I. Prelude -- About this Book -- II. School Mathematics and Its Consequences -- The Foundations of Mathematical Thinking -- Compression, Connection and Blending of Mathematical Ideas -- Set-befores, Met-befores and Long-term Learning -- Mathematics and the Emotions -- The Three Worlds of Mathematics -- Journeys through Embodiment and Symbolism -- Problem-Solving and Proof -- III. Interlude -- The Historical Evolution of Mathematics -- IV. University Mathematics and Beyond -- The Transition to Formal Knowledge -- Blending Knowledge Structures in the (...) Calculus -- Expert Thinking and Structure Theorems -- Contemplating the Infinitely Large and the Infinitely Small -- Expanding the Frontiers through Mathematical Research -- Reflections -- Appendix: Where the Ideas Came From. (shrink)
The Foundations of Mathematics (Stewart and Tall) is a horse of a different color. The writing is excellent and there is actually some useful mathematics. I definitely like this book."--The Bulletin of Mathematics Books.
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.
Take a hypothetical sequence of human beings ordered by height from tallest to shortest. Make sure there is no more than a difference of a millimeter between each person and make sure the tallest person is clearly tall and the shortest person is clearly not tall. Now consider the following argument: P1 A person of height n is tall ; P2 For any height n, if n is tall, then n–1mm is tall ; C Therefore, (...) a person of height n = 1mm is tall. P1 and P2 are intuitively true, C is intuitively false, yet the argument is deductively valid (the conclusion follows... (shrink)
In this survey paper we collect several known results on destroying tall ideals on countable sets and maximal almost disjoint families with forcing. In most cases we provide streamlined proofs of the presented results. The paper contains results of many authors as well as a preview of results of a forthcoming paper of Brendle, Guzmán, Hrušák, and Raghavan.
Hong Oak Yun is a person who is over three inches tall. And now you know who Hong Oak Yun is. For if someone were to ask you ‘Who is Hong Oak Yun?’, you could answer that Hong Oak Yun is a person who is over three inches tall, and you would know what you were saying. So you know an answer to the question ‘Who is Hong Oak Yun?’, and that is sufficient for knowing who Hong Oak (...) Yun is. Getting to know who a person is may be easier than you think. (shrink)
Many competent speakers initially judge that (i) is true and (ii) isfalse, though they know that (iii) is true. (i) Superman leaps more tallbuildings than Clark Kent. (ii) Superman leaps more tall buildings thanSuperman. (iii) Superman is identical with Clark Kent. Semanticexplanations of these intuitions say that (i) and (ii) really can differin truth-value. Pragmatic explanations deny this, and say that theintuitions are due to misleading implicatures. This paper argues thatboth explanations are incorrect. (i) and (ii) cannot differ intruth-value, (...) yet the intuitions are not due to implicatures, but ratherto mistakes in evaluating (i) and (ii). (shrink)
Phaedo. As I recall. when these points had been granted him, and it was agreed that each of the forms was something, and that the other things, partaking in them. took the name of the forms themselves, he next asked: ‘If you say that that is so, then whenever you say that Simmias is taller than Socrates but shorter than Phaedo, you mean the, don't you, that both things are in Simmias, tallness and shortness?’.
Transhumanists are ambitious. We want unlimited lifespan, unlimited intelligence, unlimited computer power. But this doesn't mean that we're ambitious about everything, for example height. Perhaps we want to be a bit taller, and we want to ensure that e.g. midgets have the opportunity to reach "normal" stature. Yet even in Second Life, or in tomorrow's immersive virtual realities, we don't for the most part want to be 1000 metres tall - despite freedom from the constraints of gravity. Of course, (...) there are some very exotic creatures in Second Life: they might say the rest of us have stunted imaginations. But intuitively, there is quite a narrow optimum for body height. Moreover, height may be regarded as what economists call a "positional good". It's socially advantageous to be slightly taller than average; but if everyone were to become taller, then no one would be better off. (shrink)
ABSTRACT The apparent consistency of Sobel sequences famously motivated David Lewis to defend a variably strict conditional semantics for counterfactuals. If Sophie had gone to the parade, she would have seen Pedro. If Sophie had gone to the parade and had been stuck behind someone tall, she would not have seen Pedro. But if the order of the counterfactuals in a Sobel sequence is reversed—in the example, if is asserted prior to —the second counterfactual asserted no longer rings (...) true. This is the Heim sequence problem. That the order of assertion makes this difference is surprising on the variably strict account. Some argue that this is reason to reject the Lewis-Stalnaker semantics outright. Others argue that the problem motivates a contextualist rendering of counterfactuals. Still others maintain that the explanation for the phenomenon is merely pragmatic. I argue that none of these is right, and I defend a novel way to understand the phenomenon. My proposal avoids the problems faced by the alternative analyses and enjoys independent support. There is, however, a difficulty for my view: it entails that many ordinarily accepted counterfactuals are not true. I argue that this cost is acceptable. (shrink)
