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)

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)

This article aims to highlight why R. S. Peters' conceptual analysis of ‘education’ was such an important contribution to the normative field of philosophy of education. In the article, I do the following: 1) explicate Peters' conception of philosophy of education as a field of philosophy and explain his approach to the philosophical analysis of concepts; 2) emphasize several (normative) features of Peters' conception of education, while pointing to a couple of oversights; and 3) suggest how Peters' analysis might be (...) used to reinvigorate a conversation on one central educational aim—that of how we might educate citizens for the 21st century. (shrink)

This article aims to highlight why R. S. Peters' conceptual analysis of ‘education’ was such an important contribution to the normative field of philosophy of education. In the article, I do the following: 1) explicate Peters' conception of philosophy of education as a field of philosophy and explain his approach to the philosophical analysis of concepts; 2) emphasize several (normative) features of Peters' conception of education, while pointing to a couple of oversights; and 3) suggest how Peters' analysis might be (...) used to reinvigorate a conversation on one central educational aim—that of how we might educate citizens for the 21st century. (shrink)

Throughout his life, John Tyler Bonner contributed to major transformations in the fields of developmental and evolutionary biology. He pondered the evolution of complexity and the significance of randomness in evolution, and was instrumental in the formation of evolutionary developmental biology. His contributions were vast, ranging from highly technical scientific articles to numerous books written for a broad audience. This historical vignette gathers reflections by several prominent researchers on the greatness of John Bonner and the implications of his work.

Humanist sociologists are activists rooted in the reality of history and change and guided by a concern for the 'real life' problems of equality, peace, and social justice. They view people as active shapers of social life, capable of creating societies in which everyone's potential can unfold. Alfred McClung Lee introduces this volume with 'Sociology: Humanist and Scientific' and develops the theme that a sociology that is humanist is also scientific. The other nine selections are grouped into four parts: 'The (...) Individual and Social Life;' 'Social Institutions: Technology, Science, and Formal Organization;' 'Political Structures: Issues of Justice and Equality;' and 'Methodological Critiques and Counterproposals.'. (shrink)

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.

The purpose of this paper is to examine the role that trustworthiness plays in the ability of teachers to function as moral role models. Through exploration of Muriel Spark’s novel, The Prime of Miss Jean Brodie, I explain some of the central features of trustworthiness as a moral virtue and suggest how these features are critical to developing moral relationships between teachers and students. I show how and why the character of Miss Jean Brodie fails to embody trustworthiness, and how (...) moral philosophy and psychological insight are bound up with teachers’ efforts to treat students well and to behave in ways that morally warrant the trust most of us typically grant them. Finally, I propose some of the important implications this analysis has for teacher education. (shrink)

Comparative studies demonstrate that homologous neural structures differ in function and that neural mechanisms underlying behavior evolved independently. A neural structure does not serve a particular function so much as it executes an algorithm on its inputs though its dynamics. Neural dynamics are altered by a neuromodulation, and species-differences in neuromodulation can account for behavioral differences.

To improve the quality and efficiency of research, groups within the scientific community seek to exploit the value of data sharing. Funders, institutions, and specialist organizations are developing and implementing strategies to encourage or mandate data sharing within and across disciplines, with varying degrees of success. Academic journals in ecology and evolution have adopted several types of public data archiving policies requiring authors to make data underlying scholarly manuscripts freely available. The effort to increase data sharing in the sciences is (...) one part of a broader “data revolution” that has prompted discussion about a paradigm shift in scientific research. Yet anecdotes from the community and studies evaluating data availability suggest that these policies have not obtained the desired effects, both in terms of quantity and quality of available datasets. We conducted a qualitative, interview-based study with journal editorial staff and other stakeholders in the academic publishing process to examine how journals enforce data archiving policies. We specifically sought to establish who editors and other stakeholders perceive as responsible for ensuring data completeness and quality in the peer review process. Our analysis revealed little consensus with regard to how data archiving policies should be enforced and who should hold authors accountable for dataset submissions. Themes in interviewee responses included hopefulness that reviewers would take the initiative to review datasets and trust in authors to ensure the completeness and quality of their datasets. We highlight problematic aspects of these thematic responses and offer potential starting points for improvement of the public data archiving process. (shrink)

