Sunday, January 13, 2008

Infinitely Large Nonexistent Strawberries

From John Venn's Symbolic Logic (p. 339):

I once had some strawberry plants furnished me which the vendor admitted would not bear many berries. But he assured me that this did not matter, since they made up in their size what they lost in their number. (He gave me in fact the hyperbolic formula, xy=c2, to connect the number and magnitude.) When summer came no fruit whatever appeared. I saw that it would be no use to complain, because the man would urge that the size of the non-existent berry was infinite, which I could not see my way to disprove. I had forgotten to bar zero values of either variable.

This little parable serves to illustrate a point about the occasional surprises that lurk in taking a formula outside familiar limits (the particular surprises he is discussing at that moment are the paradoxes of material implication).