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)
Fodor and katz criticize cavell's position on the relation between ordinary language philosophy and empirical investigations of ordinary language, In "must we mean what we say?," _inquiry, Volume 1, Pages 172-212, And "the availability of wittgenstein's later philosophy," "philosophical review", Volume 71, Pages 67-93. Cavell holds that disagreements between ordinary language philosophers over grammar and semantics are in no sense empirical. Fodor and katz show that ordinary language philosophers are engaged in empirical investigation. (staff).
Sense, Reference, and Philosophy develops the far-reaching consequences for philosophy of adopting non-Fregean intensionalism, showing that long-standing problems in the philosophy of language, and indeed other areas, that appeared intractable can now be solved. Katz proceeds to examine some of those problems in this new light, including the problem of names, natural kind terms, the Liar Paradox, the distinction between logical and extra-logical vocabulary, and the Raven paradox. In each case, a non-Fregean intentionalism provides a philosophically more satisfying solution.
In this eye-opening look at the doctor-patient decision-making process, physician and law professor Jay Katz examines the time-honored belief in the virtue of silent care and patient compliance. Historically, the doctor-patient relationship has been based on a one-way trust -- despite recent judicial attempts to give patients a greater voice through the doctrine of informed consent. Katz criticizes doctors for encouraging patients to relinquish their autonomy, and demonstrates the detrimental effect their silence has on good patient care. Seeing (...) a growing need in this age of medical science and sophisticated technology for more honest and complete communication between physician and patients, he advocates a new, informed dialogue that respects the rights and needs of both sides. In a new foreword to this edition of The Silent World of Doctor and Patient , Alexander Morgan Capron outlines the changes in medical ethics practice that have occurred since the book was first published in 1984, paying particular attention to the hotly debated issues of physician-assisted suicide and informed consent in managed care. (shrink)
The cogito ergo sum of Descartes is one of the best-known--and simplest--of all philosophical formulations, but ever since it was first propounded it has defied any formal accounting of its validity. How is it that so simple and important an argument has caused such difficulty and such philosophical controversy? In this pioneering work, Jerrold Katz argues that the problem with the cogito lies where it is least suspected--in a deficiency in the theory of language and logic that Cartesian scholars (...) have brought to the study of the cogito. Katz contends that the laws of traditional logic have distorted Descartes's reasoning so that it no longer fits either Descartes's own account of the cogito in his writings or the role he assigns it in his project. Katz proposes that the cogito can be understood as an example of "analytic entailment," a concept in the philosophy of language whereby a statement can be a formally valid inference without depending on a law of logic. Developing and defending his thesis, he shows us that by grappling with an historical philosophical problem it is possible to make an original contribution to the advance of contemporary philosopy. (shrink)
Ethical self-management; an introduction to systematic personality psychology, by M. C. Katz.--Four axiological proofs of the infinite value of man, by R. S. Hartman.--Some thoughts regarding the current philosophy of the behavioral sciences, by C. R. Rogers.--Autonomy and community, by D. Lee.--Synergy in the society and in the individual, by A. H. Maslow.--Human nature: its cause and effect; a theoretical framework for understanding human motivation, by M. C. Katz.--Mental health; a generic attitude, by G. W. Allport.--Love feelings in (...) courtship couples; an analysis, by R. P. Hattis.--Economic policies and human well-being, by W. A. Weisskopf.--The great transformation, by H. F. W. Perk.--Contingencies of reinforcement in the design of a culture, by B. F. Skinner.--For further reading (p. 238-240). (shrink)
Environmental pragmatism is a new strategy in environmental thought: it argues that theoretical debates are hindering the ability of the environmental movement to forge agreement on basic policy imperatives. This new direction in environmental philosophy moves beyond theory, advocating a serious inquiry into the practical merits of moral pluralism. Environmental pragmatism, as a coherent philosophical position, connects the methodology of classical American pragmatist thought to the explanation, solution and discussion of real issues.
In this paper, I argue for three claims. The first is that the difference between analog and digital representation lies in the format and not the medium of representation. The second is that whether a given system is analog or digital will sometimes depend on facts about the user of that system. The third is that the first two claims are implicit in Haugeland's (1998) account of the distinction.
