Never getting to zero: Elementary school students' understanding of the infinite divisibility of number and matter

Clinical interviews administered to third- to sixth-graders explored children's conceptualizations of rational number and of certain extensive physical quantities. We found within child consistency in reasoning about diverse aspects of rational number. Children's spontaneous acknowledgement of the existence of numbers between 0 and 1 was strongly related to their induction that numbers are infinitely divisible in the sense that they can be repeatedly divided without ever getting to zero. Their conceptualizing number as infinitely divisible was strongly related to their having a model of fraction notation based on division and to their successful judgment of the relative magnitudes of fractions and decimals. In addition, their understanding number as infinitely divisible was strongly related to their understanding physical quantities as infinitely divisible. These results support a conceptual change account of knowledge acquisition, involving two-way mappings between the domains of number and physical quantity.     

Hume on the Distinction of Reason

This paper concerns Hume’s treatment of the distinction of reason in the Treatise, I.i.7. Many scholars have claimed that there is a tension between his account of the distinction of reason and his commitment to his so-called separability principle. I explain why Hume’s account of the distinction of reason is fully consistent with the principle, and show how other discussions, both critical of and sympathetic to Hume, fail to appreciate the radicalness of his position. I evaluate Hume’s bold position and compare it to earlier positions (Aquinas) and current positions (Donald Davidson).     

Hume on presentation and philosophy

Most philosophers agree that an argument's presentation is relevant to its philosophical merit. This paper explains why David Hume considered presentation philosophically important. On Hume's epistemology, presentation is closely connected with two principal aims of philosophical arguments: persuasion and epistemic justification. Hume's views imply that presentation is a factor determining an argument's persuasiveness and that, by philosophical standards of justification, presentation is also a factor determining the extent to which an argument's conclusion is justified.     

休谟论道德的基础

休谟的道德哲学主要致力于对道德基础的研究。他通过对理性和情感在道德行为中的作用的分析,得出道德决定于情感的结论,这一结论可以看作理性在道德领域的自我限制,它对元伦理和规范伦理的发展都有深远的意义。     

Hume on the Standard of Virtue

Among those sympathetic to Hume'smoral philosophy, a general consensus hasemerged that his first work on the topic,A Treatise of Human Nature, is his best. Hislater work, An Enquiry Concerning thePrinciples of Morals, is regarded as scaleddown in both scope and ambition. In contrastto this standard view, I argue that Hume'slater work offers a more sophisticated theoryof moral evaluation. I begin by reviewing theTreatise theory of moral evaluation tohighlight the reasons why commentators find socompelling Hume's account of the corrections wemake to our moral sentiments. The method isendorsed by philosophers such as Henry DavidAiken and Annette C. Baier because, theyallege, it shows that moral sentiments reflecta process of judgment that includes thepossibility of corrigibility and ofjustification. But Hume's method of correctionfalls short and does not establish why thesentiments conforming to the standard of virtueshould count as moral judgments. In the secondEnquiry, Hume lays out a different set ofcriteria, including not only the need forcertain virtues of good judgment but attentionto the particular cultural and historicalorigins of the norms governing the virtues ofgood judgment. Hume's attention to diversityin evaluative outlook in his more matureposition takes seriously the relation betweenmoral authority and public debate.     

Hume and the nominalist tradition

Many of the central theses of Hume's philosophy – his rejection of real relations, universals, abstract objects and necessary causal relations – had precedents in the later medieval nominalist tradition. Hume and his medieval predecessors developed complex semantic theories to show both how ontologies are apt to become inflated and how, if we understand carefully the processes by which meaning is generated, we can achieve greater ontological parsimony. Tracing a trajectory from those medieval traditions to Hume reveals Hume to be more radical, particularly in his rejection of abstraction and abstract ideas. Hume's denial of general, abstract ideas is consistent with his philosophical principles but fails to appreciate the more sophisticated nominalist approaches to abstraction, the result of which is a theoretically impoverished account of our capacity for generalization.     

Hume and Kant on the social contract

Conclusion In this lecture I have attempted to survey the main points of contention between Hume and Kant with respect to their treatment of social or original contract theory. I have tried to show that their conflicts, running as they do to the heart of each thinker's moral philosophy, are very deep indeed. If nothing else, they show the conceptual dependence of social and political philosophy upon moral philosophy and were worth exploring at least for that reason. I have not, of course, said anything about the most important question of all: which, if either, of the two theories is true? Pehaps we can begin an exploration of that question together.A lecture presented at the David Hume Bicentennial Symposium, University of Arizona, September 17, 1976. An earlier version of the lecture was presented to the Department of Philosophy, Loyola University (Chicago), in 1975. The lecture was also presented at the 1977 meeting of the Hume Society, University of Virginia, October 29, 1977.