The way Leibniz applied his philosophy to mathematics has been the subject of longstanding debates. A key piece of evidence is his letter to Masson on bodies. We offer an interpretation of this often misunderstood text, dealing with the status of infinite divisibility in nature, rather than in mathematics. In line with this distinction, we offer a reading of the fictionality of infinitesimals. The letter has been claimed to support a reading of infinitesimals according to which they are logical fictions, (...) contradictory in their definition, and thus absolutely impossible. The advocates of such a reading have lumped infinitesimals with infinite wholes, which are rejected by Leibniz as contradicting the part–whole principle. Far from supporting this reading, the letter is arguably consistent with the view that infinitesimals, as inassignable quantities, are mentis fictiones, i.e., (well-founded) fictions usable in mathematics, but possibly contrary to the Leibnizian principle of the harmony of things and not necessarily idealizing anything in rerum natura. Unlike infinite wholes, infinitesimals—as well as imaginary roots and other well-founded fictions—may involve accidental (as opposed to absolute) impossibilities, in accordance with the Leibnizian theories of knowledge and modality. (shrink)

We analyze recent criticisms of the use of hyperreal probabilities as expressed by Pruss, Easwaran, Parker, and Williamson. We show that the alleged arbitrariness of hyperreal fields can be avoided by working in the Kanovei–Shelah model or in saturated models. We argue that some of the objections to hyperreal probabilities arise from hidden biases that favor Archimedean models. We discuss the advantage of the hyperreals over transferless fields with infinitesimals. In Paper II we analyze two underdetermination theorems by Pruss and (...) show that they hinge upon parasitic external hyperreal-valued measures, whereas internal hyperfinite measures are not underdetermined. (shrink)

ABSTRACTA probability model is underdetermined when there is no rational reason to assign a particular infinitesimal value as the probability of single events. Pruss claims that hyperreal probabilities are underdetermined. The claim is based upon external hyperreal-valued measures. We show that internal hyperfinite measures are not underdetermined. The importance of internality stems from the fact that Robinson’s transfer principle only applies to internal entities. We also evaluate the claim that transferless ordered fields may have advantages over hyperreals in probabilistic modeling. (...) We show that probabilities developed over such fields are less expressive than hyperreal probabilities. (shrink)

We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy’s foundational work associated with the work of Boyer and Grabiner; and to Bishop’s constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.

One of the most influential scientific treatises in Cauchy's era was J.-L. Lagrange's Mécanique Analytique, the second edition of which came out in 1811, when Cauchy was barely out of his teens. Lagrange opens his treatise with an unequivocal endorsement of infinitesimals. Referring to the system of infinitesimal calculus, Lagrange writes:Lorsqu'on a bien conçu l'esprit de ce système, et qu'on s'est convaincu de l'exactitude de ses résultats par la méthode géométrique des premières et dernières raisons, ou par la méthode analytique (...) des fonctions dérivées, on peut employer les infiniment petits comme un instrument sûr et commode pour abréger et simplifier les demonstrations.Lagrange's renewed enthusiasm for .. (shrink)

Cauchy’s contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the epsilon-delta doctrine in the context of an Archimedean continuum. What does one see if one refrains from viewing Cauchy as if he had read Weierstrass already? One sees, with Felix Klein, a parallel thread for the development of analysis, in the context of an infinitesimal-enriched continuum. One sees, with Emile Borel, (...) the seeds of the theory of rates of growth of functions as developed by Paul du Bois-Reymond. One sees, with E. G. Björling, an infinitesimal definition of the criterion of uniform convergence. Cauchy’s foundational stance is hereby reconsidered. (shrink)

We explore the potential of Simon Stevin’s numbers, obscured by shifting foundational biases and by 19th century developments in the arithmetisation of analysis.

Adequality, or παρισóτης (parisotēs) in the original Greek of Diophantus 1 , is a crucial step in Fermat’s method of finding maxima, minima, tangents, and solving other problems that a modern mathematician would solve using infinitesimal calculus. The method is presented in a series of short articles in Fermat’s collected works (1891, pp. 133–172). The first article, Methodus ad Disquirendam Maximam et Minimam 2 , opens with a summary of an algorithm for finding the maximum or minimum value of an (...) algebraic expression in a variable A. For convenience, we will write such an expression in modern functional notation as f (a). 3 The algorithm can be broken up into six steps in the following way:Introduce an auxiliary .. (shrink)

