Russell's PRINCIPLES OF MATHEMATICS
Bertrand Russell began work on the book we now know as The Principles of Mathematics in the beginning of 1898, when he was 27, and still under the spell of the Hegelian school of logic that dominated English philosophy at the time. Russell's purpose was to address the contradiction inherent in the nature of number. A number of things is one thing (since we talk of 'a' number or 'a' collection of things), but is also many (since it, or they, may be two or more things). He had proposed to deal with this in the Hegelian manner, by accepting the notion of quantity as inherently contradictory, then constructing a 'dialectic' around it. (A discussion of the problem survives in section 74 of the book).
After reading Whitehead's Universal Algebra, and Dedekind's Nature and Meaning of Numbers (a title which I am unable to find in any reference to Dedekind's works, but which may be Monk's translation of Was sind und was sollen die Zahlen, Meyer 1891), he became convinced that this Hegelian approach was wrong, and began work on a book to be called An Analysis of Mathematical Reasoning. He gave up this in 1900, but used the material extensively in Principles of Mathematics, a version of which was nearly complete by mid-1900.
At this point, while he now accepted Cantor's account of the continuum, he was still unable to accept Cantor's idea of an infinite whole, which underpins all of set theory. This changed when he went to present a paper at the International Congress of Philosophy at Paris in July 1899, at the invitation of Louis Couturat. Peano dominated the conference, and impressed Russell, who immediately began work on understanding the work of the Italian mathematicians. In October 1900 he rewrote large sections of the book (according to Monk, ten pages a day and 200,000 words in less than three months). He wanted to show that all mathematics could be reduced to a handful of logical concepts, and that even the concepts of 'zero', 'number' and 'successor' could be constructed out of the concept of a 'class'. He said that this period was 'the highest point of my life', and an 'intellectual honeymoon such as I have never experienced before or since'.
My sensations resembled those one has after climbing a mountain in a mist, when, on reaching the summit, the mist suddenly clears, and the country becomes visible for forty miles in every direction. For years I had been endeavouring to analyse the fundamental notions of mathematics, such as order and cardinal numbers. Suddenly, in the space if a few weeks, I discovered what appeared to be definitive answers to the problems which had baffled me for years I went about saying to myself that now at last I had done something worth doing, and I had the feeling that I must be careful not to be run over in the street before I had written it down.
He pointedly finished the manuscript on the very last day of 1900 (which, being a mathematician, he thought of as the last day of the nineteenth century).
Alas, in the Spring of 1901, he discovered what is now called 'Russell's Paradox', and which Russell himself modestly called 'The Contradiction'. Russell's description of it is in the tenth chapter of the book, here. Russell had been studying Cantor's proof that there is no greatest ordinal. Russell did not at first accept this, since he thought the class of all classes must surely have the greatest number there is. This led him (around April or May) to consider the class of all classes that are not members of themselves. Is this class a member of itself or not? If it is, it isn't, if it isn't, it is. Contradiction. The honeymoon was over.
Russell worked for at least a year on the book (Grattan Guinness (2000, 311 ff) has a detailed account of the chronology) before sending the manuscript to the publisher in early June 1902, two weeks before he notified Frege of the paradox in a letter in German.
This clearly contradicts the idea that Russell became aware of the importance of the paradox through Frege. Indeed, Russell discovered Frege fairly late, reading some of his works in detail as late as June 1902 (according to Grattan Guinness, he told Couturat of his ignorance in letters of 25 June and 2 July 1902). Hence his accounts on Frege's work appear as an appendix in sections §475 to §496 here.
Russell wrote the preface in December 1902. It appeared in May 1903, near to Russell's 31st birthday. The print-run was 1,000 copies at 12/6d each. By June 1909 the last copies were at the binders. The book had an important part in publicising the work of Cantor and Frege to the English-speaking world, and it is undoubtedly a masterpiece (Monk regards it as one of the most important philosophical works of the twentieth century). However, it is somewhat of a mess, owing to the disorderly presentation, and Russell wrote to Gilbert Murray that the book 'disgusted' him. However, the book was well received. The Spectator wrote "we should say that Mr. Russell has an inherited place in literature or statesmanship waiting for him if he will condescend to come down to the common day" (1903). Hardy thought that Russell had 'proved his point' about logicism, and confessed (interestingly) that he had never heard of Frege until Russell's book. Stout wrote to Russell that he was impressed by the book, but thought it fundamentally wrong. He asked W.E. Johnson to review it for Mind, but nothing arrived and he instead asked A.T. Shearman, who gave it 12 pages, hailing it as the most important work since Boole's Laws of Thought. Shearman also welcomed the account of Frege's work without the "extreme cumbrousness" of the Begriffschift notation.
The passages from the book given below are on the subjects of class and class-membership, the relevance of grammar to philosophy and mathematics ("Although a grammatical distinction cannot be uncritically assumed to correspond to a genuine philosophical difference, yet the one is prima facia evidence of the other, and may often be most usefully employed as a source of discovery"), the problem of the unity of the proposition, on enumerations vs. classes, classes as many vs. classes as one, the definition of an infinite set, Russell's Paradox, and Frege's contribution to logic.
THE PRINCIPLES OF MATHEMATICS
§ 21. The insistence on the distinction between e and the relation of whole and part between classes is due to Peano, and is of very great importance to the whole technical development and the whole of the applications to mathematics. In the scholastic doctrine of the syllogism, and in all previous symbolic logic, the two relations are confounded, except in the work of Frege*. The distinction is the same as that between the relation of individual to species and that of species to genus, between the relation of Socrates to the class of Greeks and the relation of Greeks to men. On the philosophical nature of this distinction I shall enlarge when I come to deal critically with the nature of classes; for the present it is enough to observe that the relation of whole and part is transitive, while e is not so; we have Socrates is a man, and men are a class, but not Socrates is a class. It is to be observed that the class must be distinguished from the class-concept or predicate by which it is to be defined: thus men are a class, while man is a class-concept. The relation e must be regarded as holding between Socrates and men considered collectively, not between Socrates and man. I shall return to this point in Chapter VI [§66-68]. Peano holds that all propositional functions containing only a single variable are capable of expression in the form "x is an a," where a is a constant class; but this view we shall find reason to doubt.
