This volume brings together those papers of mine which may be of interest not only to various specialists but also to philosophers. Many of my writings in mathematics were motivated by epistemological considerations; some papers originated in the critique of certain views that at one time dominated the discussions of the Vienna Cirele; others grew out of problems in teaching fundamental ideas of mathematics; sti II others were occasioned by personal relations with economists. Hence a wide range of subjects will (...) be discussed: epistemology, logic, basic concepts of pure and applied mathematics, philosophical ideas resulting from geometric studies, mathematical didactics and, finally, economics. The papers also span a period of more than fifty years. What unifies the various parts of the book is the spirit of searching for the elarification of basic concepts and methods and of articulating hidden ideas and tacit procedures. Part 1 ineludes papers published about 1930 which expound an idea that Carnap, after a short period of opposition in the Cirele, fully adopted ; and, under the name "Princip/e of To/erance", he eloquently formulated it in great generality in his book, Logica/ Syntax of Language, through which it was widely disseminated. "The New Logic" in Chapter 1 furthermore ineludes the first report to a larger public of Godel's epochal discovery presented among the great logic results of ali time. Chapter 2 is a translation of an often quoted 1930 paper presenting a detailed exposition and critique of intuitionism. (shrink)

One does not only talk about the length in inches of this sheet of paper but also about the length of this sheet, about length in inches and about length. A clarification of these and related concepts results from a combination of the theory of the length in a definite unit as a fluent, developed by one of the authors, with the other's concept of 2-place fluents. The length ratio L is defined by pairing a number L (α,β) to any (...) two objects of a certain kind, α,β (in a definite order). L thus may be described as the class of all pairs (α,β), L (α,β) for any objects α,β of the said kind. Length of this sheet and length in inches are specializations of this 2-place fluent. (shrink)

During the last years of his life, the mathematician Karl Menger worked on a biography of his father, the economist and founder of the Austrian School of Economics, Carl Menger. The younger Menger never finished the work. While working in the Menger collections at Duke University’s David M. Rubenstein Rare Book and Manuscript Library, we discovered draft chapters of the biography, a valuable source of information given that relatively little is known about Carl Menger’s (...) life nearly a hundred years after his death. The unfinished biography covers Carl Menger’s family background and his life through early 1889. In this article, we discuss the biography and the most valuable new insights it provides into Carl Menger’s life, including Carl Menger’s family, his childhood, his student years, his time working as a journalist and newspaper editor, his early scientific career, and his relationship with Crown Prince Rudolf. (shrink)

Although their contributions to the history of economic thought and their scholarly reputations are firmly established, relatively little is known about the relationship between Carl Menger, founder of the Austrian School of economics, and his son, Karl Menger, the mathematician, geometer, logician, and philosopher of science, whose famous Mathematical Colloquium at the University of Vienna was central to the early literature on the existence of general equilibrium and the concomitant development of mathematical economics. The present paper begins (...) to fill this gap. Karl Menger’s diaries, held in the David M. Rubenstein Rare Book and Manuscript Library at Duke University, offer insight into the intimate relationships within the Menger clan, Karl’s work and study habits, and the development of his uncommonly broad intellect, as well as on life in a vanquished city, Vienna, in the immediate wake of the humiliating defeat of the First World War and the disintegration of the Habsburg Empire. (shrink)

Karl Menger (1902-1985) was the mathematician son of the famous economist Carl Menger. When he was professor of geometry at the University of Vienna from 1927 to 1938, he joined the Vienna Circle and founded his Mathematical Colloquium. This title offers the transcription of those parts of Menger's notes.

This paper deals with the relation with Fuzzy Logic of some of the ideas of Karl Menger published between 1942 and 1966 and concerning what he called “Hazy Sets”, Probabilistic Relations and Statistical Metric Spaces. The author maintains the opinion that if Lofti A. Zadeh is actually the father of Fuzzy Logic, Menger not only was a forerunner of this field but that his ideas were and still are influential on it.

