pose that, by way of experiment, I assume that the fourth angle of my
quadrilateral will be acute, or again obtuse, will the body of
conclusions I can now deduce from my set of postulates be free from
contradictions or not? If I really give my mind to the task, cannot I
define a continuous function which is _not_ differentiable? The raising
of the first question led in fact to the discovery of what is called
'non-Euclidean' geometry, the raising of the second has banished from
the text-books of the Calculus the masses of bad reasoning which long
made that branch of mathematics a scandal to logic and led distinguished
philosophers--Kant among them--to suspect that there are hopeless
contradictions in the very foundations of mathematical science.

Now, the effect of such careful scrutiny of first principles is not, of
course, to upset any conclusions which have been correctly drawn from a
set of premisses. All that happens is that the conclusion is no longer
asserted by itself as a truth; what is asserted is that the conclusion
is true _if_ the premisses are true. Thus we no longer assert the
'theorem of Pythagoras' as a categorical proposition; what we assert is
that the theorem follows as a consequence from the assertion of some
half-dozen ultimate postulates which will be found on analysis to be the
premisses of Euclid's proof of his forty-seventh proposition.

To come back to the point I wish to illustrate. The peculiarity of the
philosopher is simply that he still goes on to 'wonder' and ask _Why_
when other persons are ready to leave off. He is less contented than
other men to take things for granted. Of course, he knows that, in the
end, you cannot get away from the necessity of taking something for
granted, but he is anxious to take for granted as few things as
possible, and when he has to take something for granted, he is
exceptionally anxious to know exactly what that something is. De Morgan
tells a story of a very pertinacious controversialist who, being asked
whether he would not at least admit that 'the whole is greater than the
part', retorted, 'Not I, until I see what use you mean to make of the
admission.' I am not sure whether De Morgan quotes this as an ensample
for our following or as a warning for our avoidance, but to my own mind
it is an excellent specimen of the philosophic temper. Until you know
what use is going to be made of your admission, you do not really know
what it is you have admitted. It