* See his Begriffschift, Halle, 1879, and Grundgesetze der Arithmetik, Jena, 1894
§ 38. We need, in fact, the notion of therefore, which is quite different from the notion of implies, and holds between different entities. In grammar, the distinction is that between a verb and a verbal noun, between, say "A is greater than B" and "A's being greater than B." In the first of these, a proposition is actually asserted, whereas in the second it is merely considered. But these are psychological terms, whereas the difference which I desire to express is genuinely logical. It is plain that, if I may be allowed to use the word assertion in a non-psychological sense, the proposition "p implies q" asserts an implication, though it does not assert p or q. The p and the q which enter into this proposition are not strictly the same as the p or the q which are separate propositions, at least, if they are true. The question is: How does a proposition differ by being actually true from what it would be as an entity if it were not true? It is plain that true and false propositions alike are entities of a kind, but that true propositions have a quality not belonging to false ones, a quality which, in a non-psychological sense, may be called being asserted. Yet there are grave difficulties in forming a consistent theory on this point, for if assertion in any way changed a proposition, no proposition which can possibly in any context be unasserted could be true, since when asserted it would become a different proposition. But this is plainly false; for in "p implies q," p and q are not asserted, and yet they may be true. Leaving this puzzle to logic, however, we must insist that there is a difference of some kind between an asserted and an unasserted proposition.
§ 46. In the present chapter, certain questions are to be discussed belonging to what may be called philosophical grammar. The study of grammar, in my opinion, is capable of throwing far more light on philosophical questions than is commonly supposed by philosophers. Although a grammatical distinction cannot be uncritically assumed to correspond to a genuine philosophical difference, yet the one is prima facia evidence of the other, and may often be most usefully employed as a source of discovery. Moreover, it must be admitted, I think, that every word occurring in a sentence must have some meaning: a perfectly meaningless sound could not be employed in the more or less fixed way in which language employs words. The correctness of our philosophical analysis of a proposition may therefore be usefully checked by the exercise of assigning the meaning of each word in the sentence expressing the proposition. On the whole, grammar seems to me to bring us much nearer to a correct logic than the current opinions of philosophers: and in what follows, grammar, though not our master, will yet be taken as our guide.
§ 52 It remains to discuss the verb, and to find marks by which it is distinguished from the adjective By transforming the verb, as it occurs in a proposition, into a verbal noun, the whole proposition can be turned into a logical subject, no longer asserted, and no longer containing in itself truth and falsehood. But here, too, there seems to be no possibility of maintaining that the logical subject which results is a different entity from the proposition. "Caesar died" and "the death of Caesar" will illustrate this point. If we ask: What is asserted in the proposition "Caesar died"? the answer must be "the death of Caesar is asserted". In that case, it would seem, it is the death of Caesar which is true or false; and yet neither truth nor falsity belongs to a mere logical subject. The answer here seems to be that the death of Caesar has an external relation to truth or falsehood whereas "Caesar died" in some way or other contains its own truth or falsehood as an element. But if this is the correct analysis, it is difficult to see how "Caesar died" differs from "the truth of Caesar's death" in the case where it is true, or "the falsehood of Caesar's death" in the other case. Yet it is quite plain that the latter, at any rate, is never equivalent to "Caesar died". There appears to be an ultimate notion of assertion, given by the verb, which is lost as soon as we substitute a verbal noun, and is lost when the proposition in question is made the subject of some other proposition. Thus the contradiction which was to have been avoided, of an entity which cannot be made a logical subject, appears to have here become inevitable. This difficulty, which seems to be inherent in the very nature of truth and falsehood, is one with which I do not know how to deal with satisfactorily. The obvious course would be to say that the difference between an asserted and an unasserted proposition is not logical but psychological. In the sense in which false propositions may be asserted, this is doubtless true. But there is another sense of assertion, very difficult to bring clearly before the mind, and yet quite undeniable, in which only true propositions are asserted. True and false propositions alike are in some sense entities, and are in some sense capable of being logical subjects; but when a proposition happens to be true, it has a further quality, over and above that which it shares with false propositions, and it is this further quality which is what I mean by assertion in a logical as opposed to psychological sense. The nature of truth, however, belongs no more to the principles of mathematics than to the principles of everything else. I therefore leave this question to the logicians with the above brief indication of a difficulty.
§ 54. The twofold nature of the verb, as actual verb and as verbal noun, may be expressed, if all verbs are held to be relations, as the difference between a relation in itself and a relation actually relating. Consider, for example, the proposition "A differs from B." The constituents of this proposition, if we analyze it, appear to be only A, difference, B. Yet these constituents, thus placed side by side, do not reconstitute the proposition. The difference which occurs in the proposition actually relates A and B, whereas the difference after analysis is a notion which has no connection with A and B. It may be said that we ought, in the analysis, to mention the relations which difference has to A and B, relations which are expressed by is and from when we say " A is different from B." These relations consist in the fact that A is referent and B relatum with respect to difference. But "A, referent, difference, relatum, B" is still merely a list of terms, not a proposition. A proposition, in fact, is essentially a unity, and when analysis has destroyed the unity, no enumeration of constituents will restore the proposition. The verb, when used as a verb, embodies the unity of the proposition, and is thus distinguishable from the verb considered as a term, though I do not know how to give a clear account of the distinction.
