Much of the mathematics with which Felix Klein and Sophus Lie are now associated (Klein’s Erlangen Program and Lie’s theory of transformation groups) is rooted in ideas they developed in their early work: the consideration of geometric objects or properties preserved by systems of transformations. As early as 1870, Lie studied particular examples of what he later called contact transformations, which preserve tangency and which came to play a crucial role in his systematic study of transformation groups and (...) differential equations. This note examines Klein’s efforts in the 1870s to interpret contact transformations in terms of connexes and traces that interpretation (which included a false assumption) over the decades that follow. The analysis passes from Klein’s letters to Lie through Lindemann’s edition of Clebsch’s lectures on geometry in 1876, Lie’s criticism of it in his treatise on transformation groups in 1893, and the careful development of that interpretation by Dohmen, a student of Engel, in his 1905 dissertation. The now-obscure notion of connexes and its relation to Lie’s line elements and surface elements are discussed here in some detail. (shrink)

This paper addresses an article by Felix Klein of 1886, in which he generalized his theory of polynomial equations of degree 5—comprehensively discussed in his Lectures on the Icosahedron two years earlier—to equations of degree 6 and 7. To do so, Klein used results previously established in line geometry. I review Klein’s 1886 article, its diverse mathematical background, and its place within the broader history of mathematics. I argue that the program advanced by this article, although historically (...) overlooked due to its eventual failure, offers a valuable insight into a time of crucial evolution of the subject. (shrink)

This paper is devoted to the 150th birthday of the mathematician Felix Klein. On this occasion, we present papers of Klein's Habilitation and give a commentary on them. They have not been analyzed until now. Especially, we have a look at some aspects which concern Klein's broader view on sciences.

The paper reproduces a hitherto unpublished report of 1902 by the mathematician of Göttingen, Felix Klein, to the Prussian ministry of education on his travels, in 1893 and 1896, to the United States. Introduction and commentary stress the relation of this document to the beginnings of German foreign cultural policy, in particular to the German-American professors' exchange program since 1905.

The extant correspondence, consisting of ten letters from the period from 1882 to 1902, from Moritz Pasch to Felix Klein is presented together with an English translation and a short introduction. These letters provide insights into the views of Pasch and Klein regarding the role of intuition and axioms in mathematics, and also into the hiring practices of mathematics professors in the 1880s.

Connections between Roman Catholic Modernism and the artistic culture of the fin de siècle have received little attention from scholars, as compared to the prominence accorded intellectual, social, and political issues. Felix Klein is one of a handful of those who worked for intellectual renewal who closely followed developments in literature and music, interpreting those developments in a way that favored an agenda of reconciling Catholicism with modernity. In two collections of essays, Nouvelles tendances en religion et en literature (...) and Autour du dilettantisme Klein expressed hopes for a return to Catholicism among a cultural elite which I turn would have a larger impact upon a broad public. (shrink)

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.

Felix Klein and Abraham Fraenkel each formulated a criterion for a theory of infinitesimals to be successful, in terms of the feasibility of implementation of the Mean Value Theorem. We explore the evolution of the idea over the past century, and the role of Abraham Robinson's framework therein.

This essay takes a close look at specially selected features of the Göttingen mathematical culture during the period 1895–1920. Drawing heavily on personal accounts and archival resources, it describes the changing roles played by Felix Klein and David Hilbert, as Göttingen's two senior mathematicians, within a fast-growing community that attracted an impressive number of young talents. Within the course of these twenty-five years Göttingen exerted a profound impact on mathematics and physics throughout the world. Many factors contributed to the (...) creation of a special atmosphere that served as a model for several other important centers for mathematical research. Göttingen exemplified a dynamic new way of doing mathematics within a highly competitive community in which the spoken word often carried more weight than did information conveyed in written texts. This oral dimension of the Göttingen culture played an important, till now overlooked role in the early development of Einstein's general theory of relativity. (shrink)

The Meno, one of the most widely read of the Platonic dialogues, is seen afresh in this original interpretation that explores the dialogue as a theatrical presentation. Just as Socrates's listeners would have questioned and examined their own thinking in response to the presentation, so, Klein shows, should modern readers become involved in the drama of the dialogue. Klein offers a line-by-line commentary on the text of the Meno itself that animates the characters and conversation and carefully probes (...) each significant turn of the argument. "A major addition to the literature on the Meno and necessary reading for every student of the dialogue."--Alexander Seasonske, Philosophical Review "There exists no other commentary on Meno which is so thorough, sound, and enlightening."-- Choice Jacob Klein was a student of Martin Heidegger and a tutor at St. John's College from 1937 until his death. His other works include Plato's Trilogy: Theaetetus, the Sophist, and the Statesman, also published by the University of Chicago Press. (shrink)

