The philosophy which I advocate is generally regarded as a species of realism, and accused of inconsistency because of the elements in it which seem contrary to that doctrine. For my part, I do not regard the issue between realists and their opponents as a fundamental one; I could alter my view on this issue without changing my mind as to any of the doctrines upon which I wish to lay stress. I hold that logic is what is fundamental in philosophy, and that schools should be characterized rather by their logic than by their metaphysic. My own logic is atomic, and it is this aspect upon which I should wish to lay stress.

Therefore I prefer to describe my philosophy as “logical atom-ism”, rather than as “realatom-ism”, whether with or without some preﬁxed adjective.

A few words as to historical development may be useful by way of preface. I came to philosophy through mathematics, or rather through the wish to ﬁnd some reason to believe in the truth of mathematics. From early youth, I had an ardent desire to believe that there can be such a thing as knowledge, combined with a great diﬃculty in accepting much that passes as know-ledge. It seemed clear that the best chance of ﬁnding indubitable truth would be in pure mathematics, yet some of Euclid’s axioms were obviously doubtful, and the inﬁnitesimal calculus,

as I was taught it, was a mass of sophisms, which I could not
bring myself to regard as anything else. I saw no reason to doubt
the truth of arithmetic, but I did not then know that arithmetic
can be made to embrace all traditional pure mathematics. At the
*age of eighteen I read Mill’s Logic, but was profoundly *
dissatis-ﬁed with his reasons for accepting arithmetic and geometry. I
had not read Hume, but it seemed to me that pure empiricism
(which I was disposed to accept) must lead to scepticism rather
than to Mill’s support of received scientiﬁc doctrines. At
*Cam-bridge I read Kant and Hegel, as well as Mr. Bradley’s Logic,*
which inﬂuenced me profoundly. For some years I was a
dis-ciple of Mr. Bradley, but about 1898 I changed my views, largely
as a result of arguments with G. E. Moore. I could no longer
believe that knowing makes any diﬀerence to what is known.

Also I found myself driven to pluralism. Analysis of mathemat-ical propositions persuaded me that they could not be explained as even partial truths unless one admitted pluralism and the reality of relations. An accident led me at this time to study Leibniz, and I came to the conclusion (subsequently conﬁrmed by Couturat’s masterly researches) that many of his most charac-teristic opinions were due to the purely logical doctrine that every proposition has a subject and a predicate. This doctrine is one which Leibniz shares with Spinoza, Hegel, and Mr. Bradley;

it seemed to me that, if it is rejected, the whole foundation for the metaphysics of all these philosophers is shattered. I therefore returned to the problem which had originally led me to phil-osophy, namely, the foundations of mathematics, applying to it a new logic derived largely from Peano and Frege, which proved (at least, so I believe) far more fruitful than that of traditional philosophy.

In the ﬁrst place, I found that many of the stock philosophical arguments about mathematics (derived in the main from Kant) had been rendered invalid by the progress of mathematics in the meanwhile. Non-Euclidean geometry had undermined the

argument of the transcendental aesthetic. Weierstrass had shown that the diﬀerential and integral calculus do not require the con-ception of the inﬁnitesimal, and that, therefore, all that had been said by philosophers on such subjects as the continuity of space and time and motion must be regarded as sheer error. Cantor freed the conception of inﬁnite number from contradiction, and thus disposed of Kant’s antinomies as well as many of Hegel’s.

Finally Frege showed in detail how arithmetic can be deduced from pure logic, without the need of any fresh ideas or axioms, thus disproving Kant’s assertion that “7+5 = 12” is synthetic—

at least in the obvious interpretation of that dictum. As all these results were obtained, not by any heroic method, but by patient detailed reasoning, I began to think it probable that philosophy had erred in adopting heroic remedies for intellectual diﬃcul-ties, and that solutions were to be found merely by greater care and accuracy. This view I had come to hold more and more strongly as time went on, and it has led me to doubt whether philosophy, as a study distinct from science and possessed of a method of its own, is anything more than an unfortunate legacy from theology.

Frege’s work was not ﬁnal, in the ﬁrst place because it
applied only to arithmetic, not to other branches of
math-ematics; in the second place because his premisses did not
exclude certain contradictions to which all past systems of
formal logic turned out to be liable. Dr. Whitehead and I in
*collaboration tried to remedy these two defects, in Principia*
*Mathematica, which, however, still falls short of ﬁnality in some*
fundamental points (notably the axiom of reducibility). But in
spite of its shortcomings I think that no one who reads this
book will dispute its main contention, namely, that from
cer-tain ideas and axioms of formal logic, by the help of the logic
of relations, all pure mathematics can be deduced, without
any new undeﬁned idea or unproved propositions. The
tech-nical methods of mathematical logic, as developed in this

book, seem to me very powerful, and capable of providing a new instrument for the discussion of many problems that have hitherto remained subject to philosophic vagueness. Dr.

*Whitehead’s Concept of Nature and Principles of Natural Knowledge may*
serve as an illustration of what I mean.

