The author of this book pursues the objective of treating the whole of pure mathematics [die ganze reine Mathematik] in four sections [Abtheilungen]. One half of the first of these sections is dedicated to arithmetic and is already available. The other half of the first section “A heuristic treatise on number [Zahlenlehre in freier Gedankenentwicklung]” which treats the same discipline is supposed to follow. The author may have opted for such an unusual separation [of the treatment of arithme..

The way in which geometrical knowledge has been obtained has always attracted the attention of philosophers. The fact that there is a science that concerns things outside our thinking and that proceeds inferentially appeared striking, and gave rise to specific theories of experience and space. Nonetheless, the geometrical method has not yet been sufficiently investigated. Philosophers who investigate the theory of knowledge discuss the question of whether geometry is an empirical science, but...

Hölder’s review of Robert Graßmann’s Theory of Number provides the first statement of Hölder’s most significant tenets concerning the distinction between the genetic and axiomatic presentation of mathematics, the mature expression of which is found in Hölder’s book The Mathematical Method of 1924. By translating Hölder’s review into English, I hope to make this unique document known to a wider public. In my introductory note I provide some context to Hölder’s paper and a few other remarks concerning the translation work.

In this paper I provide a brief reconstruction of Otto Hölder’s conception of proof. My reconstruction focuses on Hölder’s critical assessment of David Hilbert’s account of axiomatics in general, and of Hilbert’s conception of metamathematics in particular. I argue that Hölder’s analysis of Hilbert’s general methodological ideas and, more importantly, Hölder’s analysis of the logical structure of the proofs provided by Hilbert in his Grundlagen der Geometrie of 1899 are helpful in reaching a clearer understanding of van der Waerden’s (...) claim linking Hölder’s conception of proof to the tradition established by Kurt Gödel. (shrink)

L’article a pour but d’analyser la conception de la géométrie et de la mesure présentée dans Intuition et Raisonnement [Hölder 1900], « Les axiomes de la grandeur et la théorie de la mensuration » [Hölder 1901] et La Méthode mathématique [Hölder 1924]. L’article examine les relations entre a) la démarcation introduite par Hölder entre géométrie et arithmétique à partir de la notion de ‘concept donné’, b) sa position philosophique par rapport à l’apriorisme kantien et à l’empirisme et c) le choix (...) du postulat de la continuité de Dedekind parmi les axiomes de la théorie des grandeurs. L’article montre que les choix faits au niveau axiomatique sont reliés au cadre épistémologique et que l’originalité de Hölder consiste surtout dans son analyse au cas par cas des procédures déductives en mathématiques. (shrink)

La philosophie de la géométrie de Hölder, si l’on s’en tient à une lecture superficielle, est la part la plus problématique de son épistémologie. Il soutient que la géométrie est fondée sur l’expérience à la manière de Helmholtz, malgré les objections sérieuses de Poincaré. Néanmoins, je pense que la position de Hölder mérite d’être discutée pour deux motifs. Premièrement, ses implications méthodologiques furent importantes pour le développement de son épistémologie. Deuxièmement, Poincaré utilise l’opposition entre le kantisme et l’empirisme comme un (...) argument pour justifier son conventionnalisme géométrique. Cependant, Hölder montre qu’une stratégie alternative n’est pas exclue: il sait tirer parti des objections kantiennes pour développer un empirisme cohérent. En même temps, surtout dans Die mathematische Methode [Holder 1924], il adopte aussi bien les expressions que les conceptions de Kant. Dans mon article, je considère d’abord les arguments de Hölder pour la méthode déductive en géométrie dans Anschauung und Denken in der Geometrie [Holder 1900], en relation avec sa façon d’aborder la théorie de la quantité [Holder 1901]. Ensuite, j’examine son rapport avec Kant. À mon sens, les considérations méthodologiques de Hölder lui permettent de préfigurer une relativisation de l’a priori. (shrink)

The complexity approach to political economy suggests that radical uncertainty is a necessary feature of a complex and evolving socioeconomic landscape. Radical uncertainty raises various adaptive challenges that are likely to escalate in the coming decades under the “Fourth Industrial Revolution.” It jeopardizes the wellbeing of ordinary citizens, whose welfare prospects, job opportunities, and income stream are rendered insecure. It also renders precarious the robust implementation of universal human rights, including the right to social security. In fact, it will be (...) argued that human rights cannot be institutionally secured unless they are implemented in a “complexity aware” way that takes account of the uncertainty of the socioeconomic order. In particular, in order to better protect the dignity of rights holders, and to allow them to survive and thrive in the face of radical uncertainty, governments ought to explore various institutional alternatives to existing schemes. It will be suggested that most existing conditional welfare policies systematically fall short of complexity-aware institutional implementation to the extent that they rely on unreasonable assumptions regarding the availability of salient technocratic control over the complex socioeconomic order. Some economists and philosophers have suggested that UBI (Universal Basic Income) might be a viable way to implement the right to social security today. I propose that some of UBI’s most ethically and economically controversial features, such as unconditionality and universality, appear relatively appealing in light of the complexity perspective. In the absence of better alternatives, UBI might even be a requirement of a robust implementation of the human right to social security – a right that welfare states are already committed to on the level of ideal theory but predictably fail to respect in practice due to policy incoherence. However, open questions remain about whether UBI can be transformed from an ideal model into a pragmatic policy. (shrink)

