THE NEW SYMBOLIC LOGIC
=Principia Mathematica.= By =A.~ N.~ Whitehead= and
=B.~ Russell.= Vol.~ I. (Cambridge University Press.
25s.~ net.)
Perhaps twenty or thirty people in England may
be expected to read this book. It has many claims
to be widely read; all professional mathematicians,
for example, can and ought to read it; but it will have to
contend with an immense mass of prejudice and miscon-
ception, and we should probably be over-sanguine if we
supposed that there are half a dozen who will. It is a
strange and discouraging fact that mathematicians as a
class are utterly impatient of inquiries into the founda-
tions of their own subject. Natural science has always
distrusted and despised philosophy. It is hardly too
much to say that any determined attempt at accurate
thinking is likely to be distasteful to the average scientific
mind; and this, although certainly deplorable, is not so
strange as it might seem, for ingenuity and imagination,
rather than accurate thought, are the ordinary weapons
of science. The mathematician has no such excuse. He
has received an elaborate logical training and is familiar
with what is abstract and remote and unpractical; and
it is not unreasonable to expect that he should have learnt
to respect the truth and to tolerate, if not to sympathize
with, those who, like the authors of this book, seek it with a
patience and determination that no difficulties can arrest.
Any such expectation will be disappointed. In England
we find the authors regarded by mathematicians as amus-
ing cranks. In France we find the great Poincar, who has
a weakness for philosophy to which we owe several most
entertaining volumes, pouring contempt on _la Logistique,_
and preaching a form of pragmatism as hazy and elusive
as any philosopher's. Even in Germany, the home of
mathematical precision, we find the successors of Cantor
and Weierstrass protesting angrily that to ask really
fundamental questions is an indecency and an insult to
mathematics. To this sort of expert prejudice it is useless
to appeal. But philosophers are more tolerant than
mathematicians and more interested in fundamentals;
and there are no doubt many philosophers and many
laymen who would like to read this book but may be
deterred from doing so by misconceptions of a different
kind. To such it may be possible to give a little
encouragement.
Non-mathematical readers may very naturally be
frightened by an exaggerated notion of the technical
difficulty of the book. The last page is a very natural
place at which to open a book, and the appearance of the
last page, with its crazy-looking symbols, is appaling.
And it would be silly to pretend that the book is not really
difficult; some of it is very difficult indeed. It is, of
course, also true that in reading it a trained mathematician
enjoys a considerable advantage; some of the ideas will
be more or less familiar to him; he can grasp more quickly
the salient features of a symbolism; the general tone of the
book is mathematical, and there are parallels and illustra-
tions which it is difficult to appreciate without some know-
ledge of mathematics; in a word, the authors'~ minds
work naturally on mathematical lines. So much the lay
reader must be prepared to find. But he need not be too
much discouraged. In the first place, although more
knowledge of mathematics would no doubt be a great
assistance to him, no profound knowledge is required.
All that is wanted is some sort of general familiarity with
mathematical ideas. Moreover the mathematician's ad-
vantage varies very widely from page to page, and it is
just where the book is most difficult that this advantage
is least; the easy parts will be particularly easy to the
mathematician, the hard parts practically as hard to him
as to the outsider. Finally, to be frightened by the sym-
bolism is to run away from the most shadowy of bogeys. A
mathematical reader will find in a surprisingly short time
that, so far from being an additional source of
difficulty, it is extraordinarily easy to understand, to read
fluently, and even to use oneself. The language of modern
symbolic logic, vulgarly described after its originator as
`Peanese,' has been developed under the authors'~ hands
into a vehicle of astonishing flexibility and power, a
real triumph of technical skill, fairly justifying their claim
that it enables the mind `to construct trains of reasoning
in regions of thought in which the imagination would be
entirely unable to sustain itself without symbolic help.'
It is possible that the constant use of symbols has to some
extent reacted on the authors'~ mastery of their own tongue.
Certainly, in the explanatory portions (though there are
parts that are models of lucid exposition) they are often
much less clear; though sometimes, no doubt, this is due
simply to the inherent difficulties of the subject-matter,
and sometimes to condensation carried too far.
