The construction of a deductive system may be approached from two fundamentally different points of view. On the one hand the construction may be undertaken in order to see what theorems can be derived from a set of primitive propositions regarded as mutually consistent and severally independent. Such a set of primitive propositions may be called a 'a set of postulates', and the system thus constructed may be called 'a postulational system'. It is characteristic of postulational systems that the analysis involved may be circular.1 This circularity is due to the fact that the theorems and the postulates are at the same level. This means that, once the postulates have been selected, then simplicity is to be understood with reference only to the steps involved in passing from the postulates to the theorems. One theorem is simpler than another if the former is derived from the postulates in a fewer number of steps than the latter. The postulates are simple only in the sense that no steps are required to reach the postulates; on the contrary, all steps set out from them. Thus the postulates are not simpler than anything. The selection of any given postulate set may be due to a variety of reasons, but these reasons have nothing whatever to do with the system when constructed. Hilbert, Veblen, and E. V. Huntington have constructed many different postulational systems, relating to various branches of mathematics. The reference to mathematics, however, comes in only when the postulational system is interpreted. Demonstration occurs in the system, but the demonstration is relative to the initial postulates.
The construction of a postulational system may, on the other hand, be undertaken in order to analyse the entities constituting the undefined concepts and to exhibit their interrelations in an orderly manner. In such a system circularity would be a defect, since its distinguishing characteristic is that it has a direction. I propose to call a system thus constructed a 'directional system', in order to mark its difference from an ordinary postulational system. In a directional system the postulates will be properly primitive; that is to say, the distinction between the theorem and the postulates of the system will not relate merely to the fact that the latter are undemonstrated. It seems to me that Whitehead and Russell in constructing the system developed in Principia* From A Modern Introduction to Logic, Harper Torchbook, [1950]
1See C. I. Lewis, Mind and the World Order, 210.
506 Mathematica were not constructing a postulational system, but a directional system.1 They clearly state that their work 'aims at affecting the greatest possible analysis of the ideas with which it deals'. These 'ideas' are the fundamental concepts of mathematics, which they sought to reduce to their simplest elements, and thus to show that mathematical concepts are wholly reducible to the concepts of pure logic. They were not content to construct one out of a possible variety of postulational systems, any one of which would yield theorems capable of mathematical interpretation. To achieve this aim the primitive concepts cannot be taken merely as undefined, nor the primitive propositions merely as postulated; they must be fundamental in a sense which excludes arbitrary selection, and thus be simple in some relative sense.
If this view of Principia Mathematica be correct, then it is open to a criticism which is difficult to refute, and which is independent of any such defects as the assumption of the so-called Axiom of Reducibility. This criticism may be raised by considering Russell's treatment of symbolism and of the part played by definition in the system of Principia Mathematica. It was pointed out in Chapter XXII that Russell's account of definition is inconsistent and untenable.2 He wants both to regard definitions as merely 'typographical conveniences' and also to hold that definitions may 'express a notable advance' by containing an analysis of a concept. His confusion on this point is no doubt responsible for his failure to see that in defining "p É q" by "~p v q", he is, according to his own statement, merely giving a 'convenient abbreviation', whereas he treats the definition as though it afforded "an analysis of implication." In the same way he regards the notion of a propositional function, and therefore, of a class, as fundamental, yet at the same time he declares that classes are "mere symbolic conveniences'. Both these views cannot be correct. The latter is the view which Russell evidentally wishes to maintain, but he does not seem to be able to dispense with the former. His whole treatment of the notion of a class is obscured by the confusion, to which Russell is singularly prone, of the symbol with the symbolized.
This point raises the question of Russell's treatment of symbolism. Nowhere is Principia Mathematica is there any discussion of the nature and conditions of symbolism and its relation to mathematics. In the Introduction to the first edition we find the following statement: 'The use of a symbolism, other than that of words, in all parts of the book
1 For a discussion of directional analysis see my paper on 'The Method of Analysis in Metaphysics', Proc. Arist. Soc., N.S., XXIII.
2 See p. 440 above. In the rest of the discussion I shall speak of 'Russell', instead of 'Whitehead and Russell', since Vol. I is recognized to be due almost entirely to Russell, and to him the whole of the second edition of Principia Mathematica is due.
