1.4 Analytic functions and the inevitability of choice 11 x3 + ax2 + bx + c = 0 gives its three roots as − a 3 + 3 −2a3 + 9ab − 27c + (2a3 − 9ab + 27c)2 + 4(a2 + 3b)3 54 + 3 −2a3 + 9ab − 27c − (2a3 − 9ab + 27c)2 + 4(a2 + 3b)3 54 , − a 3 + −1 − i √ 3 2 3 −2a3 + 9ab − 27c + (2a3 − 9ab + 27c)2 + 4(a2 + 3b)3 54 + −1 + i √ 3 2 3 −2a3 + 9ab − 27c − (2a3 − 9ab + 27c)2 + 4(a2 + 3b)3 54 , − a 3 + −1 + i √ 3 2 3 −2a3 + 9ab − 27c + (2a3 − 9ab + 27c)2 + 4(a2 + 3b)3 54 + −1 − i √ 3 2 3 −2a3 + 9ab − 27c − (2a3 − 9ab + 27c)2 + 4(a2 + 3b)3 54 . Please notice the carefully choreographed choice of the branches of the square root √ and the cube root function 3 √ , the rhythmic dance of pluses and minuses. Without that choice, Cardano’s formula pro- duces too many values, only three of which are true roots. Indeed, if we work with multivalued functions without making any distinction between their branches, we have to accept that the superposition of an m-valued function and an n-valued function has mn values. We cannot collect like terms: an innocent looking expression like √ x + √ 9x defines, if we interpret “ √ x ” as two-valued, a function with four branches ± √ x ± √ 9x = { −4 √ x, −2 √ x, 2 √ x, 4 √ x }. It is a rigorous mathematical fact [307] that solutions of equa- tions of degree higher than two cannot be analytically expressed by choiceless multivalued formulae (even if we allow for more so- phisticated analytic functions than radicals) see a discussion of the topological nature of this fact by Vladimir Arnold [3, p. 38]. This last observation is especially interesting in the historic context. At the early period of development of symbolic algebra, mathematicians were tempted to introduce functions more general than roots. The following extract from Pierpaolo Muscharello’s Al- gorismus from 1478 is taken from Jens Høyrup [54]:

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2010 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.