_Humanizing Education_ offers historic examples of humanizing educational spaces, practices, and movements that embody a spirit of hope and change. From Dayton, Ohio, to Barcelona, Spain, this collection of essays from the _Harvard Educational Review_ carries readers to places where people have first imagined—and then organized—their own educational responses to dehumanizing practices and conditions. Contributors include Montse Sánchez Aroca, William Ayers, Kathy Boudin, Fernando Cardenal, Jeffrey M. R. Duncan-Andrade, Marco Garrido, Jay Gillen, Maxine Greene, Kathe Jervis, Nancy Uhlar Murray, Valerie (...) Miller, Wendy Ormiston, Ana Y. Ramos-Zayas, Vanessa Siddle Walker, Arthur E. Thomas, and Travis Wright. (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)
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.
Baruch Spinoza began his studies learning Hebrew and the Talmud, only to be excommunicated at the age of twenty-four for supposed heresy. Throughout his life, Spinoza was simultaneously accused of being an atheist and a God-intoxicated man. Bertrand Russell said that, compared to others, Spinoza is ethically supreme, 'the noblest and most lovable of the great philosophers'. This book is an exploration of (a) what Spinoza understood God to be, (b) how, for him, the infinite and eternal power of God (...) is expressed, and (c) how finite human beings can have a true idea of this greatest of all entities. Sherry Deveaux begins with an analytic discussion of these three questions, and an explication of three different views held by contemporary commentators on Spinoza. She then shows that the commonly held views about Spinoza are inconsistent with Spinoza's texts, especially his magnum opus, the Ethics. Next comes an analysis of topics in Spinoza that must be understood in order correctly to answer the three questions. For example, the notions of 'power' and 'true idea' are discussed, along with Spinoza's definition of the 'essence' of a thing, which is shown to be central to the discussion of Spinoza's God. Deveaux then claims that Spinoza defines God's essence as 'absolutely infinite and eternal power' and that, contrary to the commonly held view that God's essence is identical with the attributes (e.g., thought and extension), God's essence or "power" is expressed through the attributes. (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)
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)
Anthropologists have long wrestled with their impact upon the people they study. Historically, the discipline has served and subverted colonial agendas, but views itself traditionally as an advocate for the disempowered and as an instrument of public policy. Marketing is now among the pre-eminent institutions of cultural stability and change at work on the planet. Currently, ethnography is assuming a growing importance in the marketer’s effort to influence the accommodation and resistance of consumers to the neocolonial forces of globalization. The (...) ethical consequences of market-oriented ethnography are explored in this essay. (shrink)
Anthropologists have long wrestled with their impact upon the people they study. Historically, the discipline has served and subverted colonial agendas, but views itself traditionally as an advocate for the disempowered and as an instrument of public policy. Marketing is now among the pre-eminent institutions of cultural stability and change at work on the planet. Currently, ethnography is assuming a growing importance in the marketer's effort to influence the accommodation and resistance of consumers to the neocolonial forces of globalization. The (...) ethical consequences of market-oriented ethnography are explored in this essay. (shrink)
Professor Grünbaum's much-discussed refutation of Zeno's metrical paradox turns out to be ad hoc upon close examination of the relevant portion of measure theory. Although the modern theory of measure is able to defuse Zeno's reasoning, it is not capable of refuting Zeno in the sense of showing his error. I explain why the paradox is not refutable and argue that it is consequently more than a mere sophism.
I discuss John Henry Newman's correspondence with William Froude, F.R.S., (1810–79) and his family. Froude remained an unbeliever, and I argue that Newman's disputes with him about the ethics of belief and the relationship between religion and science not only reveal important aspects of his thought, but also anticipate modern discussions on foundationalism, the ethics of beliefs and scientism.
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)
Jaswal & Akhtar provide several quotes ostensibly from people with autism but obtained via the discredited techniques of Facilitated Communication and the Rapid Prompting Method, and they do not acknowledge the use of these techniques. As a result, their argument is substantially less convincing than they assert, and the article lacks transparency.
