Paradoxes

On my trip this spring to Napa Valley, I visited a winery called Paraduxx — a cute name, and the wine’s good too (and they have a very pleasant outdoor patio to have your tasting). The name got some things tumbling in my brain, and that plus a couple of recent conversations have prompted me to write a series about paradoxes.

Paradoxes are things that appear to be true, but also appear to be false. American Heritage says it this way:

An assertion that is essentially self-contradictory, though based on a valid deduction from acceptable premises.

The explanation for a paradox generally falls into one of four categories:

1. It actually is true, but just seems false or improbable.
2. It actually is not true, because it’s based on a false premise (that seems true, or that’s been lost in the weeds).
3. It has exposed an inconsistency in the logic system, and adjustments are necessary.
4. There is no explanation (or, at least, we don’t have one) — often the category that metaphysical questions fall into.

A classic example of category four is the “time-travel paradox”, often used to demonstrate that time travel (or, more accurately, changing history) is impossible because it leads to contradictions. Suppose you travel back in time and cause something to happen that prevents your parents from meeting. You would therefore not have been born, so you would not have been able to go back in time, so your parents would have met after all, so.... Or you go back to prevent Martin Luther King’s assassination. If you’re successful, your future self would not have had any reason to go back to prevent something that didn’t happen, so....

On the other hand, if you go back in time just for fun, and you don’t change anything that affects your being there, well, that ought to be possible, oughtn’t it? Well, we really don’t know. It’s something for the philosophers to ponder while the scientists work on making it happen. Meanwhile, the fiction writers have used the time-travel paradox in more stories than one can count.

The “birthday paradox” is an example of the first category. Suppose you’re in a bar with sixty other people. You put $20 on the bar and announce that you don’t know anyone else in the bar yet, but you’ll bet all takers that at least two of them share a birthday (month and day). When they all say their birthdays, indeed, you find at least one pair with the same date. You couldn’t possibly have been that sure! Could you?

In fact, you could. It only takes 23 people to make the chances about 50/50, and with 60 randomly chosen people it’s pretty close to certain. That seems so surprising that it often evokes disbelief, but it’s not hard to work out the probabilities (hint: it’s easier to work out the odds that you can not find a matching pair). The key is that neither person is chosen ahead of time (the chances that someone else in the bar shared your birthday would be much smaller, and you should not bet on that one), and the probability goes up very rapidly as more people arrive.

There are many paradoxes and related curiosities that are interesting in mathematics, and we’ll look at a few of them in future posts.
 

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The series so far:

1. Paradoxes (this entry)
2. Zeno’s Paradox
3. Russell’s Paradox
4. The Liar Paradox
5. Paradox: 1 = 0
6. Countable and uncountable sets, part 1
7. Countable and uncountable sets, part 2
8. Modus ponens, modus tollens, and proof by contradiction
9. Verrrrry interesting...