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]:
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