Aspects of Aristotelian statics in Galileo's dynamics

This paper examines geometrical arguments from Galileo's Mechanics and Two New Sciences to discern the influence of the Aristotelian Mechanical Problems on Galileo's dynamics. A common scientific procedure is found in the Aristotelian author's treatment of the balance and lever and in Galileo's rules concerning motion along inclined planes. This scientific procedure is understood as a development of Eudoxan proportional reasoning, as it was used in Eudoxan astronomy rather than simply as it appears in Euclid's Elements. Topics treated include the significance of the circle in Galileo's demonstrations, the substitution of rectilinear elements for heterogeneous factors like weight and curvilinear distance, and the way in which elements of a motion are used to measure other elements of the same motion. The indirectness of Galileo's proofs, his conception of speed as relative and comparative, and the meaning of his concept of moment all come into clearer focus. Conclusions are drawn about Galilean idealization, and also about the contrast of literal versus figural modes of explanation in Galileo's science.     

The meaning of πλασμαтιкό<Emphasis Type="Italic">ν</Emphasis> in Diophantus’ <Emphasis Type="Italic">Arithmetica</Emphasis>

Three problems in book I of Diophantus’ Arithmetica contain the adjective plasmatikon, that appears to qualify an implicit reference to some theorems in Elements, book II. The translation and meaning of the adjective sparked a long-lasting controversy that has become a nonnegligible aspect of the debate about the possibility of interpreting Diophantus’ approach and, more generally, Greek mathematics in algebraic terms. The correct interpretation of the word, a technical term in the Greek rhetorical tradition that perfectly fits the context in which it is inserted in the Arithmetica, entails that Diophantus’ text contained no (implicit) reference to Euclid’s Elements. The clause containing the adjective turns out to be a later interpolation, that cannot be used to support any algebraic interpretation of the Arithmetica.     

“... in Christophorum Clavium de Contactu Linearum Apologia” Considerazioni attorno alla polemica fra Peletier e Clavio circa l'angolo di contatto (1579–1589)

Summary In this work 1 focus my attention upon the question of the angle of tangency in the XVI^th Century, especially in the polemic between J. Peletier and Chr. Clavius (1579–1589). The interest in the question favored deliberation about the theory of proportions, the principle of Eudoxus-Archimedes and the set of angles of tangency (this is a non-Archimedian set); there were problems about logical proofs and geometrical proofs.

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Memoria presentata da H. Freudenthal     

The Mathematics of the Area Law: Kepler's Successful Proof in Epitome Astronomiae Copernicanae (1621)

Epitome V (1621), and consisted of matching an element of area to an element of time, where each was mathematically determined. His treatment of the area depended solely on the geometry of Euclid's Elements, involving only straight-line and circle propositions – so we have to account for his deliberate avoidance of the sophisticated conic-geometry associated with Apollonius. We show also how his proof could have been made watertight according to modern standards, using methods that lay entirely within his power. The greatest innovation, however, occurred in Kepler's fresh formulation of the measure of time. We trace this concept in relation to early astronomy and conclude that Kepler's treatment unexpectedly entailed the assumption that time varied nonuniformly; meanwhile, a geometrical measure provided the independent variable. Even more surprisingly, this approach turns out to be entirely sound when assessed in present-day terms. Kepler himself attributed the cause of the motion of a single planet around the Sun to a set of `physical' suppositions which represented his religious as well as his Copernican convictions; and we have pared to a minimum – down to four – the number he actually required to achieve this. In the Appendix we use modern mathematics to emphasize the simplicity, both geometrical and kinematical, that objectively characterizes the Sun-focused ellipse as an orbit. Meanwhile we highlight the subjective simplicity of Kepler's own techniques (most of them extremely traditional, some newly created). These two approaches complement each other to account for his success. (Received April 19, 2002) Published online April 2, 2003 Communicated by N. M. Swerdlow     

Superposition: on Cavalieri’s practice of mathematics

Bonaventura Cavalieri has been the subject of numerous scholarly publications. Recent students of Cavalieri have placed his geometry of indivisibles in the context of early modern mathematics, emphasizing the role of new geometrical objects, such as, for example, linear and plane indivisibles. In this paper, I will complement this recent trend by focusing on how Cavalieri manipulates geometrical objects. In particular, I will investigate one fundamental activity, namely, superposition of geometrical objects. In Cavalieri’s practice, superposition is a means of both manipulating geometrical objects and drawing inferences. Finally, I will suggest that an integrated approach, namely, one which strives to understand both objects and activities, can illuminate the history of mathematics.     

