I will present evidence that nature does not optimize in the sense of Fermat's principle of least time, contrary to what Schoemaker's unintentionally ambiguous exposition might suggest. First, Huygens' principle, an alternative nonteleological account of Snell's law, is outlined. Second, I confront Fermat's principle with a substantive conceptual problem.

In this paper we will discuss some basic aspects of the global theory of dynamical systems. Rather than entering in technical derivations, we will try to emphasize the main points of the concept of dynamical systems which lead us to the generalization presented here, as well as some results that are easily generalized. Besides, some considerations of philosophical nature will be made.

Both before and after the publication of Isaac Newton’s particulate theory of light, numerous wave theories of light were advanced by both philosophers and scientists (e.g., René Descartes, Thomas Hobbes, Robert Hooke, Francesco Grimaldi, and Christiaan Huygens). What is peculiar about this list, as frequently found in the scholarly literature on light, is that it refers to individuals who do not extend much further back than the seventeenth century. A close examination of Aristotle’s account of color and light in comparison (...) to Christiann Huygens’ theory of color and light, however, reveals that Aristotle deserves to be recognized as one who antedates and foreshadows these modern wave theorists. Indeed, we shall argue that Aristotle offered the first wave theory of color and light. On Aristotle’s theory, light is the actualization of a transparent medium such as air or water that makes possible the dissemination of color through the medium in a manner analogous to the dissemination of sound through air or water. (shrink)

A conservation principles tell us that some quantity, quality, or aspect remains constant through change. Such principles appear already in ancient and medieval natural philosophy. In one important strand of Greek cosmology, the rotatory motion of the celestial orbs is eternal and immutable. In optics, from at least the time of Euclid, the angle of reflection is equal to the angle of incidence when a ray of light is reflected. According to some versions of the medieval impetus theory of motion, (...) impetus remains in a projected body (and the associated motion persists) permanently unless the body is subject to outside interference. These examples could be multiplied. But it was in the seventeenth century that conservation principles began to play an absolutely central role in scientific theories. Each of Galileo Galilei, René Descartes, Christiaan Huygens, Gottfried Leibniz, and Isaac Newton founded his approach to physics upon the principle of inertia—that unless interfered with a body will undergo uniform rectilinear motion. A multitude of other conservation principles gained currency during the seventeenth century—some still with us, some long ago left behind. Descartes provides an interesting example of an author who attempted to derive all of his physical principles from conservation laws (Principles of Philosophy, see especially articles 36 to 42 of Part II). Descartes believed that the principles of his physics could be derived from the immutability of God, supplemented only by very weak assumptions about the existence of change in the world. He claims, in fact, that we ought to postulate the strongest conservation laws consistent with such change. These include. (shrink)

In 1672, inspired by the wave theory of Ignace Gaston Pardies, Christiaan Huygens made his first attempt to explain the sine law of refraction, but in 1673 he abandoned his plans owing to difficulties concerning double refraction. Huygens was able to explain double refraction on 6 August 1677 after his discoveries of the axis of symmetry of the crystal and of ‘Huygens's principle’. On 6 August 1679, he wrote: ‘I have found the confirmation of my theory of light and (...) of refraction’, and then described an ‘experimentum crucis’ to decide between his theory and a rival hypothesis. A study of caustics may have led Huygens to the discovery of ‘Huygens's principle’. (shrink)

Contrary to the received opinion, the fundamentals of Newton's dynamics can be set forth quite simply. In the first edition of the Principia, Newton employs a device that relates to Galileo's analysis of uniform rectilinear motion. In the second and third editions, Newton introduces an alternate device that relates to Huygens's analysis of uniform circular motion. A third device is also introduced but is hidden away as a corollary to a problem rather than set forth clearly as a theorem. Following (...) the appearance of the first edition, Newton proposed, but never published, a grand new scheme for revising his dynamics in which he clearly set forth each dynamics ratio with a complementary set of solutions to the problems. As published, however, the alternate methods are not clear and are a source more of confusion than of clarification. (shrink)