Der Band nimmt gewisse, mit dem Begriff Theurgie assoziierte Wandlungen innerhalb des spätantiken Neuplatonismus in den Blick. Theurgie wird vor allem als Suche nach immanenten Medien zur Verkörperung göttlicher Kräfte verstanden. Entgegen dem häufig kolportierten Bild vom weltflüchtigen Platonismus zielt die von Iamblichos und seinen Nachfolgern verfochtene Philosophie, so die Grundthese, auf Integration der körperlich-materiellen Wirklichkeitsebenen ab und das Endziel der Philosophie wird stärker unter dem Gesichtspunkt einer positiven Teilhabe am demiurgisch-kosmogonischen Werk begriffen. Diese Aufwertung von Körperlichkeit wird anhand verschiedener (...) Aspekte – z. B. dem Symbolbegriff, der „göttlichen Materie", aber auch des feinstofflichen Ochema-Pneuma-Konzepts – in ihrer Relevanz für das spirituelle Gesamtprofil der theurgischen Neuplatoniker erstmals umfassender untersucht. Abgerundet wird die Studie durch einen Vergleich mit ähnlichen Tendenzen im nicht-dualistischen Śivaismus von Kaschmir. (shrink)

In ”Skepticism,” Peter Klein distinguishes between the “Academic Skeptic” who proposes that we cannot have knowledge of a certain set of propositions and the “Pyrrhonian Skeptic” who refrains from opining about whether we can have knowledge. Klein argues that Academic Skepticism is plausibly supported by a “Closure Principle‐style” argument based on the claim that if x entails y and S has justification for x, then S has justification for y. He turns to contextualism to see if it can (...) contribute to the discussion between one who claims that we can have knowledge about some epistemically interesting class of propositions and the Academic Skeptic. He outlines the background of Pyrrhonian Skepticism, pointing out that the Pyrrhonist withholds assent concerning our knowledge‐bearing status because reason cannot provide an adequate basis for assent. He assesses three possible patterns of reasoning, and concludes that the Pyrrhonist view, that reason cannot resolve matters concerning the nonevident, is vindicated. (shrink)

Relying on aesthetic testimony seems problematic. For instance, it seems problematic for me to simply believe or assert that The Velvet Underground's debut album The Velvet Underground and Nico (1964) is amazing solely because you have told me so, even though I know you to be an honest and competent music critic. But why? After all, there do not seem to be similar reservations regarding testimony from many other domains. In this paper, I will argue that relying on aesthetic testimony (...) seems problematic because we are attached to an ideal of aesthetic authenticity and feel that living up to this ideal is anathema to simply relying on aesthetic testimony. (shrink)

In this paper I argue that much of the confusion and mystery surrounding the concept of "self" can be traced to a failure to appreciate the distinction between the self as a collection of diverse neural components that provide us with our beliefs, memories, desires, personality, emotions, etc (the epistemological self) and the self that is best conceived as subjective, unified awareness, a point of view in the first person (ontological self). While the former can, and indeed has, been extensively (...) studied by researchers of various disciplines in the human sciences, the latter most often has been ignored -- treated more as a place holder attached to a particular predicate of interest (e.g., concept, reference, deception, esteem, image, regulation, etc). These two aspects of the self, I contend, are not reducible -- one being an object (the epistemological self) and the other a subject (the ontological self). Until we appreciate the difficulties of applying scientific methods and analysis to what cannot be reduced to an object of inquiry without stripping it of its essential aspect (its status as subject), progress on the "self", taken as a pluralistic construct, will continue to address only one part of the problems we face in understanding this most fundamental aspect of human experience. (shrink)

