Nature Of Mathematics Ilphil28

Max Black, C. K. Ogden

• Publisher: Taylor & Francis
• Year: 2014
• Print ISBN: 9780415613828
• eBook ISBN: 9781317830504
• Edition: 1
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

First published in 2000. This is Volume I of eight in the Philosophy of Logic and Mathematics series. Written in 1933, in The Nature of Mathematics offers a critical survey the author seeks to present a considered critical exposition of Principia Mathematica and to give supplementary accounts of the formalist and intuitionist doctrines in sufficient detail to lighten the paths of all who may be provoked to read the original papers.

SECTION I: LOGISTIC

SECTION I: LOGISTIC

THIS section will be devoted to a detailed description of a group of theories concerning the nature of mathematics which assert that mathematics should be considered as a branch of logic. If this opinion is correct the distinction between the two sciences, though venerable and established, is quite arbitrary. This claim is based on proofs which seek to demonstrate in detail how the reduction of mathematics to logic is accomplished.

Any philosophy of mathematics which includes this doctrine will for convenience of reference, and with the reader’s permission, be qualified in this book by the adjective ‘logistic’. This usage of the term is frequent in the literature of the subject and it is sufficient to mention another, less frequent, use of the same term, viz. as a substantive denoting “the science which deals with types of order as such” (C. I. Lewis, Survey of Symbolic Logic, p. 3), to forestall any confusion between the two meanings. The latter use of the word is based upon and implies a distinction between logistic, the science which treats of all types of order, and symbolic logic, that section of logistic which is concerned with the specific types of order exemplified by propositions; but our use of the word will not presuppose that this distinction is recognized by the philosophers whose theories will be termed logistic.

We commence with a brief historical summary of the views under consideration.

History of Logistic Views of Mathematics

A notice of the chief logistic writers from Leibniz to Wittgenstein.

The beginnings of logistic philosophies of mathematics are to be found in the gradual application to logic of a symbolic technique modelled upon the parallel use of symbols in mathematics. In its later stages this process was accompanied by extensive alterations in the traditional Aristotelian logic, by the introduction of many more propositional forms than Aristotle or those who expounded his logic recognized. This in time presented fresh opportunities for the application of symbolic technique, until finally systems of symbols were invented of sufficient generality to be used in the attempt to reduce mathematics to logic.

A convenient starting point for the present brief mention of the landmarks of this process of development is made by Leibniz, whose technical researches in symbolism preceded and often inspired the long series of inventors who perfected the algebra of logic. His work contained the germ of the entire logistic conception; it is no mere coincidence that many of the logistic philosophers find themselves sympathetic to Leibniz and inherit the characteristic atomism of his system.¹

The significance for our purposes of Leibniz’s studies in the algebra of logic² lies in the fact that no proof with any pretensions to rigour of the thesis that mathematics can be reduced to logic is possible without a well-developed symbolism and calculus for logic itself. Statements occurring in logic must be systematically symbolized in order that their relationships to mathematical theorems should become apparent. Leibniz, a mathematician of genius as well as a philosopher, was eminently fitted to begin the task of inventing the algebra of logic and his papers² show him to have made several attempts though with other motives.

Subsequent writers, of whom the most important are De Morgan (Formal Logic, 1847), George Boole (An Investigation into the Laws of Thought, 1854), E. Schröder (Vorlesungen über die Algebra der Logik, 1890–1905), and C. S. Pierce (see bibliography), by their elaboration of the algebra of logic fulfilled Leibniz’s dream of a Characteristica Universalis, a calculus of reasoning suited for the logical analysis of concepts and the structure of scientific systems, and provided the necessary technical equipment for the logistic school. Schröder and Pierce emancipated symbolic logic not only from the Aristotelian view which permitted only the subject-predicate form for propositions but also to a great extent from the insistent preoccupation with mathematical analogies which retarded the early advance of the subject; the way is clear for the actual analysis of mathematics. The first important work of this second period was accomplished by R. Dedekind (Was sind und was sollen die Zahlen?, 1888), who supplied the now famous method of defining real numbers in the mathematical continuum in terms of the rational or fractional numbers. His work may be regarded as a continuation of Weierstrass’s movement to ‘arithmetize’ mathematics, that is to reduce all pure mathematics to the study of the properties of integers; for after Dedekind the study of irrational numbers could be replaced by the study of certain classes of fractional numbers; and the reduction of the study of fractional numbers to that of integers presents no difficulties and had already been accomplished.

The definition of real numbers by ‘Dedekind section’ as his method is called, although accepted by mathematicians and used as the very foundation of the modern theory of functions, had to meet serious criticism which subsequently led to attempts at improvement by the logistic philosophers.

