APRIL 26, 1912] 
standard. Even a slight raising of the gen- 
eral level can be accomplished only when 
some few minds have progressed far ahead 
of the average.’’ Here indeed we have, in 
these words of Professor Klein, a hint, if 
only a hint, of something better. But 
Professor Klein is not a mathematician of 
the normal type, he is hypernormal. If, in 
order to indicate the human significance 
of mathematics regarded as the science of 
the forms of thought as forms, your nor- 
mal mathematician were to say that these 
forms constitute, of themselves, an infinite 
and everlasting world whose beauty, 
though it is austere and cold, is pure, and 
in which is the secret and citadel of what- 
ever order and harmony our concrete uni- 
verse contains, it would yet be your right 
and your duty to ask, as the brillant au- 
thor of ‘‘Hast London Visions’’ once 
asked me, namely, what is the human sig- 
nificance of ‘‘this majestic intellectual 
cosmos of yours, towering up like a million- 
lustered iceberg into the arctic night,’’ 
seeing that, among mankind, none is per- 
mitted to behold its more resplendent 
wonders save the mathematician himself? 
But the normal mathematician will not say 
what I have just now supposed him to say; 
he will not say it, because he is, by hypoth- 
esis, a normal mathematician, and because, 
being a normal mathematician, he is ex- 
clusively engaged in exploring the iceberg. 
A farmer was once asked why he raised so 
many hogs. ‘‘In order,’’ he said, ‘‘to buy 
more land.’’ Asked why he desired more 
land, his answer was, ‘‘in order to raise 
more corn.’’ Being asked to say why he 
would raise more corn, he replied that he 
wished to raise more hogs. If you ask the 
normal mathematician why he explores 
the iceberg so much, his answer will be, in 
effect at least, ‘‘in order to explore it 
more.’’ In this exquisite circularity of mo- 
tive, the farmer and the normal mathema- 
SCIENCE 
641 
tician are well within their rights. They 
are within their rights just as a musician 
would be within his rights if he chanced to 
be so exclusively interested in the work of 
composition as never to be concerned with 
having his creations rendered before the 
public and never to attempt a philosophic 
estimate of the human worth of music. 
The distinction involved is not the distine- 
tion between human and inhuman, be- 
tween social and anti-social; it is the dis- 
tinction between what is human or inhu- 
man, social or anti-social, and what is 
neither the one nor the other. No one, I 
believe, may contest the normal mathema- 
tician’s right as a mathematical student or 
investigator to be quite indifferent as to 
the social value or the human worth of his 
activity. Such activity is to be prized just 
as we prize any other natural agency or 
force that, however undesignedly, yet con- 
tributes, sooner or later, directly or indi- 
rectly, to the weal of mankind. The fact is 
that, among motives in research, scientific 
curiosity, which is neither moral nor im- 
moral, is far more common and far more 
potent than charity or philanthropy or 
benevolence. But when the mathematician 
passes from the réle of student or investi- 
gator to the réle of teacher, that right of 
indifference ceases, for he has passed to an 
office whose functions are social and whose 
obligations are human. It is not his privi- 
lege to chill and depress with the encasing 
fogs of the iceberg. It is his privilege and 
his duty, in so far as he may, to disclose its 
“‘million-lustered’’ splendors in all their 
power to quicken and illuminate, to charm 
and edify, the whole mind. 
The conception of mathematics as the 
science of the forms of thought as forms, 
the conception of it as the refinement, pro- 
longation and elaboration of pure logic, is, 
as you are doubtless aware, one of the 
ereat outcomes, perhaps I should say it is 
