It is well known that Felix Klein took a decisive step in investigating the invariants of transformation groups. However, less attention has been given to Klein’s considerations on the epistemological implications of his work on geometry. This paper proposes an interpretation of Klein’s view as a form of mathematical structuralism, according to which the study of mathematical structures provides the basis for a better understanding of how mathematical research and practice develop.

This book offers a reconstruction of the debate on non-Euclidean geometry in neo-Kantianism between the second half of the nineteenth century and the first decades of the twentieth century. Kant famously characterized space and time as a priori forms of intuitions, which lie at the foundation of mathematical knowledge. The success of his philosophical account of space was due not least to the fact that Euclidean geometry was widely considered to be a model of certainty at his time. However, such (...) later scientific developments as non-Euclidean geometries and Einstein’s general theory of relativity called into question the certainty of Euclidean geometry and posed the problem of reconsidering space as an open question for empirical research. The transformation of the concept of space from a source of knowledge to an object of research can be traced back to a tradition, which includes such mathematicians as Carl Friedrich Gauss, Bernhard Riemann, Richard Dedekind, Felix Klein, and Henri Poincaré, and which finds one of its clearest expressions in Hermann von Helmholtz’s epistemological works. Although Helmholtz formulated compelling objections to Kant, the author reconsiders different strategies for a philosophical account of the same transformation from a neo-Kantian perspective, and especially Hermann Cohen’s account of the aprioricity of mathematics in terms of applicability and Ernst Cassirer’s reformulation of the a priori of space in terms of a system of hypotheses. This book is ideal for students, scholars and researchers who wish to broaden their knowledge of non-Euclidean geometry or neo-Kantianism. (shrink)

Hermann von Helmholtz’s geometrical papers have been typically deemed to provide an implicitly group-theoretical analysis of space, as articulated later by Felix Klein, Sophus Lie, and Henri Poincaré. However, there is less agreement as to what properties exactly in such a view would pertain to space, as opposed to abstract mathematical structures, on the one hand, and empirical contents, on the other. According to Moritz Schlick, the puzzle can be resolved only by clearly distinguishing the empirical qualities of spatial perception (...) from those describable in terms of axiomatic geometry. This paper offers a partial defense of the group-theoretical reading of Helmholtz along the lines of Ernst Cassirer in the fourth volume of The Problem of Knowledge of 1940. In order to avoid the problem raised by Schlick, Cassirer relied on a Kantian view of space not so much as an object of geometry, but as a precondition for the possibility of measurement. Although the concept of group does not provide a description of space, the modern way to articulate the concept of space in terms of transformation groups reveals something about the structure and the transformation of spatial concepts in mathematical and natural sciences. (shrink)

It is well known that Hermann Cohen was one of the first philosophers who engaged in the debate about non-Euclidean geometries and the concept of space. His relation to Hermann von Helmholtz, who played a major role in the same debate, is an illuminating example of how some of the leading ideas of Marburg neo-Kantianism, although motivated independently of scientific debates, naturally led to the examination of scientific works and scientists’ epistemological views. This paper deals with Cohen’s view of magnitudes (...) and measurement and with his – less known – review of Helmholtz’s paper “Zählen und Messen, erkenntnistheoretisch betrachtet”, which contains one of the first attempts to formulate a theory of measurement in the modern sense. The first part provides a brief introduction to this debate in its connection with the earlier discussion on geometrical axioms and the concept of space. The main sections deal with Helmholtz’s and Cohen’s approaches to the foundations of measurement. Cohen’s criticism of some of Helmholtz’s assumptions notwithstanding, my emphasis is on some unexpected affinities between these two approaches. In the concluding section, I rely on the constructive side of Cohen’s criticisms to reconsider the philosophical aspects of Helmholtz’s theory and draw a few comparisons with contemporary measurement theory. (shrink)

