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).