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 choice 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 formula |x| = √ x2, with the only glitch being that of the two values of the square root ± √ x2 we have to choose the positive one, namely, x2. 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 2a 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 Hobbs.7 Recall that the inverse of the square function x = y2 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 = 3 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 − 1 2 ± √ 3 2 i. The classical formula—which can be traced back to Gerolamo Car- dano (1501–1576) and Niccol` Tartaglia (1499–1557)—for the roots of the cubic equation

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