The collapse of the Soviet Union created unprecedented dilemmas for the leaders of the new independent Russia. Shedding the communist past, Boris Yeltsin embarked on an ambitious program to reorganize Russia‟s political and economic systems. Known as „shock therapy,‟ Yeltsin advocated a rapid transition from state planning to a market economy while simultaneously introducing democracy to Russia. Expecting a short period of hardship as economic reforms opened Russia to world markets, followed by prolonged growth and prosperity, Yeltsin‟s societal upheaval left (...) Russia a prostrate state, mired in a depression that left many longing for a return to socialism. (shrink)

In French mechanical treatises of the nineteenth century, Newton’s second law of motion was frequently derived from a relativity principle. The origin of this trend is found in ingenious arguments by Huygens and Laplace, with intermediate contributions by Euler and d’Alembert. The derivations initially relied on Galilean relativity and impulsive forces. After Bélanger’s Cours de mécanique of 1847, they employed continuous forces and a stronger relativity with respect to any commonly impressed motion. The name “principle of relative motions” (...) and the very idea of using this principle as a constructive tool were born in this context. The consequences of Poincaré’s and Einstein’s awareness of this approach are analyzed. Lastly, the legitimacy and significance of a relativity-based derivation of Newton’s second law are briefly discussed in a more philosophical vein. (shrink)

Summary The paper follows the lives of Mateu Orfila and François Magendie in early nineteenth-century Paris, focusing on their common interest in poisons. The first part deals with the striking similarities of their early careers: their medical training, their popular private lectures, and their first publications. The next section explores their experimental work on poisons by analyzing their views on physical and vital forces in living organisms and their ideas about the significance of animal experiments in medicine. The last part (...) describes their contrasting research on the absorption of poisons and the divergences in their approaches, methods, aims, standards of proof, and intended audiences. The analysis highlights the connections between nineteenth-century courtrooms and experimental laboratories, and shows how forensic practice not only prompted animal experimentation but also provided a substantial body of information and new research methods for dealing with major theoretical issues like the absorption of poisons. (shrink)

This volume presents a collection of refereed papers reflecting the state of the art in the area of over-constrained systems. Besides 11 revised full papers, selected from the 24 submissions to the OCS workshop held in conjunction with the First International Conference on Principles and Practice of Constraint Programming, CP '95, held in Marseilles in September 1995, the book includes three comprehensive background papers of central importance for the workshop papers and the whole field. Also included is an introduction by (...) one of the volume editors together with a bibliography listing 243 entries. All in all this is a very useful reference book relevant for all researchers and practitioners interested in hierarchical, partial, and over-constrained systems. (shrink)

The article investigates the relationship between Leibniz’s and Huygens’ theory of possibility and the principle of conservation of energy. It assumes that their criticisms of Cartesian views concerning those questions as well as their own achievements contributed to the formation of a new metaphysical basis for modern discussions on the freedom of the will. There are especially two problems whose role is crucial in this context, namely the question of God’s knowledge of future conditionals (contingentia futura) and the mind-body (...) distinction. (shrink)

Kirchhoff’s 1882 theory of optical diffraction forms the centerpiece in the long-term development of wave optics, one that commenced in the 1820s when Fresnel produced an empirically successful theory based on a reinterpretation of Huygens’ principle, but without working from a wave equation. Then, in 1856, Stokes demonstrated that the principle was derivable from such an equation albeit without consideration of boundary conditions. Kirchhoff’s work a quarter century later marked a crucial, and widely influential, point for he produced (...) Fresnel’s results by means of Green’s theorem and function under specific boundary conditions. In the late 1880s, Poincaré uncovered an inconsistency between Kirchhoff’s conditions and his solution, one that seemed to imply that waves should not exist at all. Researchers nevertheless continued to use Kirchhoff’s theory—even though Rayleigh, and much later Sommerfeld, developed a different and mathematically consistent formulation that, however, did not match experimental data better than Kirchhoff’s theory. After all, Kirchhoff’s formula worked quite well in a specific approximation regime. Finally, in 1964, Marchand and Wolf employed the transformation of Kirchhoff’s surface integral that had been developed by Maggi and Rubinowicz for other purposes. The result yielded a consistent boundary condition that, while introducing a species of discontinuity, nevertheless rescued the essential structure of Kirchhoff’s original formulation from Poincaré’s paradox. (shrink)

A rigorous wave-theoretic approach to the Michelson-Morley (M-M) experiment is presented, with special emphasis on the Huygens' principle derivation of the laws of reflection by a moving mirror. A detailed discussion of the Lorentz-Fitzgerald contraction hypothesis (CH) is included. Several mistakes appearing in the standard textbook treatments of these issues are pointed out, and a number of related historical questions are considered.

