should be tolerant of intolerance. These are all types of paradox arising from self-reference, as in the sentence “I am lying.”
Mathematicians and philosophers have long studied paradoxes. In 1901, Bertrand Russell discovered a famous paradox that shook the foundations of mathematics. It is often described informally by imagining a man in a town who is a barber. He shaves every man in the town who does not shave himself and nobody else. This causes a looped-up contradiction if we start wondering whether or not the barber shaves himself: If he doesn’t, then he does. And if he does, then he doesn’t. It’s a paradox.
This might sound pedantic and contrived, but the mathematical study of paradoxes is important for several reasons. The first is that the presence of paradoxes alerts us to the limits of our own logical thinking and points us in the direction of improving it. Russell’s paradox was formally stated in terms of sets and caused mathematicians to realize that a naive or intuitive definition of a set as “a collection of things” is not logically sound when self-referential sets become involved. Because mathematics is based on making sound logical deductions, the logic of the initial definitions is crucial. Paradoxes tell us that greater care is sometimes needed in setting out these premises.
Thinking about how to resolve paradoxes has helped mathematicians make progress throughout history. More than 2,000 years ago, the Greek philosopher Zeno pondered a range of paradoxes having to do with motion. One of them said: In order to travel from A to B, you must first cover half the distance, then half the remaining distance, then half the remaining distance and so on forever. We need to cover an infinite number of distances, but we only have finite time, so we can never arrive.
Here the conclusion is blatantly untrue: We arrive at places every day. To resolve this paradox, what was needed was a new theory of adding up infinitely many infinitely small numbers. This launched the field of calculus, which analyzes things that change continuously. And calculus, of course, would become a crucial tool of science, engineering and economics. Thus the apparently arcane study of ancient paradoxes led to the mathematical foundation of much of modern life, whether or not technology users are aware of the math involved.
Understanding how mathematicians avoid paradoxes in abstract cases can help us to work out how to deal with similar contradictions in life. For the paradox of the barber, we can simply say, “Oh well, no such barber can exist.” But Russell’s paradox is formally stated in terms of sets rather than barbers, and the paradox comes from considering the set containing “every set not containing itself.” Instead of just saying this set can’t exist, we say it exists but at a different level from ordinary sets. It is a way of breaking the self-referential loop.
This has helped me work out how to break the loop problems in life as well. I’ve made it easier to find my glasses without wearing my glasses, for instance, by always storing them in the same place. For my coffee, I’ve broken the loop by setting up my espresso machine the night before so that in the morning I just have to turn it on. In the case of tolerance, I make a distinction between people themselves and their ideas about other people, so that I can be tolerant of individuals who are intolerant but don’t have to accept their ideas.
Paradoxes are not just useful puzzles in mathematics. They can point the way to thinking more clearly about resolving complex situations in everyday life.
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