507 which aim at embodying strictly accurate demonstrative reasoning, has been forced on us by the consistent pursuit of the above three purposes'.1 But the account which follows does not explain whether symbols are essential or whether they are employed merely because human intellect, being finite, is limited in apprehension. The latter alternative is suggested, but the former seems to be the one required. If, however, symbols are essential, then in what sense have these symbols significance? A non-significant symbol is a contradiction in terms. Yet, if they have significance, whence is their significance derived? Do we not in any case require non-symbolic rules relating to the significance of the symbolic expressions? No such rules are given within the system of Principia Mathematica itself. In the Preface it is stated: 'our logical system is wholly contained in the numbered propositions, which are independent of the Introduction and the Summaries'.2 But we look in vain among the numbered propositions for any principles determining what combination of symbols are significant.
These difficulties, and others which cannot be discussed here, suggest that certain pre-mathematical notions are indispensable to the construction of a mathematical system, and that non-symbolic rules of significance are required, and must be included in the system, if the system is to be a directional system capable of yielding an analysis of the fundamental ideas of mathematics. As W. E. Johnson has pointed out: 'Even a perfectly constructed symbolic system would need to introduce some axioms, as also some propositions derived from axioms, that can only be expressed in non-symbolic terms. This necessary recourse to ordinary language in developing a deductive system shows that direct attention to meanings, presented linguistically, is entailed in the intelligent following of even a professedly symbolic exposition'.3 This is undoubtedly true of a directional system such as that presented in Principia Mathematica.
The importance of rules of significance, or, to use Johnson's expression 'attention to meanings' is recognized by Hilbert and his school, who are usually called 'Formalists'. I have space here only to consider very briefly the contrast of the Formalist conception of mathematics with that contained in Principia Mathematica. Hilbert distinguishes between mathematics and metamathematics. This distinction may be regarded as a distinction between the construction of a symbolic system involving arbitrarily selected rules, or postulates, and the statement of rules of significance. According to Hilbert the mathematician operates with marks, or numerals, i.e. physical signs on paper. These marks are as counters in a game, or as chessmen in a game of chess.1 For the statement of these three purposes, see p. 487 above.
2 This statement is not accurate, for it is not possible to understand the numbered propositions without reference to the summaries. For example in *21, it is assumed that f(y,x) shall be different from f(x,y), and mention is made of alphabetical order and of typographical order. But this explanation is contained in the Summary only, yet the sense of the relation f(x,y) is essential of the theorems.
3W. E. J., Pt. II, p. 45.
508 The rules in accordance with which certain combinations of the marks are allowed constitutes metamathematics. They cannot be developed without them. The theorems of metamathematics as the rules of chess stand to the actual positions of the chessmen in possible games of chess. As Prof. G. H. Hardy has pointed out,1 this distinction involves a sharp distinction between between two forms of proof. There is first pure demonstration, which is proof inside the system. This corresponds to games of chess. There is secondly the proof that ertain combinations of symbols cannot occur. These are the theorems of metamathematics. In the construction of these proofs there must be attention to meanings. This separation of metamathematics from mathematics seems to have for its purpose the separation of the form of the system from what would commonly be regarded as its significance. To ask whether any theorem in the purely formal system is true would be to ask a nonsensical question. We can only ask whether the rules have been followed. If they have, then the system is consistent.
There is much to be said in favor of this Formalist conception of mathematics. But it still leaves a considerable problem, which cannot be regarded as settled. This is the problem of the exact nature of the metamathematical statements, and thus of the part played by intuition in the foundation of mathematics. A word may finally be added on this important topic.
Modern logicians are agreed that spatial intuition does not enter in mathematics. Russell and Whitehead seem to recognize that intuition of the fundamental logical concepts is required. Hilbert, whilst denying that mathematical notions can be derived from purely logical notions, insists that intuition plays an important part, and that what is intuited is the actual marks, or numerals, the physical signs onpaper with which the mathematical operates. A third school, of whom Brouwer and Weyl are the most important representatives, hold that mathematics is based upon a pure intuition of the integral numbers, which are involved in the intuitions of time. None of the three views can be regarded as having been satisfactorily unless established. All that can be said here is that no view is likely to be satisfactory unless due notice is taken of the important problem of significance. Mathematics is something more than symbolic structure; it is a structure having significance. Symbolism is, indeed, essential to mathematics, and not adopted merely for convenience. The symbolism must, however, be adequate to that which is symbolized.1 "Mathematical Proof", Mind, N.S., 149. The student who is interested in the theory of the foundations of mathematics should consult this article. I should have made use of it in the first edition of this book, had it been published. I am much indebted to it now.
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