JANUARY 19, 1912] 
them. Of course, not all ideas can be defined 
—some must be assumed as a working stock— 
and those called primitive are so called merely 
because they are taken without definition; 
similarly for propositions, not all can be 
proved, and those called primitive are so 
called because they are assumed. It is not 
contended by the authors (as it was by Leib- 
niz) that there exist ideas and propositions 
that are absolutely primitive in a metaphys- 
ical sense or in the nature of things; nor do 
they contend that but one sufficient set of 
primitives (in their sense of the term) can be 
discovered. In view of the immeasurable 
wealth of ideas and propositions that enter 
logic and mathematics, the authors’ thesis is 
very imposing; and their work borrows some 
of its impressiveness from the magnificence 
of the undertaking. Jt is important to ob- 
serve that the thesis is not a thesis of logic 
or of mathematics, but is a thesis about logic 
and mathematics. It can not be proved syl- 
logistically; the only available method is that 
by which one proves that one can jump 
through a hoop, namely, by actually jumping 
through it. Jf the thesis be true, the only 
way to establish it as such is to produce the 
required primitives and then to show their 
adequacy by actually erecting upon them as a 
basis the superstructure of logic (and mathe- 
matics) to such a point of development that 
any competent judge of such architecture will 
say: “Enough! Iam convinced. You have 
proved your thesis by actually performing the 
deed that the thesis asserts to be possible.” 
And such is the method the authors have 
employed. The labor involved—or shall we 
eall it austere and exalted play?—was im- 
mense. They had predecessors, including 
themselves. Among their earlier works Rus- 
sell’s “Principles of Mathematics” and White- 
head’s “Universal Algebra” are known to 
many. The related works of their predeces- 
sors and contemporaries, modern critical 
mathematicians and modern logicians, Weier- 
strass, Cantor, Boole, Peano, Schréder, Peirce 
and many others, including their own former 
_ selves, had to be digested, assimilated and 
transcended. All this was done, in the course 
SCIENCE 
109 
of more than a score of years; and the work 
before us is a noble monument to the authors’ 
persistence, energy, acumen and idealism. A 
people capable of such a work is neither crawl- 
ing on its belly nor completely saturated with 
commercialism nor wholly philistine. There 
are preliminary explanations in ordinary lan- 
guage and summaries and other explanations 
are given in ordinary language here and there 
throughout the book, but the work proper is 
all in symbolic form. Theoretically the use 
of symbols is not necessary. A sufliciently 
powerful god could have dispensed with them, 
and, unless he were a divine spendthrift, he 
would have done so, except perhaps for the 
reason that whatever is feasible should be done 
at least once in order to complete the possible 
history of the world. But whilst the employ- 
ment of symbols is theoretically dispensable, it 
is, for man, practically indispensable. Many 
of the results in the work before us could not 
have been found without the help of symbols, 
and even if they could have been thus found, 
their expression in ordinary speech, besides 
being often unintelligible, owing to complex- 
ity and involution, would have required at 
least fifteen large volumes instead of three. 
Fortunately the symbology is both interesting 
and fairly easy to master. The difficulty in- 
heres in the subject itself. 
The initial chapter, devoted to preliminary 
explanations that any one capable of nice 
thinking may read with pleasure and profit, 
is followed by a chapter of 30 pages dealing 
with “the theory of logical types.” Mr. Rus- 
sell has dealt with the same matter in volume 
30 of the American Journal of Mathematics 
(1908). One may or may not judge the the- 
ory to be sound or adequate or necessary and 
yet not fail to find in the chapter setting it 
forth both an excellent example of analytic 
and constructive thinking and a worthy model 
of exposition. The theory, which, however, 
is recommended by other considerations, orig- 
inated in a desire to exclude from logic auto- 
matically by means of its principles what are 
called illegitimate totalities and therewith a 
subtle variety of contradiction and vicious 
circle fallacy that, owing their presence to the 
