Newton certainly regarded his second law of motion in the Principia as a fundamental axiom of mechanics. Yet the works that came after the Principia, the major treatises on the foundations of mechanics in the eighteenth century—by Varignon, Hermann, Euler, Maclaurin, d’Alembert, Euler (again), Lagrange, and Laplace—do not record, cite, discuss, or even mention the Principia’s statement of the second law. Nevertheless, the present study shows that all of these scientists do in fact assume the principle that the Principia’s second (...) law asserts as a fundamental axiom in their mechanics. (For what that second law asserts, we rely on Newton’s own testimony.) Some, like Varignon and Hermann, assume the axiom implicitly, apparently unaware that any assumption is being made, while others, like Maclaurin and Euler, assume the axiom explicitly, apparently unaware that the assertion assumed is the second law as Newton himself understood it. But in every case these scientists employ the principle asserted by the Principia’s second law fundamentally, unaware that they should be citing Neutonus, Prin., Phil. Nat. Math., Lex II. (shrink)

The first proposition of the Principia records two fundamental properties of an orbital motion: the Fixed Plane Property (that the orbit lies in a fixed plane) and the Area Property (that the radius sweeps out equal areas in equal times). Taking at the start the traditional view, that by an orbital motion Newton means a centripetal motion – this is a motion ``continually deflected from the tangent toward a fixed center'' – we describe two serious flaws in the Principia's argument (...) for Proposition 1, an argument based on a polygonal impulse approximation. First, the persuasiveness of the argument depends crucially on the validity of the Impulse Assumption: that every centripetal motion can be represented as a limit of polygonal impulse motions. Yet Newton tacitly takes the Impulse Assumption for granted. The resulting gap in the argument for Proposition 1 is serious, for only a nontrivial analysis, involving the careful estimation of accumulating local errors, verifies the Impulse Assumption. Second, Newton's polygonal approximation scheme has an inherent and ultimately fatal disability: it does not establish nor can it be adapted to establish the Fixed Plane Property. Taking then a different view of what Newton means by an orbital motion – namely that an orbital motion is by definition a limit of polygonal impulse motions – we show in this case that polygonal approximation can be used to establish both the fixed plane and area properties without too much trouble, but that Newton's own argument still has flaws. Moreover, a crucial question, haunted by error accumulation and planarity problems, now arises: How plentiful are these differently defined orbital motions? Returning to the traditional view, that Newton's orbital motions are by definition centripetal motions, we go on to give three proofs of the Area Property which Newton ``could have given'' – two using polygonal approximation and a third using curvature – as well as a proof of the Fixed Plane Property which he ``almost could have given.''. (shrink)

In this paper it is argued, firstly, that Kuhnian revolutions in mathematics are logically possible, in the sense of not being inconsistent with the nature of mathematics; and, secondly, that Kuhnian revolutions are actually possible, in the sense that a Kuhnian paradigm for mathematics can be exhibited which would, if accepted by the mathematical community, produce a full Kuhnian revolution. These two arguments depend on first proving that a shift from a classical conception of mathematics to an intuitionist conception would (...) be incommensurable, that is, that some classical statements, possessing meanings which cannot be preserved in the intuitionist language, would become unintelligible. The vague but intriguing thesis is then tentatively advanced that Kuhnian revolutions are even historically possible, in the sense that only what we might call ‘accidental’ historical factors may have prevented mathematics from undergoing just such a Kuhnian revolution in the early years of the twentieth century. (shrink)

Physicists and historians of science have always held that Isaac Newton should receive credit for the first proof that inverse-square orbits must be conics. This conviction derives from a brief argument, regarded as essentially correct, given by Newton in the Principia. Recently, however, it has been contended that this outline or sketch contains irreparable logical flaws. Here, the logical structure of this outline of Newton's, as well as the details that this outline omits, are carefully examined. We find that whilst (...) his argument does indeed contain a flaw, it is a minor omission rather than a serious logical error. Having rectified this omission, we show how Newton's outline expands into a convincing proof that inverse-square orbits must be conics. (shrink)

