The translation of Newton’s geometrical Propositions in the Principia into the language of the differential calculus in the form developed by Leibniz and his followers has been the subject of many scholarly articles and books. One of the most vexing problems in this translation concerns the transition from the discrete polygonal orbits and force impulses in Prop. 1 to the continuous orbits and forces in Prop. 6. Newton justified this transition by lemma 1 on prime and ultimate ratios which was (...) a concrete formulation of a limit, but it took another century before this concept was established on a rigorous mathematical basis. This difficulty was mirrored in the newly developed calculus which dealt with differentials that vanish in this limit, and therefore were considered to be fictional quantities by some mathematicians. Despite these problems, early practitioners of the differential calculus like Jacob Hermann, Pierre Varignon, and Johann Bernoulli succeeded without apparent difficulties in applying the differential calculus to the solution of the fundamental problem of orbital motion under the action of inverse square central forces. By following their calculations and describing some essential details that have been ignored in the past, I clarify the reason why the lack of rigor in establishing the continuum limit was not a practical problem. (shrink)

In 1662 Christiaan Huygens carried out the famous Torricelli experiment to test the existence of atmospheric pressure by inserting the apparatus in the glass receiver of a vacuum pump, and evacuating the air inside it. He reported that when the air was exhausted, a column of water remained suspended in a 4-foot tube. This unexpected result was in stark contrast with earlier experiments of Boyle and Hooke that apparently had confirmed Torricelli’s explanation that such a water column was supported by (...) outside air pressure, and would fall when the air was removed. Huygen’s “anomalous suspension” led to the continuation of controversies in the seventeenth century about the nature of the vacuum that these experiments were expected to resolve. Surprisingly, the origin of Huygens’ unexpected result has remained a puzzle up to the present time. In this paper, I discuss the dynamics of such a column of water under the experimental conditions reported by Boyle and by Huygens, that turned out to be different, and present the results of a replication of their experiments with a modern vacuum pump. Contrary to the conventional explanations of these experiments, I demonstrate that in the Boyle–Hooke version of this experiment, the water column descends initially because it is forced down by the gas pressure due to air dissolved in the water which is released inside the Torricelli tube after the external pressure is sufficiently decreased. Huygens, however, first removed this trapped air before he carried out his experiment. In the absence of this internal gas pressure, the early rudimentary vacuum pumps were inadequate to decrease the air pressure sufficiently inside the receiver to demonstrate the descent of a Torricelli column of airless water 4-foot in height or less. (shrink)

ABSTRACTThe worksheets that presumably contained Newton's early development of the fundamental concepts in his Principia have been lost. A plausible reconstruction of this development is presented based on Newton's exchange of letters with Robert Hooke in 1679, with Edmund Halley in 1686, and on some clues in the diagram associated with Proposition 1 in Book 1 of the Principia that have been ignored in the past. A graphical construction associated with this proposition leads to a rapidly convergent method to obtain (...) orbits for central forces, which elucidates how Newton may have have been led to formulate some of his most fundamental propositions in the Principia. (shrink)

SummaryIn 1693, Gottfried Wilhelm Leibniz published in the Acta Eruditorum a geometrical proof of the fundamental theorem of the calculus. It is shown that this proof closely resembles Isaac Barrow's proof in Proposition 11, Lecture 10, of his Lectiones Geometricae, published in 1670. This comparison provides evidence that Leibniz gained substantial help from Barrow's book in formulating and presenting his geometrical formulation of this theorem. The analysis herein also supports the work of J. M. Child, who in 1920 studied the (...) early manuscripts of Leibniz and concluded that he had frequently copied his diagrams from Barrow's book, but without acknowledgement. It is also shown that the diagram of Leibniz associated with his 1693 proof has often been reproduced with errors that make some aspects of his text difficult to comprehend. (shrink)

In Proposition 10, Book 2 of the Principia, Newton applied his geometrical calculus and power series expansion to calculate motion in a resistive medium under the action of gravity. In the first edition of the Principia, however, he made an error in his treatment which lead to a faulty solution that was noticed by Johann Bernoulli and communicated to him while the second edition was already at the printer. This episode has been discussed in the past, and the source of (...) Newton’s initial error, which Bernoulli was unable to find, has been clarified by Lagrange and is reviewed here. But there are also problems in Newton’s corrected version in the second edition of the Principia that have been ignored in the past, which are discussed in detail here. (shrink)

The central claim that understanding quantum mechanics requires a conscious observer, which is made by B. Rosenblum and F. Kuttner in their book “Quantum Enigma: Physics encounters consciousness”, is shown to be based on various misunderstandings and distortions of the foundations of quantum mechanics.

In Book 1 of the Principia, Newton presented two different descriptions of orbital motion under the action of a central force. In Prop. 1, he described this motion as a limit of the action of a sequence of periodic force impulses, while in Prop. 6, he described it by the deviation from inertial motion due to a continuous force. From the start, however, the equivalence of these two descriptions has been the subject of controversies. Perhaps the earliest one was the (...) famous discussion from December 1704 to 1706 between Leibniz and the French mathematician Pierre Varignon. But confusion about this subject has remained up to the present time. Recently, Pourciau has rekindled these controversies in an article in this journal, by arguing that “Newton never tested the validity of the equivalency of his two descriptions because he does not see that his assumption could be questioned. And yet the validity of this unseen and untested equivalence assumption is crucial to Newton’s most basic conclusions concerning one-body motion” (Pourciau in Arch Hist Exact Sci 58:283–321, 2004, 295). But several revisions of Props. 1 and 6 that Newton made after the publication in 1687 of the first edition of the Principia reveal that he did become concerned to provide mathematical proof for the equivalence of his seemingly different descriptions of orbital motion in these two propositions. In this article, we present the evidence that in the second and third edition of the Principia, Newton gave valid demonstrations of this equivalence that are encapsulated in a novel diagram discussed in Sect. 4. (shrink)

