Douglas Adams, Enormous Integers, and Infinity
John Baez's discussion of enormous integers and infinity:
reminded me of the following Douglas Adams quote:
Here’s a puzzle due to the logician Harvey Friedman. It too has an unexpected answer.
Say you have a finite alphabet to write with. How long can a word be if no block of letters in this word, from the nth letter to the 2nth, is allowed to appear as a subsequence of a bigger block from the mth letter to the 2mth?
If you have just one letter, this is the longest it can be:
If you have two, this is the longest it can be:
Puzzle: How long can the word be if you have three letters in your alphabet?
Friedman showed there’s still a finite upper bound on how long it can be. But, he showed it’s incomprehensibly huge!
Now Friedman is one of the world’s experts on large cardinals—large infinite numbers. So when he says a finite number is incomprehensibly huge, you sit up and listen. It’s like seeing a seasoned tiger hunter running through the jungle with his shotgun, yelling “Help! It’s a giant ant!”
I'm reminded of the saying “This is not mathematics. This is theology.” (about the more abstract parts of mathematics).
The car shot forward into the circle of light, and suddenly Arthur had a fairly clear idea of what infinity looked like.
It wasn't infinity in fact. Infinity itself looks flat and uninteresting. Looking up into the night sky is looking into infinity—distance is incomprehensible and therefore meaningless. The chamber into which the aircar emerged was anything but infinite, but it was just very very big, so big that it gave the impression of infinity far better than infinity itself.