This was written in 2009. 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!
2. The rope
For our first example, 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 one foot above the ground, all the way around. (The reason for this does not matter.)
Well, in that case, a longer rope will obviously be needed. But roughly how much longer do you think the new rope should be?
A calculation reveals a surprising answer.
The length of the original rope was π × D, where D is the Earth’s diameter. So the length of the new rope must be π × (D + 2), since the diameter has simply expanded by two feet, one on each side. But the difference between these values is simply 2π, which is about 6.28. And so the new rope needs to be only about six feet longer than the old one, which is roughly the height of a man.
Can that be right?
Most people find this very surprising, even unbelievable. Surely the new rope must be considerably longer than that? Something like a few miles longer than the old rope would seem to be more appropriate!
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 apparently unbelievable conclusion. Of course, in this case, the conclusion does not stay unbelievable for long. Once the reasoning sinks in, the conclusion comes to be accepted. This is called “swallowing the conclusion,” and is one of the standard ways to respond to a situation of this sort. It’s clearly the right way to go here since the mathematical reasoning is beyond question.
But now let’s consider a case where the opposite happens and we end up “rejecting the reasoning” instead.