The next works of historical importance are Frege’s Begriffsschrift, 1879, Grundlagen der Arithmetik, 1884, and Grundgesetze der Arithmetik, 1893–1903. The last two books completed the reduction of mathematics by defining the rational numbers in terms of logical entities. Unfortunately Frege did not use Boole’s calculus of logic, preferring an elaborate but clumsy symbolism of his own, whose intricacy prevented his work receiving the recognition it deserved; his books remained almost unknown until rediscovered by Russell after the latter’s Principles of Mathematics had been written.

While Frege had given a philosophic analysis of the concept of number, the Italian mathematician Peano and his school (Formulaire de Mathématiques, 1895–1905), in the course of extensive researches in symbolic logic, had shown that all propositions concerning the natural numbers which are required in mathematics can be deduced from a set of five axioms.

The results of Dedekind, Frege, and Peano had covered in conjunction the whole field of elementary pure mathematics,¹ and by reducing the real numbers to integers, integers to entities occurring in logic, had supplied all the materials for the logistic thesis. There was still needed a synthesis to co-ordinate these results and remedy the imperfections of these early proofs. This was begun by Bertrand Russell in Principles of Mathematics, 1903, and continued in Principia Mathematica (first edition, 1910) written in collaboration with Alfred North Whitehead. These two books are at the apex of the second period in the logistic movement; they profess to prove, rigorously and with the utmost detail, the identity of mathematics and logic.

The first is a philosophical and polemical discussion of the logistic theories; the second, written except for a minimum of incidental explanation entirely in mathematical symbols, a proof of the theories.

Since Principia Mathematica little advance has been made by the logistic school and time has shown serious defects in that work, so that the third period has been one of successive attempts to consolidate a position which at one time Whitehead and Russell appeared to have reached triumphantly.

Among the most notable of these attempts are H. Weyl’s Das Kontinuum, 1918; L. Chwistek’s Theory of Constructive Types, 1923–5; and F. P. Ramsey’s Foundations of Mathematics, 1927. All these defend a logistic position. In addition there remains the remarkable Tractatus Logico—Philosophicus, 1922, of L. Wittgenstein, a former pupil of Russell, whose conclusions, in many respects unfavourable to Principia Mathematica, should be regarded as the self-critical culmination of the logistic movement.

Tasks of a Philosophy of Mathematics

The finite and infinite problems of a philosophy of mathematics are the investigations of the notions ‘integer’ and ‘continuum’ respectively. The subsequent analysis tends to replace these unclear notions by more precise ones with the same formal properties. The plan of such analysis is outlined.

A philosophy of mathematics has two principal objects intimately connected with arithmetic and the theory of functions respectively:—

(1) To elucidate and analyze the notion of ‘integer’ or ‘natural number’,

(2) to elucidate the nature of the mathematical continuum. These are formidable tasks; ignorance of the correct answers has provided paradoxes which date back to Zeno.

For convenience of reference let these problems be called the finite and the infinite problems of mathematical philosophy respectively. They are distinct, although the solution of the second may presuppose knowledge of the solution of the first.

In spite of the contradictions which the second of these concepts appears to contain (p. 89), the notions of ‘integer’ and ‘continuum’ have been used with constant success and with such mutual agreement that the validity of proofs involving them can, with a few notable exceptions, be decided by the unanimous vote of those with sufficient mathematical training to understand them.¹

It would therefore appear that the terms ‘continuum’ and ‘integer’ have meaning for the mathematician and the same meaning for all of them,² and the natural procedure for solving both the finite and the infinite problems would seem to be to examine as closely as possible, and subsequently to analyze, the meanings of these terms. Such an approach would be bound to emphasize the ideas which mathematicians associate with the symbols they use, rather than the apparent interconnection of these symbols shown by marks on paper. And the resulting analysis would need to be such as the mathematician himself could accept as clarifications of his notions. Similar remarks are applicable to the philosophic analysis of any system of interconnected notions. Such a programme has in effect been adopted by the so-called logico-analytic school of philosophers³ who have, however, contributed but little to the analysis of mathematics, being rather concerned with the analysis of facts of everyday experience.

We will restrict ourselves to two comments on the scope of this method in the analysis of mathematical notions.

(1) In spite of the mentioned agreement between mathematicians, it seems possible to deduce contradictions from the mathematical notion of the continuum; these contradictions refer to the subject-matter of mathematics and can be deduced by formally correct mathematical reasoning (p. 89). They are sufficiently striking to have led a very celebrated living mathematician to speak of a vicious circle in present-day mathematics (Herman Weyl: “Der circulus vitiosus in der heutigen Begründung der Analysis” Jahresbericht der Deutschen Mathematiker Vereinigung, vol. xxviii, pp. 85–92, 1919). So it is not unfair to ascribe much of the agreement between mathematicians to the fa...

Table of contents

• Cover Page
• Half Title page
• Title Page
• Copyright Page
• Contents
• Preface
• Introduction
• Section I: Logistic
• Supplement A: Formalism
• Supplement B: Intuitionism
• Bibliography
• Index