In lieu of an abstract, here is a brief excerpt of the content:The humble origins of Russell's paradox by J. Alberto Coffa ON SEVERAL OCCASIONS Russell pointed out that the discovery of his celebrated paradox concerning the class of all classes not belonging to themselves was intimately related to Cantor's proof that there is no greatest cardinal. lOne of the earliest remarks to that effect occurs in The Principles ofMathematics where, referring to the universal class, the class of all classes (...) and the class of all propositions, he notes that when we apply the reasoning of his [Cantor's] proofto the cases in question we find ourselves met by definite contradictions, of which the one discussed in Chapter x is an example. (P. 362) And in a footnote he adds: "It was in this way that I discovered this contradiction". Throughout his writings Russell left a number ofhints concerning the sort of connection he had drawn between Cantor's proof and his own discovery. In fact, his suggestions are so specific that there would seem to be little room left for speculation concerning how the discovery took place.2 The picture that emerges almost immediately from Russell's observations is the following. For reasons which are 1 See, e.g., Bertrand Russell, The Principles of Mathematics, 2nd ed. (London: Allen & Unwin, 1937), §§100, 344-9; G. Frege, Wissenschaftlicher Briefwechsel (Hamburg: Felix Meiner Verlag, 1976), pp. 215-16; B. Russell, Essays in Analysis, ed. D. Lackey (New York: George Braziller, 1973), p. 139; B. Russell, Introducrion to Mathemarical Philosophy (New York: Simon and Schuster, 1971), p. 136; B. Russell, My Philosophical Development (New York: Simon and Schuster, 1959), pp. 75-6; The Autobiography of Bertrand Russell, I (New York: Bantam Books, 1968), 195. 2 See, e.g., Ch. Thiel's "Einleitung des Herausgebers" in Frege, Wissenschaftlicher Briefwechsel, pp. 203 and 216 (footnote), and 1. Grattan-Guinness, "How Bertrand Russell Discovered his Paradox", Historia Marhematica, 5 (1978),127-37. 31 32 Russell, nos. 33-4 (Spring-Summer 1979) never made clear, Russell decided to analyze Cantor's argument applying it to "large" classes such as the universal class V, and the class of all classes. When, for example, we consider V, its power set and the correlationf(x) = {x} if x is not a class, f(x) =. x otherwise, then Cantor's diagonal class D turns out to be the class of all classes not belonging to themselves. Moreover, since the element ofV which is taken toD byfisD itself(i.e., sincef(D) = D), Cantor's reasoning invites us to raise the question whether D belongs tof(D) (i.e., toD) or not; and it establishes that it does precisely if it doesn't.3 The purpose of this note is to present recently uncovered information that complements and corrects our present understanding of Russell's discovery of his paradox. As it turns out, far from originating from his desire to apply the ideas in Cantor's theorem, a version of Russell's paradox first occurred in an argument that Russell had devised, late in 1900, in order to establish the invalidity of that theorem. An appeal to "large" classes such as V or Class was, as we shall see, essential to Russell's attempted refutation. As he displayed the details of his counterexample Russell's paradox emerged, at first unrecognized, as the by-product of a project that the paradox itself would eventually undermine. In a paper written in 1901 and largely devoted to a popular exposition and defence of Cantor's theory of the infinite, Russell expressed what appeared to be a rpinor reservation to Cantor's treatment : There is a greatest of all infinite numbers, which is the number of things altogether, of every sort and kind. It is obvious that there cannot be a greater number than this, because, if everything has been taken, there is nothing left to add. Cantor has a proof that there is no greatest number, and if this proof were valid, the contradictions of infinity would reappear in a sublimated form. But in this one point, the master has been guilty of a very subtle fallacy, which I hope to explain in some future work.4 3 See Principles, §349. Perhaps I should remind... (shrink)

Scientists and laypeople alike use the sense of understanding that an explanation conveys as a cue to good or correct explanation. Although the occurrence of this sense or feeling of understanding is neither necessary nor sufficient for good explanation, it does drive judgments of the plausibility and, ultimately, the acceptability, of an explanation. This paper presents evidence that the sense of understanding is in part the routine consequence of two well-documented biases in cognitive psychology: overconfidence and hindsight. In light of (...) the prevalence of counterfeit understanding in the history of science, I argue that many forms of cognitive achievement do not involve a sense of understanding, and that only the truth or accuracy of an explanation make the sense of understanding a valid cue to genuine understanding. (shrink)

This paper examines narrativism’s claim that the historical past cannot be known once and for all because it must be continuously re-described from the standpoint of the present. We argue that this claim is based on a non sequitur. We take narrativism’s claim that the past must be re-described continuously from the perspective of the present to be the result of the following train of thought: 1) “all knowledge is conceptually mediated”; 2) “the conceptual framework through which knowledge of reality (...) is mediated changes with every new generation of historians”; therefore (narrativism’s claim) “the historical past changes with every new generation of historians”. The idea of an unchanging past, for the narrativist, requires denying premise 1 (all knowledge is conceptually mediated”) and therefore rests on a problematic commitment to the chimerical notion of the past as it is in-itself, wie es eigentlich gewesen. We argue that the narrativist’s conclusion does not follow unless one adds a further premise, namely 3) “it is not possible to view reality through the categorial framework of historical agents”. If one asserts the possibility of grasping reality through the categorial framework of others, be they contemporary or past agents (as much philosophy of history written in an idealist key does), then one no longer has to accept the narrativist’s inference that since the past cannot be known in-itself or independently of conceptual mediation, then it cannot be known as it always was for the historical agents. Narrativism’s inference that the past cannot be known as it always was does not follow from premises 1 and 2 unless one smuggles in another problematic premise, premise 3. In this paper we defend the claim that the past can be known as it always was (not as it is in-itself) by invoking a different conception of the role of conceptual mediation in historical knowledge, one which assumes the possibility of viewing reality through the categorial framework of others. This notion of the role of conceptual mediation in historical knowledge is prevalent in the idealist tradition but, in the interest of brevity, we will defend this notion of mediacy by specific reference to the idealist philosophy of R.G. Collingwood. (shrink)

