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The Principles of Mathematics Iii THE PRINCIPLES PREFACE OF MATHEMATICS The present work has two main objects. One of these, the v by proof that all pure mathematics deals exclusively with con- cepts definable in terms of a very small number of fundamen- Bertrand Russell, M. A., tal logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles, is undertaken in Parts II–VII of this Volume, and will be estab- Late Fellow of Trinity College, Cambridge lished by strict symbolic reasoning in Volume II. The demon- stration of this thesis has, if I am not mistaken, all the cer- tainty and precision of which mathematical demonstrations Vol. I are capable. As the thesis is very recent among mathemati- cians, and is almost universally denied by philosophers, I have undertaken, in this volume, to defend its various parts, as oc- casion arose, against such adverse theories as appeared most widely held or most difficult to disprove. I have also endeav- First published in 1903 by Cambridge University Press. This oured to present, in language as untechnical as possible, the online edition (version 0.16: 16 Sep 2019) is based on that edi- more important stages in the deductions by which the thesis tion, with various typographical corrections. Missing here is is established. the Introduction to the 1937 second edition, which is not yet The other object of this work, which occupies Part I, is the in the public domain. Rather than publishing a second vol- explanation of the fundamental concepts which mathemat- ume, Russell and his co-author A. N. Whitehead published ics accepts as indefinable. This is a purely philosophical task, the three volumes of Principia Mathematica in 1910–1913. and I cannot flatter myself that I have done more than indi- Original page numbers are marked in the margins. The page ci- cate a vast field of inquiry, and give a sample of the meth- tations in the Table of Contents and Index refer to these num- ods by which the inquiry may be conducted. The discussion bers. of indefinables—which forms the chief part of philosophical ii Bertrand Russell The Principles of Mathematics iii logic—is the endeavour to see clearly, and to make others see The following portions are more specially philosophical: Part clearly, the entities concerned, in order that the mind may I (omitting Chapter ii); Part II, Chapters xi, xv, xvi, xvii; Part have that kind of acquaintance with them which it has with III; Part IV, §207, Chapters xxvi, xxvii, xxxi; Part V, Chapters redness or the taste of a pineapple. Where, as in the present xli, xlii, xliii; Part VI, Chapters l, li, lii; Part VII, Chapters liii, case, the indefinables are obtained primarily as the necessary liv, lv, lvii, lviii; and the two Appendices, which belong to Part residue in a process of analysis, it is often easier to know that I, and should be read in connection with it. Professor Frege’s there must be such entities than actually to perceive them; work, which largely anticipates my own, was for the most part there is a process analogous to that which resulted in the dis- unknown to me when the printing of the present work began; covery of Neptune, with the difference that the final stage— I had seen his Grundgesetze der Arithmetik, but, owing to the the search with a mental telescope for the entity which has great difficulty of his symbolism, I had failed to grasp its im- been inferred—is often the most difficult part of the under- portance or to understand its contents. The only method, at taking. In the case of classes, I must confess, I have failed to so late a stage, of doing justice to his work, was to devote an vi perceive any concept fulfilling the conditions requisite for the Appendix to it; and in some points the views contained in the notion of class. And the contradiction discussed in Chapter x Appendix differ from those in Chapter vi, especially in §§71, proves that something is amiss, but what this is I have hitherto 73, 74. On questions discussed in these sections, I discovered failed to discover. errors after passing the sheets for the press; these errors, of The second volume, in which I have had the great good for- which the chief are the denial of the null-class, and the identi- tune to secure the collaboration of Mr A. N. Whitehead, will fication of a term with the class whose only member it is,are be addressed exclusively to mathematicians; it will contain rectified in the Appendices. The subjects treated are so dif- chains of deductions, from the premisses of symbolic logic ficult that I feel little confidence in my present opinions, and through Arithmetic, finite and infinite, to Geometry, in anor- regard any conclusions which may be advocated as essentially der similar to that adopted in the present volume; it will also hypotheses. contain various original developments, in which the method A few words as to the