Bertrand Russell, in his book The Principles of Mathematics,
proposes the following as a definition of pure
mathematics.
Pure Mathematics is the class of all propositions of
the form “p implies q,” where p and q are propositions
containing one or more variables, the same in
the two propositions, and neither p nor q contains any
constants except logical constants. And logical constants
are all notions definable in terms of the following:
Implication, the relation of a term to a class
of which it is a member, the notion of such that, the
notion of relation, and such further notions as may be
involved in the general notion of propositions of the
above form. In addition to these, mathematics uses a
notion which is not a constituent of the propositions
which it considers, namely the notion of truth.
The Princeton Companion to Mathematics could be said
to be about everything that Russell’s definition leaves
out.
Russell’s book was published in 1903, and many
mathematicians at that time were preoccupied with the
logical foundations of the subject. Now, just over a century
later, it is no longer a new idea that mathematics
can be regarded as a formal system of the kind that
Russell describes, and today’s mathematician is more
likely to have other concerns. In particular, in an era
where so much mathematics is being published that
no individual can understand more than a tiny fraction
of it, it is useful to know not just which arrangements
of symbols form grammatically correct mathematical
statements, but also which of these statements deserve
our attention.
Of course, one cannot hope to give a fully objective
answer to such a question, and different mathematicians
can legitimately disagree about what they
find interesting. For that reason, this book is far less
formal than Russell’s and it has many authors with
many different points of view. And rather than trying
to give a precise answer to the question, “What makes a
mathematical statement interesting?” it simply aims to
present for the reader a large and representative sample
of the ideas that mathematicians are grappling with
at the beginning of the twenty-first century, and to do
so in as attractive and accessible a way as possible.