The intelligibility of motion and construction: Descartes’ early mathematics and metaphysics, 1619-1637

I argue for an interpretation of the connection between Descartes’ early mathematics and metaphysics that centers on the standard of geometrical intelligibility that characterizes Descartes’ mathematical work during the period 1619 to 1637. This approach remains sensitive to the innovations of Descartes’ system of geometry and, I claim, sheds important light on the relationship between his landmark Geometry (1637) and his first metaphysics of nature, which is presented in Le monde (1633). In particular, I argue that the same standard of clear and distinct motions for construction that allows Descartes to distinguish ‘geometric’ from ‘imaginary’ curves in the domain of mathematics is adopted in Le monde as Descartes details God’s construction of nature. I also show how, on this interpretation, the metaphysics of Le monde can fruitfully be brought to bear on Descartes’ attempted solution to the Pappus problem, which he presents in Book I of the Geometry. My general goal is to show that attention to the standard of intelligibility Descartes invokes in these different areas of inquiry grants us a richer view of the connection between his early mathematics and philosophy than an approach that assumes a common method is what binds his work in these domains together.     

Legend naturalism and scientific progress: An essay on Philip Kitcher's : The advancement of science

Philip Kitcher's The Advancement of Science sets out, programmatically, a new naturalistic view of science as a process of building consensus practices. Detailed historical case studies—centrally, the Darwinian revolutio—are intended to support this view. I argue that Kitcher's expositions in fact support a more conservative view, that I dub ‘Legend Naturalism’. Using four historical examples which increasingly challenge Kitcher's discussions, I show that neither Legend Naturalism, nor the less conservative programmatic view, gives an adequate account of scientific progress. I argue for a naturalism that is more informed by psychology and a normative account that is both more social and less realist than the views articulated in The Advancement of Science.     

The Definitions of Fundamental Geometric Entities Contained in Book I of Euclids Elements

OElig;he thesis is sustained that the definitions of fundamental geometric entities which open Euclids Elements actually are excerpts from the Definitions by Heron of Alexandria, interpolated in late antiquity into Euclids treatise. As a consequence, one of the main bases of the traditional Platonist interpretation of Euclid is refuted. Arguments about the constructivist nature of Euclids mathematical philosophy are given. (Received June 6, 1997)     

Can the two-time interpretation of quantum mechanics solve the measurement problem?

Over many years, Aharonov and co-authors have proposed a new interpretation of quantum mechanics: the two-time interpretation. This interpretation assigns two wavefunctions to a system, one of which propagates forwards in time and the other backwards. In this paper, I argue that this interpretation does not solve the measurement problem. In addition, I argue that it is neither necessary nor sufficient to attribute causal power to the backwards-evolving wavefunction $〈\Phi |$ and thus its existence should be denied, contra the two-time interpretation. Finally, I follow Vaidman in giving an epistemological reading of $〈\Phi |$.     

A history of entanglement: Decoherence and the interpretation problem

This paper examines the interweaving of the history of quantum decoherence and the interpretation problem in quantum mechanics through the work of two physicists—H. Dieter Zeh and Wojciech Zurek. In the early 1970s Zeh anticipated many of the important concepts of decoherence, framing it within an Everett-type interpretation. Zeh has since remained committed to this view; however, Zurek, whose papers in the 1980s were crucial in the treatment of the preferred basis problem and the subsequent development of density matrix formalism, has argued that decoherence leads to what he terms the ‘existential interpretation’, compatible with certain aspects of both Everett's relative-state formulation and the Bohr's ‘Copenhagen interpretation’. I argue that these different interpretations can be traced back to the different early approaches to the study of environment-induced decoherence in quantum systems, evident in the early work of Zeh and Zurek. I also show how Zurek's work has contributed to the tendency to see decoherence as contributing to a ‘new orthodoxy’ or a reconstruction of the original Copenhagen interpretation.     