In the tradition of Stalnaker there is a number of well-known problems that need to be addressed, because revision of iterated belief modalities is required in this case. These problems have already been investigated in detail in recent works on DDL Leitgeb/Segerberg 2007)and DEL see e.g. Ditmarsch et. Another strategy would be to maintain and revise assumptions independently of the beliefs of an agent.I will briefly discuss the advantages and disadvantages of each of these views. In both views, assumptions constitute (...) the subjective context in which an agent interprets an utterance and encounters the world. The result of an interpretation is in turn checked against the agent’s original beliefs, and if the checking operation succeeds the agent revises his beliefs by the result in the normal way described by the AGM paradigm. The second of the above questions needs to be addressed on the basis of concrete examples. Considering utterance like David is ready’ or ‘John is tall’that from a contextualist viewpoint express semantically incomplete content in the sense of Bach is needed in order to obtain a useful model of the checking step, since fortunately not everybody believes everything that other people say. These requirements put the theory of interpretation based on assumptions in the frontline of ongoing research on the implementation of belief revision and update in dynamic logics. Such a theory might also be useful for contextualist accounts of strong knowledge, as it can be argued convincingly that when a knowledge ascription appears to be context-sensitive, this is so because the embedded proposition is context-sensitive and not because knowledge itself is context-sensitive. Hence,the context-sensitivity of embedded propositions in knowledge claims and how different agents in the same situation arrive at different assessments about them may be explained by an inferential theory of interpretation similar to the one outlined here but with another underlying concept of assumptions. Literature Alchourrón, C. E. ; Gärdenfors, P. & Makinson, D., 'On the logic of theory change: partial meet contraction and revision functions', Journal of Symbolic Logic, 510-530. Bach, K., 'Minimalism for Dummies: Reply to Cappelen and Lepore', Technical report, University of San Fransisco, Department of Philosophy. Bach, K., Context ex Machina, in án Gendler Szabó, ed.,'Semantics versus Pragmatics', Oxford UP, Oxford, pp. 16-44. Ditmarsch, H. v. ; Hoek, W. v. d. & Kooi, B., Dynamic Epistemic Logic, Kluwer. Gärdenfors, P., Knowledge in Flux, MIT Press. Leitgeb, H. & Segerberg, K., 'Dynamic doxastic logic: why, how, and where to? ', Synthese155, 167-190. Stalnaker, R., Assertion, in. Cole, ed.,'Pragmatics', Academic Press, New York, pp. 315-332. Stalnaker, R., 'Common Ground', Linguistics and Philosophy25, 701- -721. (shrink)
Just about every theorist holds that vague terms are context-sensitive to some extent. What counts as ?tall?, ?rich?, and ?bald? depends on the ambient comparison class, paradigm cases, and/or the like. To take a stock example, a given person might be tall with respect to European entrepreneurs and downright short with respect to professional basketball players. It is also generally agreed that vagueness remains even after comparison class, paradigm cases, etc. are fixed, and so this context sensitivity does (...) not solve the problems with vague terms. In this paper, I briefly sketch the main features of my broadly contextualist account of vagueness, that of the late Ruth Manor, and that of Agustín Rayo, showing how the three accounts relate to, and reinforce, each other, noting a seeming difference. A key item used to articulate my own view is David Lewis's notion of conversational score. This is used to track which borderline cases have been called in the course of a conversation. Manor alludes to Lewis's notion of accommodation, a key aspect of the kinematics of conversation, and Rayo invokes Stalnaker's notion of common ground. I'll show how the conversational score could be employed to develop and extend their views, showing how vague terms, as they construe them, are (or can be) deployed in conversation, consistent with both the underlying indeterminacy of the terms and normal communicative goals. To help develop all three views further, I invoke Craige Roberts's notion of Retrievability, a tool developed to show how definites, such as definite descriptions, singular pronouns, and proper names, are deployed in conversation, in a Lewis-style scorekeeping framework. This, I think, is exactly the right way to understand conversations in what Manor calls non-sorities situations and, to invoke my own view, in sorites situations as well. (shrink)
In 'How Many Lives Has Schrödinger's Cat?' David Lewis argues that the Everettian no-collapse interpretation of quantum mechanics is in a tangle when it comes to probabilities. This paper aims to show that the difficulties that Lewis raises are insubstantial. The Everettian metaphysics contains a coherent account of probability. Indeed it accounts for probability rather better than orthodox metaphysics does.