Twenty-first century science faces a dilemma. Two of its well-verified foundation stones - relativity and quantum theory - have proven inconsistent. Resolution of the conflict has resisted improvements in experimental precision leaving some to believe that some fundamental understanding in our world-view may need modification or even radical reform. Employment of the wave-front model of electrodynamics, as a propagation process with a Markov property, may offer just such a clarification.

Leibniz’s argument against Descartes’s conservation principle in the Brevis demonstratio (1686) has traditionally been read as passing from the premise that motive force must be conserved to the conclusion that motive force is not identical to quantity of motion and, finally, that quantity of motion is not conserved. In a lesser-known draft of the same year, Christiaan Huygens claimed that Descartes had in fact never held the view that Leibniz was attacking. Huygens is right as far as the traditional (...) reading is concerned. However, if the Brevis demonstratio is seen, as it should be, in the context of Leibniz’s attempt to ground mechanics in the foundational metaphysical axiom of the equivalence of full cause and entire effect, then Huygens’s objections can be answered. Leibniz may have committed Descartes to the foundational axiom because of the Traité de la méchanique, which was available to Leibniz at that time. Leibniz had used the equivalence of full cause and entire effect both in discovering the conservation of vis viva earlier and in various arguments for it later. Thus, the Brevis demonstratio should be viewed with Leibniz’s broader foundational mission in mind. (shrink)

Given the eikonal equation σ i=1 3 (∂ψ/∂x i ) 2 =n′ 2, we investigate the geometric structure that underlies the law of propagation of the wavefronts ψ(x 1,x 2,x 3) —ct=0. It turns out that Huygens' principle for the propagation of wavefronts is given in terms of a contact structure. Wavefronts are carried into wavefronts by contact transformations. As regards the wave-particle duality principle that arises in quantum mechanics, there is a natural geometric structure, a symplectic manifold (...) (M 2n , Ω), which “unifies” Fermat's principle and the eikonal equation (Huygens' principle). (shrink)

The pendulum clock was one of the most important metaphors for early modern philosophers. Christiaan Huygens (1629-1695) discovered his pendulum clock in 1656 based on the principle of isochronism discovered by Galileo (1564-1642). This paper aims at exploring the broad historical context of this invention, showing the role of some key figures such as Andreas Colvius (1594-1671), Elia Diodati (1576-1661), Hugo Grotius (1583-1645) and Constantijn Huygens, the father of Christiaan Huygens. Secondly, it suggests - based on this context - (...) that it is hard to believe that Huygens did not know about Galileo’s idea to construct a pendulum regulated clock. Finally, this article illustrates how the Dutch philosopher Spinoza (1632-1677) might have been inspired by Huygens’ discovery of the synchronization of the pendulum clocks in his views on the agreement between bodies in the universe. -/- Resumo: O relógio de pêndulo foi uma das metáforas mais importantes para os filósofos modernos. Christiaan Huygens (1629-1695) inventou o relógio de pêndulo em 1656 baseado no princípio do isocronismo descoberto por Galileo (1564-1642). Este artigo busca explorar o amplo contexto histórico dessa invenção, demonstrando o papel de algumas figuras-chave como Andreas Colvius (1594-1671), Elia Diodati (1576-1661), Hugo Grotius (1583-1645) e Constantijn Huygens, o pai de Christiaan Huygens. Em segundo lugar, sugere-se - baseado nesse contexto - que é difícil acreditar que Huygens não sabia da ideia de Galileo de construir um relógio regulado por um pêndulo. Por fim, este artigo ilustra como o filósofo holandês Espinosa (1632-1677) pode ter se inspirado nessa invenção de Huygens da sincronização do relógio de pêndulo em suas visões sobre o acordo entre os corpos no universo. -/- Palavras-chave: Galileo; Relógio de pêndulo; Huygens; República holandesa; Espinosa. -/- . (shrink)

