Paradox

QUINE’S QUESTION MARK 363captivating one. It is the smallest number that can beexpressed in two different ways as a sum of two cubes,”replied Ramanujan. (1,729 = 1 3 + 12 3 = 10 3 + 9 3 ). FrancoisLe Lionnais’s Nombres remarguables shows that many apparentlydull numbers are interesting. The first integer for whichhe can find no remarkable property is 39. Le Lionnais musesthat this lack of a remarkable property makes 39 interestingafter all. Just as 81 is interesting because it is the smallestsquare that can be decomposed into a sum of three squares (9 2= 1 2 + 4 2 + 8 2 ), the number 39 is interesting because it is thesmallest uninteresting integer.Mathematicians have generalized Le Lionnais’s commentinto a proof that all natural numbers are interesting. Ifthere is an uninteresting number, then there must be a firstuninteresting number. But being the first uninteresting numberwould itself be an interesting property. Therefore, allnumbers are interesting.Maybe each natural number could have some surprisingfeature that makes it interesting. Often, what appears to be adull number has proven to be “captivating.” Maybe that ishow it always is. There might even be some fancy proof toshow that all appearances of dullness are illusory. But it seemsstrange that we could prove that each and every naturalnumber is interesting by virtue of the least number theorem(which says that if any natural number has a property, thenthere is a least number that has that property). The argumentseems too simple.Consider the plight of ambivalent mathematicians whoindependently believe the conclusion of the simple argument.Since the conclusion implies each of the premises, theythink the simple argument is valid and believe the premisesand the conclusion. Despite granting that the argument is