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THE NATURE OF MATHEMATICS 85 

x n + y n z'\ if n be an integer greater than 2. This theorem 
has been proved to be true for n = 3, 4, 5, 7, and many other 
numbers, and there is no reason to doubt that it is true. But 
to this day, no general proof of it has been given. 1 This, then, 
is an example of a mathematical proposition which has been 
reached and stated as probably true by induction. 

Now, in Greek geometry, propositions were stated and 
proved by the laws of Logic helped, as we now know, by tacit 
appeals to the conclusions which common sense draws from the 
pictorial representation in the mind of geometrical figures 
about any triangles, say, or some triangles, and thus not about 
one or two particular things but about an infinity of them. 
Then, consider any two triangles ABC and DEF. It helps 
the thinking of most of us to draw pictures of particular 
triangles, but our conclusions do not hold merely for these 
triangles. If the sides BA and AC are equal in length to the 
sides ED and DP respectively, and the angle at A is equal to 
the angle at D, then BC is equal to EF. This is proved 
rather imperfectly in the fourth proposition of the first Book 
of Euclid's Elements. 

When we examine into and complete the reasonings of 
geometricians, we find that the conception of space vanishes, 
and that we are left with logic alone. Philosophers and 
mathematicians used to think and some do now that, in 
geometry, we had to do, not with the space of ordinary life in 
which our houses stand and our friends move about, and which 
certain quaint people say is "annihilated" by electric tele- 
graphs or motor cars, but an abstract form of the same thing 
from which all that is personal or material has disappeared, 
and only things like distance and order and position have re- 
mained. Indeed, some have thought that position did not 
remain ; that, in abstract space, a circle, for example, had no 
position of its own, but only with respect to other things. 
Obviously, we can only, in practice, give the position of a thing 
with respect to other things "relatively" and not "abso- 
lutely." These " relativists " denied that position had any pro- 
perties which could not be practically discovered. Relativism, 

1 This is an example of the " theory of numbers," the study of 
the properties of integers, to which the chief contributions, per- 
haps, have been made by Fermat and Gauss.