APRIL 26, 1912] 
a matter of temperament. It is much more 
a matter first of knowledge and then of the 
evaluation of the knowledge and of its sub- 
ject. To the winning of a worthy sense of 
the human significance of mathematics two 
things are indispensable, knowledge and 
reflection: knowledge of mathematics and 
reflection upon it. To the winning of such 
a sense it is essential to have the kind of 
knowledge that none but serious students 
of mathematics can gain. Equally essen- 
tial is another thing and this thing stu- 
dents of mathematics in our day do not, or 
do but seldom, gain. I mean the kind of 
insight and the liberality of view that are 
to be acquired only by prolonged contem- 
plation of the nature of mathematics and 
by prolonged reflection upon its relations 
of contrast and similitude to the other 
great forms of spiritual activity. 
The question, though it is a question 
about mathematics, is not a mathematical 
question, it is a philosophical question. 
And just because it is a philosophical ques- 
tion, mathematicians, despite the fact that 
one of the indispensable qualifications for 
considering it is possessed by them alone, 
have in general ignored it. They have, in 
general, ignored it, and their ignoring of it 
may help to explain the curious paradox 
that whilst the world, whose mathematical 
knowledge varies from little to less, has al- 
ways as if instinctively held mathematical 
science in high esteem, it has at the same 
time usually regarded mathematicians as 
eccentric and abnormal, as constituting a 
class apart, as being something more or 
something less than human. It may ex- 
plain, too, I venture to believe it does 
partly explain, both why it is that in the 
universities the number of students at- 
tracted to advanced lectures in mathe- 
matics compared with the numbers drawn 
to advanced courses in some other great 
subjects not inherently more attractive, is 
SCIENCE 
639 
so small; and why it is that, among the 
multitudes who pursue mathematics in the 
secondary schools, only a few find in the 
subject anything like delight. For I do not 
accept the traditional and still current ex- 
planation, that the phenomenon is due to 
a well-nigh universal lack of mathematical 
faculty. I maintain, on the contrary, that 
a vast majority of mankind possess 
mathematical faculty in a very consider- 
able degree. That the average pupil’s in- 
terest in mathematics is but slight, is a 
matter of common knowledge. His lack of 
interest is, in my opinion, due, not to a 
lack of the appropriate faculty in him, but 
to the circumstance that he is a human 
being, whilst mathematics, though it teems 
with human interest, is not presented to 
him in its human guise. 
If you ask the world—represented, let 
us say, by the man in the street or in the 
market place or the field—to tell you its 
estimate of the human significance of 
mathematics, the answer of the world will 
be, that mathematics has given mankind a 
metrical and computatory art essential to 
the effective conduct of daily life, that 
mathematics admits of countless applica- 
tions in engineering and the. natural sci- 
ences, and finally that mathematics is a 
most excellent instrumentality for giving 
mental discipline. Such will be the answer 
of the world. The answer is intelligible, it 
is important, and it is good so far as it 
goes; but it is far from going far enough 
and it is not intelligent. That it is far 
from going far enough will become evident 
as we proceed. That the answer is not in- 
telligent is evident at once, for the first 
part of it seems to imply that the rudi- 
mentary mathematics of the carpenter and 
the counting-house is scientific, which it is 
not; the second part of the answer is but 
an echo by the many of the voice of the 
few; and, as to the final part, the world’s 
