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SUMMARY:Impossibility and Existence - Jesper Lützen (University of Copenh
 agen)
DTSTART:20250429T091500Z
DTEND:20250429T101500Z
UID:TALK230350@talks.cam.ac.uk
DESCRIPTION: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 mathematic
 ians\, as well as non-mathematicians have had an urge to circumvent imposs
 ibilities. One way to do that\, is by inventing what Hilbert called ideal 
 elements. For example\, Cardano invented the complex numbers as a way to c
 ircumvent the impossible cases of Euclid's elliptic application of areas\,
  or equivalently quadratic equations with negative discriminants. But ther
 e have been other ways to get around impossibilities. A particularly impor
 tant example is the way the ancient Greeks got around the problem of incom
 mensurability in geometry. Their resulting pure geometry\, and in particul
 ar their theory of proportions\, influenced Western mathematics until the 
 17th century. I shall also discuss Beltrami's "real substrate" for non-Euc
 lidean 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 i
 mpossibility 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 existenc
 e theorems) attract special attention. As a conclusion I will point to asp
 ects of my talk that can be viewed as special examples of the subject of o
 ur workshop: "Modern history of mathematics".
LOCATION:Seminar Room 1\, Newton Institute
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