§ 66. It has been customary, in works on logic, to distinguish two standpoints, that of extension and that of intension. Philosophers have usually regarded the latter as more fundamental, while Mathematics has been held to deal specially with the former. .. But as a matter of fact, there are positions intermediate between pure intension and pure extension, and it is in these intermediate regions that Symbolic Logic has its lair. It is essential that the classes with which we are concerned should be composed of terms, and should not be predicates or concepts, for a class must be definite when its terms are given, but in general there will be many predicates which attach to the given terms and no others. We cannot of course attempt an intensional definition of a class as the class of all predicates attaching to the terms in question and to no others, for this would involve a vicious circle; hence the point of view of extension is to some extent unavoidable. On the other hand, if we take extension pure, our class is defined by enumeration of its terms, and this method will not allow us to deal, as Symbolic Logic does, with infinite classes. Thus our classes must in general be regarded as objects denoted by concepts, and to this extent the point of view of intension is essential. It is owing to this consideration that the theory of denoting is of such great importance.
§ 67. When an object is unambiguously denoted by a concept, I shall speak of the concept as a concept (or, loosely, as the concept) of the object in question. Thus it will be necessary to distinguish the concept of a class from a class-concept. We agreed to call man a class-concept, but man does not, in its usual employment, denote anything. On the other hand men, and all men (which I shall regard as synonyms) do denote, and I shall contend that what they denote is the class composed of all men. Thus man is the class-concept, men (the concept) is the concept of the class, and men (the object denoted by the concept men) are the class. It is no doubt confusing, at first, to use class-concept and concept of a class in different senses; but so many distinctions are required that some straining of language seems unavoidable, In the phraseology of the preceding chapter, we may say that a class is a numerical conjunction of terms
§ 68. We may, then, imagine a kind of genesis of classes, through the successive stages indicated by the typical propositions "Socrates is human", "Socrates has humanity", "Socrates is a man", "Socrates is one among men". Of these propositions, the last only, we should say, explicitly contains the class as constituent; but every subject-predicate proposition gives rise to the other three equivalent propositions, and thus every predicate gives rise to a class. This is the genesis of classes from the intensional standpoint.
On the other hand, when mathematicians deal with what they call a manifold, aggregate, Menge, ensemble, or some equivalent name, it is common, especially where the number of terms is finite, to regard the object in question (which is in fact a class) as defined by the enumeration of its terms, and as consisting possibly of a single term, which in that case is the class. Here it is not predicates and denoting that are relevant, but terms connected by the word and, in the sense in which this word stands for a numerical conjunction. Thus Brown and Jones are a class, and Brown singly is a class. This is the extensional genesis of classes.
§ 69. The best formal treatment of classes in existence is that of Peano. But in this treatment a number of distinctions of great philosophical importance are overlooked. Peano, not I think quite consciously, identifies the class with the class-concept; thus the relation of an individual to its class is, for him, expressed by is a. For him, "2 is a number" is a proposition in which a term is said to belong to the class number. Nevertheless he identifies the equality of classes, which consists in their having the same terms, with identity a proceeding which is quite illegitimate when the class is regarded as the class-concept. In order to perceive that man and featherless biped are not identical, it is quite unnecessary to take a hen and deprive the poor bird of its feathers. Or, to take a less complex instance, it is plain that even prime is not identical with integer next after 1. Thus when we identify the class with the class-concept, we must admit that two classes may be equal without being identical. Nevertheless, it is plain that when two class-concepts are equal, some identity is involved, for we say that they have the same terms. Thus there is some object which is positively identical when the two class-concepts are equal; and this object, it would seem, is more properly called the class. Neglecting the plucked hen, the class of featherless bipeds, every one would say, is the same as the class of men; the class of even primes is the same as the class of integers next after 1. Thus we must not identify the class with the class-concepts, or regard "Socrates is a man" as expressing the relation of an individual to a class of which it is a member. This has two consequences which prevent the philosophical acceptance of certain points in Peano's formalism. The first consequence is that there is no such thing as the null-class, though there are null class-concepts. The second is, that a class having only one term is to be identified, contrary to Peano's usage, with that one term. I should not propose, however, to alter his practice or his notation in consequence of either of these points; rather I should regard them as proofs that Symbolic Logic should concern itself, as far as notation goes, with class-concepts rather than with classes.
§ 70. A class, we have seen, is neither a predicate nor a class-concept, for different predicates and different class-concepts may correspond to the same class. A class also, in one sense at least, is distinct from the whole composed of its terms, for the latter is only and essentially one, while the former, where it has many terms, is, as we shall see later, the very kind of object of which many is to be asserted. The distinction of a class as many from a class as a whole is often made by language: space and points, time and instants, the army and the soldiers, the Cabinet and the Cabinet Ministers, all illustrate the distinction. The notion of a whole, in the sense of a pure aggregate which is here relevant, is, we shall find, not always applicable where the notion of the class as many applies In such cases, though terms may be said to belong to the class, the class must not be treated as itself a single logical subject [As footnote: a plurality of terms is not the logical subject when a number is asserted of it: such propositions have not one subject, but many subjects. See end of § 74]. But this never arises where a class can be generated by a predicate. Thus we may for the present dismiss this complication from our minds. In a class as many, the component terms, though they have some kind of unity, have less than is required for a whole. They have, in fact, just so much unity as is required to make them many, and not enough to prevent them from remaining many. A further reason for distinguishing wholes from classes as many is that a class as one may be one of the terms of itself as many, as in "classes are one among classes" (the extensional equivalent of "class is a class-concept"), whereas a complex whole can never be one of its own constituents.
