In June 1913 the 18-year-old Norbert Wiener presented to Harvard University a doctoral thesis comparing the logical systems of Schröder and Russell, with special reference to their treatment of relations. Shortly afterwards he visited Russell in Cambridge and showed him a copy of the thesis. Russell wrote out some comments, to which Wiener replied.None of these documents has been published. In this paper I summarise the contents of Wiener's thesis, and describe and quote from the subsequent discussion with Russell. I (...) preface the account with some remarks on the personal relationship of Wiener and Russell, and conclude it with the description of the location of the various documents cited. (shrink)
The great influence of Georg Cantor's theory of sets and transfinite arithmetic has led to a considerable interest in his life. It is well known that he had a remarkable and unusual personality, and that he suffered from attacks of mental illness; but the ‘popular’ account of his life is richer in falsehood and distortion than in factual content. This paper attempts to correct these misrepresentations by drawing on a wide variety of manuscript sources concerning Cantor's life and career, including (...) the texts of some important documents. An appendix describes the most important collection of missing manuscripts, whose location would help further the preparation of a biographical study of Cantor. (shrink)
An account is given of the emergence of the concept of work as a basic component of mechanics. It was largely an achievement of engineer savants in France during the Bourbon Restoration , with Navier, Coriolis and Poncelet playing the major roles. Some aspects of the eighteenth-century prehistory are described, and also concurrent developments in French engineering. The principal problem areas were friction, hydraulics, machine performance and ergonomics, and especially in the last context the developments became involved with social and (...) even philosophical movements in the 1820s. Education played an important role throughout; several of the principal sources are textbooks. (shrink)
Graham Priest argued that all the paradoxes of set theory and logic fall under one schema; and hence they should be solved by one kind of solution. This reply addresses both claims, and counters that in fact at least one paradox escapes the schema, and also some apparently "safe" theorems fall within it; and even for the range of paradoxes so captured by the schema, the assumption of a common solution is not obvious; each paradox surely depends upon the theory (...) and context in which it arises. Details of Priest's proposed solution are also sought. (shrink)
Although the existence of correspondence between George Boole (1815?1864) and William Stanley Jevons (1835?1882) has been known for a long time and part was even published in 1913, it has never been fully noted; in particular, it is not in the recent edition of Jevons's letters and papers. The texts are transcribed here, with indication of their significance. Jevons proposed certain quite radical changes to Boole's system, which Boole did not accept; nevertheless, they were to become well established.
This paper describes the materials in the Russell Archives relevant to Russell's work on logic and the foundations of mathematics, and suggests the kinds of information that may and may not be drawn about the historical development of his ideas. By way of illustration, a couple of episodes are described. The first concerns a logical system closely related to his theory of denoting, which preceeds the system used in Principia mathematics, while the second describes a delay in publishing the second (...) volume of that work due to the discovery by Whitehead of a conceptual error. (shrink)
In this paper I consider three mathematicians who allowed some role for menial processes in the foundations of their logical or mathematical theories. Boole regarded his Boolean algebra as a theory of mental acts; Cantor permitted processes of abstraction to play a role in his set theory; Brouwer took perception in time as a cornerstone of his intuitionist mathematics. Three appendices consider related topics.
The Companion Encyclopedia is the first comprehensive work to cover all the principal lines and themes of the history and philosophy of mathematics from ancient times up to the twentieth century. In 176 articles contributed by 160 authors of 18 nationalities, the work describes and analyzes the variety of theories, proofs, techniques, and cultural and practical applications of mathematics. The work's aim is to recover our mathematical heritage and show the importance of mathematics today by treating its interactions with the (...) related disciplines of physics, astronomy, engineering and philosophy. It also covers the history of higher education in mathematics and the growth of institutions and organizations connected with the development of the subject. Part 1 deals with mathematics in various ancient and non-Western cultures from antiquity up to medieval and Renaissance times. Part 2 treats developments in all the main areas of mathematics during the medieval and Renaissance periods up to and including the early 17th century. Parts 3-10 are divided into the main branches into which mathematics developed from the early 17th century onwards: calculus and mathematical analysis, logic and foundations, algebras, geometries, mechanics, mathematical physics and engineering, and probability and statistics. Parts 11-13 review the history of mathematics from an international perspective. The teaching of mathematics in higher education is examined in various countries, and mathematics in culture, art and society is covered. The Companion Encyclopedia features annotated bibliographies of both classic and contemporary sources; black and white illustrations, line figures and equations; biographies of major mathematicians and historians and philosophers of mathematics; a chronological table of main events in the developments of mathematics; and a fully integrated index of people, events and topics. (shrink)
A prized property of theories of all kinds is that of generality, of applicability or least relevance to a wide range of circumstances and situations. The purpose of this article is to present a pair of distinctions that suggest that three kinds of generality are to be found in mathematics and logics, not only at some particular period but especially in developments that take place over time: ‘omnipresent’ and ‘multipresent’ theories, and ‘ubiquitous’ notions that form dependent parts, or moments, of (...) theories. The category of ‘facets’ is also introduced, primarily to assess the roles of diagrams and notations in these two disciplines. Various consequences are explored, starting with means of developing applied mathematics, and then reconsidering several established ways of elaborating or appraising theories, such as analogising, revolutions, abstraction, unification, reduction and axiomatisation. The influence of theories already in place upon theory-building is emphasised. The roles in both mathematics and logics of set theory, abstract algebras, metamathematics, and model theory are assessed, along with the different relationships between the two disciplines adopted in algebraic logic and in mathematical logic. Finally, the issue of monism versus pluralism in these two disciplines is rehearsed, and some suggestions are made about the special character of mathematical and logical knowledge, and also the differences between them. Since the article is basically an exercise in historiography, historical examples and case studies are described or noted throughout. (shrink)
The Scottish logician Hugh MacColl is well known for his innovative contributions to modal and nonclassical logics. However, until now little biographical information has been available about his academic and cultural background, his personal and professional situation, and his position in the scientific community of the Victorian era. The present article reports on a number of recent findings.