Abstract.As an undergraduate at Cambridge, Newton entered into his ‘Waste Book’ an assumption that we have named the Equivalence Assumption (The Younger): ‘‘ If a body move progressively in some crooked line [about a center of motion]..., [then this] crooked line may bee conceived to consist of an infinite number of streight lines. Or else in any point of the croked line the motion may bee conceived to be on in the tangent.’’ In this assumption, Newton somewhat imprecisely describes two (...) mathematical models, a ‘‘polygonal limit model’’ and a ‘‘tangent deflected model,’’ for ‘‘one-body motion,’’ that is, for the motion of a ‘‘body in orbit about a fixed center,’’ and then claims that these two models are equivalent. In the first part of this paper, we study the Principia to determine how the elder Newton would more carefully describe the polygonal limit and tangent deflected models. From these more careful descriptions, we then create Equivalence Assumption (The Elder), a precise interpretation of Equivalence Assumption (The Younger) as it might have been restated by Newton, after say 1687. We then review certain portions of the Waste Book and the Principia to make the case that, although Newton never restates nor even alludes to the Equivalence Assumption after his youthful Waste Book entry, still the polygonal limit and tangent deflected models, as well as an unspoken belief in their equivalence, infuse Newton’s work on orbital motion. In particular, we show that the persuasiveness of the argument for the Area Property in Proposition 1 of the Principia depends crucially on the validity of Equivalence Assumption (The Elder). After this case is made, we present the mathematical analysis required to establish the validity of the Equivalence Assumption (The Elder). Finally, to illustrate the fundamental nature of the resulting theorem, the Equivalence Theorem as we call it, we present three significant applications: we use the Equivalence Theorem first to clarify and resolve questions related to Leibniz’s ‘‘polygonal model’’ of one-body motion; then to repair Newton’s argument for the Area Property in Proposition 1; and finally to clarify and resolve questions related to the transition from impulsive to continuous forces in ‘‘De motu’’ and the Principia. (shrink)

After preparing the way with comments on evanescent quantities and then Newton’s interpretation of his second law, this study of Proposition II (Book I)— Proposition II Every body that moves in some curved line described in a plane and, by a radius drawn to a point, either unmoving or moving uniformly forward with a rectilinear motion, describes areas around that point proportional to the times, is urged by a centripetal force tending toward that same point. —asks and answers the following (...) questions: When does a version of Proposition II first appear in Newton’s work? What revisions bring that initial version to the final form in the 1726 Principia? What, exactly, does this proposition assert? In particular, what does Newton mean by the motion of a body “urged by a centripetal force”? Does it assert a true mathematical claim? If not, what revision makes it true? Does the demonstration of Proposition II persuade? Is it as convincing, for example, as the most convincing arguments of the Principia? If not, what revisions would make the demonstration more persuasive? What is the importance of Proposition II, to the physics of Book III and the mathematics of Book I? (shrink)

Principia (Book 1, Sect. 6), Newton's Lemma 28 on the algebraic nonintegrability of ovals has had an unusually mixed reception. Beginning in 1691 with Jakob Bernoulli (who accepted the lemma) and Huygens and Leibniz (who rejected it and offered counterexamples), Lemma 28 has a history of eliciting seemingly contradictory reactions. In more recent times, D.T. Whiteside in 1974 gave an “unchallengeable counterexample,” while the mathematician V.I. Arnol'd in 1987 sided with Bernoulli and called Newton's argument an “astonishingly modern topological proof.” (...) This disagreement mostly stems, we argue, from Newton's vague statement of the lemma. Indeed, we identify several different interpretations of Lemma 28, any one of which Newton may have been intending to assert, and we then test a number of proposed counterexamples to see which, if any, are true counterexamples to one or more of these versions of the lemma. Following this, we study Newton's argument for the lemma to see whether and where it fails to be convincing. In the end, our study of Newton's Lemma 28 provides an answer to the question, Who is right: Huygens, Leibniz, Whiteside and the others who reject the lemma, or Bernoulli, Arnol'd, and the others who accept it? (shrink)

In this study, we test the security of a crucial plank in the Principia’s mathematical foundation, namely Newton’s path leading to his solution of the famous Inverse Kepler Problem: a body attracted toward an immovable center by a centripetal force inversely proportional to the square of the distance from the center must move on a conic having a focus in that center. This path begins with his definitions of centripetal and motive force, moves through the second law of motion, then (...) traverses Propositions I, II, and VI, before coming to an end with Propositions XI, XII, XIII and this trio’s first corollary. To test the security of this path, we answer the following questions. How far is Newton’s path from being truly rigorous? What would it take to clarify his ambiguous definitions and laws, supply missing details, and close logical gaps? In short, what would it take to make Newton’s route to the Inverse Kepler Problem completely convincing? The answer is very surprising: it takes far less than one might have expected, given that Newton carved this path in 1687.Keywords: Isaac Newton; Principia; Centripetal force; Second law of motion; Conic orbits; Inverse Kepler Problem. (shrink)

