22 
PRESIDENT’S ADDRESS — SECTION A. 
not before thought o£ between highly general and by no means funda- 
mental phenomena. In either ease at such a moment all men feel that 
a real conquest has been made. 
Now, one of the most characteristic features, though perhaps 
not universal, of mathematical processes is their precision. Mathe- 
matics is precise in its fundamental data, in its processes, and in its 
results. The mathematician shows that though his work may be full 
of pitfalls for the unwary, yet if ordinary precautions be taken his 
conquests in the realm of knowledge approach far more than those of 
any other science that ideal never reached, called certainty. 
Of the data it is the stability rather than the precision that ought 
to be insisted on. For illustration, it will be sufficient to take just 
those instances which before were used to show the other side of the 
medal. It was pointed out that we can conceive an order of things 
in which the most fundamental of our notions of number were 
violated. But the very fact that, invariably, when the view is first 
presented to an educated man he naturally recoils, and only with 
reluctance admits the new idea, is due to a profound truth. Say to 
that same man that neither he nor anybody else can be legitimately 
certain that the sun will ever rise again, and he will cheerfully assent. 
He will probably add a remark, however, that the slightness of the 
probability of such an accident warrants him in performing all his 
actions to-day on the hypothesis that there will be a to-morrow. The 
stability of our belief that the sun will rise again is enormous. 
Similar, but perhaps in a much lower degree, is the stability of the 
belief that no new human being will appear except as the offspring of 
human parents, Deductions securely derived from such premises as 
these we expect with the most absolute confidence. But the stability 
of these beliefs, and therefore all results deduced from them, sinks 
into insignificance compared with the stability of the belief that there 
is a truth of nature underlying the statement that two and two always 
were and always will be the same thing as four. Perhaps reasons can 
he assigned for the overwhelming confidence that we have in such 
data as these which lie at the base of mathematics. It is sufficient 
for present purposes to place instances of the different kinds of 
universally held beliefs side by side and appeal to the instincts of my 
hearers. You are probably all prepared to acknowledge that there 
can he nothing known if we do not know that the word “four'’ or the 
word “thousand” really means something. 
That other hackneyed illustration used above, our doubt as to 
the nature of the space in which we live, requires rather a different 
treatment. You will say that if it is now an open question whether 
or not Euclid’s ideas of space are correct, surely Euclid’s tyranny 
should he over, and every schoolboy ought to have a happy release. 
As a matter of fact, our scepticism on this point is much better 
educated than on that of the fundamental laws of number. As far as 
I am aware, nobody has yet investigated mathematically systems in 
which those laws do not hold. But with regard to space, several such 
systems have been investigated. Nevertheless, although we have no 
reason to suppose that our space belongs to one rather than another 
of such spaces, I unhesitatingly affirm that if measured by the 
standards of any science other than mathematics, our belief in the fact 
that our space is Euclidean has not received the slightest shock. By 