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)
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 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)
The case and commentaries below were developed as part of a project, Graduate Research Ethics Education, undertaken by the Association for Practical and Professional Ethics with funding from the National Science Foundation (NSF Grant No. SBR 9421897 and NSF Grant No. 9817880). The project aims at training graduate students in research ethics and building a community of scientists and engineers who are interested in and capable of teaching research ethics. As part of the project, each graduate student participant develops a (...) case for use in teaching and writes a commentary to go with the case, and then a staff member is asked to write additional commentary on the case. The case below was written in the second year of the project and was published in Research Ethics: Cases and Commentaries edited by B. Schrag, Association for Practical and Professional Ethics, Bloomington, Indiana, Vol. II (1998). Publication of these cases and commentaries will be a recurring feature of Science and Engineering Ethics. (shrink)
In “The Jesuits and the Method of Indivisibles” David Sherry criticizes a central thesis of my book Infinitesimal: that in the seventeenth century the Jesuits sought to suppress the method of indivisibles because it undermined their efforts to establish a perfect rational and hierarchical order in the world, modeled on Euclidean Geometry. Sherry accepts that the Jesuits did indeed suppress the method, but offers two objections. First, that the book does not distinguish between indivisibles and infinitesimals, and that (...) whereas the Jesuits might have reason to object to the first, the second posed no problem for them. Second, seeking an alternative explanation for the Jesuits’ hostility to the method, he proposes that its implicit atomism conflicted with the Catholic doctrine of the sacrament of the Eucharist, and was therefore heretical. In response to Sherry’s first criticism I point out that the Jesuits objected to all forms of the method of indivisibles, and that the distinction between indivisibles and infinitesimals consequently cannot explain the fight over the method in the seventeenth century. With regards to the Eucharist, I agree with Sherry that the Jesuits were indeed concerned about the method’s affinity to atomism and materialism, though for a different reason: these doctrines were antithetical to their efforts to impose divine hierarchy and order on the world. In as much as the technical details of the miracle of the Eucharist mattered, they provided no grounds for objecting to a mathematical doctrine. (shrink)
In this book, law professors Sherry F. Colb and Michael C. Dorf argue that: -/- many non-human animals, at least vertebrates, are morally considerable and prima facie wrong to harm because they are sentient, i.e., conscious and capable of experiencing pains and pleasures; most aborted human fetuses are not sentient -- their brains and nervous systems are not yet developed enough for sentience -- and so the motivating moral concern for animals doesn't apply to most abortions; later abortions affecting (...) sentient fetuses, while rare, raise serious moral concerns, but these abortions -- like all abortions -- invariably involve the interests and rights of the pregnant woman, which can make these abortions morally permissible. For a book claiming to explore the "connections" between debates about the two issues, just the summary from the book flap -- basically, what's above -- makes it appear that there really isn't much connection between the topics, at least at the core ethical level. Animals are sentient, early fetuses are not, and so the moral arguments about the two issues don't overlap or share premises. While the authors hope to use insights from one issue to shed light on the other, I find that differences in the issues limit these insights. (shrink)
Kidder's checklistfor ethical decrsion making is recommended as an addition to the existing canon of modelsfor mass media ethics. Contributions in Kidder's approach include his dichotomy between ethical dilemmas m d moral temptations, his tests for right-versus-wrong and right-versus-right issues, his framework by which to clarify values in ethical dilemmas, nnd his sequencing of the decision-making process. Kidder's model is surnmnrized nnd discussed, revisions are suggested for classroom use in medin ethics courses, nnd tke revised model is applied to media (...) ethics cases. (shrink)
In my dissertation I approach the subject of the attributes of God in Spinoza's metaphysics by considering three pivotal and closely linked problems. I discuss the problem of the relation of God to the attributes, the problem of the essence of God, and the problem of the true conception of God. ;I examine three interpretations of God and the attributes in Spinoza: that of Jonathan Bennett, according to which God is the thing that has the attributes and modes as properties, (...) that of Edwin Curley and Alan Donagan, according to which God is the collection of attributes, and that of H. F. Hallett and Steven Parchment, according to which God is the totality of attributes. I examine each stance relative to the three problems. I reveal the weaknesses of each interpretation by making explicit the problems that each interpretation is unable to deal with successfully and by exposing new problems that some of the interpretations create. I then discuss topics in Spinoza that need to be understood in order to arrive at the correct answers to the three problems. For example, I carefully examine 2D2---a definition that commentators on Spinoza generally consider imperative in the discussion of the essence of God. ;I then suggest an interpretation of God and the attributes that is motivated by Spinoza's claim in 1P34 that God's power is God's essence. I claim that an adequate idea of the essence of God is a de re conception of God; and I invoke Michael Della Rocca's account of the opacity of attribute-contexts in order to support my view of de re conceptions of God. I show that, for Spinoza, the essence of a thing is given by the definition of the thing; and I show that 1D6 correlates with his claim in 1P34 that God's power is God's essence. I argue that God's essence is absolutely infinite and eternal power, and that power is what is constituted and expressed by each of the infinite attributes. (shrink)
The Internet has drastically changed how people interact, communicate, conduct business, seek jobs, find partners, and shop. Millions of people are using social networking sites to connect with others, and employers are using these sites as a source of background information on jobapplicants.Employers report making decisions not to hire people based on the information posted on social networking sites. Few employers have policies in place to govern when and how these online character checks should be used and how to ensure (...) that the information viewed is accurate. In this article, we explore how these inexpensive, informal online character checks are harmful to society. Guidance is provided to employers on when and how to use these sites in a socially responsible manner. (shrink)
A set of six publications have introduced, commented, criticized and defended Amir Alexander’s book on infinitesimals published in 2014. The aim of the following article is to bring the various arguments together.