Atoms, complexes, and demonstration: Posterior analytics 96b15-25

There is agreement neither concerning the point that is being made in Posterior analytics 96b15–25 nor the issue Aristotle intends to address. There are two major lines of interpretation of this passage. According to one, sketched by Themistius and developed by Philoponus and Eustratius, Aristotle is primarily concerned with determining the definitions of the infimae species that fall under a certain genus. They understand Aristotle as arguing that this requires collating definitional predictions, seeing which are common to which species. Pacius, on the other hand, takes Aristotle to be saying that a genus is studied scientifically through first determining the infimae species that fall under that genus. This interpretation attributes to Aristotle a distinction between primary and derivative subjects. I argue for Pacius’s interpretation, defending it against Barnes’s objections.     

The problem of ontology for spontaneous collapse theories

The question of how to interpret spontaneous collapse theories of quantum mechanics is an open one. One issue involves what link one should use to go from wave function talk to talk of ordinary macroscopic objects. Another issue involves whether that link should be taken ontologically seriously. In this paper, I argue that the link should be taken ontologically seriously; I argue against an ontology consisting solely of the wave function. I then consider three possible links: the fuzzy link, the accessible mass density link, and the mass density simpliciter link. I show that the first two links have serious anomalies which render them unacceptable. I show that the mass density simpliciter link, in contrast, is viable.     

Incommensurabilities in the work of Thomas Kuhn

I distinguish between two ways in which Kuhn employs the concept of incommensurability based on for whom it presents a problem. First, I argue that Kuhn’s early work focuses on the comparison and underdetermination problems scientists encounter during revolutionary periods (actors’ incommensurability) whilst his later work focuses on the translation and interpretation problems analysts face when they engage in the representation of science from earlier periods (analysts’ incommensurability). Secondly, I offer a new interpretation of actors’ incommensurability. I challenge Kuhn’s account of incommensurability which is based on the compartmentalisation of the problems of both underdetermination and non-additivity to revolutionary periods. Through employing a finitist perspective, I demonstrate that in principle these are also problems scientists face during normal science. I argue that the reason why in certain circumstances scientists have little difficulty in concurring over their judgements of scientific findings and claims while in others they disagree needs to be explained sociologically rather than by reference to underdetermination or non-additivity. Thirdly, I claim that disagreements between scientists should not be couched in terms of translation or linguistic problems (aspects of analysts’ incommensurability), but should be understood as arising out of scientists’ differing judgments about how to take scientific inquiry further.     

Hermeneutical contributions to the history of science: Gadamer on ‘presentism’

This article examines how Hans G. Gadamer’s philosophical hermeneutics can contribute to contemporary debates on the concept of ‘presentism’. In the field of the history of science, this term is usually employed in two ways. First, ‘presentism’ refers to the kind of historiography which judges the past to legitimate the present. Second, this concept designates the inevitable influence of the present in the interpretation of the past. In this paper, I argue that both dimensions of the relationship between the present and the past are explored by Hans G. Gadamer in Truth and Method and other texts. In the first place, Gadamer’s critique of historicism calls into question the anti-presentist ideal of studying the past for ‘its own sake’. In the second place, Gadamer’s thesis that all understanding inevitably involves some prejudice poses the question of the inherent “present-centredness” of historical interpretations. By examining Gadamer’s hermeneutics, I seek to provide historians with new arguments and perspectives on the question of ‘presentism’.     

Galileo’s quanti: understanding infinitesimal magnitudes

In On Local Motion in the Two New Sciences, Galileo distinguishes between ‘time’ and ‘quanto time’ to justify why a variation in speed has the same properties as an interval of time. In this essay, I trace the occurrences of the word quanto to define its role and specific meaning. The analysis shows that quanto is essential to Galileo’s mathematical study of infinitesimal quantities and that it is technically defined. In the light of this interpretation of the word quanto, Evangelista Torricelli’s theory of indivisibles can be regarded as a natural development of Galileo’s insights about infinitesimal magnitudes, transformed into a geometrical method for calculating the area of unlimited plane figures.     

Making sense of Day 1 of the Two New Sciences: Galileo’s Aristotelian-inspired agenda and his Jesuit readers

This study proposes an explanation for the choice of topics Galileo addressed in Day 1 of his 1638 Two New Sciences, a section of the work which has long puzzled historians of science. I argue that Galileo’s agenda in Day 1, that is the topics he discusses and the questions he poses, was shaped by contemporary teaching commentaries on Books 3 through 8 of Aristotle’s Physics. Building on the insights and approach of theorists of reader reception, I confirm this interpretation by examining the response of professors of natural philosophy at the Jesuit Collegio Romano to Galileo’s text.     