Philosophers agree that scientific explanations aim to produce understanding, and that good ones succeed in this aim. But few seriously consider what understanding is, or what the cues are when we have it. If it is a psychological state or process, describing its specific nature is the job of psychological theorizing. This article examines the role of understanding in scientific explanation. It warns that the seductive, phenomenological sense of understanding is often, but mistakenly, viewed as a cue of genuine understanding. (...) The article closes with a discussion of several new paths of research that tie the psychology of scientific explanation to cognate notions of learning, testimony, and understanding. (shrink)

At some distance from a high-rate source in an elastic half-space, the dominant wave motion at the free surface is a Rayleigh surface wave. The calculation of surface waves generated by a concentrated force in a half-space is a basic problem in elastodynamics. By straightforward manipulations, the result can be used to obtain surface waves for other kinds of wave-generating body-force arrangements. For example, appropriate combinations of double-forces (or dipoles) can be used to represent the surface loading due to laser (...) irradiation, or due to acoustic emission from the opening of a sub-surface crack or from sliding over a fault surface. The surface wave motion is usually obtained by the application of integral transform techniques and the subsequent extraction of the surface waves as the contributions from poles in the integral for the inverse transform. In this paper, we use a much simpler approach based on the elastodynamic reciprocity theorem. We consider a transversely isotropic solid whose axis of symmetry is normal to the surface of the half-space. Surface wave pulses generated by a single force, by irradiation from a laser source, and by opening of a crack, have been determined. (shrink)

There is now a considerable literature on Michel Foucault but this is the first monograph which explicitly addresses his influence and impact upon education. Personal autonomy has been seen as a major aim, if not the aim of liberal education. But if Foucault is correct that personal autonomy and the notion of the autonomous person are myths, then the pursuit of such an aim by educationalists is misguided. The author develops this critique of personal autonomy and liberal education from the (...) writings of Foucault, and also considers Foucault's own educational practices. The author, James Marshall, who lives in New Zealand, has already written several articles for academic journals on Foucault. (shrink)

A fresh, daring, and genuine alternative to the traditional story of scientific progress Explaining the world around us, and the life within it, is one of the most uniquely human drives, and the most celebrated activity of science. Good explanations are what provide accurate causal accounts of the things we wonder at, but explanation's earthly origins haven't grounded it: we have used it to account for the grandest and most wondrous mysteries in the natural world. Explanations give us a sense (...) of understanding, but an explanation that feels right doesn't mean it is true. For every true explanation, there is a false one that feels just as good. A good theory's explanations, though, have a much easier path to truth. This push for good explanations elevated science from medieval alchemy to electro-chemistry, or a pre-inertial physics to the forces underlying nanoparticles. And though the attempt to explain has existed as long as we have been able to wonder, a science timeline from pre-history to the present will reveal a steep curve of theoretical discovery that explodes around 1600, primarily in the West. Ranging over neuroscience, psychology, history, and policy, Wondrous Truths answers two fundamental questions-Why did science progress in the West? And why so quickly? J.D. Trout's answers are surprising. His central idea is that Western science rose above all others because it hit upon successive theories that were approximately true through an awkward assortment of accident and luck, geography and personal idiosyncrasy. Of course, intellectual ingenuity partially accounts for this persistent drive forward. But so too does the persistence of the objects of wonder. Wondrous Truths recovers the majesty of science, and provides a startling new look at the grand sweep of its biggest ideas. (shrink)

Scientific realism has been advanced as an interpretation of the natural sciences but never the behavioral sciences. This book introduces a novel version of scientific realism, Measured Realism, that characterizes the kind of theoretical progress in the social and psychological sciences that is uneven but indisputable. It proposes a theory of measurement, Population-Guided Estimation, that connects natural, psychological, and social scientific inquiry. Presenting quantitative methods in the behavioral sciences as at once successful and regulated by the world, the book will (...) engage philosophers of science, historians of science, sociologists of science, and scientists interested in the foundations of their own disciplines. Part I introduces measured realism beginning with a novel realist theory of measurement (Population-Guided Estimation, or PGE). It examines two other versions of scientific realism — minimal (Hacking) and robust (Boyd, Putnam, and Newton-Smith) — and advances measured realism as best supported by the evidence in the behavioral sciences. It also critically evaluates the anti-realist arguments of van Fraassen, Laudan, and Fine. Part II formulates and develops a version of epistemological naturalism most appropriate to measured realism. Drawing upon the realist theory of measurement developed earlier, it favorably appraises a variety of statistical methods in the behavioral sciences against selected criticisms. It concludes that traditional, narrative methods in the behavioral and social sciences are unreliable by the measures of epistemological naturalism. (shrink)

Bayesian probabilistic explication of inductive inference conflates neutrality of supporting evidence for some hypothesis H ("not supporting H") with disfavoring evidence ("supporting not-H"). This expressive inadequacy leads to spurious results that are artifacts of a poor choice of inductive logic. I illustrate how such artifacts have arisen in simple inductive inferences in cosmology. In the inductive disjunctive fallacy, neutral support for many possibilities is spuriously converted into strong support for their disjunction. The Bayesian "doomsday argument" is shown to rely entirely (...) on a similar artifact. Copyright 2010 by the Philosophy of Science Association. All rights reserved. (shrink)

(2002). Philosophy of Science 72 (January), 198-208.