## Saturday, November 15th — The Tristram Shandy Paradox

### November 16, 2008

**Today I Learned…**

…about the Tristram Shandy paradox, originally formulated by philosopher, logician, and all around smart guy Bertrand Russell. It is as follows:

Tristram Shandy is writing a detailed autobiography, where each day of his life is documented in extraordinary detail. However, he is a remarkably slow writer, and it takes him a whole year to write about one day. The question is this: given infinite time, will Shandy ever finish his autobiography if his continues at this rate?

You’d think at first inspection that he wouldn’t, because he just keeps falling further and further behind. However, this is the strange thing about infinity — *he will finish*. The crux of the proof is as follows: imagine two lines, one of which has a point on it for every day that Shandy has to write about, the other with a point for every year he needs to actually do the writing. You can construct a set of ordered pairs {(d,y)} where d is a point on the first line and y is a point on the second establishing a one-to-one correspondance between the two sets. Therefore, even though both sets are infinitely large, and it seems that the set of days is more dense than the set of years, the cardinalities of the sets are equal — they are both aleph-null. That’s just one of the weird things about infinity.

November 16, 2008 at 5:26 pm

One of those cases where math proofs kind of miss the point. While this setup is true and provable, it only works if you accept Russell’s initial layout. I don’t. All he’s proven – and think about this for a minute – is that there will be one page for every day Shandy lives, which was in his initial premise. That’s all he has truly demonstrated – what he started with.

I’m going to instead put both sets onto a single line, representing time elapsed. For every integral point D, representing a number of days since the beginning of events in Shandy’s book, there is a corresponding point Y, representing when he’ll finish writing about it, that is 365 (.25, but who’s counting?) times farther down the timeline. The longer he goes, the farther behind he gets. Given infinite time the distance between any point D and its corresponding Y will become infinite.

On the plus side, it’s not as bad as Zeno’s paradox, which I can disprove by walking a few steps.

November 16, 2008 at 9:05 pm

I agree that the notion of ‘finish’ here is a bit hazy, so the discussion is a bit ungrounded. For Zeno’s “paradox”, doesn’t everything make perfect sense if you think of motion (a concept philosophers apparently had a hard time with) as a path in four dimensions?

November 18, 2008 at 5:03 pm

Fascinating post – you have an interesting blog here that compares well in terms of interest with many commercial online “zines” that I have seen. Good work 🙂