It is widely known that one of the projects that was most preoccupying the late, and much regretted, Alexander Kazhdan in his latter years was a study of Byzantine literature that would bring this field into the modern era. For far too long Byzantine literature, he asserted, had been encased in the strait-jacket imposed by Krumbacher's magisterial Geschichte. Byzantinists were constrained by a Handbuch mentality whose bonds had been confirmed by the three volumes that replaced Krumbacher's single tome: Beck on (...) ‘Theologische Literatur’ and ‘Volksliteratur’ and Hunger on high-style secular literature. However humane the tone might appear at times in these reference works , nothing could disguise the fact that these books were taxonomic catalogues. Byzantium's literary output was categorised by genre, genres largely imposed by modern historians and not necessarily recognisable to any Byzantine: authors were disposed under a series of at times arbitrary divisions and it was virtually impossible to have a sense of any individual's total output; Psellos was perhaps the most notoriously splintered example. Furthermore, the content of these catalogues was resolutely pragmatic: names, dates, works, editions, contents. Qualitative assessments were eschewed, there was little interpretation of literary movements. (shrink)
Since our visual perception of physical things essentially involves our identifying objects by their colours, any theory of visual perception must contain some account of the colours of things. The central problem with colour has to do with relating our normal, everyday colour perceptions to what science, i.e. physics, teaches us about physical objects and their qualities. Although we perceive colours as categorical surface properties of things, colour perceptions are explained by introducing physical properties like reflectance profiles or dispositions to (...) cause certain experiences in normal human perceivers. Hence, it seems as if colours as they are experienced by us have no place in the physical world, because they are fundamentally different from the properties which we ascribe to physical objects in scientific accounts of colour perceptions. This special issue on perspectives on colour perception presents new suggestions to solve to this major problem. (shrink)
Roslynn D. Haynes, From Faust to Strangelove: Representations of the Scientist in Western Literature. Baltimore: Johns Hopkins University Press, 1994. Pp. ix+417. ISBN 0-8018-4801-6, £16.50.George Levine , Realism and Representation: Essays on the Problem of Realism in Relation to Science, Literature and Culture. Madison: University of Wisconsin Press, 1993. Pp. xiii+330. ISBN 0-229-13630-2, £40.00 ; 0-229-13634-5, £19.00 .Sherry Turkle, Life on the Screen: Identity in the Age of the Internet. Cambridge, MA: Simon and Schuster, 1995. Pp. 347. ISBN 0-297-81514-8. (...) No price given.Despite the alarming drop in numbers of students studying science throughout the Western world today there is no more important subject in our time than science broadly construed, and these three books provide some of the reasons. Their diversity indicates the shape of the debates occurring about the scientist in Western culture, science's tortured philosophical realism and representation as troubled categories, and, most predictably, life on the screen in the age of the Internet. (shrink)
A contemporary of Giordano Bruno and Galileo, Tommaso Campanella was a controversial philosopher, theologian, astrologer, and poet who was persecuted during the Inquisition and spent much of his adult life imprisoned because of his heterodox views. He is best known today for two works: _The City of the Sun_, a dialogue inspired by Plato’s _Republic_, in which he prophesies a vision of a unified, peaceful world governed by a theocratic monarchy; and his well-meaning _Defense of Galileo_, which may have done (...) Galileo more harm than good because of Campanella’s previous conviction for heresy. But Campanella’s philosophical poems are where his most forceful and undiluted ideas reside. His poetry is where his faith in observable and experimental sciences, his astrological and occult wisdom, his ideas about deism, his anti-Aristotelianism, and his calls for religious and secular reform most put him at odds with both civil and church authorities. For this volume, Sherry Roush has selected Campanella’s best and most idiosyncratic poems, which are masterpieces of sixteenth-century Italian lyrics, displaying a questing mind of great, if unorthodox, brilliance, and showing Campanella’s passionate belief in the intrinsic harmony between the sacred and secular. (shrink)
We make the case that the Prisoner’s Dilemma, notwithstanding its fame and the quantity of intellectual resources devoted to it, has largely failed to explain any phenomena of social scientific or biological interest. In the heart of the paper we examine in detail a famous purported example of Prisoner’s Dilemma empirical success, namely Axelrod’s analysis of WWI trench warfare, and argue that this success is greatly overstated. Further, we explain why this negative verdict is likely true generally and not just (...) in our case study. We also address some possible defenses of the Prisoner’s Dilemma. (shrink)