§ 71 Class may be defined either extensionally or intensionally. That is to say, we may define the kind of object which is a class, or the kind of concept which denotes a class: this is the precise meaning of the opposition of extension and intension in this connection. But although the general notion can be defined in this two-fold manner, particular classes, except when they happen to be finite, can only be defined intensionally, i.e. as the objects denoted by such concepts. I believe this distinction to be purely psychological: logically, the extensional definition appears to be equally applicable to infinite classes, but practically, if we were to attempt it, Death would cut short our laudable endeavour before it had attained its goal. Logically, therefore, extension and intension seem to be on a par. I will begin with the extensional view.
When a class is regarded as defined by the enumeration of its terms, it is more naturally called a collection. I shall for the moment adopt this name, as it will not prejudge the question whether the objects denoted by it are truly classes or not. Buy a collection I mean what is conveyed by "A and B", or "A and B and C", or any other enumeration of definite terms. The collection is defined by the actual mention of the terms, and the terms are connected by and. It would seem that and represents a fundamental way of combining terms, and that just this way of combination is essential if anything is to result of which a number other than 1 can be asserted. Collections do not presuppose numbers, since they result simply from the terms together with and: they could only presuppose numbers in the particular case where the terms of the collection themselves presupposed numbers. There is a grammatical difficulty which, since no method exists of avoiding it, must be pointed out and allowed for. A collection, grammatically, is singular, whereas A and B, A and B and C, etc. are essentially plural. This grammatical difficulty arises from the logical fact (to be discussed presently) that whatever is many in general forms a whole which is one; it is, therefore, not removable by a better choice of technical terms.
The notion of and was brought into prominence by Bolzano*. In order to understand what infinity is, he says,
we must go back to one of the simplest conceptions of our understanding, in order to reach an agreement concerning the word we are to use to denote it. This is the conception which underlies the conjunction and, which, however, if it is to stand out as clearly as is required, in many cases, both by the purposes of mathematics and by those of philosophy, I believe to be best expressed by the words: 'A system (Inbegriff) of certain things,' or ' a whole consisting of certain parts.' But we must add that every arbitrary object A can be combined in a system with any others B, C, D, , or (still speaking more correctly) already forms a system by itself i.e. the combination of A with B, C, D already forms a system], of which some more or less important truth can be enunciated, provided only that each of the presentations A, B, C, D, in fact represents a different object, or in so far as none of the propositions 'A is the same as B,' 'A is the same as C,' 'A is the same as D,' etc., is true. For if e.g. A is the same as B, then it is certainly unreasonable to speak of a system of the things A and B.
The above passage, good as it is, neglects several distinctions which we have found necessary. First and foremost, it does not distinguish the many from the whole which they form. Secondly, it does not appear to observe that the method of enumeration is not practically [EDB: my emphasis] applicable to infinite systems. Thirdly, and this is connected with the second point, it does not make any mention of intensional definition nor of the notion of a class. What we have to consider is the difference, if any, of a class from a collection on the one hand, and from the whole formed of the collection on the other. But let us examine further the notion of and.
Anything of which a finite number other than 0 or 1 can be asserted would commonly be said to be many, and many, it might be said, are always of the form "A and B and C and " Here A, B, C, are each one and are all different. To say that A is one seems to amount to much the same as to say that A is not of the form "A1 and A2 and A3 and " To say that A, B, C, are all different seems to amount only to a condition as regards the symbols: it should be held that "A and A" is meaningless, so that diversity is implied by and, and need not be specially stated.
A term A which is one may be regarded as a particular case of a collection, namely as a collection of one term. Thus every collection which is many presupposes many collections which are each one: A and B presupposes A and presupposes B. Conversely some collections of one term presuppose many, namely those which are complex: thus "A differs from B" is one, but presupposes A and difference and B. But there is no symmetry in this respect, for the ultimate presuppositions of anything are always simple terms.
[ A and B may be fictional but have Being]
The question may now be asked: What is meant by A and B? Does this mean anything more than the juxtaposition of A with B? That is, does it contain any element over and above that of A and that of B? To either answer there are objections. In the first place, and, we might suppose, cannot be a new concept, for if it were, it would have to be some kind of relation between A and B; A and B would then be a proposition, or at least a propositional concept, and would be one, not two. Moreover, if there are two concepts, there are two, and no third mediating concept seems necessary to make them two. Thus and would seem meaningless. But it is difficult to maintain this theory. To begin with, it seems rash to hold that any word is meaningless. When we use the word and, we do not seem to be uttering idle breath, but some idea seems to correspond to the word. Again some kind of combination seems to be implied by the fact that A and B are two, which is not true of either separately. When we say "A and B are yellow," we can replace the proposition by "A is yellow" and "B is yellow"; but this cannot be done for "A and B are two"; on the contrary, A is one and B is one. Thus it seems best to regard and as expressing a definite unique kind of combination, not a relation, and not combining A and B into a whole, which would be one. This unique kind of combination will in future be called addition of individuals. It is important to observe that it applies to terms, and only applies to numbers in consequence of their being terms. Thus for the present, 1 and 2 are two, and 1 and 1 is meaningless.
As regards what is meant by the combination indicated by and, it is indistinguishable from what we before called a numerical conjunction. That is, A and B is what is denoted by the concept of a class of which A and B are the only members. If u be a class-concept of which the propositions "A is a u" "B is a u" are true, but of which all other propositions of the same form are false, then "all u's" is the concept of a class whose only terms are A and B; this concept denotes the terms A, B combined in a certain way, and "A and B" are those terms combined in just that way. Thus "A and B" are the class, but are distinct from the class-concept and from the concept of the class.