Hermann von Helmholtz has been widely acknowledged as one of the forerunners of contemporary theories of measurement. However, his conception of measurement differs from later, representational conceptions in two main respects. Firstly, Helmholtz advocated an empiricist philosophy of arithmetic as grounded in some psychological facts concerning quantification. Secondly, his theory implies that mathematical structures are common to both subjective experiences and objective ones. My suggestion is that both of these differences depend on a classical approach to measurement, according to which (...) the arithmetic laws of addition define what is measurable as a particular domain for their application, and, at the same time, the extensibility of these laws to all known physical processes works as a heuristic principle for empirical research. Such an approach is worth reconsidering, not only because it lends plausibility to some of the controversial aspects of Helmholtz’s theory, but also because it offers a philosophical perspective on quantification problems that originated in the nineteenth-century.This paper draws insights on Helmholtz’s philosophical views from his engagement with the measurability of sensations via Fechner’s psychophysical law. This seems to be in contrast with the fact that the reception of Helmholtz’s theory culminated with the formulation of the theory of extensive measurement. I will contend that Helmholtz reached a no less important standpoint in the nineteenth-century debate on whether sensations are different (i.e., intensive) kinds of magnitudes and on how, if at all, they can be measured. (shrink)

When non-Euclidean geometry was developed in the nineteenth century, both scientists and philosophers addressed the question as to whether the Kantian theory of space ought to be refurbished or even rejected. The possibility of considering a variety of hypotheses regarding physical space appeared to contradict Kant’s supposition of Euclid’s geometry as a priori knowledge and suggested the view that the geometry of space is a matter for empirical investigation. In this article, I discuss two different attempts to defend the Kantian (...) theory of space against geometrical empiricism. Both Cohen and Riehl defended the apriority of geometrical axioms. Cohen pointed out that knowledge necessarily requires both sensibility and understanding. Therefore, he maintained that space can be proved to be a condition of experience without admitting that some statements about it are intrinsically necessary. By contrast, Riehl emphasized universality and necessity as intrinsic properties of a priori knowledge. He was one of the first philosophers to discuss the space problem in detail and to distinguish between a priori properties of space and empirical properties. Nevertheless, I suggest that Cohen developed a more plausible view because he was not committed to any spatial structure independently of empirical science. (shrink)

Cassirer’s neo-Kantian epistemology has become a classical reference in contemporary history and philosophy of science. However, the historical aspects of his thought are sometimes seen to be in some tension with his defence of a priori elements of knowledge. This paper reconsiders Cassirer’s strategy to address this tension by positing functional dependencies at the core of the notion of objectivity. This requires the epistemologist to account for the determination of the objects of knowledge within given scientific theories, but also for (...) conceptual changes over time. (shrink)

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)

Several studies have emphasized the limits of invariance-based approaches such as Klein’s and Cassirer’s when it comes to account for the shift from the spacetimes of classical mechanics and of special relativity to those of general relativity. Not only is it much more complicated to find such invariants in the case of general relativity, but even if local invariants in Weyl’s fashion are admitted, Cassirer’s attempt at a further generalization of his approach to the spacetime structure of general relativity seems (...) to obscure the fact that the determination of this structure requires a completely different approach that is derived from Riemann rather than Klein. This paper aims to reconsider these issues by drawing attention to the combination of subtractive and additive strategies (more in line with Riemann’s) in Klein’s own considerations about the determination of the structure of spacetime from 1897 to 1910. I will point out that Cassirer relied on Klein’s argument in some central passages from Substance and Function (1910), and elaborated further on his combined approach in Einstein’s Theory of Relativity (1921), also taking into account the application of Riemannian geometry in general relativity. My suggestion is that an appreciation of Cassirer’s continuing commitment to a variety of geometrical traditions may shed light on his particular understanding of a priori elements of knowledge and avoid some of the classical objections to the idea of a relativization of the a priori. (shrink)

