Jean Van Heijenoort Research Articles

Logic as Calculus and Logic as Language: Too Suggestive to be Truthful?

The paper focuses on the inferential role of quantifiers in Frege, Peano and Russell. Two aspects of the early years of mathematical logic are discussed: the gradual perfection of the principles of reasoning with quantifiers, and the presumed conceptual impossibility of posing metatheoretical questions, as embodied in Jean van Heijenoort’s well-known dictum about “logic as calculus and logic as language.”

Жан ван Хейеноорт как историк логики

В статье предпринимается попытка достаточно лаконичного обзора жизни и творчества Жана ван Хейеноорта (1912–1986) в области истории логики. Приводится информация биографического характера, в которой отмечается отношение ученого к марксизму и его тесное в определенный период сотрудничество с Л.Д. Троцким, разочарование в ключевых принципах марксистской доктрины и ее практической реализации, но сохранение в целом интереса к этому политическому течению. На основании некоторых неопубликованных материалов, полученных из архива американской математики в Остине, штат Техас, описываются работы ван Хейеноорта в области истории логики. Отмечается, что математическое образование он получил как геометр и тополог, но его интересы перенеслись в логику, которая по природе своей доказательности и канонам строгости рассуждений была близка геометрии. Особый акцент в статье делается на обстоятельствах работы над фундаментальной антологией развития логической мысли "От Фреге до Гёделя", составленной и прокомментированной ван Хейеноортом и рядом его коллег. Обращается внимание на то, что вне поля зрения антологии осталось алгебраическое направление развития математической логики. Описываются труды ван Хейеноорта, относящиеся к теореме о неполноте и первым двум томам собрания сочинений К. Гёделя в начале 1980-х годов, прервавшиеся в результате гибели ван Хейеноорта. Приводятся оценки его коллег работ ван Хейеноорта по истории логики.

The Grundgesetze

176 Reviews c:\users\ken\documents\type3402\rj 3402 050 red.docx 2015-02-04 9:19 PM THE GRUNDGESETZE Nicholas Griffin Russell Research Centre / McMaster U. Hamilton, on, Canada l8s 4l6 ngriffin@mcmaster.ca Gottlob Frege. Basic Laws of Arithmetic. Derived Using Concept-script. Volumes i and ii. Translated and edited by Philip A. Ebert and Marcus Rossberg with Crispin Wright. Oxford: Oxford U. P., 2013. Pp. xxxix + xxxii + 253 + xv + 285 + A–42 + i–11. isbn 978-0-19-928174-9. £60; us$110. iven the steadily rising interest in Frege’s work since the 1950s, it is surprising that his Grundgesetze der Arithmetik, the work he thought would be the crowning achievement of his career, has not previously been fully translated into English. Other Frege works have long been available in English. The earlier Grundlagen der Arithmetik was nicely, if not always entirely accurately , translated by J. L. Austin and published in 1950.1 Translations of Frege’s other philosophical writings began to appear in influential editions about the same time,2 initially a small corpus that has been gradually added to over the years to include works from his Nachlass and letters from his correspondence .3 Somewhat later, the Begriffsschrift, Frege’s momentous first 1 The Foundations of Arithmetic (Oxford: Blackwell, 1950; 2nd edn. 1953). The later translation under the same title by Dale Jacquette (London: Longman, 2007) is much less satisfactory. 2 The first importance source for these was Peter Geach and Max Black’s Translations from the Philosophical Writing of Gottlob Frege (1952; 2nd edn. 1960; 3rd edn. 1980), though it included reprints of some much earlier translations. 3 The two most comprehensive English sources for the shorter writings are now Brian d= Reviews 177 c:\users\ken\documents\type3402\rj 3402 050 red.docx 2015-02-04 9:19 PM contribution to modern logic, the work which Quine 4 said had made logic a great subject, was translated in full despite the difficulties of Fregean notation .5 But through all this the Grundgesetze remained largely (though not entirely ) untranslated. The view seems, for a long time, to have been that Frege’s contributions to logic were to be found in the Begriffsschrift and his contributions to philosophy in the Grundlagen and a small number of seminal articles. The Grundgesetze, on the other hand, was widely thought not to have added much to these magni ficent achievements, but to have employed them in a project that was doomed by the discovery of Russell’s paradox; namely, the attempt to show that arithmetic could be rigorously derived from purely logical principles. In the bleak aftermath of Russell’s paradox it came to seem as if whatever was of value in the Grundgesetze had already been published elsewhere, while what was new in it was mistaken.6 That assessment of the book’s value, combined with the huge difficulties of typesetting Frege’s idiosyncratic two-dimensional notation, and the sheer scale of the work (even without the anticipated third volume, which was abandoned in the face of Russell’s paradox, it was, by far, Frege’s largest work) made it seem as though a translation of the whole Grundgesetze was not only a prohibitively large undertaking, but also one which was not really necessary. And so, through much of the twentieth century , the Grundgesetze vied with Principia Mathematica as the world’s bestknown but least-read philosophical book. But this understanding of the Grundgesetze’s importance could not withstand the development of neo-logicism, brought to prominence by Crispin Wright’s Frege’s Conception of Numbers as Objects.7 The key insight of neoMcGuinness , ed., Frege, Collected Papers on Mathematics, Logic, and Philosophy (1984); and Michael Beaney, ed., The Frege Reader (1997). 4 At least in early editions of Mathematical Logic. The remark was removed in later ones in deference to Boole. 5 By Stefan Bauer-Mengelberg in Jean van Heijenoort, ed., From Frege to Gödel (1967), pp. 5–82. 6 Not surprisingly the one part of the Grundgesetze which has appeared most frequently in translation is the “Nachwort” responding to Russell’s paradox which Frege added as the book was in press. Geach...