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)
Engineers, architects, and other technological professionals designed the genocidal death machines of the Third Reich. The death camp operations were highly efficient, so these technological professionals knew what they were doing: they were, so to speak, good engineers. As an educator at a technological university, I need to explain to my students—future engineers and architects—the motivations and ethical reasoning of the technological professionals of the Third Reich. I need to educate my students in the ethical practices of this hellish regime (...) so that they can avoid the kind of ethical justifications used by the Nazi engineers. In their own professional lives, my former students should not only be good engineers in a technical sense, but good engineers in a moral sense. In this essay, I examine several arguments about the ethical judgments of professionals in Nazi Germany, and attempt a synthesis that can provide a lesson for contemporary engineers and other technological professionals. How does an engineer avoid the error of the Nazi engineers in their embrace of an evil ideology underlying their technological creations? How does an engineer know that the values he embodies through his technological products are good values that will lead to a better world? This last question, I believe, is the fundamental issue for the understanding of engineering ethics. (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)
Contemporary philosophy standardly accepts Frege's conceptions of sense as the determiner of reference and of analyticity as (necessary) truth in virtue of meaning. This paper argues that those conceptions are mistaken. It develops referentially autonomous notions of sense and analyticity and applies them to the semantics of natural kind terms. The arguments of Donnellan, Putnam, and Kripke concerning natural kind terms are widely taken to refute internalist and rationalist theories of meaning. This paper shows that the counter-intuitive consequences about the (...) reference of natural kind terms depend as much on Frege's conceptions of sense and analyticity as on what such theories of meaning say about the senses of natural kind terms. Rather than refuting the internalist and rationalist theories of meaning, the arguments of Donnellan, Putnam, and Kripke are best recast as refutations of their own Fregean assumptions. The paper also shows how autonomous notions of sense and analyticity enable us to reconstruct such theories, formulate an internalist/ rationalist account of semantic knowledge, and preserve Donnellan's, Putnam's, and Kripke's insights about reference. (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)
This paper considers the question of whether there are truths independent of God's power. It defends a traditional conception of divine power, according to which God's power does not extend to logically necessary truths, such as those of logic and mathematics, against Cartesian voluntarism, here taken as the doctrine that every truth falls within the compass of God's creative will. The paper argues that the voluntarist position is internally inconsistent. It concludes that if God is an absolute, unconditioned reality, then (...) there must be truths that are independent of God's power. (shrink)
Pleasure, in the inclusive usages most important in moral psychology, ethical theory, and the studies of mind, includes all joy and gladness — all our feeling good, or happy. It is often contrasted with similarly inclusive pain, or suffering, which is similarly thought of as including all our feeling bad. Contemporary psychology similarly distinguishes between positive affect and negative affect.[1..
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)
This paper deals with the two-envelope paradox. Two main formulations of the paradoxical reasoning are distinguished, which differ according to the partition of possibilities employed. We argue that in the first formulation the conditionals required for the utility assignment are problematic; the error is identified as a fallacy of conditional reasoning. We go on to consider the second formulation, where the epistemic status of certain singular propositions becomes relevant; our diagnosis is that the states considered do not exhaust the possibilities. (...) Thus, on our approach to the paradox, the fallacy, in each formulation, is found in the reasoning underlying the relevant utility matrix; in both cases, the paradoxical argument goes astray before one gets to questions of probability or calculations of expected utility. (shrink)
In "Literal Meaning," John Searle claims to refute the view that sentences of a natural language have a meaning independent of the social contexts in which their utterances occur. The present paper is a reply on behalf of this view. In the first section, I show that the issue is not a parochial dispute within a narrow area of the philosophy of language, of interest only to specialists in the area, but is at the heart of a wide range of (...) important philosophical problems, those on which the recent linguistic turn in philosophy has properly taken a grammatical perspective. In the second section, I reply to Searle's criticisms of the view. (shrink)
Although Levinas talks about ethics as a response to the other, most scholars assume that this "response" is not something tangible—it is not an actual giving of food or providing of shelter and clothing. But there is evidence in Levinas's own writings that indicate he does intend for a positive response to the Other. In any event, while he acknowledges that the other is the sole person I wish to kill, killing the other, within an ethical framework would be a (...) violation of that response. The failure to respond to the other ethically requires us to ask if Levinas's project needs an educational philosophy or a model of moral cultivation to supplement it. This essay explores this question by putting into conversation Levinas's ethical project and his interest in Jewish education with John Dewey's philosophy of education and its relationship to the political community. This exploration will help us see what this field of research might offer in promoting the cultivation of ethical response as Levinas envisions it and what its limits are. (shrink)
Rolls's preliminary definitions of emotion and speculative restriction of consciousness, including emotional sentience, to humans, display behaviorist prejudice. Reinforcement and causation are not by themselves sufficient conceptual resources to define either emotion or the directedness of thought and motivated action. For any adequate definition of emotion or delimitation of consciousness, new physiology, such as Rolls is contributing to, and also the resources of other fields, will be required.
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)