Is the procreation asymmetry intuitively supported? According to a recent article in this journal, an experimental study suggests the opposite. Dean Spears (2020) claims that nearly three-quarters of participants report that there is a reason to create a person just because that person’s life would be happy. In reply, I argue that various confounding factors render the study internally invalid. More generally, I show how one might come to adopt the procreation asymmetry for the wrong reasons by misinterpreting one’s (...) intuitions. (shrink)

Szabó follows Heim in viewing familiarity, rather than uniqueness, as the essence of the definite article, but attempts to derive both familiarity and uniqueness implications pragmatically, assigning a single semantic interpretation to both the definite and indefinite articles. I argue that if there is no semantic distinction between the articles, then there is no way to derive these differences between them pragmatically.

A set of replies to critics of my 2021 book 'The Fiery Test of Critique: A Reading of Kant's Dialectic' (OUP). -/- The criticisms are based on talks given at an Author-meets-critics symposium at Princeton University on April 22nd, 2023. The critics are: Beatrice Longuenesse, Patricia Kitcher, Allen Wood, Des Hogan, and Anja Jauernig.

Reply to commentators on the paper 'Realistic monism: why physicalism entails panpsychism'.

Healey (2019a) makes four comments on my (Park, 2019a) objections to van Fraassen’s positions. The four comments concern the issues of whether ‘disbelief’ is appropriate or inappropriate to characterize van Fraassen’s position, what the relationship between a theory and models is for van Fraassen, whether he believes or not that a theory is empirically adequate, and whether destructive empiricism is tenable or not. I reply to those comments in this paper.

"The Uneasy Heirs of Acquaintance" is my first-round contribution to a 4-way exchange with Bill Brewer, Anil Gupta, and John McDowell. In the first round, each of us writes a commentary on the other three, and in the second round we reply to each other's first-round contributions. This is my second-round contribution.

Reply to criticsThe replies in this symposium are some of the most insightful contributions to contemporary metaphilosophy I have read. I wish I had seen them before I wrote Philosophy without Intuitions . It would have made it a better book. I also wish I had space to explore all the important issues raised, but unfortunately, the focus here will have to be on points of disagreement. The replies build on each other—I draw on material from the earlier replies (...) in the later ones. It is possible to read each reply in isolation, but they are best read in sequence.Socratic knowledge and its role in philosophy: Reply to WeathersonWeatherson presents one of the most interesting accounts of the role of intuitions in philosophy that I have encountered. For Weatherson, the role is limited, fragile, and elusive .One of the most interesting ideas in the paper is that “the important intuitions are the ones you bar .. (shrink)

강병호는 자신의 논문에서 호네트의 인정도덕 구상이 드러나는 모든 글을 체계적으로 검토하여, 칸트의 의무론적 요소를 발굴하고, 칸트의 존중개념을 통해 이를 의무론적으로 재구조화하겠다고 한다. 그리고 강병호는 이를 국내에서도 국외에서도 드문 시도라고 자평하면서, 호네트를 비롯한 국내 연구자들에게 대한 신랄한 비판을 가한다. 본 논문에서는 강병호의 주장과 비판이 과연 정당한지를 검토하기 위해, 첫째, 강병호의 비판 대상이 되는 호네트의 인정도덕 구상이 어떤 것인지 소개할 것이다. 그리고 이에 이어서 둘째, 호네트가 칸트의 의무론적 요소를 경시했고, 도덕에 대한 기능주의적 파악과 규범적 정당화를 범주적으로 혼동했다는 강병호의 비판이 과연 정당한지를 살펴볼 (...) 것이다. 그리고 끝으로 강병호가 칸트의 존중 개념을 통해 인정이론을 재구조화한다는 것이 과연 무엇을 의미하는지를 검토하면서 그 타당성을 따져 볼 것이다. (shrink)

Here I reply to criticisms by Jeroen de Ridder and S. Kate Devitt to my "Collective (Telic) Virtue Epistemology".

In these replies, I shall respond to criticisms offered by Kaczor and Rodger to my article titled “Ectogestation and the Problem of Abortion.” In the process, I shall also try to bring into focus why the possibility of ectogestation will radically alter the shape of the abortion debate.

This article responds to Tim Hsiao's "The Moral Case for Gun Ownership".

A reply to Fine’s critique of Aboutness. Fine contrasts two notions of truthmaker, and more generally two notions of “state.” One is algebraic; states are sui generis entities grasped primarily through the conditions they satisfy. The other uses set theory; states are sets of worlds, or, perhaps, collections of such sets. I try to defend the second notion and question some seeming advantages of the first.

I reply to two commentaries—one by Johanna Thoma and Jonathan Weisberg and one by James M. Joyce—concerning how risk-weighted expected utility theory handles the Allais preferences and Dutch books.

I argue that Alvin Goldman has failed to save process reliabilism from my critique in earlier work of consequentialist or teleological epistemic theories. First, Goldman misconstrues the nature of my challenge: two of the cases he discusses I never claimed to be counterexamples to process reliabilism. Second, Goldman’s reply to the type of case I actually claimed to be a counterexample to process reliabilism is unsuccessful. He proposes a variety of responses, but all of them either feature an implausible (...) restriction on process types, or fail to rule out cases with the sort of structure that generates the worry, or both. (shrink)