Implicit definitions have been much discussed in the history and philosophy of science in relation to logical positivism. Not only have the logical positivists been influential in establishing this notion, but they have addressed the main problems connected with the use of such definitions, in particular the question whether there can be such definitions, and the problem of delimiting their scope. This paper aims to draw further insights on implicit definitions from the development of this notion from its first occurrence (...) in German language in Enriques’s “Principles of Geometry” (1907) to Schlick’s General Theory of Knowledge (1918). Enriques was one of the first to acknowledge that implicit definitions in mathematics are possible only for higher-order entities or structures, which can have infinitely many interpretations in terms of physical objects. While Schlick introduced coordinating principles to account for the scientific interpretations of implicit definitions, Enriques addressed the problem of bridging the gap between abstract and concrete terms in a different way: He identified, within mathematics, structural patterns that provide a clarification of conceptual relations, and so also serve (indirectly) the purposes of applied mathematics. My suggestion is that Enriques’s analysis of these patterns deserves deeper consideration also from a contemporary perspective on mathematical concept formation. (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)

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)

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 Authors [2017]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected] book Space, Number, and Geometry from Helmholtz to Cassirer is a reworked version of Francesca Biagioli’s PhD thesis. It aims ‘[to offer] a reconstruction of the debate on non-Euclidean geometry in neo-Kantianism between the second half of the nineteenth century and the first decades of the twentieth century’. More precisely, Biagioli concentrates on how the Marburg school of neo-Kantianism dealt with (...) what may be called ‘Helmholtz’s challenge’ to the classical Kantian thesis that the geometry of space is necessarily Euclidean. To be precise, besides the Marburg current, SNG also deals with some other neo-Kantians, among them the realist neo-Kantian Alois Riehl and the South-West German neo-Kantian Bruno Bauch.According to SNG, the Marburg revision of the Kantian theory of space opened the door to a new way... (shrink)

Francesca Biagioli’s Space, Number, and Geometry from Helmholtz to Cassirer is a substantial and pathbreaking contribution to the energetic and growing field of researchers delving into the physics, physiology, psychology, and mathematics of the nineteenth and twentieth centuries. The book provides a bracing and painstakingly researched re-appreciation of the work of Hermann von Helmholtz and Ernst Cassirer, and of their place in the tradition, and is worth study for that alone. The contributions of the book go far beyond (...) that, however. It is a clear, accurate, and deep account of fascinating and philosophically momentous implications of the move to relativity theory. (shrink)

Until recently, dance was not considered to warrant copyright protection because it existed only as a live performance that was not fixed in a ‘tangible medium of expression’. Not being an object, it could not be property. But the more we try to fold dance into existing modes of copyright and conventional notions of property, the more it resists, upsetting the core assumptions of Locke's social contract theory. Legal scholars argue that the expansion of copyright protection shrinks the public domain. (...) While copyright has become more important for dancers and choreographers who wish to control the appropriation of their work that is now made available to millions of end-users online, it also potentially restricts them from engaging in a dialog with other dancers or building on inspiring dance moves across communities. This paper investigates notions of property that rely on both the commons and individual personhood in the context of dance. (shrink)

While the initial reception of Alfred Hitchcock's Vertigo was unspectacular, it made its presence felt in a host of other films – from Chris Marker's Sans Soleil, to Brian De Palma's Obsession, and David Lynch's Mulholland Dr.. What seemed to have eluded the critics at the time is that Vertigo is a film about being haunted: by illusive images, turbulent emotions, motion and memory, the sound and feeling of falling into the past, into a nightmare. But it is also a (...) shrewdly reflexive film that haunts filmmakers, critics, and artists alike, raising fundamental questions about the ontology of moving images and the regime of fascination that churns them out. Douglas Gordon's Feature Film, D.N. Rodowick's The Wanderers, and Lynn Hershman's VertiGhost are contemporary examples of how the appropriation and contemplation of some the film's most iconic motifs, themes and strategies ask us to rethink the relation of fetishism to fabulation, and supplementarity to dissimulation and social engineering. Feature Film, The Wanderers, and VertiGhost are supplementary works, but like the original film they are about duplicity, doppelgänger, and dissimulation. What interests us is how they challenge the authority over, or even proximity to, that which returns in the form of the supplement. And ultimately, attaching themselves to the chain of forgers and forgeries, these supplementary works take their place in the vertiginous sequence of substitutions the film established: a neat allegory for a reign of the digital ghosting that Hitchcock could never have anticipated. (shrink)