Since the introduction of mathematical population genetics, its machinery has shaped our fundamental understanding of natural selection. Selection is taken to occur when differential fitnesses produce differential rates of reproductive success, where fitnesses are understood as parameters in a population genetics model. To understand selection is to understand what these parameter values measure and how differences in them lead to frequency changes. I argue that this traditional view is mistaken. The descriptions of natural selection rendered by population genetics models are (...) in general neither predictive nor explanatory and introduce avoidable conceptual confusions. I conclude that a correct understanding of natural selection requires explicitly causal models of reproductive success. *Received May 2006; revised December 2006. †To contact the author, please write to: Department of Philosophy, Kansas State University, 201 Dickens Hall, Manhattan, KS 66506; e‐mail: [email protected] . (shrink)

The most successful theory in all of science--and the basis of one third of our economy--says the strangest things about the world and about us. Can you believe that physical reality is created by our observation of it? Physicists were forced to this conclusion, the quantum enigma, by what they observed in their laboratories. Trying to understand the atom, physicists built quantum mechanics and found, to their embarrassment, that their theory intimately connects consciousness with the physical world. Quantum Enigma explores (...) what that implies and why some founders of the theory became the foremost objectors to it. Schrodinger showed that it "absurdly" allowed a cat to be in a "superposition" simultaneously dead and alive. Einstein derided the theory's "spooky interactions." With Bell's Theorem, we now know Schrodinger's superpositions and Einstein's spooky interactions indeed exist. Authors Bruce Rosenblum and Fred Kuttner explain all of this in non-technical terms with help from some fanciful stories and bits about the theory's developers. They present the quantum mystery honestly, with an emphasis on what is and what is not speculation. Physics' encounter with consciousness is its skeleton in the closet. Because the authors open the closet and examine the skeleton, theirs is a controversial book. Quantum Enigma's description of the experimental quantum facts, and the quantum theory explaining them, is undisputed. Interpreting what it all means, however, is controversial. Every interpretation of quantum physics encounters consciousness. Rosenblum and Kuttner therefore turn to exploring consciousness itself--and encounter quantum physics. Free will and anthropic principles become crucial issues, and the connection of consciousness with the cosmos suggested by some leading quantum cosmologists is mind-blowing. Readers are brought to a boundary where the particular expertise of physicists is no longer a sure guide. They will find, instead, the facts and hints provided by quantum mechanics and the ability to speculate for themselves. (shrink)

Offers a compelling vision of how to achieve and conduct a liberal but democratic society through the ideal of Neutrality--between people and ideas of the good--and using the tool of Neutral dialogue.

Prabhpal Singh has recently defended a relational account of the difference in moral status between fetuses and newborns as a way of explaining why abortion is permissible and infanticide is not. He claims that only a newborn can stand in a parent–child relation, not a fetus, and this relation has a moral dimension that bestows moral value. We challenge Singh’s reasoning, arguing that the case he presents is unconvincing. We suggest that the parent–child relation is better understood as an extension (...) of an existing relationship formed during the gestational period. The change in this relation at birth is not sufficient to justify the radical change in moral status required to rule out infanticide while accepting the permissibility of abortion. Given that the moral status of orphans is also problematic under Singh’s account, we conclude that Singh has not shown that a newborn has greater moral worth than a fetus. (shrink)

The Great Psychotherapy Debate: Models, Methods, and Findings comprehensively reviews the research on psychotherapy to dispute the commonly held view that the benefits of psychotherapy are derived from the specific ingredients contained in a given treatment (medical model). The author reviews the literature related to the absolute efficacy of psychotherapy, the relative efficacy of various treatments, the specificity of ingredients contained in established therapies, effects due to common factors, such as the working alliance, adherence and allegiance to the therapeutic protocol, (...) and effects that are produced by different therapists. In each case, the evidence convincingly corroborates the contextual model and disconfirms the prevailing medical model. (shrink)

The second edition of The Great Psychotherapy Debate has been updated and revised to include a history of healing practices, medicine, and psychotherapy, an expanded theoretical presentation of the contextual model, an examination of therapist effects, and a thorough review of the research on common factors such as the alliance, expectations, and empathy.

