  `Categorification' is the process of replacing equations by isomorphisms. We
describe some of the ways a thoroughgoing emphasis on categorification can
simplify and unify mathematics. We begin with elementary arithmetic, where the
category of finite sets serves as a categorified version of the set of natural
numbers, with disjoint union and Cartesian product playing the role of addition
and multiplication. We sketch how categorifying the integers leads naturally to
the infinite loop space Omega^infinity S^infinity, and how categorifying the
positive rationals leads naturally to a notion of the `homotopy cardinality' of
a tame space. Then we show how categorifying formal power series leads to
Joyal's `especes des structures', or `structure types'. We also describe a
useful generalization of structure types called `stuff types'. There is an
inner product of stuff types that makes the category of stuff types into a
categorified version of the Hilbert space of the quantized harmonic oscillator.
We conclude by sketching how this idea gives a nice explanation of the
combinatorics of Feynman diagrams.