The notion of and, however, does not enter into the meaning of a class, for a single term is a class, although it is not a numerical conjunction. If u be a class-concept, and only one proposition of the form "x is a u" be true, then "all u's" is a concept denoting a single term, and this term is the class of which "all u's" is a concept. Thus what seems essential to a class is not the notion of and, but the being denoted by some concept of a class. This brings to the intensional view of classes.
* Paradoxien des Unendlichen, Leipzig, 1854 (2nd ed., Berlin 1889)
§ 72. We agreed in the preceding chapter that there are not different ways of denoting, but only different kinds of denoting concepts and correspondingly different kinds of denoted objects. We have discussed the kind of denoted object which forms a class; we have now to consider the kind of denoting concept.
The consideration of classes which results from denoting concepts is more general than the extensional consideration, and that in two respects. In the first place it allows, what the other practically excludes, the admission of infinite classes; in the second place it introduces the null concept of a class. But, before discussing these matters, there is a purely logical point of some importance to be examined.
If u be a class-concept, is the concept "all u's" analyzable into two constituents, all and u, or is it a new concept, defined by a certain relation to u, and no more complex than u itself? We may observe, to begin with, that "all u's" is synonymous with "u's" at least according to a very common use of the plural. Our question is, then, as to the meaning of the plural. The word all has certainly some definite meaning, but it seems highly doubtful whether it means more than the indication of a relation. "All men" and "all numbers" have in common the fact that they both have a certain relation to a class-concept, namely to man and number respectively. But it is very difficult to isolate any further element of all-ness which both share, unless we take as this element the mere fact that both are concepts of classes. It would seem, then, that "all u's" is not validly analyzable into all and u, and that language, in this case as in some others, is a misleading guide. The same remark will apply to every, any, some, a, and the.
It might perhaps be thought that a class ought to be considered, not merely as a numerical conjunction of terms, but as a numerical conjunction denoted by the concept of a class. This complication, however, would serve no useful purpose, except to preserve Peano's distinction between a single term and the class whose only term it is a distinction which is easy to grasp when the class is identified with the class-concept, but which is inadmissible in our view of classes. It is evident that a numerical conjunction considered as denoted is either the same entity as when not so considered, or else is a complex of denoting together with the object denoted; and the object denoted is plainly what we mean by a class.
With regard to infinite classes, say the class of numbers, it is to be observed that the concept all numbers, though not itself infinitely complex, yet denotes an infinitely complex object. This is the inmost secret of our power to deal with infinity. An infinitely complex concept, though there may be such, can certainly not be manipulated by the human intelligence; but infinite collections, owing to the notion of denoting, can be manipulated without introducing any concepts of infinite complexity. Throughout the discussion of infinity in later Parts of the present work, this remark should be borne in mind: if it is forgotten, there is an air of magic which causes the results obtained to seem doubtful.
§ 74. A question which is very fundamental in the philosophy of Arithmetic must now be discussed in a more or less preliminary fashion. Is a class which has many terms to be regarded as itself one or many? Taking the class as equivalent simply to the numerical conjunction "A and B and C and etc.," it seems plain that it is many; yet it is quite necessary that we should be able to count classes as one each, and we do habitually speak of a class. Thus classes would seem to be one in one sense and many in another.
There is a certain temptation to identify the class as many and the class as one, e.g., all men and the human race. Nevertheless, wherever a class consists of more than one term, it can be proved that no such identification is permissible. A concept of a class, if it denotes a class as one, is not the same as any concept of the class which it denotes. That is to say, classes of all rational animals, which denotes the human race as one term, is different from men, which denotes men, i.e. the human race as many. But if the human race were identical with men, it would follow that whatever denotes the one must denote the other, and the above difference would be impossible. We might be tempted to infer that Peano's distinction, between a term and a class of which the said term is the only member, must be maintained, at least when the term in question is a class [as footnote: this conclusion is actually drawn by Frege from an analogous argument: Archiv fur Systematische Philosophie I (1895) p. 444]. But it is more correct, I think, to infer an ultimate distinction between a class as many and a class as one, to hold that the many are only many, and are not also one. The class as one may be identified with the whole composed of the terms of the class, i.e. in the case of men, the class as one will be the human race.
But can we now avoid the contradiction always to be feared, where there is something that cannot be made a logical subject? I do not myself see any way of eliciting a precise contradiction in this case. In the case of concepts, we were dealing with what was plainly one entity; in the present case, we are dealing with a complex essentially capable of analysis into units. In such a proposition as "A and B are two" there is no logical subject: the assertion is not about A, not about B, nor about the whole composed of both, but strictly and only about A and B. Thus it would seem that assertions are not necessarily about single subjects, but may be about many subjects; and this removes the contradiction which arose, in the case of concepts, from the impossibility of making assertions about them unless they were turned into subjects. This impossibility being here absent, the contradiction which was to be feared does not arise.
§ 75. We may ask, as suggested by the above discussion, what is to be said of the objects denoted by a man, every man, some man, and any man. Are these objects one or many or neither? Grammar treats them all as one. But to this view, the natural objection is, which one? Certainly not Socrates, nor Plato, nor any other particular person. Can we conclude that no one is denoted? As well might we conclude that every one is denoted, which in fact is true of the concept every man. I think one is denoted in every case, but in an impartial distributive manner. Any number is neither 1 nor 2 nor any other particular number, whence it is easy to conclude that any number is not any one number, a proposition at first sight contradictory, but really resulting from an ambiguity in any, and more correctly expressed by "any number is not some one number." There are, however, puzzles in this subject which I do not yet know how to solve.
