10 1 A Taste of Things to Come
despite the immense complexity and power of the brain, the men-
tal processes of mathematics appear to be surprisingly resource-
limited. Therefore I have a feeling that branches of logic developed
for the needs of complexity theory might provide better metaphors
than the general theory of computation.
1.4 Analytic functions and the inevitability of
AEROFLOT flight attendant: “Would you like a dinner?”
Passenger: “And what’s the choice?”
Flight attendant: “Yes—or no.”
We have mentioned at the beginning of our discussion that |x|
is a non-analytic function. It can be written by a single algebraic
|x| =

with the only glitch being that of the two values of the square root

x2 we have to choose the positive one, namely,

One may argue that in the case of the absolute value function
the choice is artificial and is forced on us by the function’s awkward
definition. But let us turn to solutions of algebraic equations, which
give more natural examples of the inevitability of choice.
Chris Hobbs,
aged 6
The classical formula
x1,2 =
−b ±

b2 4ac
for the roots of the quadratic equation is the limit of what
we can do with analytic functions without choosing branches
of multivalued analytic functions—but even here, beware of
complications and read an interesting comment from Chris
Recall that the inverse of the square function x =
is a two-valued function y = ±

x whose graph has two
branches, positive y =

x and negative y =

x. Similarly,
the cube root function y =

x has three distinct branches,
but they become visible only in the complex domain, since
only one cube root of a real number is real; the other two are
obtained from it by multiplying it by complex factors


The classical formula—which can be traced back to Gerolamo Car-
dano (1501–1576) and Niccol` o Tartaglia (1499–1557)—for the roots
of the cubic equation
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