as led
to the distinction between the philosopher and the man of science, a
_practical_ distinction between the two makes its appearance. It is
_convenient_ that our knowledge of detail should be steadily extended by
considering the consequences which follow from a given set of postulates
without waiting for the solution of the more strictly philosophical
questions whether our postulates have been reduced to the simplest and
most unambiguous expression, whether the list might not be curtailed by
showing that some of its members which have been accepted on their own
merits can be deduced from the rest, or again enlarged by the express
addition of principles which we have all along been using without any
actual formulation of them. The point may be illustrated by considering
the set of 'postulates' explicitly made in the geometry of Euclid. We
cannot be said to have made geometry thoroughly scientific until we know
whether the traditional list of postulates is complete, whether some of
the traditional postulates might not be capable of demonstration, and
whether geometry as a science would be destroyed by the denial of one or
more of the postulates. But it would be very undesirable to suspend
examination of the consequences which follow from the Euclidean
postulates until we have answered all these questions. Even in pure
mathematics one has, in the first instance, to proceed tentatively, to
venture on the work of drawing inferences from what seem to be plausible
postulates before one can pass a verdict on the merits of the postulates
themselves. The consequence of this tentative character of our
inquiries is that, so far as there is a difference between Philosophy
and Science at all, it is a difference in _thoroughness_. The more
philosophic a man's mind is, the less ready will he be to let an
assertion pass without examination as obviously true. Thus Euclid makes
a famous assumption--the 'parallel-postulate'--which amounts to the
assertion that if three of the angles of a rectilinear quadrilateral are
right angles, the fourth will be a right angle. The mathematicians of
the eighteenth and early nineteenth centuries, again, generally assumed
that if a function is continuous it can always be differentiated. A
comparatively unphilosophical mind may let such plausible assertions
pass unexamined, but a more philosophical mind will say to itself, when
it comes across them, 'You great duffer, aren't you going to ask _Why_?'
Sup