Strategy and Tactics in a Revolutionary Period: U. S. Trotskyism and the European Revolution, 1943–1946

The outbreak of the Second World War found U. S. Trotskyism divided into two organizations: the Socialist Workers Party led by James Cannon, and the Workers Party led by Max Shachtman. The downfall of Mussolini on July 24, 1943 led to the appearance of a third current: a minority within the SWP led by Felix Morrow, Jean van Heijenoort and Albert Goldman. Confronting the SWP leaders' line, according to which U. S. imperialism would operate in Europe through “Franco-type governments,” the minority argued that it would rely on democratic regimes to stem the advance of the revolution, propping them up with economic aid, and that it would be helped in this task by the Socialist and Communist Parties, which would revive the policy of class collaboration known as Popular Front. The task of the European Trotskyists was therefore to wrest control of the masses from those parties through democratic and transitional demands (a Democratic Republic, a Constituent Assembly, etc.) which would help the workers discover the anti-socialist agenda of their mass organizations through their own experience. The Morrow–Goldman–Heijenoort tendency's inglorious ending precluded any serious analysis of the dire consequences of the policies pursued by the SWP leadership.

In Defense of Logical Universalism: Taking Issue with Jean van Heijenoort

Van Heijenoort’s main contribution to history and philosophy of modern logic was his distinction between two basic views of logic, first, the absolutist, or universalist, view of the founding fathers, Frege, Peano, and Russell, which dominated the first, classical period of history of modern logic, and, second, the relativist, or model-theoretic, view, inherited from Boole, Schroder, and Lowenheim, which has dominated the second, contemporary period of that history. In my paper, I present the man Jean van Heijenoort (Sect. 1); then I describe his way of arguing for the second view (Sect. 2); and finally I come down in favor of the first view (Sect. 3). There, I specify the version of universalism for which I am prepared to argue (Sect. 3, introduction). Choosing ZFC to play the part of universal, logical (in a nowadays forgotten sense) system, I show, through an example, how the usual model theory can be naturally given its proper place, from the universalist point of view, in the logical framework of ZFC; I outline another, not rival but complementary, semantics for admissible extensions of ZFC in the very same logical framework; I propose a way to get universalism out of the predicaments in which universalists themselves believed it to be (Sect. 3.1). Thus, if universalists of the classical period did not, in fact, construct these semantics, it was not that their universalism forbade them, in principle, to do so. The historical defeat of universalism was not technical in character. Neither was it philosophical. Indeed, it was hardly more than the victory of technicism over the very possibility of a philosophical dispute (Sect. 3.2).

Jean van Heijenoort’s Contributions to Proof Theory and Its History

Jean van Heijenoort was best known for his editorial work in the history of mathematical logic. I survey his contributions to model-theoretic proof theory, and in particular to the falsifiability tree method. This work of van Heijenoort’s is not widely known, and much of it remains unpublished. A complete list of van Heijenoort’s unpublished writings on tableaux methods and related work in proof theory is appended.

Scholarly Publications of Jean van Heijenoort Compiled by lrving H. Anellis

Scholarly Publications of Jean van Heijenoort Compiled by lrving H. Anellis

Jean van Heijenoort and the Gödel Editorial Project

A colleague’s personal recollections of Jean van Heijenoort’s contributions to the editing of volumes I–III of Godel’s Collected Works and of his interactions with the other editors.

Unexpected Inhibition of Peptidoglycan LD-Transpeptidase from Enterococcus faecium by the β-Lactam Imipenem

The beta-lactam antibiotics mimic the D-alanyl(4)-D-alanine(5) extremity of peptidoglycan precursors and act as "suicide" substrates of the DD-transpeptidases that catalyze the last cross-linking step of peptidoglycan synthesis. We have previously shown that bypass of the dd-transpeptidases by the LD-transpeptidase of Enterococcus faecium (Ldt(fm)) leads to high level resistance to ampicillin. Ldt(fm) is specific for the L-lysyl(3)-D-alanine(4) bond of peptidoglycan precursors containing a tetrapeptide stem lacking D-alanine(5). This specificity was proposed to account for resistance, because the substrate of Ldt(fm) does not mimic beta-lactams in contrast to the D-alanyl(4)-D-alanine(5) extremity of pentapeptide stems used by the DD-transpeptidases. Here, we unexpectedly show that imipenem, a beta-lactam of the carbapenem class, totally inhibited Ldt(fm) at a low drug concentration that was sufficient to inhibit growth of the bacteria. Peptidoglycan cross-linking was also inhibited, indicating that Ldt(fm) is the in vivo target of imipenem. Stoichiometric and covalent modification of Ldt(fm) by imipenem was detected by mass spectrometry. The modification was mapped into the trypsin fragment of Ldt(fm) containing the catalytic Cys residue, and the Cys to Ala substitution prevented imipenem binding. The mass increment matched the mass of imipenem, indicating that inactivation of Ldt(fm) is likely to involve rupture of the beta-lactam ring and acylation of the catalytic Cys residue. Thus, the spectrum of activity of beta-lactams is not restricted to transpeptidases of the DD-specificity, as previously thought. Combination therapy with imipenem and ampicillin could therefore be active against E. faecium strains having the dual capacity to manufacture peptidoglycan with transpeptidases of the LD- and DD-specificities.

Logic as a Universal Medium or Logic as a Calculus? Husserl and the Presuppositions of “the Ultimate Presupposition of Twentieth Century Philosophy”

AbstractThis paper discusses Jean van Heijenoort's (1967) and Jaakko and Merrill B. Hintikka's (1986, 1997) distinction between logic as a universal language and logic as a calculus, and its applicability to Edmund Husserl's phenomenology. Although it is argued that Husserl's phenomenology shares characteristics with both sides, his view of logic is closer to the model‐theoretical, logic‐as‐calculus view. However, Husserl's philosophy as transcendental philosophy is closer to the universalist view. This paper suggests that Husserl's position shows that holding a model‐theoretical view of logic does not necessarily imply a calculus view about the relations between language and the world. The situation calls for reflection about the distinction: It will be suggested that the applicability of the van Heijenoort and the Hintikkas distinction either has to be restricted to a particular philosopher's views about logic, in which case no implications about his or her more general philosophical views should be inferred from it; or the distinction turns into a question of whether our human predicament is inescapable or whether it is possible, presumably by means of model theory, to obtain neutral answers to philosophical questions. Thus the distinction ultimately turns into a question about the correct method for doing philosophy.

Anita Burdman Feferman and Solomon Feferman. Alfred Tarski: Life and Logic. Cambridge: Cambridge University Press, 2004. Pp. vi + 435. ISBN 0-521-80240-7.

Alfred Tarski (1901–1983) was considered by many to be one of the top two logicians of the twentieth century. It was reported to the authors by the logician John Corcoran that Tarski once referred to himself as ‘the greatest living sane logician’, presumably so as to eliminate from contention his chief rival, Kurt Gödel. This quotation is typical of the items of gossip that academics love to hear and that are served up plentifully in this lively biography, along with a large number of interesting photographs. Tarski was born and grew up in Warsaw, where he obtained his Ph.D. in mathematics in 1924 with a thesis, ‘On the Primitive Term of Logistic’, having to do with a logical system of his advisor, Stanislaus Leśniewski. For the next fifteen years, he pursued his research while earning a modest living from his positions as a teacher in a Warsaw lycée and an adjunct professor at the university and from research grants. When Germany began World War II with the invasion of Poland in 1939, Tarski was stranded in the United States during a brief tour to give lectures and to attend a Unity of Science Conference in Cambridge. Although this probably saved his life, it turned out that he would be separated from his wife and two small children in Poland until 1946. After a term teaching at CCNY and a year doing research at the Institute of Advanced Study in Princeton, he was appointed in 1942 to the Department of Mathematics of the University of California at Berkeley, where he remained, except for a few visiting positions at other universities, until his death forty-one years later. The authors are able to give a thorough and intimate picture of Tarski's academic career, research, and private life because Sol Feferman attended Berkeley from 1948 through 1955, where he studied with Tarski, Leon Henkin, and others and officially obtained his doctorate under Tarski's supervision in 1957. At that time, he was already on the faculty at Stanford, where he has distinguished himself as one of the leading American logicians. His co-author and wife, Anita, is a writer, who has previously published a biography of the logician and political activist, Jean van Heijenoort. The book is made accessible to the general reader by virtue of the fact that the authors concentrate on the personal and academic aspects of Tarski's life, rather than technical details. The principal exceptions involve his work on the completeness and decidability of elementary algebra and geometry and his work on the concept of truth, which the authors regard as his most important contributions, but even here the subjects are not treated in great depth. Of course, the general description of Tarski's activities often requires mention of other research in which he was engaged, and the authors also emphasize the broad range of his interests (for example, botanical, artistic, and political). He generally avoided issues in the philosophy of mathematics, except with respect to the concept of truth. Although staying away from philosophical discussions in his papers, he privately expressed nominalist views (p. 52), characterizing classical logic and set theory as ‘fairy tales’ and his work on them as a ‘hobby’.

The Road to Modern Logic—An Interpretation

AbstractThis paper aims to outline an analysis and interpretation of the process that led to First-Order Logic and its consolidation as a core system of modern logic. We begin with an historical overview of landmarks along the road to modern logic, and proceed to a philosophical discussion casting doubt on the possibility of a purely rational justification of the actual delimitation of First-Order Logic. On this basis, we advance the thesis that a certain historical tradition was essential to the emergence of modern logic; this traditional context is analyzed as consisting in some guiding principles and, particularly, a set of exemplars (i.e., paradigmatic instances). Then, we proceed to interpret the historical course of development reviewed in section 1, which can broadly be described as a two-phased movement of expansion and then restriction of the scope of logical theory. We shall try to pinpoint ambivalencies in the process, and the main motives for subsequent changes. Among the latter, one may emphasize the spirit of modern axiomatics, the situation of foundational insecurity in the 1920s, the resulting desire to find systems well-behaved from a proof-theoretical point of view, and the metatheoretical results of the 1930s. Not surprisingly, the mathematical and, more specifically, the foundational context in which First-Order Logic matured will be seen to have played a primary role in its shaping.Mathematical logic is what logic, through twenty-five centuries and a few transformations, has become today. (Jean van Heijenoort)

Collected works, Volume II, Publications 1938–1974, by Kurt Gödel, edited by Solomon Feferman, John W. DawsonJr. , Stephen C. Kleene, Gregory H. Moore, Robert M. Solovay, and Jean van Heijenoort, Oxford University Press, New York and Oxford 1990, xv + 407 pp.

Collected works, Volume II, Publications 1938–1974, by Kurt Gödel, edited by Solomon Feferman, John W. DawsonJr. , Stephen C. Kleene, Gregory H. Moore, Robert M. Solovay, and Jean van Heijenoort, Oxford University Press, New York and Oxford 1990, xv + 407 pp.

Solomon Feferman. Gödel's life and work. Collected Works, Volume I, Publications 1929–1936, by Kurt Gödel, edited by Solomon Feferman, John W. DawsonJr., Stephen C. Kleene, Gregory H. Moore, Robert M. Solovay, and Jean van Heijenoort, Clarendon Press, Oxford University Press, New York and Oxford1986, pp. 1–36.

