In Through the Looking Glass by Lewis Carroll we find the indomitable twins Tweedledee and Tweedledum. I have noted their presence before in an earlier blog post, but I have another point to make about this pair in what is usually regarded as a child’s story — unless you count the horror story made about Wonderland by Hollywood (which I don’t). I also discount Walt Disney’s abortive attempt to capture the magic of Carroll’s masterpiece. One really has to read the original and delight in the remarkable drawings by John Tenniel.
In any event, Alice comes across the pair in a thick woods and they stand arm-in-arm looking sideways at Alice as she struggles to tell them apart. They form what geometers call “enantiomorphs,” mirror-image forms of each other. This is appropriate since they are encountered in looking-glass world where everything is a bit backwards and “it takes all the running you can do to keep in the same place. If you want to get somewhere else, you must run twice as fast as that.” — as the Red Queen explains to Alice earlier in the story. (Reminds us all of the condition of politics in this country at the moment — or is that the caucus race?)
But the Tweedle twins also pose another philosophical conundrum, which may have been in the back of Carroll’s mind. Logicians call it the “identity of indiscernables.” The two are identical and differ only with respect to place. One is to the left of the other. This suggests that unless we regard space as a feature of a thing’s essence, the two twins would not be two but only one. If the only respect in which they differ is that one is to the left of the other — and vice versa — then, if they are identical in every other respect, they are one and the same person! But this seems absurd. We must therefore conclude that the thing’s position in space is essential to it’s being what it is. But that is also absurd. That would mean that every time you move you would be someone else — because your spatial location (which we have assumed is part of who you are) has changed. In fact, no one would ever be the same person in one place that she is in another. This paradox is embraced by such thinkers as Hegel, but most philosophers refuse to regard spacial location as in any way a defining characteristic of who they are. It is a mere accident.
Thus, it would seem, Twedledee and Tweedledum are the same person. Or so it would seem.
The delightful thing about Carroll’s tale is not only that it is a chess game (which it is) but it is also filled with logical puzzles like this one and we are reminded on nearly every page that the man was a mathematician with the most remarkable imagination. What a delight!