After a short preface, the first of the three sections of this paper is devoted to historical and philosophic aspects of categoricity. The second section is a self-contained exposition, including detailed definitions, of a proof that every mathematical system whose domain is the closure of its set of distinguished individuals under its distinguished functions is categorically characterized by its induction principle together with its true atoms (atomic sentences and negations of atomic sentences). The third section deals with applications especially those (...) involving the distinction between characterizing a system and axiomatizing the truths of a system. (shrink)

Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre (...) Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations. (shrink)

Here the author of How to Solve It explains how to become a "good guesser." Marked by G. Polya's simple, energetic prose and use of clever examples from a wide range of human activities, this two-volume work explores techniques of guessing, inductive reasoning, and reasoning by analogy, and the role they play in the most rigorous of deductive disciplines.

This article provides an English translation of a historic discussion on the foundations of mathematics, during which Kurt GÖdel first announced his incompleteness theorem to the mathematical world. The text of the discussion is preceded by brief background remarks and commentary.

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.

This is an extensively revised edition of Mr. Quine's introduction to abstract set theory and to various axiomatic systematizations of the subject.

/ Part INTRODUCTION Fuzziness is not a priori an obvious concept and demands some explanation. "Fuzziness" is what Black (NF) calls "vagueness" when ...

Presents general information about meteorology, weather, and climate and includes more than thirty activities to help study these topics, including making a ...

The Protean Character of Mathematics SAUNDERS MAC LANE (Chicago) 1. Introduction The thesis of this paper is that mathematics is protean. ...

SummaryIn 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 (England) 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)

Articles by two American mathematicians, E. V. Huntington and Oswald Veblen, are discussed as examples of a movement in foundational research in the period 1900-1930 called American postulate theory. This movement also included E. H. Moore, R. L. Moore, C. H. Langford, H. M. Sheffer, C. J. Keyser, and others. The articles discussed exemplify American postulate theorists' standards for axiomatizations of mathematical theories, and their investigations of such axiomatizations with respect to metatheoretic properties such as independence, completeness, and consistency.

Articles by two American mathematicians, E. V. Huntington and Oswald Veblen, are discussed as examples of a movement in foundational research in the period 1900-1930 called American postulate theory. This movement also included E. H. Moore, R. L. Moore, C. H. Langford, H. M. Sheffer, C. J. Keyser, and others. The articles discussed exemplify American postulate theorists' standards for axiomatizations of mathematical theories, and their investigations of such axiomatizations with respect to metatheoretic properties such as independence, completeness, and consistency.

François Rostand. CHAPITRE II LES APPELS INTUITIFS Une proposition, située à un point donné du schéma déductif, appartient au raisonnement par son rôle : elle est prémisse, ou conclusion. Aussi distinguerons-nous a priori deux sortes ...