When pure mathematics is organized as a deductive sys-tem—i.e. as the set of all those propositions that can be deduced from an assigned set of premisses—it becomes obvi-ous that, if we are to believe in the truth of pure mathematics, it cannot be solely because we believe in the truth of the set of premisses. Some of the premisses are much less obvious than some of their consequences, and are believed chieﬂy because of their consequences. This will be found to be always the case when a science is arranged as a deductive system. It is not the logically simplest propositions of the system that are the most obvious, or that provide the chief part of our reasons for believing in the system. With the empirical sciences this is evident. Electro-dynamics, for example, can be concentrated into Maxwell’s equations, but these equations are believed because of the observed truth of certain of their logical con-sequences. Exactly the same thing happens in the pure realm of logic; the logically ﬁrst principles of logic—at least some of them—are to be believed, not on their own account, but on account of their consequences. The epistemological question:

“Why should I believe this set of propositions?” is quite
diﬀerent from the logical question: “What is the smallest and
logically simplest group of propositions from which this set
of propositions can be deduced?” Our reasons for believing
logic and pure mathematics are, in part, only inductive and
*probable, in spite of the fact that, in their logical order, the*
propositions of logic and pure mathematics follow from the
premisses of logic by pure deduction. I think this point
important, since errors are liable to arise from assimilating the
logical to the epistemological order, and also, conversely, from

assimilating the epistemological to the logical order. The only
way in which work on mathematical logic throws light on the
truth or falsehood of mathematics is by disproving the supposed
*antinomies. This shows that mathematics may be true. But to*
*show that mathematics is true would require other methods and*
other considerations.

One very important heuristic maxim which Dr. Whitehead and I found, by experience, to be applicable in mathematical logic, and have since applied to various other ﬁelds, is a form of Occam’s Razor. When some set of supposed entities has neat logical properties, it turns out, in a great many instances, that the supposed entities can be replaced by purely logical structures composed of entities which have not such neat properties. In that case, in interpreting a body of propositions hitherto believed to be about the supposed entities, we can substitute the logical structures without altering any of the detail of the body of pro-positions in question. This is an economy, because entities with neat logical properties are always inferred, and if the proposi-tions in which they occur can be interpreted without making this inference, the ground for the inference fails, and our body of propositions is secured against the need of a doubtful step.

The principle may be stated in the form: “Wherever possible, substitute constructions out of known entities for inferences to unknown entities.”

The uses of this principle are very various, but are not intelli-gible in detail to those who do not know mathematical logic.

The ﬁrst instance I came across was what I have called “the
principle of abstraction”, or “the principle which dispenses
with abstraction”.^{1} This principle is applicable in the case of
any symmetrical and transitive relation, such as equality. We are
apt to infer that such relations arise from possession of some
common quality. This may or may not be true; probably it is true

1*External World, p. 42.*

in some cases and not in others. But all the formal purposes of
a common quality can be served by membership of the group of
terms having the said relation to a given term. Take magnitude,
for example. Let us suppose that we have a group of rods, all
equally long. It is easy to suppose that there is a certain quality,
called their length, which they all share. But all propositions in
which this supposed quality occurs will retain their truth-value
*unchanged if, instead of “length of the rod x” we take *
*“member-ship of the group of all those rods which are as long as x”. In*
various special cases—e.g. the deﬁnition of real numbers—a
simpler construction is possible.

A very important example of the principle is Frege’s deﬁn-ition of the cardinal number of a given set of terms as the class of all sets that are “similar” to the given set—where two sets are

“similar” when there is a one-one relation whose domain is the one set and whose converse domain is the other. Thus a cardinal number is the class of all those classes which are similar to a given class. This deﬁnition leaves unchanged the truth-values of all propositions in which cardinal numbers occur, and avoids the inference to a set of entities called “cardinal numbers”, which were never needed except for the purpose of making arithmetic intelligible, and are now no longer needed for that purpose.

Perhaps even more important is the fact that classes
them-selves can be dispensed with by similar methods. Mathematics
is full of propositions which seem to require that a class or an
aggregate should be in some sense a single entity—e.g. the
*proposition “the number of combinations of n things any*
number at a time is 2* ^{n}*”. Since 2

^{n}*is always greater than n, this*proposition leads to diﬃculties if classes are admitted because the number of classes of entities in the universe is greater than the number of entities in the universe, which would be odd if classes were some among entities. Fortunately, all the proposi-tions in which classes appear to be mentioned can be interpreted

without supposing that there are classes. This is perhaps the most
*important of all the applications of our principle. (See Principia*
*Mathematica, *20.)*

