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-
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-
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-
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).