While many books have been written about Bertrand Russell's philosophy and some on his logic, I. Grattan-Guinness has written the first comprehensive history of the mathematical background, content, and impact of the mathematical logic and philosophy of mathematics that Russell developed with A. N. Whitehead in their Principia mathematica (1910-1913).? This definitive history of a critical period in mathematics includes detailed accounts of the two principal influences upon Russell around 1900: the set theory of Cantor and the mathematical logic of (...) Peano and his followers. Substantial surveys are provided of many related topics and figures of the late nineteenth century: the foundations of mathematical analysis under Weierstrass; the creation of algebraic logic by De Morgan, Boole, Peirce, Schröder, and Jevons; the contributions of Dedekind and Frege; the phenomenology of Husserl; and the proof theory of Hilbert. The many-sided story of the reception is recorded up to 1940, including the rise of logic in Poland and the impact on Vienna Circle philosophers Carnap and Gödel. A strong American theme runs though the story, beginning with the mathematician E. H. Moore and the philosopher Josiah Royce, and stretching through the emergence of Church and Quine, and the 1930s immigration of Carnap and GödeI. Grattan-Guinness draws on around fifty manuscript collections, including the Russell Archives, as well as many original reviews. The bibliography comprises around 1,900 items, bringing to light a wealth of primary materials. Written for mathematicians, logicians, historians, and philosophers--especially those interested in the historical interaction between these disciplines--this authoritative account tells an important story from its most neglected point of view. Whitehead and Russell hoped to show that (much of) mathematics was expressible within their logic; they failed in various ways, but no definitive alternative position emerged then or since. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:RUSSELL AND G. H. HARDY: A STUDY OF THEIR RELATIONSHIP I. GRATTAN-GUINNESS Faculty of Science, Engineering and Mathematics Middlesex Polytechnic Enfield, Middlesex EN3 45F, England I. INTRODUCTION Prom time to time the name of Hardy turns up in Russell's career: a common interest in set theory and the philosophy of mathematics, similar political and religious sentiments, and certain matters of mutual concern arising at Trinity College Cambridge and in (...) the university in general. However, there is no connected account of their contacts. The purpose of this article is to fill a gap in the literature on both men. After enduring private instruction at home, Bertrand Russell gained a minor scholarship in mathematics at Trinity College in 1890; he took Part I of the Mathematical Tripos until 1893 and then Part II of the Moral Sciences Tripos in the following year. In 1895 he gained a Prize Fellowship, which he held until 1901; in 19IO.he was awarded a college lectureship which he lost six years later under circumstances noted in section IV below. Much of the intervening time was spent in Oxford or London; however, he spent part of each year in Cambridge.1 The later years are brieHy recorded in section IV. Born into an intellectual family ofmodest means in 1877 (five years younger than Russell, therefore), Godfrey Harold Hardy won a scholarship to Winchester College and then went to Trinity College in I Standard biographical information on Russell is easily available; see, for example, R. W. Clark, The Life ofBertrand Russel~ specific phases and events are dealt with in succeeding footnotes. The Autobiography ofBertrand Russell is of lower quality than one might imagine; for example, Hardy is not mentioned at all! rus.ell: the Journal of the Bertrand Russell Archives McMasrer Unive~ity Library Press noS. 11 (wimer 1991-92): 165-79 ISSN 0036-01631 166 I. GRATTAN-GUINNESS 1896, just after Russell had gained his fellowship. He was elected to the Apostles (the "Society") in 1898, the year after Russell "took wings" after five years' active membership; he seems to have been less active than Russell as a speaker during his own three years there.2 At some stage he also joined the Cambridge University Moral Sciences Club, which Russell had joined in. 1891; a surprising contribution will be described in section IV.3 Graduating as fourth wrangler in Part I of the Mathematical Tripos in 1898 and gaining a high place in Part II in 1900, Hardy received at once a Prize Fellowship (like Russell). Upon its termination six years later, he obtained a colleg~ lectureship, which he held until 1919. He was elected a Fellow of the Royal Society of London in 1910 (two years after Russell),4 and in the following year he began his life-long collaboration with his younger colleague ]. E. Littlewood (1885-1977), also a Trinity College graduate and then Fellow, and also a friend of Russell.5 The next stages of Hardy's career will be described in section IV: his personality is assessed in a concluding section v. 1 On Russell'~ and Hardy's roles in the Apostles, see passim in P. Levy, Moore: G. E Moore and the Cambridge Apostles (London: Weidenfeld and Nicolson, 1979). 3 The chief obituary of Hardy is by E. C. Titchmarsh, published in slightly different forms in Obituary Notices ofthe Royal Society ofLondon, 6 (1949); 447-61, and Journal ofthe London Mathematical Society, 25 (1950): 81-101 (followed by more technical surveys of his mathematics by Titchmarsh and eight other authors on pp. 102-38). The first version is reprinted with a few changes in Hardy's Collected Papers, 7 vols. (Oxford: Clarendon P., 1966-79), I: 1-12. No major study has been written on him since. 4 Hardy was n~minated for a Royal Sociery fellowship in 1907 by A. R. Forsyth, seconded by J. W L. Glaisher, and supported "on personal knowledge" by W Burnside, E. W Hobson, P. A. MacMahon, A. N. Whitehead, A. E. H. Love, E. B. Elliott and T.]. d'A. Bromwich; the proposal was "suspended" (put up for voring) in 1908, 1909 and 1910 (Royal Sociery Cettificates, 12: foJ. 311). Although Russell was... (shrink)