12 1 A Taste of Things to Come Pronic root is as you say, 9 times 9 makes 81. And now take the root of 9, which is 3, and this 3 is added above 81, so that the pronic root of 84 is said to be 3. In effect, Muscharello wanted to introduce the inverse of the func- tion z → z4 + z. Arnold’s theorem explains why such tricks could not lead to an easy solution of cubic and quadric equations and why it had been aban- doned. 1.5 You name it—we have it This section is more technical and can be skipped. As I have already said on several occasions, this book is about simple atomic objects and processes of mathematics. However, mathematics is huge and immensely rich even the simplest ob- servations about its simplest objects may already have been de- veloped into sophisticated and highly specialized theories. Mathe- matics’ astonishing cornucopian richness and its bizarre diversity are not frequently mentioned in works on philosophy and method- ology of mathematics—but this point has to be emphasized, since its makes the question about unity of mathematics much more in- teresting. In this section, I will briefly describe a “mini-mathematics”, a mathematical theory concerned with a close relative of the abso- lute value function, the maximum function of two variables z = max(x, y). Of course, the absolute value function |x| can be expressed as |x| = max(x, −x). Similarly, the maximum max(x, y) can be expressed in terms of the absolute value |x| and arithmetic operations—I leave it to the reader as an exercise. [?] Oh yes, do it. The theory is known by the name of tropical mathematics. The strange name has no deep meaning: the adjective “tropical” was coined by French mathematicians in honor of their Brazil- ian colleague Imre Simon, one of the pioneers of the new disci- pline. Tropical mathematics works with the usual real numbers but uses only two operations: addition, x + y, and taking the maximum, max(x, y)—therefore it is one of the extreme cases of “switch-flipping”, choice-based mathematics. Notice that addition is distributive with respect to taking maximum: a + max(b, c) = max(a + b, a + c).

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