Paradox

362 A BRIEF HISTORY OF THE PARADOXThe Structure of Appearances implies an answer to “Is thenumber of individuals in the universe odd or even?” Sincethere are only finitely many atoms and each individual isidentical to a combination of atoms, there are exactly as manyindividuals as there are combinations of atoms. If there are natoms, there are 2 n - 1 combinations of individuals. No matterwhich number we choose for n, 2 n -1 is an odd number.Therefore, the number of individuals in the universe is odd!The exclamation point is not for the oddness per se. Asidefrom those who think the universe is infinite, people agree thatthe universe contains either an odd number of individuals oran even number of individuals. What they find absurd is thatthere could be a proof that the number of individuals is odd.“Is the number of individuals in the universe odd oreven?” illustrates the possibility of one good answer being toomany. Our expectation is that this question is unanswerable.The lone good answer confounds beliefs about what argumentscan accomplish. Here the excess is a top-down judgment.(More commonly, the overabundance is a bottom-upverdict: a good answer clashes with another good answer.)INTERESTING NUMBERSOur meta-argumentative expectations can also be frustratedby how something is proved rather than the sheer fact that itis proved. Consider the question of whether all naturalnumbers are interesting. When G. H. Hardy visited themathematical genius Ramanujan as he lay dying in a sanitarium,he was at a loss for what to say. So Hardy mentionedthat the taxi that he had hired to take him to the sanitariumhad the rather dull number 1729. “Oh no, Hardy. It is a