We seek to elucidate the philosophical context in which one of the most important conceptual transformations of modern mathematics took place, namely the so-called revolution in rigor in infinitesimal calculus and mathematical analysis. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind,and Weierstrass. The dominant current of philosophy in Germany at the time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical. Our (...) main thesis is that Marburg neo-Kantian philosophy formulated a sophisticated position towards the problems raised by the concepts of limits and infinitesimals. The Marburg school neither clung to the traditional approach of logically and metaphysically dubious infinitesimals, nor whiggishly subscribed to the new orthodoxy of the “great triumvirate” of Cantor, Dedekind, and Weierstrass that declared infinitesimals conceptus nongrati in mathematical discourse. Rather, following Cohen’s lead, the Marburg philosophers sought to clarify Leibniz’s principle of continuity, and to exploit it in making sense of infinitesimals and related concepts. (shrink)

The Mathematical Intelligencer recently published a note by Y. Sergeyev that challenges both mathematics and intelligence. We examine Sergeyev’s claims concerning his purported Infinity computer. We compare his grossone system with the classical Levi-Civita fields and with the hyperreal framework of A. Robinson, and analyze the related algorithmic issues inevitably arising in any genuine computer implementation. We show that Sergeyev’s grossone system is unnecessary and vague, and that whatever consistent subsystem could be salvaged is subsumed entirely within a stronger and (...) clearer system. Lou Kauffman, who published an article on a grossone, places it squarely outside the historical panorama of ideas dealing with infinity and infinitesimals. (shrink)

A construction of the real number system based on almost homomorphisms of the integers $\mathbb {Z}$ was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Even the maximal (i.e., On -saturated) hyperreal number system described by Kanovei and Reeken (2004) and independently (...) by Ehrlich (2012) can be obtained in this fashion, albeit not in NBG . In NBG , it can be obtained via a one-step construction by means of a definable ultrapower (modulo a suitable definable class ultrafilter). (shrink)

The widespread idea that infinitesimals were “eliminated” by the “great triumvirate” of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum (...) with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d’Alembert, Cauchy, and others. (shrink)

The Dirac δ function has solid roots in nineteenth century work in Fourier analysis and singular integrals by Cauchy and others, anticipating Dirac’s discovery by over a century, and illuminating the nature of Cauchy’s infinitesimals and his infinitesimal definition of δ.

Foundations of Science recently published a rebuttal to a portion of our essay it published 2 years ago. The author, G. Schubring, argues that our 2013 text treated unfairly his 2005 book, Conflicts between generalization, rigor, and intuition. He further argues that our attempt to show that Cauchy is part of a long infinitesimalist tradition confuses text with context and thereby misunderstands the significance of Cauchy’s use of infinitesimals. Here we defend our original analysis of various misconceptions and misinterpretations concerning (...) the history of infinitesimals and, in particular, the role of infinitesimals in Cauchy’s mathematics. We show that Schubring misinterprets Proclus, Leibniz, and Klein on non-Archimedean issues, ignores the Jesuit context of Moigno’s flawed critique of infinitesimals, and misrepresents, to the point of caricature, the pioneering Cauchy scholarship of D. Laugwitz. (shrink)

Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for (...) solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework. (shrink)

We apply a framework developed by C. S. Peirce to analyze the concept of clarity, so as to examine a pair of rival mathematical approaches to a typical result in analysis. Namely, we compare an intuitionist and an infinitesimal approaches to the extreme value theorem. We argue that a given pre-mathematical phenomenon may have several aspects that are not necessarily captured by a single formalisation, pointing to a complementarity rather than a rivalry of the approaches.

To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on pro- cedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If, as we argue here, first order logic is indeed suitable for developing modern proxies for the inferential moves found in Leibnizian infinitesimal (...) calculus, then modern infinitesimal frameworks are more appropriate to interpreting Leibnizian infinitesimal calculus than modern Weierstrassian ones. (shrink)

Many historians of the calculus deny significant continuity between infinitesimal calculus of the seventeenth century and twentieth century developments such as Robinson’s theory. Robinson’s hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, (...) Robinson regards Berkeley’s criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley’s criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz’s infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz’s defense of infinitesimals is more firmly grounded than Berkeley’s criticism thereof. We show, moreover, that Leibniz’s system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz’s strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity. (shrink)

We seek to elucidate the philosophical context in which the so-called revolution of rigor in inifinitesimal calculus and mathematical analysis took place. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind, and Weierstrass. The dominant current of philosophy in Germany at that time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical. Our main thesis is that Marburg Neo-Kantian philosophy formulated a sophisticated (...) position towards the problems raised by the concepts of limits and infinitesimals. The Marburg school neither clung to the traditional approach of logically and metaphysically dubious infinitesimals, nor whiggishly subscribed to the new orthodoxy of the "great triumvirate" of Cantor, Dedekind, and Weierstrass. Expressed in terms of modern mathematics, the Marburg philosophers saw the introduction of both infinitesimals and limits as completions whose prototype was Dedekind's of the rational number system resulting in the real numbers. At least partially,, this idea of "completions" can be captured in terms of a category-theoretical description of the conceptual development of modern mathematics. The feasibility of such a modern reformuation may be taken as evidence that the philosophical resources of Marburg neo-Kantianism may be of interest even for contemporary philosophy of mathematics. (shrink)

