Imagining Numbers (Particularly the Square Root of Minus Fifteen)

Abstract

Steven G. Krantz Harvard University, founded in 1636, is America’s oldest institution of higher learning. It is the wellspring of many of our intellectual traditions. One of my favorite of these is the ritual of various Harvard faculty from the humanities and the sciences and the social studies getting together once per month or so to exchange ideas. It is a fascinating exercise: the humanist trying to explain to the cosmologist the current issues of deconstructionism; the homotopy theorist explaining to the philologist about toposes; the philosopher informing the geneticist about logical positivism. Barry Mazur is evidently the product of this crucible of erudition. His work, obviously a popular math book, is not the mindless gibbering of 1089 and All That [ACH], nor is it the self-important bombast of Chaos [GLE]. Barry Mazur has a mission: he wishes to explain to a humanist or a social theorist or a poet what √−15 is. This is a remarkable quest, and I am quite sure that I do not know how to carry it out myself. Bear in mind that I am a professional mathematician, an accomplished expositor, and in fact I am a complex analyst. I am supposed to know what √−15 is. But in fact I do not. The casual reader might conclude that this is what is wrong with the tenure system: Irresponsible faculty who are accountable to nobody. But that is not really the nub of the matter.