The sixth chapter focuses on the evolution of Cartesianism in the last quarter of the seventeenth century in Leiden and Amsterdam, against the background of the emergence of alternative views in natural philosophy capable of replacing it as a dominant paradigm, namely, the experimental philosophy of Robert Boyle and the mathematical-experimental approach of Huygens and Newton. The last evolution of Cartesianism is reconstructed in this chapter by considering the ‘Cartesian empiricism’ of Burchard de Volder, and the reflections on the language (...) of philosophy and practical disciplines by De Raey in Amsterdam, where he moved in 1669. In 1675 De Volder established the Leiden Theatrum physicum. There he performed experiments in order to teach the principles of a mechanical philosophy largely inspired by Descartes but open to the use of experimental and mathematical evidences in the formulation of natural laws. A similar approach was in fact assumed in the same years by Wolferd Senguerd, who used the Theatrum to perform experiments in pneumatics to teach some of the principles of his eclectic worldview, encompassing some Cartesian principles (such as that of the circularity of movement), but also rejecting the vortex theory. Yet, only De Volder developed a foundational theory for the basic principles of mechanicism, namely, the assumption that every phenomenon can be explained by the notions of matter and movement alone. This was the result of a movement internal to Cartesianism, as De Volder not only reacted (positively) to the emergence of an experimental-mathematical natural philosophy, but was also involved in the defence of Cartesianism against the Censura philosophiae cartesianae of PierreDaniel Huet (1689), in which Descartes’s metaphysics is rejected as inconsistent given its very foundation on doubt and cogito. The intermingling of these different issues resulted, in De Volder’s hands, in a further de-metaphysicisation of physics – as metaphysics cannot provide a justification for the laws of motion – and in the narrowing of the scope of foundationalism, which can only sanction the psychological character of clarity and distinction as a criterion for internal truth, defined in terms of indubitability only. Accordingly, for De Volder metaphysics cannot demonstrate, on a Cartesian basis, that phenomena are actually ruled by the principles of mechanism: insofar as, for him, metaphysics has a prominent reflective role, and loses its status as justification of the absolute truth of scientific statements. This process can be labelled as the transition from foundationalism to philosophy of science and does not characterise only his ‘Cartesian empiricism’. In Amsterdam, De Raey was, over the same years, developing his Cogitata de interpretatione (1692), embodying one of the first, self-standing philosophical considerations of language. Still maintaining his separation thesis, and attacking Hobbes and the radical Cartesians, in this text he aimed to clarify how words meaning sensory data and abstract notions (such as those of mathematics) can be used in philosophy. Rather than setting a method for how to use mental faculties, De Raey aimed, at the end of his career, to provide an updating of the linguistic meanings of scientific vocabulary, namely, a reflection rather than a justification of science. (shrink)

Algebraic equations in the tradition of Descartes and Frans Van Schooten accompany Christiaan Huygens’s early work on collision, which later would be reorganized and presented as De motu corporum ex percussione. Huygens produced the equations at the same time as his announcement of his rejection of Descartes’s rules of collision. Never intended for publication, the equations appear to have been used as preliminary scaffolding on which to build his critiques of Descartes’s physics. Additionally, Huygens used algebraic equations of this (...) form to accurately predict the speeds of bodies after collision in experiments carried out at the Royal Society. Despite their deceptive simplicity, Huygens’s algebraic equations pose significant conceptual problems both mathematically and for their physical interpretation especially for negative speeds; they may very well have been the source of a new principle, the conservation of quantity of motion with direction. (shrink)