In his 1676 text De Quadratura Arithmetica, Leibniz distinguished infinita terminata from infinita interminata. The text also deals with the notion, originating with Desargues, of the point of intersection at infinite distance for parallel lines. We examine contrasting interpretations of these notions in the context of Leibniz’s analysis of asymptotes for logarithmic curves and hyperbolas. We point out difficulties that arise due to conflating these notions of infinity. As noted by Rodríguez Hurtado et al., a significant difference exists between the (...) Cartesian model of magnitudes and Leibniz’s search for a qualitative model for studying perspective, including ideal points at infinity. We show how respecting the distinction between these notions enables a consistent interpretation thereof. (shrink)

Currently the two popular ways to practice Robinson’s nonstandard analysis are the model-theoretic approach and the axiomatic/syntactic approach. It is sometimes claimed that the internal axiomatic approach is unable to handle constructions relying on external sets. We show that internal frameworks provide successful accounts of nonstandard hulls and Loeb measures. The basic fact this work relies on is that the ultrapower of the standard universe by a standard ultrafilter is naturally isomorphic to a subuniverse of the internal universe.

The first half of the 17th century was a time of intellectual ferment when wars of natural philosophy were echoes of religious wars, as we illustrate by a case study of an apparently innocuous mathematical technique called adequality pioneered by the honorable judge Pierre de Fermat, its relation to indivisibles, as well as to other hocus-pocus. André Weil noted that simple applications of adequality involving polynomials can be treated purely algebraically but more general problems like the cycloid curve cannot be (...) so treated and involve additional tools–leading the mathematician Fermat potentially into troubled waters. Breger attacks Tannery for tampering with Fermat’s manuscript but it is Breger who tampers with Fermat’s procedure by moving all terms to the left-hand side so as to accord better with Breger’s own interpretation emphasizing the double root idea. We provide modern proxies for Fermat’s procedures in terms of relations of infinite proximity as well as the standard part function. (shrink)

We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S , all definable sets of reals are (...) Lebesgue measurable, suggesting that Connes views a theory as being “virtual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace −∫ (featured on the front cover of Connes’ magnum opus) and the Hahn–Banach theorem, in Connes’ own framework. We also note an inaccuracy in Machover’s critique of infinitesimal-based pedagogy. (shrink)

The small, the tiny, and the infinitesimal have been the object of both fascination and vilification for millenia. One of the most vitriolic reviews in mathematics was that written by Errett Bishop about Keisler’s book Elementary Calculus: an Infinitesimal Approach. In this skit we investigate both the argument itself, and some of its roots in Bishop George Berkeley’s criticism of Leibnizian and Newtonian Calculus. We also explore some of the consequences to students for whom the infinitesimal approach is congenial. The (...) casual mathematical reader may be satisfied to read the text of the five act play, whereas the others may wish to delve into the 130 footnotes, some of which contain elucidation of the mathematics or comments on the history. (shrink)

We apply Benacerraf’s distinction between mathematical ontology and mathematical practice to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like (...) Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves. (shrink)

Did Leibniz exploit infinitesimals and infinities à la rigueur or only as shorthand for quantified propositions that refer to ordinary Archimedean magnitudes? Hidé Ishiguro defends the latter position, which she reformulates in terms of Russellian logical fictions. Ishiguro does not explain how to reconcile this interpretation with Leibniz’s repeated assertions that infinitesimals violate the Archimedean property (i.e., Euclid’s Elements, V.4). We present textual evidence from Leibniz, as well as historical evidence from the early decades of the calculus, to undermine Ishiguro’s (...) interpretation. Leibniz frequently writes that his infinitesimals are useful fictions, and we agree, but we show that it is best not to understand them as logical fictions; instead, they are better understood as pure fictions. (shrink)

