Abstract

In a recently published book I have dealt with the history of impossibility in mathematics. In the talk I will combine this subject with the other side of the coin namely the subject of existence in mathematics. I shall discuss a series of examples from different time periods showing how impossibility and existence have gone hand in hand. Indeed mathematicians, as well as non-mathematicians have had an urge to circumvent impossibilities. One way to do that, is by inventing what Hilbert called ideal elements. For example, Cardano invented the complex numbers as a way to circumvent the impossible cases of Euclid’s elliptic application of areas, or equivalently quadratic equations with negative discriminants. But there have been other ways to get around impossibilities. A particularly important example is the way the ancient Greeks got around the problem of incommensurability in geometry. Their resulting pure geometry, and in particular their theory of proportions, influenced Western mathematics until the 17th century. I shall also discuss Beltrami’s “real substrate” for non-Euclidean geometry. This “model”, to use a later phrase, was interpreted by some mathematicians as a proof that non-Euclidean geometry exists and by others as a proof that it does not exist. I shall end by discussing Blake’s existence theorem concerning voting procedures and Arrow’s more famous impossibility theorem. In this connection I will try to answer the question: When did impossibility theorems achieve full citizenship in mathematics, and when did existence theorems (in particular non-constructive existence theorems) attract special attention. As a conclusion I will point to aspects of my talk that can be viewed as special examples of the subject of our workshop: “Modern history of mathematics”.