A logical difficulty remains in regard to the nature of the whole composed of all the terms of a class. Two propositions appear self-evident: (1) Two wholes composed of different terms must be different; (2) A whole composed of one term only is that one term. It follows that the whole composed of a class considered as one term is that class considered as one term, and is therefore identical with the whole composed of the terms of the class; but this result contradicts the first of our supposed self-evident principles. The answer in this case, however, is not difficult. The first of our principles is only universally true when all the terms composing our two wholes are simple. A given whole is capable, if it has more than two parts, of being analyzed in a plurality of ways; and the resulting constituents, so long as analysis is not pushed as far as possible, will be different for different ways of analyzing. This proves that different sets of constituents may constitute the same whole, and thus disposes of our difficulty.
§ 76. Something must be said as to the relation of a term to a class of which it is a member, and as to the various allied relations. One of the allied relations is to be called e, and is to be fundamental in Symbolic Logic. But it is to some extent optional which of them we take as symbolically fundamental
Logically, the fundamental relation is that of subject and predicate, expressed in "Socrates is human" a relation which, as we saw in Chapter IV, is peculiar in that the relatum cannot be regarded as a term in the proposition. The first relation that grows out of this is the one expressed by "Socrates has humanity," which is distinguished by the fact that here the relation is a term. Next comes "Socrates is a man." This proposition, considered as a relation between Socrates and the concept man, is the one which Peano regards as fundamental; and his e expresses the relation is a between Socrates and man. So long as we use class-concepts for classes in our symbolism, this practice is unobjectionable; but if we give e this meaning, we must not assume that two symbols representing equal class-concepts both represent one and the same entity. We may go on to the relation between Socrates and the human race, i.e. between a term and its class considered as a whole: this is expressed by "Socrates belongs to the human race." This relation might equally well be represented by e. It is plain that, since a class, except when it has one term, is essentially many, it cannot be as such represented by a single letter: hence in any possible Symbolic Logic the letters which do duty for classes cannot represent the classes as many, but must represent either class-concepts, or the wholes composed of classes, or some other allied single entities. And thus e cannot represent the relation of a term to its class as many; for this would be a relation of one term to many terms, not a two-term relation such as we want. This relation might be expressed by "Socrates is one among men"; but this, in any case, cannot be taken to be the meaning of e.
§ 77. A relation which, before Peano, was almost universally confounded with e, is the relation of inclusion between classes, as e.g. between men and mortals. This is a time-honoured relation, since it occurs in the traditional form of the syllogism: it has been a battleground between intension and extension, and has been so much discussed that it is astonishing how much remains to be said about it. Empiricists hold that such propositions mean an actual enumeration of the terms of the contained class, with the assertion, in each case, of membership of the containing class. They must, it is to be inferred, regard it as doubtful whether all primes are integers, since they will scarcely have the face to say that they have examined all primes one by one. Their opponents have usually held, on the contrary, that what is meant is a relation of whole and part between the defining predicates, but turned in the opposite sense from the relation between the classes: i.e. the defining predicate of the larger class is part of that of the smaller. This view seems far more defensible than the other; and wherever such a relation does hold between the defining predicates, the relation of inclusion follows. But two objections may be made, first, that in some cases of inclusion here is no such relation between the defining predicates, and secondly, that in any case what is meant is a relation between the classes, not a relation of their defining predicates. The first point may be easily established by instances.
The concept even prime does not contain as a constituent the concept integer between 1 and 10; the concept "English King whose head was cut off" does not contain the concept "people who died in 1649"; and so on through innumerable obvious cases. This might be met by saying that, though the relation of the defining predicates is not one of whole and part, it is one more or less analogous to implication, and is always what is really meant by propositions of inclusion. Such a view represents, I think, what is said by the better advocates of intension, and I am not concerned to deny that a relation of the kind in question does always subsist between defining predicates of classes one of which is contained in the other. But the second of the above points remain valid as against any intensional interpretation. When we say that men are mortals, it is evident that we are saying something about men, not about the concept man or the predicate human. The question is, then, what exactly are we saying?
Peano held, in earlier editions of his Formulaire, that what is asserted is the formal implication "x is a man implies x is a mortal." This is certainly implied, but I cannot persuade myself that it is the same proposition. For in this proposition, as we saw I n Chapter III, it is essential that x should take all values, and not only such as are men. But when we say "all men are mortals, " it seems plain that we are only speaking of men, and not of all other imaginable terms
Chapter X "The Contradiction"
§ 100. It is necessary to examine in more detail the singular contradiction, already mentioned [i.e. Russell's paradox], with regard to predicates not predicable of themselves. I may mention that I was led to it in the endeavour to reconcile Cantor's proof that there can be no greatest cardinal number with the very plausible supposition that the class of all terms has necessarily the greatest possible number of members.
§ 101. If x be a predicate, x may or may not be predicable of itself. Let us assume that "not predicable of oneself" is a predicate. Then to suppose either that this predicate is, or that it is not, predicate of itself, is self-contradictory. The conclusion, in this case, seems obvious: "not predicable of oneself" is a not predicate.
Let us now state the same contradiction in terms of class-concepts. A class-concept may or may not be a term of its own extension. "Class-concept which is not a term of its own extension" appears to be a class-concept. But if it is a term of its own extension, it is a class-concept which is not a term of its own extension, and vice versa. Thus we must conclude, against appearances, that "class-concept which is not a term of its own extension" is not a class-concept.
§ 118 Among finite classes, if one is a proper part of another, the one has a smaller number of terms than the other. (A proper part is a part not the whole). Bur among infinite classes, this no longer holds. This distinction is, in fact, an essential part of the above definitions of the finite and the infinite .
we may define a0 as follows. Let P be a transitive and asymmetrical relation, and let any two different terms of the field of P have the relation P or its converse. Further let any class u contained in the field of P and having successors (i.e. terms to which every term of u has the relation P) have an immediate successor, i.e. a term whose predecessors either belong to u or precede some term of u; let there be one term of the field of P which has no predecessors, but let every term which has predecessors have successors and also have an immediate predecessor; then the number of terms in the field pf P is a0. Other definitions may be suggested, but as all are equivalent it is not necessary to multiply them. The following characteristic is important: Every class whose number is a0 can be arranged in a series having consecutive terms, a beginning but no end, and such that the number of predecessors of any term of the series is finite; and any series having these characteristics has the number a0.