Nous analysons la réponse qu’a donnée Otto Hölder à la célèbre critique de la preuve archimédienne de la loi du levier par Ernst Mach et nous la situons dans sa pensée epistémologique. Pour Hölder, la preuve de la loi du levier fournit une étude de cas sur la déduction en mécanique. Considérant la preuve comme un processus constructif et synthétique montant d’un niveau conceptuel à un niveau supérieur, Hölder parvient à défendre la preuve d’Archimède contre la critique de Mach, (...) pour qui au contraire une preuve n’amène qu’à une vérité analytique. (shrink)

1 The epistemology of Otto Hölder This special issue is devoted to the philosophical ideas developed by Otto Hölder (1859-1937), a mathematician who made important contributions to analytic functions and group theory. Hölder’s substantial work on the foundations of mathematics and the general philosophical conception outlined in this work are, however, still largely unknown. Up to the present, philosophical interest in Hölder’s work has been limited to his axiomatic formulation of a theory of..

In 1870, Hermann von Helmholtz criticized the Kantian conception of geometrical axioms as a priori synthetic judgments grounded in spatial intuition. However, during his dispute with Albrecht Krause (Kant und Helmholtz über den Ursprung und die Bedeutung der Raumanschauung und der geometrischen Axiome. Lahr, Schauenburg, 1878), Helmholtz maintained that space can be transcendental without the axioms being so. In this paper, I will analyze Helmholtz’s claim in connection with his theory of measurement. Helmholtz uses a Kantian argument that can be (...) summarized as follows: mathematical structures that can be defined independently of the objects we experience are necessary for judgments about magnitudes to be generally valid. I suggest that space is conceived by Helmholtz as one such structure. I will analyze his argument in its most detailed version, which is found in Helmholtz (Zählen und Messen, erkenntnistheoretisch betrachtet 1887. In: Schriften zur Erkenntnistheorie. Springer, Berlin, 1921, 70–97). In support of my view, I will consider alternative formulations of the same argument by Ernst Cassirer and Otto Hölder. (shrink)

Concerns a problem posed, but not solved, by Frege in part III of his Grundgesetze. As a preliminary to defining ‘real number’, Frege attempts to analyse the notion of a quantitative domain. He was unaware of the previous attempt of Otto Holder to do this; it is remarked how much weaker Frege's assumptions were in deriving theorems than Holder's. Frege deals with groups on which there is a right‐invariant semilinear ordering, although he does not use this terminology. He is (...) uncertain whether it can be deduced that the ordering is also left‐invariant, and proves as much as possible without invoking the assumption that it is. The independence proof, due to Dr Peter Neumann, establishes that it could not be deduced. (shrink)

This article overviews the scholarly history of research centred on the Socratic problem. Eight breaking points in this history are identified and the developments emerging from them are analysed. 1) The pre-history of the Socratic problem began with the varying and conflicting portrayals of Socrates that sprang up during his lifetime and shortly after his death. 2) The problem was first explicitly admitted in the 18 th century with the earliest historical investigations critically pursuing the truth behind the fiction. 3) (...) The problem was then established in research in the 19 th century as a result of the influential remarks of Schleiermacher and Hegel on the discrepancies in Plato’s and Xenophon’s portrayals of Socrates. 4) In the beginning of the 20 th century, an emerging disapproval of Xenophon as a reliable source was accompanied by an elevation of the source value of Plato. 5) The crisis of the mid-20 th century took the form of a denial of the very possibility of reaching genuine knowledge of the historical Socrates. 6) The reaction to the crisis assumed the two-folded shape of a broadened study of the sources and a partial reappraisal of Plato’s trustworthiness. 7) Beginning from the 1990s, an optimistic program claiming that all of Plato’s early dialogues recreates the philosophy of the historical Socrates gained ground. 8) Today’s research climate is characterised by a renewed pessimism regarding the possibility of approaching the historical Socrates. (shrink)

Anknupfend an die platonische Begriffsbildung der "Kalokagathie" ("Schongute") wird seit dem 19. Jahrhundert diskutiert, inwieweit asthetischen Kategorien Aussagekraft auch fur Politik und Staat zukommt.