The article investigates the mathematical and philosophical backdrop of the digital ocean as contemporary model, moving from the digitalized ocean of Georg Cantor’s set theory to that of Alan Turing’s computation theory. It examines in Cantor what is arguably the most rigorous historical attempt to think the structural essence of the continuum, in order to clarify what disappears from the computational paradigm once Turing begins to advocate for the structural irrelevance of this ancient ground.

This highly original history of ideas considers the impact of Hegel on French philosophy from the 1920s to the present. As Baugh's lucid narrative makes clear, Hegel's influence on French philosophy has been profound, and can be traced through all the major intellectual movements and thinkers in France throughout the 20th Century from Jean Wahl, Sartre, and Bataille to Foucault, Deleuze, and Derrida. Baugh focuses on Hegel's idea of the "unhappy consciousness," and provides a bold new account of Hegel's early (...) reception in French intellectual history. (shrink)

Mass literacy, mass communication, and the Internet have all increased the amount of information available. But false knowledge still abounds. Taking cues from Sir Thomas Browne, the English Renaissance skeptic, this title examines a host of contemporary errors in thinking and offers a powerful explanation of why they occur.

Since its publication in 1982, _Beyond Positivism _has become established as one of the definitive statements on economic methodology. The book’s rejection of positivism and its advocacy of pluralism were to have a profound influence in the flowering of work methodology that has taken place in economics in the decade since its publication. This edition contains a new preface outlining the major developments in the area since the book’s first appearance. The book provides the first comprehensive treatment of twentieth century (...) philosophy of science which emphasizes the issues relevant to economics. It proceeds to demonstrate this relevance by reviewing some of the key debates in the area. Having concluded that positivism has to be rejected, the author examines possible alternative bases for economic methodology. Arguing that there is no best method, he advocates methodological pluralism. (shrink)

Ranging from Joseph Bellamy to Hilary Putnam, and from early New England Divinity Schools to contemporary university philosophy departments, historian Bruce Kuklick recounts the story of the growth of philosophical thinking in the United States. Readers will explore the thought of early American philosphers such as Jonathan Edwards and John Witherspoon and will see how the political ideas of Benjamin Franklin, Thomas Paine and Thomas Jefferson influenced philosophy in colonial America. Kuklick discusses The Transcendental Club (members Henry David Thoreau, (...) Ralph Waldo Emerson) and describes the rise of pragmatism centered on Metaphysical Club of Cambridge (and members William James, Oliver Wendell Holmes, and Charles Peirce). He examines the profound impact Darwinism had on American philosophy and looks at Idealists such as the Kantian Josiah Royce and the Hegelian John Dewey. The book shows how, in the twentieth century, the Nazi conquest of Europe unleashed a flood of European intellectuals onto these shores, including such major thinkers as Theodore Adorno, Erich Fromm, Rudolph Carnap, and Alfred Tarski. Finally, Kuklick examines the contributions of such contemporary philosophers as Sidney Hook and Willard Quine and such books as John Rawl's A Theory of Justice and Herbert Marcuse's One Dimensional Man. Kuklick pulls no punches in portraying the state of American philosophy today and its contested role in the intellectual life of the nation and the world. The range of philosophical thought in our nation's history has been great, from Edwards's Religious Affections to Kuhn's The Structure of Scientific Revolutions, and Bruce Kuklick has captured it all in a book that blends intricate details with sweeping vision. (shrink)

Contemporary thought has been profoundly shaped by the early-twentieth-century turn toward synchronic models of explanation, which analyze phenomena as they appear at a single moment, rather than diachronically as they develop through time. But the relationship between time and system remains unexplained by the standard account of this shift. Through a new history of systematic thinking across the humanities and sciences, The Writing of Spirit argues that nineteenth-century historicism wasn't simply replaced by a more modern synchronic perspective. The structuralist revolution (...) consisted rather in a turn toward time's absolutely minimal conditions, and thus also toward a new theory of diachrony. Pourciau arrives at this surprising and powerful conclusion through an analysis of language-scientific theories over the course of two centuries, associated with thinkers from Jacob Grimm and Richard Wagner to the Russian Futurists, in domains as disparate as historical linguistics, phonology, acoustics, opera theory, philosophy, poetics, and psychology. The result is a novel contribution to a pressing contemporary question--namely, what role history should play in the interpretation of the present. (shrink)

Voting Institutions Justifications Notes.