This paper looks at how the idea of pointless topology itself evolved during its pre-localic phase by analyzing the definitions of the concept of topological space of Menger and Nöbeling. Menger put forward a topology of lumps in order to generalize the definition of the real line. As to Nöbeling, he developed an abstract theory of posets so that a topological space becomes a particular case of topological poset. The analysis emphasizes two points. First, Menger's geometrical perspective (...) was superseded by an algebraic one, a lattice-theoretical one to be precise. Second, Menger's bottom–up approach was replaced by a top–down one. (shrink)

In the 1950s and 1960s, the search for the mechanism of oxidative phosphorylation by biochemists paralleled the description of mitochondrial form by George Palade and Fritiof Sjöstrand using electron microscopy. This paper explores the extent to which biochemists studying oxidative phosphorylation took mitochondrial form into account in the formulation of hypotheses, design of experiments, and interpretation of results. By examining experimental approaches employed by the biochemists studying oxidative phosphorylation, and their interactions with Palade, I suggest that use of mitochondrial form (...) as a guide to experimentation and interpretation varied considerably among investigators. Most notably, Peter Mitchell, whose chemiosmotic hypothesis was ultimately the basis of the correct mechanism of oxidative phosphorylation, incorporated crucial aspects of mitochondrial form into his model that others failed to recognize. I discuss these historical observations in terms of the background and training of the biochemists, as well as a proposed heuristic of form, whose use may increase the possibility that biologically meaningful molecular mechanisms will be discovered. (shrink)

Farmers of the Peruvian Andesmake use of seed-size variation as a source offlexibility in the production of ``nativecommercial'' farmer varieties of Andeanpotatoes and ulluco. In a case study of easternCuzco, the use of varied sizes of seed tubers isfound to underpin versatile farm strategiessuited to partial commercialization (combinedwith on-farm consumption and the next season'sseed). Use of seed-size variation also providesadaptation to diverse soil-moistureenvironments. The importance and widespread useof seed-size variation among farmers isdemonstrated in the emphasis and consistency oflinguistic expressions about (...) this trait. Smalland small-medium seed is typically sown in thecommunity's ``Hill'' unit of sub-humid,upper-elevation agriculture. Seed tubers ofmedium-size and larger are needed fordrought-stressed locales in lower-elevationlandscape units. Farm-level preferences for theseed-size of tubers also suggest potentialrelations to resource endowments of farmhouseholds and gender-related management,although these tendencies were notstatistically significant in the study. Anintra-varietal, landscape-environmentalperspective on seed-size management, whichincludes an emphasis on within-fieldversatility, helps to strengthen the researchsupport of local seed production in policiesand programs aiming for in situagrobiodiversity conservation, marketingcapacity, and food security. (shrink)

Present-day speculation on extraterrestrial intelligence, taking its cue from a number of programmes currently "listening" for electro-magnetic signals from space, have a prehistory dating back to the immediate aftermath of the Copernican Revolution. Throughout the centuries of the Early Modern Period, it was not only theological but also more or less secular philosophical concerns that informed the hopes and the fears that writers of all stripes associated with the tentative notion that "We are not alone." It was not until the (...) "Romantic" period, around 1800, that a reaction set in: as Goethe put it, there is also a universe within us-one worthier of our attention than the cosmos. (shrink)

Present-day speculation on extraterrestrial intelligence, taking its cue from a number of programmes currently "listening" for electro-magnetic signals from space, have a prehistory dating back to the immediate aftermath of the Copernican Revolution. Throughout the centuries of the Early Modern Period, it was not only theological but also more or less secular philosophical concerns that informed the hopes and the fears that writers of all stripes associated with the tentative notion that "We are not alone." It was not until the (...) "Romantic" period, around 1800, that a reaction set in: as Goethe put it, there is also a universe within us-one worthier of our attention than the cosmos. (shrink)

Globalization offers a mix of new trajectories for agriculture, livelihoods, resource use, and environmental conservation. The papers in this issue share elements that advance our understanding of these new trajectories. The shared elements suggest an approach that places stress on: (i) the common ground of theoretical concepts (local-global interactions), methodologies (case study design), and analytical frameworks (spatio-temporal emphasis); (ii) farm-level economic diversification and the dynamics of agricultural intensification-disintensification; (iii) the pervasive role of agricultural as well as environmental institutions, organizations, and (...) governance issues; (iv) the ‹agency of nature’ that blends the roles of non-human organisms and the cultural and social practices of people both at the local scale and beyond; (v) the framing of sustainability initiatives and outcomes through the perspective of historical change; (vi)␣spatial environmental dynamics of the ‹new geographics of environmental conservation’ that impact agriculture, food production, and resource management; and (vii) successful and promising policies, projects, and developments mapping out possible spaces of hope for agricultural sustainability, aquitable development, and food security. The adoption and application of these elements is successful also in avoiding the tendency toward just-so accounts or overly simplified stories of agrarian and environment successes amid the often grim realities of globalization and its impacts. (shrink)