We trust, therefore, that no one who is prepared to
recognize the value of work in logic and who appreciates
the enormous importance in logic and general philosophy
of mathematical concepts, _function, class, number, magni-
tude,_ and so forth, will be too diffident of his ability to cope
with the difficulties of this book. It is not a book that many
will read right through. But it will be a foolish philosopher
who will not make a serious effort to master the most essen-
tial sections. The time has passed when a philosopher
can afford to be ignorant of mathematics, and a little perse-
verance will be well rewarded. It will be something to
learn how many of the spectres that have haunted philo-
sophers modern mathematics has finally laid to rest. And
it is as bracing as a cold bath to turn from the muddy
pragmatisms of current philosophy to clear-cut and dis-
passionate discussions in which it is recognized that words
have definite meanings, and that premises imply conclu-
sions, and that careful reasoning is the only method by
which we can hope to arrive at the truth.
The first volume of `Principia Mathematica,' which is
all that has appeared so far, contains most of the work
that is likely to be of interest to the general reader. It
is true that the two later volumes will include the mathe-
matical theories of infinity and continuity, topics which
have distracted philosophers in the past and will no doubt
continue to do so until philosophers make a habit of learn-
ing a little mathematics. But these theories are by now
classical and easily accessible to any one who wishes to
become acquainted with them. And this volume contains
the part of the work most likely to excite general interest
and controversy, an introduction of ninety pages written
in the ordinary tongue and summarizing all that is most
novel in the general doctrines of the book. There is no
doubt that this introduction, dealing as it does with all the
most obscure and controversial portions of the subject,
is from the standpoint of the general reader too short.
Its style is in places almost painfully condensed, and there
are passages that are hardly intelligible until we refer for-
ward to the detailed symbolic development which comes
later. Moreover, the authors, in their desire to `avoid
controversy and general philosophy,' are apt to leave the
reader uncertain as to what (if anything) in the way of general
philosophy their whole treatment presupposes. They
would probably say that the mathematical edifice is inde-
pendent of the precise material with which we fill in the
philosophical foundations; and no doubt in this they would
be right. But they have naturally found it impossible to
carry out their intention of `avoiding general philosophy'
quite consistently; and the result is sometimes very puzzling.
To take a definite instance -- `proposition' must be re-
garded, for the main purposes of the book, as ultimate and
unanalysable. Any analysis of the _meaning_ of `pro-
position' is prior to mathematics. So much the reader
will cheerfully accept. But it will come as an unpleasant
surprise to him (at any rate if he is not familiar with Mr.~
Russell's `Philosophical Essays') to be told (p.~ 46) that
`what we call a `proposition' is not a single entity at
all...; the phrase which expresses a proposition...
does not have a meaning in itself.' It would have been
better, we think, if the authors had made up their minds
to `face the music,' and had begun with a definitely
philosophical excursus -- a mere ha'porth of tar in the outfit
of such a leviathan.
The main thesis of the book is the same as that of Mr.~
Russell's `Principles of Mathematics' -- the thesis that
pure mathematics involves no axioms or indefinables
beyond those of formal logic. And its cardinal doctrines
group themselves roughly into three divisions -- the
general theory of the variable and the proposi-
tional function, the theory of incomplete symbols, the
doctrine of types. None of these doctrines will be found
entirely novel by students of Mr.~ Russell's writings; all
have been formulated or foreshadowed in the `Principles'
or elsewhere. Here they are stated for the first time
with an air of finality and as a connected whole.
The theory of propositional functions shows in a striking
way how certain kinds of ultimate philosophical inquiry
may be irrelevant to mathematics. It is beyond doubt that
the `propositional function' is a notion of extraordinary
logical importance -- mathematics, one may say, is the
science of propositional functions. And it is perfectly easy
to recognize a propositional function when we see it; `_x_
is _x_' is a propositional function; when _x_ is determined --
when for _x_ we substitute Socrates or Plato -- we obtain a
proposition. But _what_ is a propositional function? The
question is `by no means an easy one.' When we say
`_x_ is _x_'
it is plain that, regarded psychologically, we have here a single
judgment. But what are we to say of the object of the judg-
ment? We are not judging that Socrates is Socrates, nor that
Plato is Plato, nor any other of the definite judgments that are
instances of the law of identity,
for we may be quite capable of judging `_x_ is _x_' even
if we have never heard of Socrates or Plato. But we cannot
follow the discussion further now. Our object is to point
out, on the one hand, that the reader must not expect to
find this book free from ultimate doubts, and, on the other,
that the persistence of such doubts need not imperil the
logical superstructure.
