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THURSDAY, DECEMBER 18, 10913. 
PRINCIPLES OF MATHEMATICS. 
Principia Mathematica. By Dr. A. N. White- 
head, F.R.S., and Bertrand Russell, F.R.S. 
Vol. iii. Pp. x+491. (Cambridge: University 
Press, 1913.) Price 21s. net. 
HE third volume of this work has followed 
very closely upon the second, which was 
only published last year, and is in every respect 
a worthy successor to it. It is mainly concerned 
with the theory of series, which was begun in the 
second volume, and then proceeds to the theory 
of measurement. A further and final volume will 
deal with geometry. To some extent the treat- 
ment has been influenced by the coming volume, 
especially in the section devoted to the theory of 
measurement. For the same reason, a special 
section is included, containing the theory of cyclic 
families, such as the angles about a given point 
in a given plane. 
That the monumental task which the authors 
have undertaken should already have reached this 
stage is almost incredible. For they are in effect 
creating a new science, with a symbolism of its 
own, quite foreign to mathematics, which develops 
naturally as the work proceeds. It is scarcely 
necessary to point out that the Principia does 
not concern itself with the development of mathe- 
matics, as understood by the mathematician, but 
solely with the logical deduction of the proposi- 
tions of mathematics from merely logical founda- 
tions. It represents, in one aspect, the culmina- 
tion of the movement, which has swept over 
mathematics of late years, towards a rigorous 
examination of its fundamental premises. To such 
a work there is always the disadvantage inherent 
in a new symbolism, but a symbolism is essential 
to its development, and the authors employ the 
method which inflicts the minimum of labour on 
the reader: no symbol or abbreviation is em- 
ployed until it becomes essential, and then its very 
recurrence fixes it in the mind of the reader. 
The general scope of the volume has been indi- 
cated already, and it only remains to consider the 
detailed treatment adopted. Well-ordered series 
are considered first, as possessing many impor- 
tant properties not shared by series in general. 
In particular, they obey a process of transfinite 
induction, which is an extended form of mathe- 
matical induction, differing, however, in the fact 
that it deals with the successors of classes instead 
of single terms. On the whole, Cantor’s’ treat- 
ment is followed closely, but an exception is made 
in dealing with Zermolo’s theorem, and in the 
NO. 2303, VOL. 92] 
cases where Cantor assumes the multiplicative 
axiom. The writers emphasise the dubious char- 
acter of much of the ordinary theory of trans- 
finite ordinals, depending on the fact that it is 
founded on a proposition requiring this axiom. 
Ordinal numbers are defined as the relation- 
numbers of well-ordered series, after Cantor, 
serial numbers being the relation-numbers of series 
in general. Products of an ordinal number of 
ordinal numbers are not in general ordinal num- 
bers, although the sums are. The treatment of 
sums and products contains much new matter. 
Perhaps the most interesting part of the work is 
the authors’ solution of the paradox proposed by 
Burali-Forti in 1897, relating to the greatest 
ordinai number. It appears that in any one type 
there is no greatest ordinal number, and that all 
the ordinal numbers of a given type are exceeded 
by those of higher types. 
An important section is concerned with the dis- 
tinction of finite and infinite as applied to series 
and ordinals. The distinguishing properties of 
finite ordinals are then established. It does not 
appear that a proof can be found of the existence 
of alephs or ’s with infinite suffixes. For the 
type increases with each successive existence- 
theorem, and infinite types appear to have no 
meaning. The treatment of the theory of ratio 
and measurement is quite new. The quantities 
are regarded as “vectors” in a generalised sense, 
so that ratios can hold between relations. The 
hypothesis that the vectors concerned in any con- 
text form a group is not prominent. The theory 
of measurement is a combination of two other 
theories, one a pure arithmetic of ratios and real 
numbers, and the other a pure theory of vectors. 
If the axiom of infinity is assumed, great diff- 
culties in connection with the existence-theorems 
are avoided. But the authors have endeavoured to 
get rid of the assumption, for, as they point out, 
it does not seem proper to make the theory of a 
simple ratio like 2/3 depend on the fact that the 
universe contains an infinite number of objects. 
The theory of ratio and measurement is actually 
| the most important part of the volume, but it is 
impossible in a brief review to do justice to it. 
Yet it must be said that the publication of this 
volume is a landmark in the theory, and the 
authors have earned the sincere thanks of all 
mathematicians who are interested in the logical 
foundations of their subject. The printing must 
have been a peculiarly difficult task, on account 
of the nature of the symbols, and the Cambridge 
University Press is to be congratulated on, the 
| manner in which this work, like its predecessor, 
has been produced. 
Kk 