Farmers’ markets (FMs) represent a crucial player in urban food systems, being the interconnection of local agricultural production and consumption, and serving as spaces for both economic exchange and community building. Despite their transformative potential, there is a scarcity of research that comprehensively investigates the dynamics of FMs network structure and the influence of the actors shaping FMs. The present article delves into the network of relationships within FMs in the Italian city of Bologna. This study adopts the Social Network (...) Analysis (SNA) methodology applied with the Net-Map tool. The research objectives are to visualise the underlying network structure, map the dynamics, and identify the key actors who play pivotal roles in Bologna’s FMs and their level of influence. The research carried out interviews with ten FMs stakeholders, revealing the network of relationships between 54 actors, divided into three categories—civil society, food economy and public administration. Actors were linked by a total of 428 relationships across three types of networks: support, economic, and hindering. Findings indicate a strong support network structure characterised by varying degrees of centrality among different actors. Farmers emerge as a central node due to their pivotal role in providing fresh, local produce. Additionally, local institutions contribute significantly to FMs resilience and growth. Our research demonstrates the importance of recognising the embeddedness of FMs within the local context. By understanding the network structure and influential actors in FMs, policymakers can devise more effective policies for promoting local agriculture, and enhancing the sustainability of urban–rural exchanges. In conclusion, the present study offers valuable insights into the network dynamics of FMs, highlighting their crucial role in the sustainable development of urban and local food systems. (shrink)

In health communication, metaphor can be considered as a reasoning device to let people understand an abstract concept in terms of a concrete one (Lakoff and Johnson 1980; Bowdle and Gentner 2005). Both the positive and negative communicative effects of metaphors have been largely pointed out in a variety of medical fields, from oncology (Semino et al. 2016, 2018) to mental health (Frezza and Zoccolotti 2019). The use of metaphors in vaccine communication has been less considered, though it might be (...) crucial to let people understand vaccination as an important collective health phenomenon. A previous study (Scherer et al. 2015) focused on the communicative effects of conventional metaphors for vaccination, but not specifically on novel metaphors deliberately used to explain the concept of population immunity in vaccination. We hypothesized that metaphors used to explain vaccination as a collective phenomenon could improve the communicative aims of pro-vaccination texts in many respects, ranging from understandability to trust in expertise. We designed a study to test the hypothesis: two groups of participants were presented with a text about vaccination as a collective phenomenon, described in either literal or metaphorical terms. They were asked to evaluate the text on different communicative aspects from both an individual and a collective point of view. The results confirmed the hypothesis that metaphor is an effective reasoning device and that it also enhances the overall communicative effects of the message, in terms of understandability, persuasion, feeling of control over the health situation, trust in expertise, and uptake of experts’ advice. However, the results show that this effect is significantly nuanced in the individual vs. collective response type. (shrink)

Kant’s essay on the question of literary piracy has so far been read as a foundational text in the history of literary property. When Kant refers to the book as a “mute instrument,” scholars of intellectual property already know how to interpret that formulation because they presume the distinction that the contemporary jurisprudence of intellectual property makes between matter and form and its concomitant assumption that print is just an inert, nonagentive medium. In fact, Kant begins his analysis of unauthorized (...) publication not with the question What is an author? but with the question What is publication?. His insight was that unauthorized publication revealed a structural feature of publication in general. The effect of the interposition of print, or the fact that speech begins in and with print, is that the author is structurally alienated from his or her speech. Kant was not an Enlightenment media theorist, but he recognized that speech had to be mediated and therefore delivered to the reader and that it was the medium itself that determined the conditions under which authors and readers could exist and be present to one another and under which speech could become an authorship. In order to create a language channel for public reason, Kant had to take a detour through the legal fiction of agency. Speech might have been an action that could have its existence “only in a person,” but this supposedly innate bond was made by a legal fiction of representation, according to which one person could truly speak “as if” they were another. Kant’s answer to the question What is a book? developed a law that was not so much a law of literary property as a law that sought to suspend the alienating effects of print so as to restore the Enlightenment ideal of communication. (shrink)

