Galileo, to the positions of Plato. I do not mean that the whole
programme has been completely executed and that there is nothing for a
successor of Frege or Russell to do. It is instructive to observe that
at the very end of the great work on arithmetic to which I have referred
Frege found himself compelled by difficulties which had been overlooked
until Russell called attention to them to add an appendix confessing
that there was a single important flaw in his elaborate logical
construction of the principles of arithmetic. He had shown that if there
are certain things called 'integers', defined as he had defined them,
the whole of arithmetic follows. But he had not shown that there _is_
any object answering to his definition of an integer, and the logical
researches of Russell had thrown some doubt on the point. This proved
that some restatement of the initial assumptions of the theory was
needed. Since the date of Frege's appendix (1903), Mr. Russell and
others have done something towards the necessary rectification, and the
resulting 'Theory of Types' is pretty certainly one of the most
important contributions ever made to logical doctrine, but it may still
be reasonably doubted whether the 'Theory of Types', as expounded by
Whitehead and Russell in their _Principia Mathematica_, is the last word
required. At any rate, it seems clear that it is a great step on the
right road to the solution of a most difficult problem.

There still remains the greatest problem of all, the harmonization of
Science and Life. I cannot believe that this problem is an illegitimate
one, or that we must sit down content to accept the severance of 'fact'
and 'value' as final for our thought. Even the unification of the
sciences itself remains imperfect so long as we treat it as merely
something which 'happens to be the case' that there are many things and
many kinds of things in the universe and also a number of relations in
which they 'happen' to stand. It is significant that in his later