646 
come what they are not. The mathe- 
matical concept of invariance and that of 
infinitude, especially the imposing doc- 
trines that explain their meanings and 
bear their names—what are they but 
mathematicizations of that which has ever 
been the chief of life’s hopes and dreams, 
of that which has ever been the object of 
its deepest passion and of its dominant 
enterprise, I mean the finding of worth 
that abides, the finding of permanence in 
the midst of change, and the discovery of 
the presence, in what has seemed to be a 
finite world, of being that is infinite? It 
is needless further to multiply examples 
of a correlation that is so abounding and 
complete as indeed to suggest a doubt 
whether it be juster to view mathematics 
as the abstract idealization of life than to 
regard life as the concrete realization of 
mathematics. 
Finally, I wish to emphasize the fact 
that the great concepts out of which the 
so-called higher mathematical branches 
have grown—the concepts of variable and 
constant, of function, class and relation, 
of transformation, invariance, and group, 
of finite and infinite, of discreteness, limit, 
and continuity—I wish, in closing, to em- 
phasize the fact that these great ideas of 
the higher mathematics, besides penetra- 
ting life, as we have seen, in all its com- 
plexity and all its dimensions, are omni- 
present, from the very beginning, in the 
elements of mathematics as well. ‘The no- 
tion of group, for example, finds easy and 
beautiful illustration, not only among the 
simpler geometric motions and configura- 
tions, but even in the ensemble of the very 
integers with which we count. The lke 
is true of the distinction of finite and in- 
finite, and of the ideas of transformation, 
of invariant, and nearly all the rest. Why 
should the presentation of them have to 
await the uncertain advent of graduate 
SCIENCE 
[N.S. Vou. XXXV. No. 904 
years of study? For life already abounds, 
and the great ideas that give it its inter- 
ests, order and rationality, that is to say, 
the focal concepts of the higher mathe- 
matics, are everywhere present in the ele- 
ments of the science as glistening bassets 
of gold. It is our privilege, in teaching 
the elements, to avail ourselves of the 
higher conceptions that are present in 
them; it is our privilege to have and to 
give a lively sense of their presence, their 
human significance, their beauty and their 
light. I do not advocate the formal pres- 
entation, in secondary schools, of the 
higher conceptions, in the way of printed 
texts, for the printed text is apt to be arid 
and the letter killeth. What I wish to 
recommend is the presentation of them, as 
opportunity may serve, in Greek fashion, 
by means of dialectic, face to face, voice 
answering to voice, animated with the 
varying moods and motions and accents of 
life—laughter, if you will, and the light- 
ning of wit to cheer and speed the slower 
currents of sober thought. Of dialectic 
excellence, Plato at his best, as in 
““Pheedo’’ or the ‘‘Republic,’’ gives us the 
ideal model and eternal type. But Plato’s 
ways are frequently circuitous, weari- 
some and long. They are ill suited to the 
manners of a direct and undeliberate age; 
and we must find, each for himself, a 
shorter course. Somebody imbued with 
the spirit of the matter, possessed of ample 
knowledge and having, besides, the re- 
quisite skill and verve ought to write a 
book showing, in so far as the printed page 
can be made to show, how naturally and 
swiftly and with what.a delightful sense of 
emancipation and power thought may pass 
by dialectic paths from the traditional 
elements of mathematics both to its larger 
concepts and to a vision of their bearings 
on the higher interests of life. I need not 
say that such a handling of ideas implies 
