1 The objection is sometimes raised that without certain types of experience, such as encountering several objects of the same kind, the integers and the arithmetical operations with them would never have been invented, and that therefore the propositions of arithmetic do have an empirical basis. This type of argument, however, involves a confusion of the logical and the psychological meaning of the term "basis." It may very well be the case that certain experiences occasion psychologically the formation of arithmetical ideas and in this sense form an empirical "basis" for them; but this point is entirely irrelevant for the logical questions as to the grounds on which the propositions of arithmetic may be accepted as true. The point made above is that no empirical "basis" or evidence whatever is needed to establish the truth of the propositions of arithmetic.

2 A precise account of the definition and the essential characteristics of the identity relation may be found in A. Tarski, Introduction to Logic, New York, 1941, ch. III.

3 For a lucid and concise account of the axiomatic method, see A. Tarski, loc. cit., ch. VI.

4 Bertrand Russell, Introduction to Mathematical Philosophy, New York and London, 1919, p. 71.

5 For a more detailed account of the construction of the number system on Peano's basis, cf. Bertrand Russell, loc. cit., esp. chs. I and VII. A rigorous and concise presentation of that construction, beginning, however, with the set of all integers rather than that of the natural numbers, may be found in G. Birkhoff and S. MacLane, A Survey of Modern Algebra, New York, 1941, chs. I, II, III, V. For a general survey of the construction of the number system, cf. also J. W. Young, Lectures on the Fundamental Concepts of Algebra and Geometry, New York, 1911, esp. lectures X, XI, XII.

6 As a result of very deep-reaching investigations carried out by K. Godel it is known that arithmetic, and a fortiori mathematics, is an incomplete theory in the following sense: While all those propositions which belong to the classical systems of arithmetic, algebra, and analysis can indeed be derived, in the sense characterised above, from the Peano postulates, there exist nevertheless other propositions which can be expressed in purely arithmetical terms, and which are true, but which cannot be derived from the Peano system. And more generally: For any postulate system of arithmetic (or of mathematics for that matter) which is not self-contradictory, there exist propositions which are true, and which can be stated in purely arithmetical terms, but which cannot be derived from that postulate system. In other words, it is impossible to construct a postulate system which is not self-contradictory, and which contains among its consequences all true propositions which can be formulated within the language of arithmetic.

This fact does not, however, affect the result outlined above, namely, that it possible to deduce, from the Peano postulates and the additional definitions of non-primitive terms, all those propositions which constitute the classical theory of arithmetie, algebra, and analysis; and it is to these propositions that I refer above and subsequently as the propositions of mathematics.

7 For a more detailed discussion, cf. Russell, loc. cit., chs. II, III, IV. A complete technical development of the idea can be found in the great standard work in mathematical logic, A. N. Whitehead and B. Russell, Principia Mathematica, Cambridge, England, 1910-1913. For a very precise development of the theory, see W. V. O. Quine, Mathematical Logic, New York, 1940. A specific discussion of the Peano system and its interpretations from the viewpoint of semantics is included in R. Carnap, Foundations of Logic and Mathematics, International Encyclopedia of Unified Science, Vol. I, no. 3. Chicago, 1939; especially sections 14, 17, 18.

8 The assertion that the definitions given above state the "customary" meaning of the arithmetical terms involved is to be understood in the logical, not the psychological sense of the term "meaning." It would obviously be absurd to claim that the above definitions express "what everybody has in mind" when talking about numbers and the various operations that can be performed with them. What is achieved by those definitions is rather a "logical reconstruction" of the concepts of arithmetic in the sense that if the definitions are accepted, then those statements in science and everyday discourse which involve arithmetical terms can be interpreted coherently and systematically in such a manner that they are capable of objective validation. The statement about the two persons in the office provides a very elementary illustration of what it meant here.

9 Cf. Bertrand Russell, loc. cit., p. 24 and ch. XIII.

10 The principles of logic developed in modern systems of formal logic embody certain restrictions as compared with those logical rules which had been rather generally accepted as sound until about the turn of the 20th century. At that time, the discovery of the famous paradoxes of logic, especially of Russell's paradox (cf. Russell, loc. cit., ch. XIII), revealed the fact that the logical principles implicit in customary mathematical reasoning involved contradictions and therefore had to be curtailed in one manner or another.

11 Note that we may say "hence" by virtue of the rule of substitution, which is one of the rules of logical inference.

12 For a more detailed discussion of this point, cf. the next article in this collection.