Symbolic logics tend to be too mathematical for the philosophers and too philosophical for the mathematicians; and their history is too historical for most mathematicians, philosophers and logicians. This paper reflects upon these professional demarcations as they have developed during the century.
Among the papers left by Bertrand Russell (1872?1970) and now held at the Russell Archives at McMaster University, is a large quantity of material on mathematical logic and the foundations of mathematics. This paper is a provisional survey of their extent and content. Some indications are given of their historical significance, and a discussion is added to the possible modes of their publication in the edition of Russell's Collected papers, currently in progress.
Este artículo presenta un alnplio panorama histórico de las conexiones existentes entre ramas de las matematícas y tipos de lógica durante el periodo 1800-1914. Se observan dos corrientes principales,bastante diferentes entre sí: la lógica algebraica, que hunde sus raíces en la logique yen las algebras de la época revolucionaria francesa y culmina, a través de Boole y De Morgan, en los sistemas de Peirce y de Schröder; y la lógica matematíca, que tiene una fuente de inspiraeión en el analisis matemático (...) de Cauchy y de Weierstrass y culmina, a través de las inieiativas de Peano y de la teoria de conjuntos deCantor, en la obra de Russell. Se extraen algunas conclusiones generales, con referencias relativas a la situaeión posterior a 1914.This article contains a broad historical survey of the connections made between branches of mathematics and types of logic during the period 1800-1914. Two principal streams are noted, rather different from each other: algebraic logic, rooted in French Revolutionary logique and algebras and culminating, via Boole and De Morgan, in the systems of Peirce and Schröder; and mathematical logic, inspired by the mathematical analysis of Cauchy and Weierstrass and culminating, via the initiatives of Peano and the set theory of Cantor, in the work of Russell. Some general conclusions are drawn, with examples given of the state of affairs after 1914. (shrink)
Post's Nachlass has recently been made available to the public in an archive in the U.S.A. After a short summary of his life and career, this article indicates the character and content of the manuscripts, and their significance is assessed. Two short passages are transcribed; and. as a separate item, a paper of the 1930s on the paradoxes is reproduced.
I became the editor of this journal in 1974, under rather strange circumstances and with no prior warning. The circumstances and their background are described here.
Russell argued against the Great War, but he also wanted to drop atomic bombs on the Soviet Union after World War II, and later he advocated nuclear disarmament. How could a great logician accommodate such inconsistencies? How, as a private citizen, did he make such a world-wide impact in his late years?
The main text of Popper’s Logik der Forschung consists of seven chapters outlining the main features of his falsificationist philosophy of science, followed by two involving probability theory and quantum mechanics, and finally one on corroboration, which is the main concern here., noting its newer material when appropriate.) The chapter begins with a section on non-verifiability of theories, which would have been better placed in chapter 6 on testability, and then two sections on the probability of events and of hypotheses (...) and on probability logic, which belong rather in chapter 8. Only then come four short sections on corroboration, the property of a theory that it passes its tests. The chapter is placed last because of a link between corroboration and probability; he even claimed that there was no difference between the two in that both are appraisals. In a later appendix *ix, and in more detail in other work, he offered quantitative measures of corroboration involving probabilities, but at the end of the latter text he almost dismissed the enterprise; and in similar vein, I shall not consider it further, since the measures depend upon the tests which happen to have been carried out and so are not very significant. Otherwise, he did not discuss corroboration very much in his writings, and it is not well handled by commentators;1 but this rather low status contrasts markedly with the actual practice of science, where scientists and engineers regularly work with highly corroborated theories, even to the extent that ‘reliability science’ is a subject in its own right. (shrink)
The main text of Popper’s Logik der Forschung consists of seven chapters outlining the main features of his falsificationist philosophy of science, followed by two involving probability theory and quantum mechanics, and finally one on corroboration, which is the main concern here., noting its newer material when appropriate.) The chapter begins with a section on non-verifiability of theories, which would have been better placed in chapter 6 on testability, and then two sections on the probability of events and of hypotheses (...) and on probability logic, which belong rather in chapter 8. Only then come four short sections on corroboration, the property of a theory that it passes its tests. The chapter is placed last because of a link between corroboration and probability; he even claimed that there was no difference between the two in that both are appraisals. In a later appendix *ix, and in more detail in other work, he offered quantitative measures of corroboration involving probabilities, but at the end of the latter text he almost dismissed the enterprise; and in similar vein, I shall not consider it further, since the measures depend upon the tests which happen to have been carried out and so are not very significant. Otherwise, he did not discuss corroboration very much in his writings, and it is not well handled by commentators;1 but this rather low status contrasts markedly with the actual practice of science, where scientists and engineers regularly work with highly corroborated theories, even to the extent that ‘reliability science’ is a subject in its own right. (shrink)
A non-standard contribution to Mozart's bicentenary year is made by showing that he was a refined numerologist, especially in the opera Die Zauberflöte , but also in some other works of his maturity. An extensive analysis of this opera is furnished, showing that the numerology is evident not only in the structure of the work and the design of melodies and repetitions of musical and literary motives, but even in the timing and staging of the first performance. The numerology is (...) shown to be inherited from the Masonic tradition, to which he and his librettist Schikaneder adhered. The suggestion is made that a significant but largely unnoticed aspect of Mozart's thought should become a normal part of professionalized musicological study of Mozart. The chances for such a development are appraised sceptically in the final section. (shrink)