This volume provides an extensive overview of the Ethics of Artificial Intelligence for the Sustainable Development Goals. The authors are experts contributing with perspectives from different fields. The comprehensive collection of chapters illustrates the pressing governance problems related to using AI for the SDGs, and case studies describing how AI is advancing and can advance the achievement of the Goals. Students, scholars, and practitioners working on AI for SDGs, the ethical governance of AI, sustainability, and the fourth revolution can find (...) this book a helpful reference. (shrink)

This annual edited volume presents an overview of cutting-edge research areas within digital ethics as defined by the Digital Governance Research Group of the University of Oxford. It identifies new challenges and opportunities of influence in setting the research agenda in the field. The 2022 edition of the Yearbook presents research on the following topics: autonomous weapons, cyber weapons, digital sovereignty, smart cities, artificial intelligence for the Sustainable Development Goals, vaccine passports, and sociotechnical pragmatism as an approach to technology. This (...) text appeals to students, researchers, and professionals in the field. (shrink)

Digital cameras and motion capture technologies that document and share creative practices have transformed the way we think about dance as an embodied knowledge as well as the way we experience it bodily. Computational media, which not only records and archives but also calculates, analyses and models dance, further complicates its ontological status. This move to document and inscribe dance in a tangible medium marks a shift from understanding dance as an ungraspable event towards conceiving of dance as a tangible (...) process that can be collected, transmitted, repeated and packaged or archived as an object. I look at two dance works – Matthias Sperling’s Loop Atlas and Siobhan Davies and Anri Sala’s Solo in the Doldrums – that trouble conventional notions of embodiment and its relationship to both presence and absence, at the same time they challenge how computational media fixates on figures and gestures that can be traced, mapped and calibrated. (shrink)

This paper studies internal (or intra-)mathematical explanations, namely those proofs of mathematical theorems that seem to explain the theorem they prove. The goal of the paper is a rigorous analysis of these explanations. This will be done in two steps. First, we will show how to move from informal proofs of mathematical theorems to a formal presentation that involves proof trees, together with a decomposition of their elements; secondly we will show that those mathematical proofs that are regarded as having (...) explanatory power all display an increase of conceptual complexity from the assumptions to the conclusion. (shrink)

Through this contribution I aim to explore the horizons and limits of digital medicine in light of an embodied approach to the issue of care. I will sketch the historical background of digital medicine and show the contemporary status of this interdisciplinary field, as well as its applications and outcomes. Then, I will address a critique of the computational theory of mind (CTM) upon which many contemporary mental health apps are designed. This approach to the mind is inscribed into the (...) modern trend of neuromania, which conceives the embodied, living human being as a mere series of data analyses and algorithms, with the brain as the seat of the person. Drawing on continental philosophy and classical phenomenology I will develop a notion of health conceived as a moral enterprise, showing how wellbeing and vitality are not reducible to a product made by technologies or doctors, but rather they are the results of an embodied encounter. As such, empathy and person-centered care cannot be achieved by digital medicine. They are the result of a journey into the regions of recognition, embodied presence and the aliveness constituting the person. Finally, I will show how the issue of care cannot be separated from an embodied encounter between people, especially in case of dementia. (shrink)

Criminal liability for acts committed by AI systems has recently become a hot legal topic. This paper includes three different contributions. The first contribution is an analysis of the extent to which an AI system can satisfy the requirements for criminal liability: accomplishing an actus reus, having the corresponding mens rea, possessing the cognitive capacities needed for responsibility. The second contribution is a discussion of criminal activity accomplished by an AI entity, with reference to a recent case involving an online (...) bot, the Random Darknet Shopper. This discussion will provide the context for the analysis of commonalities and differences between criminal activities by humans and by artificial systems. The third contribution concerns the evaluation of different ways of addressing criminal activities by AI systems in a regulatory perspective. (shrink)