origin of the present work may serve of Professor Peano, as supplemented by the Logic of Relations, to show the importance of the questions discussed. About six has shown itself a powerful instrument of mathematical inves- years ago, I began an investigation into the philosophy of Dy- tigation. namics. I was met by the difficulty that, when a particle is sub- The present volume, which may be regarded either as a com- ject to several forces, no one of the component accelerations vii mentary upon, or as an introduction to, the second volume, actually occurs, but only the resultant acceleration, of which is addressed in equal measure to the philosopher and to the they are not parts; this fact rendered illusory such causation mathematician; but some parts will be more interesting to the of particulars by particulars as is affirmed, at first sight, by the one, others to the other. I should advise mathematicians, un- law of gravitation. It appeared also that the difficulty in re- less they are specially interested in Symbolic Logic, to begin gard to absolute motion is insoluble on a relational theory of with Part IV, and only refer to earlier parts as occasion arises. space. From these two questions I was led to a re-examination iv Bertrand Russell The Principles of Mathematics v of the principles of Geometry, thence to the philosophy of ical questions, since mathematical space and motion are held continuity and infinity, and thence, with a view to discover- by many, perhaps most, philosophers to be self-contradictory, ing the meaning of the word any, to Symbolic Logic. The final and therefore necessarily different from actual space and mo- outcome, as regards the philosophy of Dynamics, is perhaps tion, whereas, if the views advocated in the following pages rather slender; the reason of this is, that almost all the prob- be valid, no such self-contradictions are to be found in mathe- lems of Dynamics appear to me empirical, and therefore out- matical space and motion. But extra-mathematical considera- side the scope of such a work as the present. Many very in- tions of this kind have been almost wholly excluded from the teresting questions have had to be omitted, especially in Parts present work. VI and VII, as not relevant to my purpose, which, for fear of On fundamental questions of philosophy, my position, in viii misunderstandings, it may be well to explain at this stage. all its chief features, is derived from Mr G. E. Moore. I have When actual objects are counted, or when Geometry and accepted from him the non-existential nature of propositions Dynamics are applied to actual space or actual matter, or when, (except such as happen to assert existence) and their indepen- in any other way, mathematical reasoning is applied to what dence of any knowing mind; also the pluralism which regards exists, the reasoning employed has a form not dependent the world, both that of existents and that of entities, as com- upon the objects to which it is applied being just those objects posed of an infinite number of mutually independent enti- that they are, but only upon their having certain general prop- ties, with relations which are ultimate, and not reducible to erties. In pure mathematics, actual objects in the world of exis- adjectives of their terms or of the whole which these com- tence will never be in question, but only hypothetical objects pose. Before learning these views from him, I found myself having those general properties upon which depends what- completely unable to construct any philosophy of arithmetic, ever deduction is being considered; and these general proper- whereas their acceptance brought about an immediate liber- ties will always be expressible in terms of the fundamental con- ation from a large number of difficulties which I believe to cepts which I have called logical constants. Thus when space be otherwise insuperable. The doctrines just mentioned are, or motion is spoken of in pure mathematics, it is not actual in my opinion, quite indispensable to any even tolerably sat- space or actual motion, as we know them in experience, that isfactory philosophy of mathematics, as I hope the following are spoken of, but any entity possessing those abstract general pages will show. But I must leave it to my readers to judge how properties of space or motion that are employed in the reason- far the reasoning assumes these doctrines, and how far it sup- ings of geometry or dynamics. The question whether these ports them. Formally, my premisses are simply assumed; but properties belong, as a matter of fact, to actual space or ac- the fact that they allow mathematics to be true, which most tual motion, is irrelevant to pure mathematics, and therefore current philosophies do not, is surely a powerful argument in to the present work, being, in my opinion, a purely empirical their favour.
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