THE mathematical treatment of the principles of mathematics, which is the subject of the present work, has arisen from the conjunction of two
different studies, both in the main very modern. On the one hand we have
the work of analysts and geometers, in the way of formulating and systematising
their axioms, and the work of Cantor and others on such matters as the theory
of aggregates. On the other hand we have symbolic logic, which, after a
necessary period of growth, has now, thanks to Peano and his followers,
acquired the technical adaptability and the logical comprehensiveness that are
essential to a mathematical instrument for dealing with what have hitherto
been the beginnings of mathematics. From the combination, of these two
studies two results emerge, namely (1) that what were formerly taken, tacitly
or explicitly, as axioms, are either unnecessary or demonstrable; (2) that the
same methods by which supposed axioms are demonstrated will give valuable
results in regions, such as infinite number, which had formerly been regarded
as inaccessible to human knowledge. Hence the scope of mathematics is
enlarged both by the addition of new subjects and by a backward extension
into provinces hitherto abandoned to philosophy.