108 
he meets an allusion to the foundation of the 
theory of the real variable, or to the founda- 
tion of Euclidean geometry or of projective 
geometry or of Mengenlehre or of some other 
branch of mathematics. The founding, in the 
sense indicated, of various distinct branches 
of mathematics is one of the great outcomes 
of a century of critical activity in the science. 
It has engaged and still engages the best 
efforts of men of genius and men of talent. 
Such activity is commonly described as fun- 
damental. It is very important, but funda- 
mental in a strict sense it is not. For one no 
sooner examines the foundations that have 
been found for various mathematical branches 
and thereby as well as otherwise gains a deep 
conviction that these branches are constitu- 
ents of something different from any one of 
them and different from the mere sum or 
collection of all of them than the question 
supervenes whether it may not be possible to 
discover a foundation for mathematics itself 
such that the above-indicated branch founda- 
tions would be seen to be, not fundamental 
to the science itself, but a genuine part of the 
superstructure. That question and the at- 
tempt to answer it are fundamental strictly. 
The question was forced upon mathematicians 
not only by developments within the tradi- 
tional field of mathematics, but also inde- 
pendently from developments in a field long 
regarded as alien to mathematics, namely, the 
field of symbolic logic. The emancipation of 
logic from the yoke of Aristotle very much 
resembles the emancipation of geometry from 
the bondage of Euclid; and, by its subsequent 
growth and diversification, logic, less abun- 
dantly perhaps but not less certainly than 
geometry, has illustrated the blessings of free- 
dom. When modern logic began to learn from 
such a man as Leibniz (who with the most 
magnificent expectations devoted much of his 
life to researches in the subject) the immense 
advantage of the systematic use of symbols, 
it soon appeared that logic could state many 
of its propositions in symbolic form, that it 
could prove some of these, and that the dem- 
onstration could be conducted and expressed 
in the language of symbols. Evidently such a 
SCIENCE 
[N.S. Vou. XXXV. No. 896 
logic looked like mathematics and acted like it. 
Why not call it mathematics? Evidently it 
differed from mathematics in neither spirit 
nor form. If it differed at all, it was in re- 
spect of content. But where was the decree 
that the content of mathematics should be 
restricted to this or that, as number or space? 
No one could find it. If traditional mathe- 
matics could state and prove propositions 
about number and space, about relations of 
numbers and of space configurations, about 
classes of numbers and of geometric entities, 
modern logic began to prove propositions about 
propositions, relations and classes, regardless 
of whether such propositions, relations and 
classes have to do with number and space or 
any other specific kind of subject. At the 
same time what was admittedly mathematics 
was by virtue of its own inner developments 
transcending its traditional limitations to 
number and space. The situation was un- 
mistakable: traditional mathematics began to 
look like a genuine part of logic and no longer 
like a separate something to which another 
thing called logic applied. And so modern 
logicians by their own researches were forced 
to ask a question, which under a thin disguise 
is essentially the same as that propounded by 
the bolder ones among the critical mathema- 
ticians, namely, is it not possible to discover 
for logic a foundation that will at the same 
time serve as a foundation for mathematics as 
a whole and thus render unnecessary (and 
strictly impossible) separate foundations for 
separate mathematical branches? 
It is to answer that great question that 
Messrs. Whitehead and Russell have written 
“Principia Mathematica ”—a work consisting 
of three large volumes, the first being in 
hand, the second and third soon to appear— 
and the answer is affirmative. The thesis is: 
it is possible to discover a small number of 
ideas (to be called primitive ideas) such that 
all the other ideas in logic (including mathe- 
matics) shall be definable in terms of them, 
and a small number of propositions (to be 
called primitive propositions) such that all 
other propositions in Jogic (including mathe- 
matics) can be demonstrated by means of 
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