Another important example concerns what I call “deﬁnite
descriptions”, i.e. such phrases as “the even prime”, “the present
King of England”, “the present King of France”. There has
always been a diﬃculty in interpreting such propositions as “the
present King of France does not exist”. The diﬃculty arose
through supposing that “the present King of France” is the
sub-ject of this proposition, which made it necessary to suppose that
he subsists although he does not exist. But it is diﬃcult to
attrib-ute even subsistence to “the round square” or “the even prime
greater than 2”. In fact, “the round square does not subsist” is
just as true as “the present King of France does not exist”. Thus
the distinction between existence and subsistence does not help
us. The fact is that, when the words “the so-and-so” occur in
a proposition, there is no corresponding single constituent of
the proposition, and when the proposition is fully analysed the
words “the so-and-so” have disappeared. An important
con-sequence of the theory of descriptions is that it is meaningless
*to say “A exists” unless “A” is (or stands for) a phrase of the*
*form “the and-so”. If the and-so exists, and x is the *
*so-and-so, to say “x exists” is nonsense. Existence, in the sense in*
which it is ascribed to single entities, is thus removed altogether
from the list of fundamentals. The ontological argument and
most of its refutations are found to depend upon bad grammar.

*(See Principia Mathematica, *14.)*

There are many other examples of the substitution of con-structions for inferences in pure mathematics, for example, series, ordinal numbers, and real numbers. But I will pass on to the examples in physics.

Points and instants are obvious examples: Dr. Whitehead has shown how to construct them out of sets of events all of which have a ﬁnite extent and a ﬁnite duration. In relativity theory, it is

not points or instants that we primarily need, but event-particles,
which correspond to what, in older language, might be described
as a point at an instant, or an instantaneous point. (In former
days, a point of space endured throughout all time, and an
instant of time pervaded all space. Now the unit that
mathemat-ical physics wants has neither spatial nor temporal extension.)
Event-particles are constructed by just the same logical process
by which points and instants were constructed. In such
con-structions, however, we are on a diﬀerent plane from that of
constructions in pure mathematics. The possibility of
construct-ing an event-particle depends upon the existence of sets of events
with certain properties; whether the required events exist can
*only be known empirically, if at all. There is therefore no a priori*
reason to expect continuity (in the mathematical sense), or to
feel conﬁdent that event-particles can be constructed. If the
quantum theory should seem to demand a discrete space-time,
our logic is just as ready to meet its requirements as to meet
those of traditional physics, which demands continuity. The
question is purely empirical, and our logic is (as it ought to be)
equally adapted to either alternative.

Similar considerations apply to a particle of matter, or to a piece of matter of ﬁnite size. Matter, traditionally, has two of those “neat” properties which are the mark of a logical construc-tion; ﬁrst, that two pieces of matter cannot be at the same place at the same time; secondly, that one piece of matter cannot be in two places at the same time. Experience in the substitution of constructions for inferences makes one suspicious of anything so tidy and exact. One cannot help feeling that impenetrability is not an empirical fact, derived from observation of billiard-balls, but is something logically necessary. This feeling is wholly justi-ﬁed, but it could not be so if matter were not a logical construc-tion. An immense number of occurrences coexist in any little region of space-time; when we are speaking of what is not logical construction, we ﬁnd no such property as impenetrability, but,

on the contrary, endless overlapping of the events in a part of space-time, however small. The reason that matter is impene-trable is because our deﬁnitions make it so. Speaking roughly, and merely so as to give a notion of how this happens, we may say that a piece of matter is all that happens in a certain track in space-time, and that we construct the tracks called bits of matter in such a way that they do not intersect. Matter is impenetrable because it is easier to state the laws of physics if we make our constructions so as to secure impenetrability. Impenetrability is a logically necessary result of deﬁnition, though the fact that such a deﬁnition is convenient is empirical. Bits of matter are not among the bricks out of which the world is built. The bricks are events, and bits of matter are portions of the structure to which we ﬁnd it convenient to give separate attention.

In the philosophy of mental occurrences there are also
*opport-unities for the application of our principle of constructions versus*
inferences. The subject, and the relation of a cognition to what is
known, both have that schematic quality that arouses our
suspi-cions. It is clear that the subject, if it is to be preserved at all,
must be preserved as a construction, not as an inferred entity;

the only question is whether the subject is suﬃciently useful to
be worth constructing. The relation of a cognition to what is
known, again, cannot be a straightforward single ultimate, as I
at one time believed it to be. Although I do not agree with
pragmatism, I think William James was right in drawing
atten-tion to the complexity of “knowing”. It is impossible in a
gen-eral summary, such as the present, to set out the reasons for this
view. But whoever has acquiesced in our principle will agree that
*here is prima facie a case for applying it. Most of my Analysis of*
*Mind consists of applications of this principle. But as psychology*
is scientiﬁcally much less perfected than physics, the
opportun-ities for applying the principle are not so good. The principle
depends, for its use, upon the existence of some fairly reliable
body of propositions, which are to be interpreted by the logician

in such a way as to preserve their truth while minimizing the element of inference to unobserved entities. The principle there-fore presupposes a moderately advanced science, in the absence of which the logician does not know what he ought to construct.

Until recently, it would have seemed necessary to construct geo-metrical points; now it is event-particles that are wanted. In view

Until recently, it would have seemed necessary to construct geo-metrical points; now it is event-particles that are wanted. In view