Attempting to answer the question "what is a variable?," menger discusses the following topics: (1) numerical variables and variables in the sense of the logicians, (2) variable quantities, (3) scientific variable quantities, (4) functions, And (5) variable quantities and functions in pure and applied analysis. (staff).

Die von Karl Menger und seinen Mitarbeitern (darunter Kurt Gödel) herausgegebenen "Ergebnisse eines Mathematischen Kolloquiums" zählen zu den wichtigsten Quellenwerken der Wissenschafts- und Geistesgeschichte der Zwischenkriegszeit, mit bahnbrechenden Beiträgen von Menger, Gödel, Tarski, Wald, John von Neumann und vielen anderen. In diesem Band liegt der Inhalt erstmals gesammelt vor. Der Nobelpreisträger Gerard Debreu schrieb die Einleitung, die Kommentare wurden vom Logiker und Gödel-Biographen John Dawson jr., dem Topologen Ryszard Engelking und dem Wirtschaftstheoretiker Werner Hildenbrand verfasst. Außerdem enthält (...) der Band einen biographischen Aufsatz über Karl Menger sowie einen von Menger verfassten Überblick über die wichtigsten topologischen und geometrischen Arbeiten des Kolloquiums. (shrink)

The rapid development of physics, the result of observations made and ideas introduced within the last few decades, has brought about a change in the whole system of physical concepts. This fact is common knowledge, and has already attracted the attention of philosophers. It is less well known that geometry too has had its crises, and undergone a reconstruction. For centuries, so-called “geometrical intuition” was used as a method of proof. In geometrical demonstrations, certain steps were allowed because they were (...) “self-evident,” because the correctness of the conclusion was “shown by the adjoined diagram,” etc. A crisis occurred in geometry because such intuition proved to be untrustworthy. Many of the propositions regarded as self-evident or based upon the consideration of diagrams turned out to be false. And so Euclidean geometry was reconstructed by methods free from all intuitive elements and strictly logical in nature. Moreover, for more than a century, various other geometries have been devised purely as logical constructions. Since they start with assumptions different from those of Euclid, and lead to conclusions partly in contradiction with his theorems, they are called “non-Euclidean.” Nevertheless, each one of these geometries is a closed system of propositions exempt from contradiction. Recently, some of them have even found application in physics. (shrink)

Am Wiener Kreis scheiden sich die Geister, trat er doch mit dem dezidierten Anspruch auf, mit den Mitteln der modernen Logik den metaphysischen Schutt von Jahrtausenden aus dem Weg zu räumen. Statt einer homogenen Bewegung, die sich empiristischen Dogmen verschrieb, erscheint der Wiener Kreis in der philosophischen Forschung jedoch heute als eine heterogene Gruppe von eigenständigen Denkern, die gemeinsam die Grundlagen der modernen Wissenschaftstheorie legten. In jeweils spezifischer Weise setzten sie sich von der philosophischen Tradition ab oder versuchten, einzelne Teile (...) davon in die wissenschaftliche Weltauffassung zu integrieren. Der Band enthält eine repräsentative Auswahl von Aufsätzen der Mitglieder des Wiener Kreises. Sie beschränkt sich mit Ausnahme dreier Frühschriften der Gründer bewußt auf die Jahre des historischen Kreises vom Beginn der Treffen 1924 bis zur Ermordung Schlicks 1936, obwohl einige der Hauptideen erst in den USA ihre volle Wirkung entfalteten und von dort Ende der sechziger Jahre wieder in die deutschsprachige Philosophie zurückwirkten. In ihrer Einleitung skizzieren die Herausgeber die historische Entwicklung des Kreises und erläutern die zentralen thematischen Fragestellungen. Die Anmerkungen identifizieren die internen wie externen Opponenten und erklären den Kontext der zeitgenössischen Wissenschaft. (shrink)