Simon Blackburn can be seen as challenging those committed to sui generis moral facts to explain the supervenience of the moral on the descriptive. We hold that normative facts in general are sui generis. We also hold that the normative supervenes on the descriptive, and we here endeavour to answer the generalization of Blackburn's challenge. In the course of pursuing this answer, we suggest that Frank Jackson's descriptivism rests on a conception of properties inappropriate to discussions of normativity, and we (...) see reason to reject descriptivism generally. We also discuss the views of David Brink, Jonathan Dancy and Bernard Williams in this area. (shrink)
A cardinal κ is tall if for every ordinal θ there is an embedding j: V → M with critical point κ such that j > θ and Mκ ⊆ M. Every strong cardinal is tall and every strongly compact cardinal is tall, but measurable cardinals are not necessarily tall. It is relatively consistent, however, that the least measurable cardinal is tall. Nevertheless, the existence of a tall cardinal is equiconsistent with the existence of (...) a strong cardinal. Any tall cardinal κ can be made indestructible by a variety of forcing notions, including forcing that pumps up the value of 2κ as high as desired. (shrink)
It is widely assumed that the normativity of conceptual judgement poses problems for naturalism. Thus John McDowell urges that 'The structure of the space of reasons stubbornly resists being appropriated within a naturalism that conceives nature as the realm of law' (1994, p 73). Similar sentiments have been expressed by many other writers, for example Robert Brandom (1994, p xiii) and Paul Boghossian (1989, p 548).
David and Mary Norton present the definitive scholarly edition of Hume's Treatise, one of the greatest philosophical works ever written. This set comprises the two volumes of texts and editorial material, which are also available for purchase separately. -/- David Hume (1711 - 1776) is one of the greatest of philosophers. Today he probably ranks highest of all British philosophers in terms of influence and philosophical standing. His philosophical work ranges across morals, the mind, metaphysics, epistemology, religion, and (...) aesthetics; he had broad interests not only in philosophy as it is now conceived but in history, politics, economics, religion, and the arts. He was a master of English prose. -/- The Clarendon Hume Edition will include all of his works except his History of England and minor historical writings. It is the only thorough critical edition, and will provide a far more extensive scholarly treatment than any previous editions. This edition (which has been in preparation since the 1970s) offers authoritative annotation, bibliographical information, and indexes, and draws upon the major advances in textual scholarship that have been made since the publication of earlier editions - advances both in the understanding of editorial principle and practice and in knowledge of the history of Hume's own texts. (shrink)
Second part of the translation into Spanish of David Lewis' "New Work for a Theory of Universals", corresponding to the last sections of the original paper. || Segunda parte de la traducción al español del trabajo de David Lewis "New Work for a Theory of Universals", correspondiente a últimas secciones del artículo original. Artículo original publicado en: Australasian Journal of Philosophy, Vol. 61, No. 4, Dec. 1983, pp. 343-377.
David and Mary Norton present the definitive scholarly edition of Hume's Treatise, one of the greatest philosophical works ever written. This volume contains their account of how the Treatise was written and published; an explanation of how they established the text; an extensive set of annotations; and a detailed bibliography and index.
David and Mary Norton present the definitive scholarly edition of Hume's Treatise, one of the greatest philosophical works ever written. The first volume contains the critical text of David Hume's Treatise of Human Nature, followed by the shortand concluding with A Letter from a Gentleman to his Friend in Edinburgh.
David and Mary Norton present the definitive scholarly edition of Hume's Treatise, one of the greatest philosophical works ever written. This set comprises the two volumes of texts and editorial material, which are also available for purchase separately.
First part of the translation into Spanish of David Lewis' "New Work for a Theory of Universals", corresponding to the introduction and the first two sections of the original paper. || Primera parte de la traducción al español del trabajo de David Lewis "New Work for a Theory of Universals", correspondiente a la introducción y las dos primeras secciones del artículo original. Artículo original publicado en: Australasian Journal of Philosophy, Vol. 61, No. 4, Dec. 1983, pp. 343-377.
David Hume's writings on history, politics and philosophy have shaped thought to this day. His bold scepticism ranged from common notions of the 'self' to criticism of standard theistic proofs. He insisted on grounding understandings of popular religious beliefs in human psychology rather than divine revelation, and he aimed to disentangle philosophy from religion in order to allow the former to pursue its own ends. In this book, Professors David W Purdie and Peter S Fosl decipher some of (...) Hume's most challenging texts for the modern reader, while preserving the sharp intellect and undaunted nerve for which Hume is famous. Hume's spirit is brought alive for contemporary times and his writing is made accessible for its intended audience: the general public."-- Back cover. (shrink)
David and Mary Norton present the definitive scholarly edition of Hume's Treatise, one of the greatest philosophical works ever written. The first volume contains the critical text of David Hume's Treatise of Human Nature , followed by the shortin which Hume set out the key arguments of the larger work; the volume concludes with A Letter from a Gentleman to his Friend in Edinburgh , Hume's later defence of the Treatise.
We construct two models containing exactly one supercompact cardinal in which all non-supercompact measurable cardinals are strictly taller than they are either strongly compact or supercompact. In the first of these models, level by level equivalence between strong compactness and supercompactness holds. In the other, level by level inequivalence between strong compactness and supercompactness holds. Each universe has only one strongly compact cardinal and contains relatively few large cardinals.