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)

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)

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).

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)

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)

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)

There is a phenomenon which has from of old and in a peculiar degree attracted the attention of social philosophers and practical economists, the fact of certain commodities (these being in advanced civilizations coined pieces of gold and silver, together subsequently with documents representing those coins) becoming universally acceptable media of exchange. It is obvious even to the most ordinary intelligence, that a commodity should be given up by its owner in exchange for another more useful to him. But that (...) every economic unit in a nation should be ready to exchange his goods for little metal disks apparently useless as such, or for documents representing the latter, is a procedure so opposed to the ordinary course of things, that we cannot well wonder if even a distinguished thinker like Savigny finds it downright 'mysterious.' It must not be supposed that the form of coin, or document, employed as current money, constitutes the enigma in this phenomenon. We may look away from these forms and go back to earlier stages of economic development, or indeed to what still obtains in countries here and there, where we find the precious metals in a uncoined state serving as the medium of exchange, and even certain other commodities, cattle, skins, cubes of tea, slabs of salt, cowrie shells, etc.; still we are confronted by this phenomenon, still we have to explain why it is that the economic man is ready to accept a certain kind of commodity, even if he does not need it, or if his need of it is already supplied, in exchange for all the goods he has brought to market, while it is none the less what he needs that he consults in the first instance, with respect to the goods he intends to acquire in the course of his transactions. And hence there runs, from the first essays of reflective contemplation of a social phenomena down to our own times, an uninterrupted chain of disquisitions upon the nature and specific qualities of money in its relation to all that constitutes traffic.. (shrink)

IntroductionHead and neck cancer diagnosis and treatment can be a significant life trauma. Some HNC survivors experience post-traumatic growth, which has been linked with better health-related quality-of-life. Empirical research on PTG, and theoretical models, point to the importance of being able to purposely make sense of the traumatic experience. Intrusive rumination, by contrast, is linked to poorer outcomes. This study explored HNC survivors’ experiences of rumination.MethodsTwenty HNC survivors between 9 months and 5 years post-diagnosis were recruited. They had a range (...) of HNC sub-types and cancer treatments. Participants underwent a semi-structured interview about their cancer diagnosis and treatment. Reflexive thematic analysis identified themes and sub-themes around rumination.ResultsFour themes with linked subthemes on the content and process of rumination were identified. Theme 1 was rumination and worry related to diagnosis. Here, survivors discussed how the HNC diagnosis and plans for treatment had dominated their initial thoughts. Theme 2 was processing the trauma of HNC. This theme reflected rumination on the traumatic experience of diagnosis and treatment and how the participant was reacting to it. Theme 3 was considering the impact. This theme encompassed retrospective thinking and comparisons between the participant now versus the early days after diagnosis. Theme 4 was continued rumination. This theme included ongoing intrusive and distressing rumination about the trauma and impact of cancer. Those who expressed ongoing rumination revisited fears or returned to negative experiences.ConclusionThis study uniquely describes the nature and content of rumination following HNC. Early intrusive rumination is common and may reflect perceptions of cancer as an existential threat. Over time, rumination can become more reflective and move towards deliberate meaning-making. Some HNC survivors may benefit from interventions to reduce barriers to this transition. The content of distressing and difficult to control rumination helps to identify those who may benefit from more directed psychological support. (shrink)

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