640 
conception of intellectual discipline is 
neither profound nor well informed but is 
itself in sorry need of discipline. 
If, turning from the world to a normal 
mathematician, you ask him to explain to 
you the human significance of mathematics, 
he will repeat to you the answer of the 
world, of course with far more apprecia- 
tion than the world has of what the answer 
means, and he will supplement the world’s 
response by an important addition. He 
will add, that is, that mathematics is the 
exact science, the science of exact thought 
or of rigorous thinking. By this he will 
not mean what the world would mean if 
the world employed, as sometimes it does 
employ, the same form of words. He will 
mean something very different. Especially 
if he be, as I suppose him to be, a normal 
mathematician of the modern critical type, 
he will mean that mathematics is, in the 
oft-cited language of Benjamin Peirce, 
“‘the science that draws necessary conclu- 
sions;’’ he will mean that, in the felicitious 
words of William Benjamin Smith, 
‘‘mathematies is the universal art apo- 
dictic ;’’ he will mean that mathematics is, 
in the nicely technical phrase of Pieri, ‘‘a 
hypothetico-deductive system.’’ If», you 
ask him whether mathematics is the science 
of rigorous thinking about all the things 
that engage the thought of mankind or 
only about a few of them, such as numbers, 
fizures, certain operations, and the like, the 
answer he will give you depends. If he be 
a normal mathematician of the elder school, 
he will say that mathematics is the science 
of rigorous thinking about only a relatively 
few things and that these are such as you 
have exemplified. And if now, with a 
little Socratic persistence, you press him 
to indicate the human significance of a 
science of rigorous thinking about only a 
few of the countless things that engage 
human thought, his answer will give you 
SCIENCE 
[N.S. Vou. XXXV. No. 904 
but little beyond a repetition of the above- 
mentioned answer of the world. But if he 
be a normal mathematician of the modern 
critical type, he will say that mathematics 
is the science of rigorous thinking about all 
the things that engage human thought, 
about all of them, he will mean, in the 
sense that thinking, as it approaches per- 
fection, tends to assume certain definite 
forms, that these forms are the same what- 
ever the subject matter of the thinking may 
be, and that mathematics is the science of 
these forms as forms. If you respond, as 
you well may respond, that, in accordance 
with this ontological conception of mathe- 
matics, this science, instead of thinking 
about all, thinks about none, of the con- 
erete things of interest to human thought, 
and that accordingly Mr. Bertrand Rus- 
sell was right in saying that ‘‘mathematics 
is the science in which one never knows 
what one is talking about nor whether 
what one says is true’’—if you respond 
that, from the point of view above assumed, 
that delicious mot of Mr. Russell’s must be 
solemnly held as true, and then if, in ac- 
cordance with your original purpose, you 
once more press for an estimation of the 
human significance of such a science, I fear 
that the reply, if your interlocutor is a 
mathematician of the normal type, will con- 
tain little that is new beyond the assertion 
that the science in question is very inter- 
esting, where, by interesting, he means, of 
course, interesting to mathematicians. It 
is true that Professor Klein has said: 
““Apart from the fact that pure mathe- 
matics can not be supplanted by anything 
else as a means for developing the purely 
logical faculties of the mind, there must 
be considered here as elsewhere the neces- 
sity of the presence of a few individuals in 
each country developed in a far higher de- 
gree than the rest, for the purpose of keep- 
ing up and gradually raising the general 