In relation to a thesis put forward by Marx Wartofsky, we seek to show that a historiography of mathematics requires an analysis of the ontology of the part of mathematics under scrutiny. Following Ian Hacking, we point out that in the history of mathematics the amount of contingency is larger than is usually thought. As a case study, we analyze the historians’ approach to interpreting James Gregory’s expression ultimate terms in his paper attempting to prove the irrationality of \. Here (...) Gregory referred to the last or ultimate terms of a series. More broadly, we analyze the following questions: which modern framework is more appropriate for interpreting the procedures at work in texts from the early history of infinitesimal analysis? As well as the related question: what is a logical theory that is close to something early modern mathematicians could have used when studying infinite series and quadrature problems? We argue that what has been routinely viewed from the viewpoint of classical analysis as an example of an “unrigorous” practice, in fact finds close procedural proxies in modern infinitesimal theories. We analyze a mix of social and religious reasons that had led to the suppression of both the religious order of Gregory’s teacher degli Angeli, and Gregory’s books at Venice, in the late 1660s. (shrink)

We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to (...) teach us the opposite lesson, namely that the castle is floating in midair. Halmos’ realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians’ concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson’s framework is “unnecessary” but Henson and Keisler argue that Robinson’s framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos’ criticisms. (shrink)

Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy’s proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy’s proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy’s proof closely and show that it finds closer proxies in a different modern framework.

The aim of the dissertation is to formulate a research program in moral cognition modeled on aspects of Universal Grammar and organized around three classic problems in moral epistemology: What constitutes moral knowledge? How is moral knowledge acquired? How is moral knowledge put to use? Drawing on the work of Rawls and Chomsky, a framework for investigating -- is proposed. The framework is defended against a range of philosophical objections and contrasted with the approach of developmentalists like Piaget and Kohlberg. (...) ;One chapter consists of an interpretation of the analogy Rawls draws in A Theory of Justice between moral theory and generative linguistics. A second chapter clarifies the empirical significance of Rawls' linguistic analogy by formulating a solution to the problem of descriptive adequacy with respect to a class of commonsense moral intuitions, including those discussed in the trolley problem literature originating in the work of Foot and Thomson. Three remaining chapters defend Rawls' linguistic analogy against some of its critics. In response to Hare's objection that Rawls' conception of moral theory is too empirical and insufficiently normative, it is argued that Hare fails to acknowledge both the centrality of the problem of empirical adequacy in the history of moral philosophy and the complexity of Rawls' approach to the problem of normative adequacy. In response to Nagel's claim that the analogy between moral theory and linguistics is false because whatever native speakers agree on is English, but whatever ordinary men agree in condemning is not necessarily wrong, it is argued that the criticism ignores both Rawls' use of the competence-performance distinction and the theory-dependence of the corresponding distinction in linguistics. In response to Dworkin's claim that Rawls' conception of moral theory is incompatible with naturalism and presupposes constructivism, it is argued that Dworkin's distinction between naturalism and constructivism represents a false antithesis; neither is an accurate interpretation of the model of moral theory Rawls describes in A Theory of Justice. The thesis concludes by situating Rawls' linguistic analogy within the context of broader debates in metaethics, democratic theory, natural law theory, and the theory of moral development. (shrink)

In this tribute to Steven T. Katz on the occasion of his seventieth birthday, Michael Zank and Ingrid Anderson present sixteen original essays written by senior and junior scholars in comparative religion, philosophy of religion, modern Judaism, and theology after the Holocaust, fields of inquiry where Steven Katz made major contributions over the course of his distinguished scholarly career. The authors of this volume, specialists in Jewish history, especially the modern experience, and Jewish thought from the Bible to (...) Buber, offer theoretical and practical observations on the value of the particular. Contributions range from Tim Knepper's reevaluation of the ineffability discourse to the particulars of the Settlement Cookbook, examined by Nora Rubel as an American classic. (shrink)

Bringing Human Rights Education to US Classrooms presents ten research-based human rights projects powerfully implemented in a range of U.S. classrooms, from elementary school through community college and university. In these classrooms, the students--primarily young people of color who have experienced or witnessed human rights abuses such as discrimination and poverty--are exposed for the first time to thinking about their own lives and the world through an empowering human rights lens. Unique in integrating theory and classroom practice, and in addressing (...) human rights issues with special relevance for communities of color in the US, Katz and Spero provide indispensable guidance for those studying and teaching human rights. (shrink)

This book offers an interdisciplinary approach to religion and psychology, including a compelling denouement that reveals new narratives about longstanding rumours in the early history of the psychoanalytic movement. This volume demonstrates that the first generation of psychoanalysts succeeded in writing themselves into the history of religious thought and sacralizing the origins of psychoanalysis.

Yeshayahu Leibowitz (1903–1994) was an Israeli philosopher and scientist. For decades, his thinking and persona were the embodiment of a Judaism in Israel. Getting to know him is getting to know a great Israeli thinker and also invites a window into the life of Israel and its problems.

