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The Paradox of Tristram Shandy

This paradox, formulated by Bertrand Russell, is based on the 18th century novel The Life and Opinions of Tristram Shandy, Gentleman, by Laurence Sterne. Here is Russell's statement of the paradox:

"Tristram Shandy, as we know, took two years writing the history of the first two days of his life, and lamented that, at this rate, material would accumulate faster than he could deal with it, so that he could never come to an end. Now I maintain that, if he had lived for ever, and not wearied of his task, then, even if his life had continued as eventfully as it began, no part of his biography would have remained unwritten."

Suppose Tristram Shandy continued at the painfully slow rate at which he started, so that he took a full year to write about each day of his life. In spite of this, there is a one-to-one correspondence between each year that he writes in and each day he writes about. Therefore, no matter what day of his life you care to consider, there will eventually come a year in which he will be able to write about it. There is no part of his life that can never be written down. Nevertheless, he gets further and further behind!

It's interesting to reverse this paradox and consider what would happen if Tristram had already been writing for an infinite amount of time. It at first may seem that the two temporal directions might be mirror images of one another. In that case, just as he might begin to write at time t in the original paradox, in the reversed version it would seem he might have finished his task at t. But a little reflection shows that that is impossible. If he had just finished writing his autobiography, then he would have just written about the most recent day of his life. But since it takes him a year to write about each day, he would have had to start writing about this most recent day 364 days before the day started! Thus, unless Tristram can foretell the future, he cannot have finished writing yet, even though he has already spent an infinite amount of time on the task.

Suppose Tristram has in fact been writing forever and has just finished describing another day. When might have been the day he just finished writing about? As we've just seen, he could not have been writing about today, for he would have had to start writing about it a year ago. So it seems that the most recent day he could have been writing about is a year ago today. But then what was he writing about in the previous year? He would have been writing about a year ago yesterday. But that too is impossible, for he would once again have had to start 364 days too soon. Repeated application of this argument shows that, no matter what date in the past one chooses, Tristram could not already have written about it. He therefore can only have finished writing about a day that lies in the infinitely remote past!

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