Are field quanta real objects? Some remarks on the ontology of quantum field theory

One of the key philosophical questions regarding quantum field theory is whether it should be given a particle or field interpretation. The particle interpretation of QFT is commonly viewed as being undermined by the well-known no-go results, such as the Malament, Reeh-Schlieder and Hegerfeldt theorems. These theorems all focus on the localizability problem within the relativistic framework. In this paper I would like to go back to the basics and ask the simple-minded question of how the notion of quanta appears in the standard procedure of field quantization, starting with the elementary case of the finite numbers of harmonic oscillators, and proceeding to the more realistic scenario of continuous fields with infinitely many degrees of freedom. I will try to argue that the way the standard formalism introduces the talk of field quanta does not justify treating them as particle-like objects with well-defined properties.     

The problem of grounding natural modality in Kant's account of empirical laws of nature

One of the central problems of Kant's account of the empirical laws of nature is: What grounds their necessity? In this article I discuss the three most important lines of interpretation and suggest a novel version of one of them. While the first interpretation takes the transcendental principles as the only sources of the empirical laws' necessity, the second interpretation takes the systematicity of the laws to guarantee their necessity. It is shown that both views involve serious problems. The third interpretation, the “causal powers interpretation”, locates the source of the laws' necessity in the properties of natural objects. Although the second and third interpretations seem incompatible, I analyse why Kant held both views and I argue that they can be reconciled, because the metaphysical grounding project of the laws' necessity is accounted for by Kant's causal powers account, while his best system account explains our epistemic access to the empirical laws. If, however, causal powers are supposed to fulfil the grounding function for the laws' natural modality, then I suggest that a novel reading of the causal powers interpretation should be formulated along the lines of a genuine dispositionalist conception of the laws of nature.     

A similarity-based approach of Kuhn’s no-overlap principle and anomalies

In their book Cognitive Structure of Scientific Revolutions, Hanne Andersen, Peter Barker, and Xiang Chen reconstruct Kuhn’s account of conceptual structure and change, based on the dynamic frame model. I argue against their reconstruction of anomalies and of the no-overlap principle and propose a competing model, based on the similarity relation. First, I introduce the concept of psychological distance between objects, and then I show that the conceptual structure of a theory consists of a set of natural families, separated by a significant empty space. I argue that, in such a conceptual structure, the ES condition, according to which the distance between natural families should be greater than the distance between any two objects belonging to the same natural family, is satisfied. Anomalous objects lead to the violation of this condition. I argue that in a conceptual structure satisfying the ES condition, a similarity relation could be defined, so that natural families would be similarity classes, satisfying the no-overlap principle. In a structure not satisfying this principle, such similarity classes could not be delimited.     

Between Viète and Descartes: Adriaan van Roomen and the <Emphasis Type="Italic">Mathesis Universalis</Emphasis>

Adriaan van Roomen published an outline of what he called a Mathesis Universalis in 1597. This earned him a well-deserved place in the history of early modern ideas about a universal mathematics which was intended to encompass both geometry and arithmetic and to provide general rules valid for operations involving numbers, geometrical magnitudes, and all other quantities amenable to measurement and calculation. ‘Mathesis Universalis’ (MU) became the most common (though not the only) term for mathematical theories developed with that aim. At some time around 1600 van Roomen composed a new version of his MU, considerably different from the earlier one. This second version was never effectively published and it has not been discussed in detail in the secondary literature before. The text has, however, survived and the two versions are presented and compared in the present article. Sections 1–6 are about the first version of van Roomen’s MU the occasion of its publication (a controversy about Archimedes’ treatise on the circle, Sect. 2), its conceptual context (Sect. 3), its structure (with an overview of its definitions, axioms, and theorems) and its dependence on Clavius’ use of numbers in dealing with both rational and irrational ratios (Sect. 4), the geometrical interpretation of arithmetical operations multiplication and division (Sect. 5), and an analysis of its content in modern terms. In his second version of a MU van Roomen took algebra into account, inspired by Viète’s early treatises; he planned to publish it as part of a new edition of Al-Khwarizmi’s treatise on algebra (Sect. 7). Section 8 describes the conceptual background and the difficulties involved in the merging of algebra and geometry; Sect. 9 summarizes and analyzes the definitions, axioms and theorems of the second version, noting the differences with the first version and tracing the influence of Viète. Section 10 deals with the influence of van Roomen on later discussions of MU, and briefly sketches Descartes’ ideas about MU as expressed in the latter’s Regulae.