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]: