Van Heijenoort Research Articles

Growing Up with Mies and Saarinen

Paul Guyer interviews his cousin Laure van Heijenoort, who as a teenager lived for several years with her parents in one of Mies van der Rohe's Lafayette Park townhouses in Detroit and attended school at Eliel Saarinen's Cranbrook campus. She reports on her experiences of living and study in both, experiences still vivid to her and now to a broader audience.

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.”

LINGUA CHARACTERICAANDCALCULUS RATIOCINATOR: THE LEIBNIZIAN BACKGROUND OF THE FREGE-SCHRÖDER POLEMIC

AbstractAfter the publication ofBegriffsschrift, a conflict erupted between Frege and Schröder regarding their respective logical systems which emerged around the Leibnizian notions oflingua charactericaandcalculus ratiocinator. Both of them claimed their own logic to be a better realisation of Leibniz’s ideal language and considered the rival system a merecalculus ratiocinator. Inspired by this polemic, van Heijenoort (1967b) distinguished two conceptions of logic—logic as language and logic as calculus—and presented them as opposing views, but did not explain Frege’s and Schröder’s conceptions of the fulfilment of Leibniz’s scientific ideal.In this paper I explain the reasons for Frege’s and Schröder’s mutual accusations of having created a merecalculus ratiocinator. On the one hand, Schröder’s construction of the algebra of relatives fits with a project for the reduction of any mathematical concept to the notion of relative. From this stance I argue that he deemed the formal system ofBegriffsschriftincapable of such a reduction. On the other hand, first I argue that Frege took Boolean logic to be an abstract logical theory inadequate for the rendering of specific content; then I claim that the language ofBegriffsschriftdid not constitute a completelingua charactericaby itself, more being seen by Frege as a tool that could be applied to scientific disciplines. Accordingly, I argue that Frege’s project of constructing alingua charactericawas not tied to his later logicist programme.

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.

¿Qué es el absolutismo lógico? Respuesta a Alejandro Tomasini

<p class="p1">El presente texto es una respuesta a la reseña que el profesor Tomasini ha hecho de mi libro Lógica, lenguaje y realidad. Argumento que su crítica está equivocada en al menos cinco puntos importantes: 1) Descuida las diferencias entre la distinción de van Heijenoort y la de Hintikka, entre la lógica como lenguaje y el lenguaje como medio universal; y de paso, también la mía. 2) Su demanda por una definición clara de la noción de absolutismo lógico se encuentra desencaminada. 3) Confunde mi interpretación del absolutismo de Wittgenstein con la llamada teoría pictórica. 4) La interpretación heredada sobre la aritmética de Wittgenstein, sostenida por Tomasini, es errónea en varios sentidos y no puede usarse contra la interpretación que él opone; esto es, contra el logicismo. 5) Por último, la discusión de Tomasini sobre la noción de analiticidad de Frege se encuentra simplemente fuera de lugar.</p>

Solution, extensions and applications of the Schauder's 54th problem in Scottish book

This paper presents the Axiom of Infinite Choice: Given any set P, there exist at least countable choice functions or there exist at least finite choice functions. The author continues herein with the further study of two papers of the Axiom of Choice in order by E. Zermelo [Neuer Beweis für die Möglichkeit einer Wohlordung, Math. Annalen, 65 (1908), 107-128; translated in van Heijenoort 1967, 183-198], and by M. Taskovic [The axiom of choice, fixed point theorems, and inductive ordered sets, Proc. Amer. Math. Soc., 116 (1992), 897-904]. Fredholm and Leray-Schauder alternatives are two direct consequences of the Axiom of Infinite Choice! This paper presents applications of the Axiom of Infinite Choice to the Fredholm and Leray-Schauder theory. In this sense, I give a solution and some extensions of Schauder's problem (in Scottish book, problem 54). This paper presents some new mathematical n-person games. In the theory of n-person games, there have been some further developments in the direction of transversal games and mathematical alternative theory.

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.

Many projectively unique polytopes

We construct an infinite family of 4-polytopes whose realization spaces have dimension smaller or equal to $$96$$ . This in particular settles a problem going back to Legendre and Steinitz: whether and how the dimension of the realization space of a polytope is determined/bounded by its $$f$$ -vector. From this, we derive an infinite family of combinatorially distinct 69-dimensional polytopes whose realization is unique up to projective transformation. This answers a problem posed by Perles and Shephard in the sixties. Moreover, our methods naturally lead to several interesting classes of projectively unique polytopes, among them projectively unique polytopes inscribed to the sphere. The proofs rely on a novel construction technique for polytopes based on solving Cauchy problems for discrete conjugate nets in $$S^d$$ , a new Alexandrov–van Heijenoort Theorem for manifolds with boundary and a generalization of Lawrence’s extension technique for point configurations.

