Mathematical induction is a beautiful tool by which one is able to
prove infinitely many things with a finite amount of paper and ink. It
works by exploiting underlying structure: a complex and unwieldy
problem can sometimes be broken apart along its fault lines so as to
leave behind many smaller problems, each of which is more easily
solved. Induction is one method of finding such fault lines and
organizing the smaller pieces of a larger problem. Often, the simpler
structure of the small pieces permeates the whole, and a complicated
structure can be seen to operate based on the same simple rules that
govern its pieces. This can lead to results that are both powerful and
counterintuitive.
Induction
is only one of many techniques through which one may attempt to wrestle
with infinity in finite terms (which is to say: with home field
advantage), but it holds a rather distinguished position in
mathematics. Conveniently, it requires very little background knowledge
to learn, and for this reason it is often taught in high school and
could reasonably be included in an elementary school curriculum. Its
home is in the natural
numbers
: 1, 2, 3, 4, . . ., which are, barring geometrical objects, arguably
the most intuitive of all mathematical ob ects. Despite its
apparent simplicity, its use in contemporary mathematics is widespread.
But perhaps most tellingly, a casual lunchtime conversation with my
colleagues about induction revealed that everyone seemed to have their
own “induction story”, a tale of their first encounter with or first
appreciation of mathematical induction. It is clear that induction
holds a special place in the mathematician’s heart, and so it is no
surprise that it can be the source of so much beauty, confusion, and
surprise.