This contribution to part 4 of the Common Knowledge symposium “Fuzzy Studies: On the Consequence of Blur” shows how the reputedly radical position that history is not about eternal truths but about the creative construction of a convincing narrative of past events is not an argument of recent vintage. In the days when postmodernism was a technical term used mainly by scholars of art and architecture—and indeed, decades before then—professional historians were grappling with the incapacity of facts to write themselves (...) into a universally satisfying, single version of history. Successive presidents of the American Historical Association such as Andrew Dickson White, Carl Becker, Charles Beard, and William McNeill admitted that writing history is a desperate attempt at pattern recognition in a fuzzy discipline. Pattern recognition is a tool, valuable as a stage in historical thinking, but destined ultimately to come undone. What remains, then, is fuzzy thinking. It can be a good thing, the article concludes, to let our thinking about history remain a blur or, at least, to bear in mind that any patterns we recognize in the past are liable to dissolve under a different light. (shrink)

The essays in this volume are a contribution to this process of reappraisal, focusing specifically on the phenomena of scepticism and millenarianism, especially ...

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)

Chomsky's criticism of Bloomfieldian structuralism's conception of linguistic reality applies equally to his own conception of linguistic reality. There are too many sentences in a natural language for them to have either concrete acoustic reality or concrete psychological or neural reality. Sentences have to be types, which, by Peirce's generally accepted definition, means that they are abstract objects. Given that sentences are abstract objects, Chomsky's generativism as well as his psychologism have to be given up. Langendoen and Postal's argument in (...) The Vastness of Natural Languages to show that there are more than denumerably many sentences is flawed. But, with the view that sentences are abstract objects, the flaws can be corrected. Once psychologism and generativism are abandoned, the revolution against Bloomfieldian structuralism can be brought to completion and linguistics can be put on a sound philosophical basis. (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)