§ 119 If n be any finite number, the number obtained by adding 1 to n is also finite, and is different from n. Thus beginning with 0 we can form a series of numbers by successive additions of 1. We may define finite numbers, if we choose, as those numbers that can be obtained from 0 by such steps, and that obey mathematical induction. That is, the class of finite numbers is the class of numbers which is contained in every class s to which belongs 0 and the successor of every number belonging to s, where the successor of a number is the number obtained by adding 1 to the given number. Now a0 is not such a number, since, in virtue of propositions already proved, no such number is similar to part of itself. Hence also no number greater than a0 is finite according to the new definition. But it is easy to prove that every number less than a0 is finite with the new definition as with the old. Hence the two definitions are equivalent. Thus we may define finite numbers either as those that can be reached my mathematical induction, starting from 0 and increasing by 1 at each step, or as those of classes which are not similar to the parts of themselves obtained by taking away single terms. These two definitions are both frequently employed, and it is important to realise that either is a consequence of the other.
§ 180 at this point, the philosopher is apt to step in, and to declare that, by all true philosophic principles, every well-defined series of terms must have a last term. If he insists on creating this last term, and calling it infinity, he easily deduces intolerable contradictions, from which he infers the inadequacy of mathematics to deduce absolute truth.
§ 183 Of all the philosophers who have inveighed against infinite number, I doubt whether there is one who has known the difference between finite and infinite numbers. The difference is simply this. Finite numbers obey the law of mathematical induction; infinite numbers do not It is in this alone, and in its consequences, that finite and infinite numbers differ.
The principle may otherwise be stated thus: If every proposition which holds concerning 0, and also holds concerning the immediate successor of every number of which it holds, holds concerning the number n, then n is finite; if not, not. This is the precise sense of what may be popularly expressed by saying that every finite number can be reached from 0 by successive steps, or by successive additions of 1. This the principle which the philosopher must be held to lay down as obviously applicable to all numbers, though he will have to admit that the more precisely his principle is stated, the less obvious it becomes.
§ 184 It may be worth while to show exactly how mathematical induction enters into the above proofs. Let us take the proof of (a) [that there are consecutive magnitudes], and suppose there are n magnitudes between A and B. Then to begin with, we supposed these magnitudes capable of enumeration, i.e. of an order in which there consecutive terms and a first term, and a term immediately preceding any tem except the first. This property presupposes mathematical induction, and in fact was the very property in dispute. Hence we must not presuppose the possibility of enumeration, which would be a petitio principii.
Every series having a finite number of terms can be shown by mathematical induction to have a first and a last term; but no way exists of proving this concerning other series, or of proving that all series are finite The philosopher's finitist arguments, therefore, rest on a principle of which he is ignorant, which there is no reason to affirm, and every reason to deny.
§ 292. we may define the infinite ordinals or cardinals as those which, starting from 0 or 1, can be reached by mathematical induction. This principle, therefore, is not to be taken as an axiom or postulate, but as the definition of finitude. It is to be observed that, in virtue of the principle that every number has an immediate successor, we can prove that any assigned number, say 10,397, is finite provided, of course, that the number assigned is a finite number. That is to say, every proposition concerning 10,397 can be proved without the use of mathematical induction, which, as most of us can remember, was not mentioned in the Arithmetic of our childhood. There is therefore no kind of logical error in using the principle as a definition of the class of finite numbers, nor is there a shadow of a reason for supposing that the principle applies to all ordinal or all cardinal numbers.
We can now define finite numbers either by the fact that mathematical induction can reach them or by the fact that they are the numbers of collections such that no proper part of them has the same number as the whole. These two conditions may be easily shown to be equivalent. But they alone precisely distinguish the finite form the infinite, and any discussion off infinity which neglects them must be more or less frivolous.
.. it is capable of formal proof that the number of the finite numbers themselves cannot be a term in the progression of finite numbers. A number not belonging to this progression is called infinite.
§ 339. That there are infinite classes is so evident that it will scarcely be denied. Since, however, it is capable of formal proof, it may be as well to prove it. A very simple proof is that suggested in the Parmenides, which is as follows. Let it be granted that there is a number 1. Then 1 is, or has Being, and therefore there is Being. But 1 and Being are two: hence there is a number 2 and so on. Formally, we have proved that 1 is not the number of numbers; we prove that n is the number of numbers from 1 to n, and that these numbers together with Being form a class which has a new finite number, so that n is not the number of finite numbers. Hence the finite numbers, by mathematical induction, are all contained in the class of things which are not the number of finite numbers. Since the relation of similarity is reflexive for classes, every class has a number; therefore the class of finite numbers has a number which, not being finite, is infinite. A better proof, analogous to the above, is derived from the fact that, if n be any finite number, the number of numbers from 0 up to and including n is n + 1, whence it follows that n is not the number of numbers. Again, it may be proved directly, by the correlation of whole and part, that the number of propositions or concepts is infinite [footnote: Cf. Bolzano Paradoxien des Unendlichen, § 13; Dedekind, Was sind und was sollen die Zahlen? No. 66]
§ 343. It is very important to realise, as regards w or a0, that neither has a number immediately preceding it. This characteristic they share with all limits, for the limit of a series is never immediately preceded by any term of the series which it limits When it is forgotten that w has no immediate predecessor, all sorts of contradictions emerge. For suppose n to be the last number before w; then n is a finite number, and the number of finite numbers is n+1. In fact, to say that w has no predecessor is merely to say that the finite numbers have no last term. Though w is preceded by all finite numbers, it is not preceded immediately by any of them: there is none next to w.
