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
1External 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 2n”. Since 2n 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