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...

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..

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)

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.

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)

Recent years have featured a leap in academic and public interest in Corporate Social Responsibility (CSR) activities and related corporate reporting. Two main themes in this literature are the exploration of management incentives to engage in and disclose this information, and of the use and value of this information to market participants. We extend the second theme by examining the interest that specific investor classes have in the use of CSR information. We rely on feminist intersectionality, which suggests that gender (...) intersects with other identities to yield different values, experiences, and opportunities that can lead to gender-based preferences for CSR information. Based upon a survey of 750 US-based retail investors, we find that female retail investors have a greater interest in the use of CSR information, relative to male retail investors. Women express greater anticipated future demand for this information than do men. Further, the magnitude of the increase from current use to anticipated future demand is greater for women than men. Age is a significant modifying factor in that the discrepancy between women of any age and older men is greater than that between women and younger men. Finally, women also exhibit greater demand for streamlining of the information flow, consistent with pressures induced by time poverty. It appears that current disclosure practices provide a less than optimal match with the needs of the information consumers that are primarily interested in using this information. This mismatch may result in a systematic disenfranchisement of female investing classes which suggests an ethical need to level the playing field. (shrink)

The logical pattern underlying twentieth century science is strikingly uniform from physics through biology and psychology to social science. Our purpose will be to analyze and illustrate this pattern, to trace its development, especially from the Middle Ages to the present time, and to suggest some possible consequences for the future.

İnsan Dasein’ının varoluş felsefesindeki[1] anlayışı, bir iç zorunlulukla ölüm sorusuna çıkmak durumundadır ve bu anlayış öncelikle kendisinin son belirlenmişliğinde karşılanabilir. İnsan Dasein’ının sonluluğu belli bir konuma fırlatılmışlık içerisinde sorumluluğu, nasıl ki sınır durumdaki deneyimlere yüklemede etkisini gösteriyorsa, sonluluk da zorunlu biçimde insan Dasein’ının en son ve en uç sınırı olan ölümü ortaya çıkartır. İnsanın kendini bir hiçliğin önünde durmuş ve gerçek bir olabilmeye davet edilmiş bulduğu ruh hali olarak korku içinde bulunuş, ölümle karşılaştırıldığında insanın en uç korkusuna da işaret ediyor (...) olmalıdır. Öyle ki bir iç zorunluluktan hareketle en uç gerçek olarak ölüm varoluş felsefesi tarafından, insan kavrayışının odağına yerleştirilmelidir. Bu, Dasein’ın en kaçınılmaz gerçekliğidir. Ölüm, modern felsefenin genel tarihinde neredeyse önemsiz bir rol oynadıktan sonra, bugün artık insan Dasein’ının bütünü için belirleyici önemiyle varoluş felsefesi tarafından yeniden keşfedilmiştir. (shrink)