§ 348 classes of propositions are only some among objects, yet Cantor's argument shows that there are more of them than there are propositions. Again, we can easily prove that there are more propositional functions than objects. For suppose a correlation of all objects and some propositional functions to have been affected [sic], and let fx be the correlate of x. Then "not-fx(x)," i.e. "fx does not hold of x" is a propositional function not contained in this correlation; for it is true or false of x according as fx is false or true of x, and therefore it differs from fx for every value of x.
§ 475 to § 496 are "The Logical and Arithmetical doctrines of Frege"
[Disagreements with Frege] all result from difference on three points: (1) Frege does not think that there is a contradiction in the notion of concepts which cannot be made logical subjects (see § 49 supra); (2) he thinks that, if a term a occurs in a proposition, the proposition can always be analysed into a and an assertion about a (see Chapter VII); (3) he is not aware of the contradiction [the Paradox] discussed in Chapter X.
§ 478. There are some difficulties in the above theory which it will be well to discuss. In the first place, it seems doubtful whether the introduction of truth-values marks any real analysis. If we consider, say, "Caesar died," it would seem that what is asserted is the propositional concept "the death of Caesar," not "the truth of the death of Caesar." This latter seems to be merely another propositional concept, asserted in "the death of Caesar is true," which is not, I think, the same proposition as "Caesar died." There is great difficulty in avoiding psychological elements here, and it would seem that Frege has allowed them to intrude in describing judgment as the recognition of truth (Gg p. x). The difficulty is due to the fact that there is a psychological sense of assertion, which is what is lacking to Meinong's Annahmen, and that this does not run parallel with the logical sense. Psychologically, any proposition, whether true or false, may be merely thought of, or may be actually asserted: but for this possibility, error would be impossible. But logically, true propositions only are asserted, though they may occur in an unasserted form as parts of other propositions. In "p implies q," either or both of the propositions p, q may be true, yet each, in this proposition, is unasserted in a logical, and not merely in a psychological sense. Thus assertion has a definite place among logical notions, though there is a psychological notion of assertion to which nothing logical corresponds. But assertion does not seem to be a constituent of an asserted proposition, although it is, in some sense, contained in an asserted proposition. If p is a proposition, "p's truth" is a concept which has being even if p is false, and thus "p's truth" is not the same as p asserted. Thus no concept can be found which is equivalent to p asserted, and therefore assertion is not a constituent in p asserted. Yet assertion is not a term to which p, when asserted, has an external relation; for any such relation would need to be itself asserted in order to yield what we want. Also a difficulty arises owing to the apparent fact, which may however be doubted, that an asserted proposition can never be part of another proposition: thus, if this be a fact, where any statement is made about p asserted, it is not really about p asserted, but only about the assertion of p. This difficulty becomes serious in the case of Frege's one and only principle of inference (Bs p.9): "p is true and p implies q; therefore q is true." Here it is quite essential that there should be three actual assertions, otherwise the assertion of propositions deduced from asserted premisses would be impossible; yet the three assertions together form one proposition, whose unity is shown by the word therefore, without which q would not have been deduced, but would have been asserted as a fresh premiss.
It is also impossible, at least to me, to divorce assertion from truth, as Frege does. An asserted proposition, it would seem, must be the same as a true proposition. We may allow that negation belongs to the content of a proposition (Bs p.4), and regard every assertion as asserting something to be true. We shall then correlate p and not-p as unasserted propositions, and regard "p is false" as meaning "not-p is true." But to divorce assertion from truth seems only possible by taking assertion in a psychological sense.
§ 478. (Classes) the extension of a Begriff, Frege says, is the range (Werthvelauf) of a function whose value for every argument is a truth-value (FuB. p.16). Ranges are things, whereas functions are not (p.19). There would be no null-class, if classes were taken in extension; for the null-class is only possible if a class is not a collection of terms (KB. pp 436-7). If x be a term, we cannot identify x, as the extensional view requires, with the class whose only member is x; for suppose x to be a class having more than one member, and let y, z be two different members of x; then if x is identical with the class whose only member is x, y and z will both be members of this class, and will therefore be identical with x and with each other, contrary to the hypothesis (p. 444). The extension of a Begriff has its being in the Begriff itself, not in the individuals falling under the Begriff (p. 451). When I say something about all men, I say nothing about some wretch in the centre of Africa, who is in no way indicated* (Bedeutet?) and does not belong to the indication (Bedeutung?) of man (p. 454). Begriffe are prior to their extension, and it is a mistake to attempt, as Schroeder does, to base extension on individuals; this leads to the calculus of regions (Gebiete) not to Logic (p. 455).
* Russell translates Bedeutung as indication.
Bs = Begriffschift
FT = Uber formale Theorien der Arithmetik
FuB = Function und Begriff
BuG Uber Begriff und Gegenstand
SuB Sinn und Bedeutung
KB = "A critical elucidation of some points in E. Schroeder"s Vorlesungen Ueber Die Algebra der Logik", Archiv fur systematische Philosophie 1895, pp 433-456.
BP "Ueber die Begriffschrift des Herrn Peano und meine eigene" (Berichte der math.-physiscen Classe der Koenigl. Saechse Gesellschaft der Wissenschaften zu Leipzig (1896)
Grattan-Guinness, I. The Search for Mathematical Roots, 1870-1940, Princeton 2000.
Monk, R.., Bertrand Russell: The Spirit of Solitude, London 1996
Shearman, A.T. Review of Russell's Principle's of Mathematics. Mind ns 16, pp. 254-65.
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