Note: Paper to appear in special issue of the journal Psychology of Consciousness: Theory, Research, and Practice, on the evolution of consciousness //// The explanatory challenge of sentience is known as the “hard problem of consciousness”: How does subjective experience arise from physical objects and their relations? Despite some optimistic claims, the perennial struggle with this question shows little evidence of imminent resolution. In this article I focus on the “why” rather than on the “how” of sentience. Specifically, why did (...) sentience evolve in organic lifeforms? From an evolutionary perspective this question can be framed: “What adaptive problem(s) did organisms face in their evolutionary past and how were those challenges met? I argue that sentience was a critical component of the adaptive solution (i.e., adopting an agentic stance) to increasingly complex and unpredictable demands placed on vertebrates approximately 500 million years ago (the so-called Cambrian explosion). One consequence of taking an agentic stance is that it freed the organism from its neural moorings, positioning it within phenomenal space outside its brain. (shrink)

The epistemic position of an agent often depends on their position in a larger network of other agents who provide them with information. In general, agents are better off if they have diverse and independent sources. Sullivan et al. [19] developed a method for quantitatively characterizing the epistemic position of individuals in a network that takes into account both diversity and independence; and presented a proof-of-concept, closed-source implementation on a small graph derived from Twitter data [19]. This paper reports on (...) an open-source reimplementation of their algorithm in Python, optimized to be usable on much larger networks. In addition to the algorithm and package, we also show the ability to scale up our package to large synthetic social network graph profiling, and finally demonstrate its utility in analyzing real-world empirical evidence of ‘echo chambers’ on online social media, as well as evidence of interdisciplinary diversity in an academic communications network. (shrink)

This book is the most comprehensive and rigourous critique of the ways disciplinary boundaries still inhibit knowledge-production and integration.

This is an important work of scholarship with regard to Machiavelli and the development of political thought in England. It charts the reactions of successive English thinkers to Machiavelli’s challenge, and the different aspects of Machiavelli’s thought which were perceived in the changing context of English history. There is the Machiavelli of Catholic and Protestant reformers, the Machiavelli of Raleigh and Bacon, of the royalist Clarendon and the republican Harrington. Through their eyes the reader can see the gradual process whereby (...) the atheistical monster repudiated by the subjects of Henry VIII was quietly absorbed by the politically sophisticated subjects of William III. (shrink)

Recently, Jennifer Saul (“Racial Figleaves, the Shifting Boundaries of the Permissible, and the Rise of Donald Trump”, 2017; “Racist and Sexist Figleaves”, 2021) has explored the use of what she calls “figleaves” in the discourse on race and gender. Following Saul, a figleaf is an utterance that, for some portion of the audience, blocks the conclusion that some other utterance, R, or the person who uttered R is racist or sexist. Such racial and gender figleaves are pernicious, says Saul, because, (...) among other things, they can shift the boundaries of what is deemed acceptable to think or say. This paper expands on Saul’s picture in a threefold way: It is first argued that appeals to statistics can function as figleaves. It is then argued that—as far as figleaves go—there is reason to believe that statistics-as-figleaves are especially pernicious. Finally, the paper explores some strategies for counterspeech. (shrink)

Should refugees govern refugee camps? This paper argues that they should. It draws on normative political thought in consulting the all-subjected principle and an instrumental defense of democratic rule. The former holds that all those subjected to rule in a political unit should have a say in such rule. Through analyzing the conditions that pertain in refugee camps, the paper demonstrates that the all-subjected principle applies there, too. Refugee camps have developed as near distinct entities from their host states. They (...) have formed their own economic, legal and even political systems within which refugees are subjected to political rule. The paper then demonstrates that democratic rule should be preferred over any other decision-making procedure. No amount of experts can replace the institutions that would lead to the accountability of decision-makers and to the incorporation of refugees as situated and epistemically diverse knowers of the problems they face and the solutions that would work best. (shrink)

Neural engineers and clinicians are starting to translate advances in electrodes, neural computation, and signal processing into clinically useful devices to allow control of wheelchairs, spellers, prostheses, and other devices. In the process, large amounts of brain data are being generated from participants, including intracortical, subdural and extracranial sources. Brain data is a vital resource for BCI research but there are concerns about whether the collection and use of this data generates risk to privacy. Further, the nature of BCI research (...) involves understanding and making inferences about device users’ mental states, thoughts, and intentions. This, too, raises privacy concerns by providing otherwise unavailable direct or privileged access to individuals mental lives. And BCI-controlled prostheses may change the way clinical care is provided and the type of physical access caregivers have to patients. This, too, has important privacy implications. I In this chapter we examine several of these privacy concerns in light of prominent views of the nature and value of privacy. We argue that increased scrutiny needs to be paid to privacy concerns arising from Big Data and decoding of mental states, but that BCI research may also provide opportunity for individuals to enhance their privacy. (shrink)