Our understanding of the nature and processing of figurative language is central to several important issues in cognitive science, including the relationship of language and thought, how we process language, and how we comprehend abstract meaning. Over the past fifteen years, traditional approaches to these issues have been challenged by experimental psychologists, linguists, and other cognitive scientists interested in the structures of the mind and the processes that operate on them. In Figurative Language and Thought, internationally recognized experts in the (...) field of figurative language, Albert Katz, Mark Turner, Raymond W. Gibbs Jr., and Cristina Cacciari, provide a coherent and focused debate on the subject. The book's authors discuss a variety of fundamental questions, including: What can figures of speech tell us about the structure of the conceptual system? If and how should we distinguish the literal from the nonliteral in our theories of language and thought? Are we primarily figurative thinkers and consequently figurative language users or the other way around? Why do we prefer to speak metaphorically in everyday conversation, when literal options may be available for use? Is metaphor the only vehicle through which we can understand abstract concepts? What role do cultural and social factors play in our comprehension of figurative language? These and related questions are raised and argued in an integrative look at the role of nonliteral language in cognition. This volume, a part of Counterpoints series, will be thought-provoking reading for a wide range of cognitive psychologists, linguists, and philosophers. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Levinas—Between Philosophy and Rhetoric:The “Teaching” of Levinas’s Scriptural ReferencesClaire Elise KatzIn an interview titled "On Jewish Philosophy," Emmanuel Levinas illuminates the connection that he sees between philosophical discourse and the role of midrash in interpreting the Hebrew scriptures. His interviewer immediately expresses surprise at Levinas's comments that suggested he saw the traditions of philosophy and biblical theology as in some sense harmonious (quoted in Robbins 2001, 239). Levinas responds (...) by elaborating on this connection he sees. For him, the lived experience of Judaism is above all a sense of belonging to humanity and, in turn, to a supreme order of responsibility (240). Although his intellectual life focused on the sacred texts of Judaism, this life included non-Jewish books, which he believed expressed a similar concern or a similar responsibility—a concern for the meaning of life. This list of such books includes Hamlet, Macbeth, King Lear, The Miser, and The Misanthrope, in addition to the novels of Tolstoy and Dostoyevsky.In Levinas's view, these novels are not exercises of philosophyper se; as mentioned above, they express the same concerns that occupy philosophy, but they express these concerns in a different idiom. They do not offer a philosophical argument; instead, they use figurative language and literary narrative to convey their messages. By drawing this distinction between philosophy and literature, Levinas appears to have divided the written word into two categories: philosophy, on the one hand, and literary texts, on the other. Yet readers of his philosophical work cannot help but notice his frequent references to literary sources, including the numerous citations taken from the Jewish holy texts. His use of these citations encourages us to ask how we are to classify these books in general and how to classify, in particular, the Scriptures and the holy texts in Judaism. With regard to the holy texts in Judaism, should we classify them as simply a [End Page 159] literary device? Or, do they constitute a third category that falls somewhere in between philosophy and rhetoric? If the latter, then what is their status relative to these larger categories?Although Levinas insists on separating his two bodies of work—his philosophical writings and his Jewish writings—the distinction between them becomes increasingly blurred with his frequent introduction of citations from the Hebrew texts into his philosophy and by inflecting his Jewish writings with philosophical interpretations.1 In what follows, I examine the unique and significant role of Scriptural references within Levinas's philosophical work. My claim is that the Scriptural references are in fact rhetorical, but not "simply" rhetorical; they do pedagogical work for Levinas that other literary devices cannot accomplish. They engage his readers such that they are invited to grapple with the Scriptures in a manner potentially unfamiliar to them, and this invitation, in turn, has the pedagogical effect of attuning his readers to his texts and to his ethical project—even, one might say, to the other—and inviting them into an experience of transcendence.In the interview mentioned above, Levinas tells his interlocutor that for the Jews the Scriptures do not exist apart from the interpretations that accompany them. These interpretations are found in the collections of midrashim and the Talmud. One need know only a little bit about Hebrew and the construction of the Hebrew texts in order to see why these interpretations are significant. There are three grammatical concerns with the Hebrew language: first, biblical Hebrew does not contain vowels, though it does employ vowel aspirations; second, it is a language based on root letter combinations, and the meaning of these combinations alters, depending on the vowel aspirations that are inserted. For example, "Kiddish," the Hebrew blessing over wine, is based on a combination of Hebrew letters that are transliterated into English as KDSH. The blessing changes if we insert different vowel aspirations: "Kaddish" is the blessing said for those who have died. Third, biblical Hebrew lacks punctuation. Owing to the absence of vowels and punctuation, interpretation difficulties immediately emerge. The rabbinic model of midrash addresses these problems by offering multiple explanations and interpretations.Levinas, however, is not concerned with this aspect of the written language. Rather, he emphasizes... (shrink)

Liz JacksonEducation University of Hong KongNel Noddings is known around the world for her contributions to philosophy and philosophy of education. Her work on caring and relational ethics broke ne...

This article analyzes the question of how the size of bribes should impact criminal sanctions. In contrast to the commonly held view that punishment should increase with the size of the bribe, we argue to the contrary: that the punishment of the bribee should decrease with the size of the bribe. Our conclusion is based both on a philosophical argument and an economic argument. We argue that all else being equal, as an agent’s reservation price for selling public interests decreases, (...) the culpability of the agent willing to receive a bribe increases. In addition, from an economic perspective, the expected social harm of an official acting with a low reservation price for bribes is much greater than one acting with a high reservation price: both the susceptibility of being bribed as well as the potential for social harm is much greater when the reservation price is low. (shrink)

A Conversation with Claire Katz about her book, Levinas and the Crisis of Humanism.

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.

This Article takes an experimental approach to test whether the salience of the law as a system that governs an interaction affects people’s preferences. I find that when the law is made salient in an interaction people’s preferences are altered: they express more future-oriented preferences and donate less money to charity, as compared to when the law is not salient in an otherwise identical interaction. When the law is salient in an interaction people also prefer ‘products’ over experiences, but this (...) gap is only marginally significant. The findings suggest that the framing of an interaction as legal tends to evoke cultural scripts and implicit rules of behavior (“common knowledge”) that incorporate the shared assumptions in society about the law. In response, participants interpret the interaction as more rational and instrumental and express preferences accordingly. (shrink)