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I. Grattan-Guinness

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"I know of no comparably comprehensive treatment of the history of this important period in modern logic. There is a large body of historical literature that is in need of just the kind of synthesis and masterly overview that this work provides. Though most people recognize mathematics as a principal motivating force behind the development of modern logic, the influences on and from mathematics have been largely ignored or minimized. The Search for Mathematical Roots acts as a guide through that challenging mathematical thicket."--Albert C. Lewis, chief editor of The History of Mathematics from Antiquity to the Present

"Ivor Grattan-Guinness provides a marvelous, comprehensive overview of the history of efforts to come to an understanding of mathematical logic and its relation to mathematics in the period 1870-1940. Given its rich detail and inclusion of under-appreciated figures who deserve to be better known, this is an especially important and useful book."--Joseph Dauben, author of George Cantor: His Mathematics and Philosophy of the Infinite

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The Search for Mathematical Roots, 1870–1940

PRINCETON UNIVERSITY PRESS

Contents

BIBLIOGRAPHY..............................................................................................594INDEX.....................................................................................................671

Chapter One

1.1 Sallies

Language is an instrument of Logic, but not an indispensable instrument.

Boole 1847a, 118

We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of exact science are mathematics and logic; the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it sees better with one eye than with two.

De Morgan 1868a, 71

That which is provable, ought not to be believed in science without proof.

Dedekind 1888a, preface

If I compare arithmetic with a tree that unfolds upwards in a multitude of techniques and theorems whilst the root drives into the depths [...]

Frege 1893a, xiii

Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians.

Russell 1903a, 451

1.2 Scope and Limits of the Book

1.2.1 An outline history. The story told here from §3 onwards is regarded as well known. It begins with the emergence of set theory in the 1870s under the inspiration of Georg Cantor, and the contemporary development of mathematical logic by Gottlob Frege and especially Giuseppe Peano. A cumulation of these and some related movements was achieved in the 1900s with the philosophy of mathematics proposed by Alfred North Whitehead and Bertrand Russell. They claimed that "all" mathematics could be founded on a mathematical logic comprising the propositional and predicate calculi including a logic of relations, with set theory providing many techniques and various other devices to hand, especially to solve the paradoxes of set theory and logic which Russell discovered or collected. Their position was given a definitive presentation in the three volumes of Principia mathematica (1910–1913). The name 'logicism' has become attached to this position; it is due in this sense of the word to Abraham Fraenkel (§8.7.6) and especially Rudolf Carnap (§8.9.3) only in the late 1920s, but I shall use it throughout.

Various consequences followed, especially revised conceptions of logic and/or logicism from Russell's followers Ludwig Wittgenstein and Frank Ramsey, and from his own revisions of the mid 1920s. Then many techniques and aims were adopted by the Vienna Circle of philosophers, affirmatively with Carnap but negatively from Kurt Gödel in that his incompletability theorem of 1931 showed that the assumptions of consistency and completeness intuitively made by Russell and by most mathematicians and logicians of that time could not be sustained in the form intended. No authoritative position, either within or outside logicism, emerged: after 1931 many of the main questions had to be re-framed, and another epoch began.

The tale is fairly familiar, but mostly for its philosophical content; here the main emphasis is laid on the logical and mathematical sides. The story will now be reviewed in more detail from these points of view.

1.2.2 Mathematical aspects. First of all, the most pertinent parts of the prehistory are related in §2. The bulk of the chapter is given over to developments of new algebras in France in the early 19th century and their partial adoption in England; and then follow the contributions of George Boole and Augustus De Morgan (§2.4–5), who each adapted one of these algebras to produce a mathematicised logic. The algebras were not the same, so neither were the resulting logics; together they largely founded the tradition of algebraic logic, with some adoption by others (§2.6). By contrast, the prehistory of mathematical logic lies squarely in mathematical analysis, and its origins in Augustin-Louis Cauchy and extension led by Karl Weierstrass are recalled in §2.7, the concluding section of this chapter, to lead in to the main story which then follows. A common feature of both traditions is that their practitioners handled collections in the traditional way of part-whole theory, where, say, the sub-collection of Englishmen is part of the collection of men, and membership to it is not distinguished from inclusion within it.

The set theory introduced in §3 is the 'Mengenlehre' of Georg Cantor, both the point set topology and transfinite arithmetic and the general theory of sets. In an important contrast with part-whole theory, an object was distinguished from its unit set, and belonged to a set S whereas sub-sets were included in S: for example, object a belongs to the set {a, b, c} of objects while sets {a} and {a, b} are subsets of it. The appearance of both approaches to collections explains the phrase 'set theories' in the sub-title of this book.

Next, §4 treats a sextet of related areas contemporary with the main themes outlined above, largely over the period 1870–1900. Firstly, §4.2 records the splitting in the late 1890s of Cantor's Mengenlehre into its general and its topological branches, and briefly describes measure theory and functional analysis. Next, §4.3–4 outlines the extension of algebraic logic by Ernst Schroder and Charles Sanders Peirce, where in particular the contributions of Boole and De Morgan were fused in a Boolean logic of relations; Peirce also introduced quantification theory, which Schröder developed. All this work continued within part-whole theory. §4.5 outlines the creation of a version of mathematical logic by Frege, highly regarded today but as will be explained modestly noted in his own time; it included elements of set theory. Then follows §4.6 on the first stages in the development of phenomenological logic by Edmund Husserl. Finally, §4.7 notes the early stages of David Hilbert's proof theory not yet his meta. mathematics, and of American work in model theory influenced by E. H. Moore.

Then §5 describes the work of Peano and his followers who were affectionately known as the 'Peanists', which gained the greatest attention of mathematicians. Inspired by Weierstrass's analysis and Mengenlehre, this 'mathematical logic' Peano's name was used to express quite a wide range of mathematical theories in terms of proportional and predicate calculi with quantification but the latter now construed in terms of members of sets rather than part-whole theory. The period covered runs from 1888 to 1900, when Russell and Whitehead became acquainted with the work of the Peanists and were inspired by it to conceive of logicism.

Russell's career in logic is largely contained within the next two chapters. First, §6 begins with his début in both...