William Simkulet has recently criticised Colgrove et al’s defence against what they have called inconsistency arguments—arguments that claim opponents of abortion (OAs) act in ways inconsistent with their underlying beliefs about human fetuses (eg, that human fetuses are persons at conception). Colgrove et al presented three objections to inconsistency arguments, which Simkulet argues are unconvincing. Further, he maintains that OAs who hold that the fetus is a person at conception fail to act on important issues such as the plight of (...) frozen embryos, poverty and spontaneous abortion. Thus, they are morally negligent. In response, we argue that Simkulet has targeted a very narrow group of OAs, and so his criticisms are inapplicable to most OAs. We then explain why his responses to each of Colgrove et al’s objections do not succeed, even for this restricted group. Finally, we note that Simkulet fails to provide evidence for his claims regarding OAs’ supposed failures to act, and we show that OAs veritably do invest resources into these important issues. We conclude that Colgrove et al’s reasons for rejecting inconsistency arguments (en masse) remain intact. (shrink)

In a book that is part memoir and part writing guide, the author discusses how he used a journal to enhance his experiences on the yoga mat and then explains how readers can best start and maintain their own yoga journal. Original.

An indispensable collection that re-examines what it means to belong in the world. "Where are you from?" The word cosmopolitan was first used as a way of evading exactly this question, when Diogenes the Cynic declared himself a “kosmo-polites,” or citizen of the world. Cosmopolitanism displays two impulses—on the one hand, a detachment from one’s place of origin, while on the other, an assertion of membership in some larger, more compelling collective. Cosmopolitanisms works from the premise that there is more (...) than one kind of cosmopolitanism, a plurality that insists cosmopolitanism can no longer stand as a single ideal against which all smaller loyalties and forms of belonging are judged. Rather, cosmopolitanism can be defined as one of many possible modes of life, thought, and sensibility that are produced when commitments and loyalties are multiple and overlapping. Featuring essays by major thinkers, including Homi Bhabha, Jean Bethke Elshtain, Thomas Bender, Leela Gandhi, Ato Quayson, and David Hollinger, among others, this collection asks what these plural cosmopolitanisms have in common, and how the cosmopolitanisms of the underprivileged might serve the ethical values and political causes that matter to their members. In addition to exploring the philosophy of Kant and the space of the city, this volume focuses on global justice, which asks what cosmopolitanism is good for, and on the global south, which has often been assumed to be an object of cosmopolitan scrutiny, not itself a source or origin of cosmopolitanism. This book gives a new meaning to belonging and its ground-breaking arguments call for deep and necessary discussion and discourse. (shrink)

Micheline Sauvage of the French National Scientific Research Centre traces for us the story of this great Athenian and great philosopher, as seen both by his contemporaries and by the European philosophers who followed after him.

Editors provide a substantive introduction to the history and theories of perfectionism and neutrality, expertly contextualizing the essays and making the collection accessible.

I hope to persuade Charles Fried to think again about his developing views on distributive justice. Since I live at a certain remove from Cambridge, the best I can offer is a hypothetical dialogue with an imaginary person whose views seem, to me at least, of a Friedian inspiration. My central question deals with the way Fried establishes his rights to things he candidly concedes he does not deserve. To present my problems, I shall begin with a simpler case than (...) those – involving kidneys and talents – that Fried makes central to his discussion. Rather than starting with these rather special goods, I find it clarifying to focus first on more garden variety commodities – which, to emphasize their character, I shall call apples. (shrink)

In Philosophy as Frustration: Happiness Found and Feigned from Greek Antiquity to Present Bruce Silver argues that traditional philosophical views of happiness, as well as recent psychological theories of happiness, are at odds with themselves and with important accounts of a truly happy life.

Karl Popper, in "The Logic of Scientific Discovery" Section *vii, argues that if you find that some objecta a,b, c ... have a specific property P, then this discovery by itself does not increase the probability that some other object also has P. He concludes that there can be no effective principle of induction. My paper disproves Popper's claim, using very elementary considerations..