The Lantibiotic NAI-107 Binds to Bactoprenol-bound Cell Wall Precursors and Impairs Membrane Functions

The lantibiotic NAI-107 is active against Gram-positive bacteria including vancomycin-resistant enterococci and methicillin-resistant Staphylococcus aureus. To identify the molecular basis of its potency, we studied the mode of action in a series of whole cell and in vitro assays and analyzed structural features by nuclear magnetic resonance (NMR). The lantibiotic efficiently interfered with late stages of cell wall biosynthesis and induced accumulation of the soluble peptidoglycan precursor UDP-N-acetylmuramic acid-pentapeptide (UDP-MurNAc-pentapeptide) in the cytoplasm. Using membrane preparations and a complete cascade of purified, recombinant late stage peptidoglycan biosynthetic enzymes (MraY, MurG, FemX, PBP2) and their respective purified substrates, we showed that NAI-107 forms complexes with bactoprenol-pyrophosphate-coupled precursors of the bacterial cell wall. Titration experiments indicate that first a 1:1 stoichiometric complex occurs, which then transforms into a 2:1 (peptide: lipid II) complex, when excess peptide is added. Furthermore, lipid II and related molecules obviously could not serve as anchor molecules for the formation of defined and stable nisin-like pores, however, slow membrane depolarization was observed after NAI-107 treatment, which could contribute to killing of the bacterial cell.

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).

Editor’s Introduction to Jean van Heijenoort, Historical Development of Modern Logic

Van Heijenoort’s account of the historical development of modern logic was composed in 1974 and first published in 1992 with an introduction by his former student. What follows is a new edition with a revised and expanded introduction and additional notes.

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.

Logic as a Science and Logic as a Theory: Remarks on Frege, Russell and the Logocentric Predicament

Since its publication in 1967, van Heijenoort’s paper, “Logic as Calculus and Logic as Language” has become a classic in the historiography of modern logic. According to van Heijenoort, the contrast between the two conceptions of logic provides the key to many philosophical issues underlying the entire classical period of modern logic, the period from Frege’s Begriffsschrift (1879) to the work of Herbrand, Godel and Tarski in the late 1920s and early 1930s. The present paper is a critical reflection on some aspects of van Heijenoort’s thesis. I concentrate on the case of Frege and Russell and the claim that their philosophies of logic are marked through and through by acceptance of the universalist conception of logic, which is an integral part of the view of logic as language. Using the so-called “Logocentric Predicament” (Henry M. Sheffer) as an illustration, I shall argue that the universalist conception does not have the consequences drawn from it by the van Heijenoort tradition. The crucial element here is that we draw a distinction between logic as a universal science and logic as a theory. According to both Frege and Russell, logic is first and foremost a universal science, which is concerned with the principles governing inferential transitions between propositions; but this in no way excludes the possibility of studying logic also as a theory, i.e., as an explicit formulation of (some) of these principles. Some aspects of this distinction will be discussed.

On Rereading van Heijenoort’s Selected Essays

This is a critical reexamination of several pieces in van Heijenoort’s Selected Essays that are directly or indirectly concerned with the philosophy of logic or the relation of logic to natural language. Among the topics discussed are absolutism and relativism in logic, mass terms, the idea of a rational dictionary, and sense and identity of sense in Frege.

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.

How Peircean was the “‘Fregean’ Revolution” in Logic?

The work in logic of Charles Peirce is surveyed in light of the characteristics enumerated by historian of logic J. van Heijenoort as defining the original innovations in logic of Frege and which together are said to be the basis of what has come to be called the “Fregean revolution” in logic and which are said to constitute the elements of Frege’s Begriffsschrift of 1879 as the “founding” document of modern logic.

The axiom of infinite choice

In this paper we present the Axiom of Infinite Choice: Given any set P , there exist at least countable choice functions or there exist at least finite choice functions. This paper continues the study of the Axiom of Choice by E. Z e r m e l o [Neuer Beweis für die Möglichkeit einer Wohlordung, Math. Annalen, 65 (1908), 107-128; translated in van Heijenoort 1967, 183-198], and by M. T a s k o v i ć [The axiom of choice, fixed point theorems, and inductive ordered sets, Proc. Amer. Math. Soc., 116 (1992), 897-904]. Fredholm and Leray-Schauder alternatives are two direct consequences of the Axiom of Infinite Choice!.

The Versatility of Universality in Principia Mathematica

In this article, I examine the ramified-type theory set out in the first edition of Russell and Whitehead's Principia Mathematica. My starting point is the ‘no loss of generality’ problem: Russell, in the Introduction (Russell, B. and Whitehead, A. N. 1910. Principia Mathematica, Volume I, 1st ed., Cambridge: Cambridge University Press, pp. 53–54), says that one can account for all propositional functions using predicative variables only, that is, dismissing non-predicative variables. That claim is not self-evident at all, hence a problem. The purpose of this article is to clarify Russell's claim and to solve the ‘no loss of generality’ problem. I first remark that the hierarchy of propositional functions calls for a fine-grained conception of ramified types as propositional forms (‘ramif-types’). Then, comparing different important interpretations of Principia’s theory of types, I consider the question as to whether Principia allows for non-predicative propositional functions and variables thereof. I explain how the distinction between the formal system of the theory, on the one hand, and its realizations in different epistemic universes, on the other hand, makes it possible to give us a more satisfactory answer to that question than those given by previous commentators, and, as a consequence, to solve the ‘no loss of generality’ problem. The solution consists in a substitutional semantics for non-predicative variables and non-predicative complex terms, based on an epistemic understanding of the order component of ramified types. The rest of the article then develops that epistemic understanding, adding an original epistemic model theory to the formal system of types. This shows that the universality sought by Russell for logic does not preclude semantical considerations, contrary to what van Heijenoort and Hintikka have claimed.