This was written in 2009. (Revised 2022.) It explains the notion of a logical paradox in a way that comes naturally to me. The conclusion drawn at the end is probably the main thing for me (“We are blinder than we can imagine in matters of straightforward logical reasoning”) but it is not always emphasized by others!

For our first example of an “easily resolved” paradox, imagine that a long rope has been laid snug around the Earth’s equator. (Pretend the Earth is a smooth sphere.)

It’s an engineering project, say, and the men involved have just finished the job.

To their chagrin, the foreman now tells them that a mistake has been made. The rope should not lie snug on the ground like that, but should rather be raised

Well, in that case, a

A calculation reveals a surprising answer.

The length of the original rope was π ×

Can that be right?

Most people find this result to be very surprising, even “unbelievable.” Surely the new rope must be considerably longer than that? Something on the order of

And yet the reasoning above is hard to fault: go through it again carefully if necessary!

This is a simple example of a “compelling” piece of reasoning that leads to an “unbelievable” conclusion. The rational mind, as we have seen, cannot tolerate such an abominable tension and will strive to find a resolution. In this example, as I hope you’ll agree, the conclusion does not

This is called “swallowing the conclusion,” and is one of the standard ways to respond to a rational tension of this sort. (The other way is to “reject the reasoning,” which I’ll demonstrate in the next section.) Swallowing the conclusion is clearly the right way to go here since the mathematical reasoning above is beyond question.

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Let’s consider another case in which a paradox is quickly resolved by “swallowing the conclusion.” Our next example is often called the birthday paradox and has to do with the following simple question. What are the chances that, in a random group of people, at least two people will have the same birthday?

The answer to this question depends, of course, on how large the group is: the more people there are, the likelier the possibility of a shared birthday. For example, if the group contained just three people, then a shared birthday would be very unlikely, whereas if the group contained 366 people (or more), then it would be

Very well, but now consider this. How large does the group have to be if you merely require there to be an

The birthday paradox is that you need to assemble just twenty-three people to have an even chance of a shared birthday, which is a surprisingly small number! Many people find this hard to believe. Consider the photograph above, for instance, which contains twenty-four people. You’d think that the chances of two of them having the same birthday would be rather low. After all, there are 365 possible birthdays and only twenty-four of them! In fact, however, the chances of a shared birthday between them are greater than even.

The proof of this “unbelievable” result is as follows. Suppose there are

\( \frac{365}{365} \times \frac{364}{365} \times \frac{363}{365} \times \ldots \) (*N* terms)

where there are exactly

Notice also that, as

The photograph is actually of the attendees of the Fourth Solvay Conference in Physics, held in 1924 in Brussels. I found this photograph at random on the web and was relieved to discover that, indeed, both Marie Curie and Edwin Herbert Hall, seated next to each other in the front row (Hall is holding a hat), have the same birthday of November 7.

The larger point is that this is just another case of a “compelling” piece of reasoning that leads to an “unbelievable” conclusion. The rational mind cannot bear this abominable tension and must either reject the reasoning or swallow the conclusion. In this case, as I hope you’ll again agree, we should swallow the conclusion: despite what may be natural to believe, twenty three people really

Let’s now see how we can resolve a paradox in the

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