# The Search for Mathematical Roots, 1870-1940: Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Godel

I. GRATTAN-GUINNESS
Pages: 624
https://www.jstor.org/stable/j.ctt7rp8j

1. Front Matter
(pp. i-iv)
(pp. v-2)
3. CHAPTER 1 Explanations
(pp. 3-13)

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...

4. CHAPTER 2 Preludes: Algebraic Logic and Mathematical Analysis up to 1870
(pp. 14-74)

The story begins in French mathematics and philosophy in the late 18th century: specifically the semiotic ‘logique’ of Condillac and Condorcet and the connections with the algebraic theories, especially the calculus, developed by Lagrange (§2.2). Then it moves to England, for both topics: the adoption of Lagrangian mathematics by Babbage and Herschel, and the revival of logic (although not after the French model) in the 1820s (§2.3). Next come the two principal first founders of algebraic logic, De Morgan and Boole (§2.4–5). The main initial reactions to Boole are described in §2.6.

In a change of topic, §2.7 also...

5. CHAPTER 3 Cantor: Mathematics as Mengenlehre
(pp. 75-125)

After summarising Cantor’s life and career in the next section, the story is told of his creation of the branch of mathematics which we call ‘set theory’; but when I wish to refer to his version of it I shall conserve even in translations the word ‘Mengenlehre’ which he used especially in his final years of the mid 1890s and which became the most common name among German-writing authors thereafter (§4.2.1). First, §3.2–3 covers its founding between 1870 and 1885, and §3.4 treats the final papers. Important concurrent work of Dedekind is also included: on irrational numbers in §3.2.4,...

6. CHAPTER 4 Parallel Processes in set Theory, Logics and Axiomatics, 1870s–1900s
(pp. 126-218)

In this chapter are collected six concurrent developments of great importance which, with one exception, ran alongside mathematical logic rather than within it. It is largely a German story, with some important American ingredients; among the main general sources is the reviewingJahrbuch über die Fortschritte der Mathematik. Set theory is the main common thread, and §4.2 deals with the growth of interest in it, both as CantorianMengenlehre, and more generally.

Next, §4.3 describes the contributions to algebraic logic made by C. S. Peirce and some followers at Johns Hopkins University. The union of Boole’s algebra with De Morgan’s...

7. CHAPTER 5 Peano: the Formulary of Mathematics
(pp. 219-267)

Giuseppe Peano was an important contributor to mathematical analysis and a principal founder of mathematical logic, as well as the leader of a school of followers in Italy. Our concern here is with their work until around 1900, when Russell met Peano; their later contributions will be noted in subsequent chapters.

The account focuses upon logic and the foundations of arithmetic and analysis, including set theory. Peano’s own writings are the main concern; they seem to have gained the main reaction at the time, not only with Russell. §5.2 traces his initial contributions to mathematical analysis and acquaintance with logic...

8. CHAPTER 6 Russell’s Way In: From Certainty to Paradoxes, 1895–1903
(pp. 268-332)

This chapter and its successor treat Russell’s career in logic from 1897 to 1913. The point of division lies in 1903, when he published the bookThe principles of mathematics, where he expounded in detail the first version of his logicist thesis. This chapter traces the origins of that enterprise in his student ambitions at Cambridge University from 1890 to 1894 followed by six years of research under a Prize Fellowship at Trinity College and then a lectureship there;The principleswas the principal product.

In addition to the birth of logicism, we shall record the growing positive role of...

9. CHAPTER 7 Russell and Whitehead Seek the Principia Mathematica, 1903–1913
(pp. 333-410)

This chapter covers the period during which Whitehead and Russell collaborated to work out their logicistic programme in detail. Mostly they preparedPrincipia mathematicaat their respective homes at Grantchester near Cambridge and Bagley Wood near Oxford; thus much discussion was executed in letters, of which several survive at Russell’s end.

This chapter divides into two halves around 1906 and 1907 because of their change of strategy. After accumulating more paradoxes and axioms, and much work on denoting (§7.2–§7.4.5), Russell developed intensively a logical system which he called ‘the substitutional theory’ (§7.4.6–8); but then he abandoned it and...

10. CHAPTER 8 The Influence and Place of Logicism, 1910–1930
(pp. 411-505)

The reception ofPMfrom its publication to around 1940 is covered in this and the next chapters, with the break coming around 1930. There was a wide range of reactions both to the logical calculus ofPMand to logicism; some striking similarities arose from different backgrounds. I associate each main philosophy with a ‘school’, in contrast to the ‘traditions’ of algebraic and mathematical logics.

This chapter falls into two roughly equal parts, with Anglo-Saxon attitudes followed by reactions elsewhere. §8.2 surveys Whitehead’s and Russell’s very different transitions from logic to philosophy till around 1916. Whitehead did not adopt...

11. CHAPTER 9 Postludes: Mathematical Logic and Logicism in the 1930s
(pp. 506-555)

As noted in §8.1, this chapter contains rather less detail than before; for example, most details after 1935 round off old stories rather than launch new ones. Other philosophical schools, especially formalism (where I follow authors who used the word) and intuitionism, are treated lightly.

To start, §9.2 focuses upon Gödel’s incompletability theorem and corollary (as his second result is often called), and the first reactions to them. §9.3 takes the Vienna Circle: Carnap dominates, but we find two new figures and two others already met. §9.4 covers the U.S.A., whither several Circle members had to emigrate after 1933, and...

12. CHAPTER 10 The Fate of the Search
(pp. 556-573)

A story as rich and interconnected as this one could generate masses of meta-consideration; but I avoid the temptation, especially as various ‘‘local’’ conclusions have been drawn and summaries madeen route. After a general comparison of algebraic and mathematical logics, the focus falls mainly upon Russell, and is mainly organised before, during and afterPM; it ends with several appraisals of logic(ism) in the U.S.A. in the early 1940s. The chapters ends with a flow-chart for the whole story and some notes on formalism and intuitionism,¹ before locating symbolic logic in mathematics and philosophy in general, and emphasising the...

13. CHAPTER 11 Transcription of Manuscripts
(pp. 574-593)

Most of the manuscripts transcribed here are (parts of) letters to or from Russell, both ways in §11.8: they are ordered chronologically. Unless otherwise stated, the originals are kept in the Russell Archives; permissions to publish have been recorded in §1.6. Orthography has been followed, including the rendering of underlinings and capital letters, and the presence or absence of punctuation and inverted commas (single or double). The layout of formulae has been followed as closely as practicable. But deletions and slips in writing have been ignored when they are insignificant; and opening flourishes and signatures are omitted. Editorial insertions or...

14. BIBLIOGRAPHY
(pp. 594-670)
15. INDEX
(pp. 671-690)