Solomon Feferman. Gödel's life and work. Collected Works, Volume I, Publications 1929–1936, by Kurt Gödel, edited by Solomon Feferman, John W. DawsonJr., Stephen C. Kleene, Gregory H. Moore, Robert M. Solovay, and Jean van Heijenoort, Clarendon Press, Oxford University Press, New York and Oxford1986, pp. 1–36.

John W. Dawson Jr. A Gödel chronology. Collected Works, Volume I, Publications 1929– 1936, by Kurt Gödel, edited by Solomon Feferman, John W. DawsonJr., Stephen C. Kleene, Gregory H. Moore, Robert M. Solovay, and Jean van Heijenoort, Clarendon Press, Oxford University Press, New York and Oxford1986, pp. 37– 43.

John W. Dawson Jr. A Gödel chronology. Collected Works, Volume I, Publications 1929– 1936, by Kurt Gödel, edited by Solomon Feferman, John W. DawsonJr., Stephen C. Kleene, Gregory H. Moore, Robert M. Solovay, and Jean van Heijenoort, Clarendon Press, Oxford University Press, New York and Oxford1986, pp. 37– 43. - Volume 55 Issue 1

W. V. Quine. Introductory note. From Frege to Gödel, A source book in mathematical logic, 1879–1931, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1967, pp. 216–217. - Alfred North Whitehead and Bertrand Russell. Incomplete symbols: Descriptions. Reprinted from 1947, pp. 66–71. Incomplete symbols: Descriptions. Reprinted from 1947, pp. 217–223.

W. V. Quine. Introductory note. From Frege to Gödel, A source book in mathematical logic, 1879–1931, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1967, pp. 216–217. - Alfred North Whitehead and Bertrand Russell. Incomplete symbols: Descriptions. Reprinted from 1947, pp. 66–71. Incomplete symbols: Descriptions. Reprinted from 1947, pp. 217–223. - Volume 40 Issue 3

Jean van Heijenoort. Introductory note. From Frege to Gödel, A source book in mathematical logic, 1879–1931, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1967, pp. 1–5. Reprinted in Frege and Gödel, Two fundamental texts in mathematical logic, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1970, pp. 1–5. - Gottlob Frege. Begriffsschrift, a formula language,

Jean van Heijenoort. Introductory note. From Frege to Gödel, A source book in mathematical logic, 1879–1931, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1967, pp. 1–5. Reprinted in Frege and Gödel, Two fundamental texts in mathematical logic, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1970, pp. 1–5. - Gottlob Frege. Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought. English translation of 491 by Stefan Bauer-Mengelberg. From Frege to Gödel, A source book in mathematical logic, 1879–1931, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1967, pp. 5–82. Reprinted in Frege and Gödel, Two fundamental texts in mathematical logic, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1970, pp. 5–82. - Jean van Heijenoort. Introductory note. From Frege to Gödel, A source book in mathematical logic, 1879–1931, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1967, pp. 592–595. Reprinted in Frege and Gödel, Two fundamental texts in mathematical logic, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1970, pp. 83–86. - Kurt Gödel. Some metamathematical results on completeness and consistency. English translation of 4181 by Stefan Bauer-Mengelberg. From Frege to Gödel, A source book in mathematical logic, 1879–1931, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1967, pp. 595–596. Reprinted in Frege and Gödel, Two fundamental texts in mathematical logic, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1970, pp. 86–87. - Kurt Gödel. On formally undecidable propositions of Principia mathematica and related systems I. English translation of 4183 by Jean van Heijenoort. From Frege to Gödel, A source book in mathematical logic, 1879–1931, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1967, pp. 596–616. Reprinted in Frege and Gödel, Two fundamental texts in mathematical logic, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1970, pp. 87–107. - Kurt Gödel. On completeness and consistency. English translation of 4188 by Jean van Heijenoort. From Frege to Gödel, A source book in mathematical logic, 1879–1931, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1967, pp. 616–617. Reprinted in Frege and Gödel, Two fundamental texts in mathematical logic, edited by Jean van Heijenoort, Harvard University Press, Cambridge, Mass., 1970, pp. 107–108. - Volume 37 Issue 2

Book Reviews

Book reviewed in this article:Learn Basic Slide Rule on Your Own: Programed Instruction Manual together with a slide rule and case.From Frege to Godel—A Source Book in Mathematical Logic, 1879–1931 by Jean van Heijenoort.