The theory of `incomplete symbols' is one of the
authors'~ triumphs; it could hardly be clearer, nor, once
understood, more consonant with common sense. It
cannot be illustrated better than by the old puzzle --
old, that is to say, to readers of Mr.~ Russell -- Of George IV.~
and Scott. George IV.~ wanted to know whether Scott
was the author of `Waverley,' and in point of fact he was.
It seems, therefore, that what George IV.~ really wanted to
know was whether Scott was Scott. For, if `the author of
Waverley' is a definite object _a_, `Scott is the author
of Waverley' means `Scoot is _a._' And this proposition
is either trivial or false; trivial if _a_ is Scott, when `Scott
is _a_' reduces to `Scott is Scott,' and false if _a_ is anything
but Scott. On the other hand, it is perfectly obvious that
`Scott is the author of Waverley' is neither trivial nor
false. From this dilemma there is only one way of escape --
namely, by denying that `the author of Waverley' _is_
a definite object _a_. This is to deny that `the author of
Waverley' means anything, and seems at first paradoxical.
The paradox is one which soon disappears; the point is,
of course, that `the author of Waverley,' and `descrip-
tions' in general, mean nothing _by themselves,_ though
_phrases containing them_ often have a perfectly definite
meaning. This idea is familiar enough to mathematicians,
and in it we have in germ the whole theory of `incom-
plete symbols,' a theory applied by the authors now not
only to descriptions but also to classes and relations, though
here the theory becomes a little more complicated and
elusive.
With the doctrine of types we come to the most difficult,
and perhaps the most controversial, theory of the book.
This doctrine has been invented for the express purpose of
solving a class of puzzles which has tormented generations
of logicians. The classical example is the `Epimenides.'
In its most modern form this paradox is as follows. If
I say `I am lying,' then, if my statement is true, I _am_
lying, and therefore it is false; and if it is false, I am _not_
lying, and therefore it is true. Such paradoxes cannot
(as is often supposed) be accounted for by holding up
one's hands and saying `How absurd!' The doctrine
of types, in which the authors find rest from all these
puzzles, is in a way the least fundamental, and will pro-
bably be found the least satisfying, part of the book; and the
authors are careful not to claim too much finality for their
solution. There can be no doubt, too, that it does involve
consequences likely to startle common sense; it forces us,
for example, to believe that a whole series of common
words, _true_ and _false_ among them, have infinitely many
different meanings. And there are important points about
which the authors leave us doubtful. Are there `infinite
types'? Can one tell a lie `of infinite order'? Is
it really true, as theologians tell us, that `the finite cannot
comprehend the infinite,' or is that a mere logical super-
stition, as De Morgan held? But of two things the authors
have convinced us. One is that _some_ form of some such
doctrine as the doctrine of types is logically indispensable;
the other is that there really _are_ different meanings of
`truth' -- that when I say `it is true that `_x_ is identical
with _x_, for all values of _x_'' and when I say `it is true
that `this is yellow,'' I do not in the two cases mean the
same by _true_. This, we think, they show not only to follow
from their premises, but to be convincing to `expert
common sense'; and when we have admitted this, we have
admitted that the chief paradox of their doctrine has
disappeared.
It would be insulting to affix the ordinary labels of praise
to a book conceived with so far-reaching an object and
on so vast a scale. We may perhaps venture to pick out
a minor feature of the book for commendation. It is easy
to think, but hard to joke, in symbols; and this volume
has not the consistent humour of the `Principles of Mathe-
matics.' Still, considering the difficulty of the medium,
some of the jokes are very good. The best is that perpe-
trated at the expense of the law of contradiction. But
it would be unfair to the circulation of the book that a
reviewer should repeat them; and we leave the reader to
discover them for himself.