This chapter contains sections titled: I The Nature of Philosophy of Education and Conceptual Analysis II Peters' Conception of Education and some of its Normative Implications III Peters' Indeterminacy of Educational Aims and the Need for Sustained Dialogue References.

Depending upon their epistemological foundations philosophical systems can be divided into five types: empiricist (Locke), rationalist (Descartes), intuitivist (Bergson), traditionalist (Confucius), and scriptural (Aquinas). In the history of philosophy there were five major waves of scriptural reasoning-Hindu, Buddhist, Jewish, Christian, and Muslim. In this context Bah ' philosophy represents the sixth wave, and it finds itself in a fruitful dialogue not only with the traditional forms of religious philosophy but also with modern Western thought which is based solely on reason (...) and empirical observation. In this collection the reader will find articles on various aspects of Bah ' philosophy-philosophical anthropology, philosophy of science, philosophy of culture, epistemology, ethics, philosophy of religion, comparative and political philosophy, and, finally, history of philosophy. The authors of those articles-Bah ' and non-Bah ' scholars-reflect global aspirations of this religion by representing diverse countries of the world-USA, Canada, France, Italy, and Sweden. This collection of articles also introduces a Global Faith Book Series which publishes works on various aspects of the Bah ' Faith and globalization. (shrink)

The expression “linguistic Kantianism” is widely used to refer to ideas about thought and cognition being determined by language — a conception characteristic of 20th century analytic philosophy. In this article, I conduct a comparative analysis of Kant’s philosophy and views falling under the umbrella expression “linguistic Kantianism.” First, I show that “linguistic Kantianism” usually presupposes a relativistic conception that is alien to Kant’s philosophy. Second, I analyse Kant’s treatment of linguistic determinism and the place of his ideas in the (...) 18th century intellectual milieu and provide an overview of relevant contemporary literature. Third, I show that authentic Kantianism and “linguistic Kantianism” belong to two different types of transcendentalism, to which I respectively refer as the “transcendentalism of the subject” and the “transcendentalism of the medium.” The transcendentalism of the subject assigns a central role to the faculties of the cognising subject. The transcendentalism of the medium assigns the role of an “active” element neither to the external world nor to the faculties of the cognising subject, but to something in between — language, in the case of “linguistic Kantianism.” I conclude that the expression “linguistic Kantianism” can be misleading when it comes to the origins of this theory. It would be more appropriate to refer to this theory by the expression “linguistic transcendentalism,” thus avoiding an incorrect reference to Kant. (shrink)

This book is not only a major twentieth-century contribution to Dostoevsky’s studies, but also one of the most important theories of the novel produced in our century. As a modern reinterpretation of poetics, it bears comparison with Aristotle.“Bakhtin’s statement on the dialogical nature of artistic creation, and his differentiation of this from a history of monological commentary, is profoundly original and illuminating. This is a classic work on Dostoevsky and a statement of importance to critical theory.” Edward Wasiolek“Concentrating on the (...) particular features of ‘Dostoevskian discourse,’ how Dostoevsky structures a hero and a plot, and what it means to write dialogically, Bakhtin concludes with a major theoretical statement on dialogue as a category of language. One of the most important theories of the novel in this century.” The Bloomsbury Review. (shrink)

Is the science of moral cognition usefully modelled on aspects of Universal Grammar? Are human beings born with an innate 'moral grammar' that causes them to analyse human action in terms of its moral structure, with just as little awareness as they analyse human speech in terms of its grammatical structure? Questions like these have been at the forefront of moral psychology ever since John Mikhail revived them in his influential work on the linguistic analogy and its implications for (...) jurisprudence and moral theory. In this seminal book, Mikhail offers a careful and sustained analysis of the moral grammar hypothesis, showing how some of John Rawls' original ideas about the linguistic analogy, together with famous thought experiments like the trolley problem, can be used to improve our understanding of moral and legal judgement. (shrink)

This groundbreaking work by one of the world's foremost theoreticians of Russian literature, culture, and thought gives for the first time an extensive and detailed examination of the development of Russian thought during the late Soviet period. Countering the traditional view of an intellectual wilderness under the Soviet regime, Mikhail Epstein offers a systematic account of Russian thought in the second half of the 20th century. In doing so, he provides new insights into previously ignored areas such as Russian (...) liberalism, personalism, structuralism, neo-rationalism, and culturology. Epstein shows how Russian philosophy and culture has long been trapped in an intellectual prison of its own making as it sought to create its own utopia. However, he demonstrates that it is time to reappraise Russian philosophical thought and cultural theory, now freed from the bonds of totalitarianism. We are left with not only a new and exciting interpretation of Russian thought, but also an opportunity to rethink our own intellectual heritage. (shrink)