Eine kollektive Intelligenz kann dann gelingen, wenn sich ihre einzelnen Bestandteile in Kollektiv-Visionen auf sie beziehen. Diese Vorstellung prägt Utopien und Beschreibungen, in denen Medien als Hilfsmittel verteilter Intelligenz vorkommen. In einer Betrachtung der Medialität dieser Medien zeigt sich hingegen die Flüchtigkeit und Instabilität einer Formierung von kollektiver Intelligenz. Kollektiv-Visionen lassen sich einerseits als Strategien der Stabilisierung beobachten, andererseits als Darstellungen, in denen sich die Möglichkeiten kollektiver Intelligenz abzeichnen. Collective intelligence can be successful if each of its components relates to (...) it in collective visions. This idea characterizes utopia and descriptions, in which media appear as means of distributed intelligence. A consideration of the mediality of these media, however, highlights the volatility and instability of a formation of collective intelligence. Collective visions can be seen both as a strategy of stabilization and as representations from which the possibilities of collective intelligence emerge. (shrink)

Problematischer Inhalt: Zugriff gesperrt Dieser Titel aus dem De Gruyter-Verlagsarchiv ist digitalisiert worden, um ihn der wissenschaftlichen Forschung zugänglich zu machen. Eine digitale Version dieses erstmals im Nationalsozialismus publizierten Titels stellen wir nur auf Nachfrage ausschließlich im Kontext konkreter wissenschaftlicher Forschungsaufgaben zur Verfügung. Mehr erfahren Sie hier.

Der Dialog »Menon« (399 v. Chr.) handelt von der Erkennbarkeit und Lehrbarkeit der Tugend. Ausgehend von dieser Fragestellung begründet und entwickelt Platon die Form und Methode hypothetischer Erkenntnis; der Dialog bietet daher auch eine gute Einführung in die Philosophie Platons.

Enth. u.a.: ".eine Form und ein Name: Tristan". Strukturelle und semantische Untersuchungen an Hans Werner Henzes Préludes für Klavier, Tonbänder und Orchester / Peter Petersen ; Prélude à l'après-midi d'un faune. Mallarmés Gedicht und Debussys Musik / József Ujfalussy ; Fragmente - Stille, An Diotima. Ikonographische Untersuchungen an Luigi Nonos Streichquartett / Georg Friedrich Haas.

The Current Legal Problems lecture series and annual volume was established around sixty years ago at the Faculty of Laws, University College London and has long been recognized as a major reference point for legal scholarship. The continuing strength of Current Legal Problems is its representation of a broad range of legal scholarship opinion, theory, methodology, and subject matter, with an emphasis upon contemporary developments of law. Contributions to the 61st volume in the series include an analysis of war as (...) crime by Professor Gerry Simpson, an assessment of the problems in building a realistic climate change regime by Professor Daniel Cole, and an exploration of the problems of character evidence by Dr Mike Redmayne. (shrink)

Excerpt from Aristoteles als Padagog und Didaktiker ln einer Sammlung von Monographien iiber Die groBen Erzieher darf Aristoteles, der Lehrer und Mentor des grossen Alexander, der Begrunder des Lyzeums, der Methodiker par excellerzce nicht fehlen. Wenn hier eine Darstellung seiner Erziehungsund Unterrichtslehre unternommen wird, so wird damit zugleich eine alte schuld an den groBen Denker und Lehrer abgetragen, denn bisher ist der Gegenstand nur in kleineren Schriften oder kurzen Partien groBerer Werke behandelt worden, etwa abgesehen von dem Buche Alex. Kapps: (...) Aristoteles' staatspadagogik, Hamm 1837, dessen Titel schon die einseitige Auffassung des Verfassers erkennen laotx Aristoteles ist weit weniger staatspadagog als Plato; besser wurde ihn der Titel Sozialpadagogsc charakterisieren, allein auch dieser erschopft das Wesen seiner Erziehungslehre nicht, die mit demselben Rechte Individualpadagogik genannt werden konnte. Es steht bei ihm das individuale und das soziale Element im Gleichgewichte, und darum sind seine Lehren fiir die Gegenwart, welche ein solches Gleichgewicht sucht, von geradezu aktuellem Interesse, das ihnen, wie sich zeigen wird, noch in anderem Betrachte zukommt. Ware es bei uns Brauch, wie es solcher bei den Romern war, dem Andenken Verstorbener Widmungen zu machen, so wurde ich mein Buch den Manen Adolf Trendelenburg's zueignen, der mehr als jeder andere fiir die Neubelebung der Aristotelesstudien gewirkt hat, und den ich vor nunmehr funfzig Jahren zum Lehrer zu erhalten, das Gliick hatte. Den Hilfsmitteln zu diesen Studien, an deren spitze die Gaben des Berliner Aristotelilcers stehen, ein neues, nicht ganz unbrauchbares und die alten wieder in Erinnerung bringendes, angereiht zu haben, wird mir als der Lohn der nicht eben miihelosen Arbeit gelten. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.". (shrink)