638 
enough to regard as having some sort of 
likeness to a message. 
My conviction is, that hope of improve- 
ment in mathematics teaching, whether in 
secondary schools or in colleges, lies mainly 
in the possibility of humanizing it. It is 
worth while to remember that our pupils 
are human beings. What it means to be a 
human being we all of us presumably 
know pretty well; indeed we know it so 
well that we are unable to tell it to one 
another adequately; and, just because we 
do so well know what it means to be a hu- 
man being, we are prone to forget it as we 
forget, except when the wind is blowing, 
that we are constantly immersed in the 
earth’s atmosphere. To humanize the 
teaching of mathematics means so to pre- 
sent the subject, so to interpret its ideas 
and doctrines, that they shall appeal, not 
merely to the computatory faculty or to the 
logical faculty but to all the great powers 
and interests of the human mind. That 
mathematical ideas and doctrines, whether 
they be more elementary or more advanced, 
admit of such a manifold, liberal and stim- 
ulating interpretation, and that therefore 
the teaching of mathematics, whether in 
secondary schools or in colleges, may be- 
come, in the largest and best sense, human, 
IT have no doubt. That mathematical ideas 
and doctrines do but seldom receive such 
interpretation and that accordingly the 
teaching of mathematics is but seldom, in 
the largest and best sense, human, I be- 
lieve to be equally certain. That the indi- 
cated humanization of mathematical teach- 
ing, the bringing of the matter and the 
spirit of mathematics to bear, not merely 
upon certain fragmentary faculties of the 
mind, but upon the whole mind, that this 
is a great desideration is, I assume, beyond 
dispute. 
How can such humanization be brought 
about? The answer, I believe, is not far 
SCIENCE 
[N.S. Vou. XXXV. No. 904 
to seek. I do not mean that the answer is 
easy to discover or easy to eommunicate. 
I mean that the game is near at hand and 
that it is not difficult to locate it, though it 
may not be easy to capture it. The diffi- 
culty inheres, I believe, in our conception 
of mathematics itself; not so much in our 
conception of what mathematics, in a defi- 
nitional sense, is, for that sense of what 
mathematics is has become pretty clear in 
our day, but in our sense or want of sense 
of what mathematics, whatever it may be, 
humanly signifies. In order to humanize 
mathematical teaching it is necessary, and 
I believe it is sufficient, to come under the 
control of a right conception of the human 
significance of mathematics. It is suffi- 
cient, I mean to say, and it is necessary, 
greatly to enlarge, to enrich and to vitalize 
our sense of what mathematics, regarded as 
human enterprise, signifies. 
What does mathematics, regarded as an 
enterprise of the human spirit, signify? 
What is a just and worthy sense of the hu- 
man significance of mathematics? 
To the extent in which any of us really 
succeeds in answering that question worth- 
ily, his teaching will have the human qual- 
ity, in so far as his teaching is, in point of 
external circumstance, free to be what it 
would. I believe it is important to put the 
question, and it is with the putting of it 
rather than with the proposing of an an- 
swer to it that I am here at the outset 
mainly concerned. For any one who is 
really to acquire possession of an answer 
that is worthy must win the answer for 
himself. I need not say to you that such 
an acquisition as a worthy answer to this 
kind of question does not belong to the 
category of things that may be lent or bor- 
rowed, sold or bought, donated or acquired 
by gift. No doubt the answers we may 
severally win will differ as our tempera- 
ments differ. Yet the matter is not solely 