Gottfried Wilhelm Leibniz possessed one of history's great minds. The German philosopher, mathematician, and logician invented calculus. His metaphysics bequeathed a set of problems and approaches that drove the course of Western philosophy, from Kant's eighteenth century until the present day. For over fifty years, the study of Leibniz has been a consistent passion for distinguished philosopher Nicholas Rescher. _On Leibniz_ offers eleven of his essays, written with signature clarity, exploring the aspects of Leibniz's work and life that still resonate (...) in the discipline of philosophy. Rescher's essays are snapshots of Leibniz, lucidly drawn case studies that explain the fundamentals of his ontology: the theory of possible worlds, the world's contingency, space-time frameworks, and intermonadic relationships. Several illuminating pieces reveal Leibniz as a substantial contributor to theories of knowledge. Discussions of his epistemology and methodology, its relationship to John Maynard Keynes and Talmudic scholarship, broaden the traditional view of Leibniz as a uniquely metaphysical thinker. Rescher also explores, in four absorbing biographical essays, Leibniz's scholarly development and professional career in historical context. As a "philosopher courtier" to the Hanoverian court, Leibniz was associated with the leading intellectuals and politicians of his era, including Spinoza, Huygens, Newton, Queen Sophie Charlotte, and Czar Peter the Great. A concluding essay holds up Leibniz's mode of philosophy as a role model for today's scholars. Rescher argues that many current problems can be effectively addressed with principles of process philosophy along lines inspired by Leibniz's monadology. _On Leibniz_ is essential reading for students of Leibniz and Rescher alike. (shrink)

Ever since Newton created dynamics, there has been controversy about its foundations. Are space and time absolute? Do they form a rigid but invisible framework and container of the universe? Or are space, time, and motion relative? If so, does Newton's 'framework' arise through the influence of the universe at large, as Ernst Mach suggested? Einstein's aim when creating his general theory of relativity was to demonstrate this and thereby implement 'Mach's Principle'. However, it is widely believed that he (...) achieved only partial success. This question of whether motion is absolute or relative has been a central issues in philosophy; the nature of time has perennial interest. Current attempts to create a quantum description of the whole universe keep these issues at the cutting edge of modern research. Written by the world's leading expert on Mach's Principle, The Discovery of Dynamics is a highly original account of the development of notions about space, time, and motion. Widely praised in its hardback version, it is one of the fullest and most readable accounts of the astronomical studies that culminated in Kepler's laws of planetary motion and of the creation of dynamics by Galileo, Descartes, Huygens, and Newton. Originally published as Absolute or Relative Motion?, Vol. 1: The Discovery of Dynamics, The Discovery of Dynamics provides the technical background to Barbour's recently published The End of Time, in which he argues that time disappears from the description of the quantum universe. (shrink)

Ever since Newton created dynamics, there has been controversy about its foundations. Are space and time absolute? Do they form a rigid but invisible framework and container of the universe? Or are space, time, and motion relative? If so, does Newton's 'framework' arise through the influence of the universe at large, as Ernst Mach suggested? Einstein's aim when creating his general theory of relativity was to demonstrate this and thereby implement 'Mach's Principle'. However, it is widely believed that he (...) achieved only partial success. This question of whether motion is absolute or relative has been a central issues in philosophy; the nature of time has perennial interest. Current attempts to create a quantum description of the whole universe keep these issues at the cutting edge of modern research. Written by the world's leading expert on Mach's Principle, The Discovery of Dynamics is a highly original account of the development of notions about space, time, and motion. Widely praised in its hardback version, it is one of the fullest and most readable accounts of the astronomical studies that culminated in Kepler's laws of planetary motion and of the creation of dynamics by Galileo, Descartes, Huygens, and Newton. Originally published as Absolute or Relative Motion?, Vol. 1: The Discovery of Dynamics, The Discovery of Dynamics provides the technical background to Barbour's recently published The End of Time, in which he argues that time disappears from the description of the quantum universe. (shrink)

THE ART OF CAUSAL CONJECTURE Glenn Shafer Table of Contents Chapter 1. Introduction........................................................................................ ...........1 1.1. Probability Trees..........................................................................................3 1.2. Many Observers, Many Stances, Many Natures..........................................8 1.3. Causal Relations as Relations in Nature’s Tree...........................................9 1.4. Evidence............................................................................................ ...........13 1.5. Measuring the Average Effect of a Cause....................................................17 1.6. Causal Diagrams..........................................................................................20 1.7. Humean Events............................................................................................23 1.8. Three Levels of Causal Language................................................................27 1.9. An Outline of the Book................................................................................27 Chapter 2. Event Trees............................................................................................... .....31 2.1. Situations and Events...................................................................................32 2.2. The Ordering of Situations and Moivrean Events.......................................35 2.3. Cuts................................................................................................ ..............39 2.4. Humean Events............................................................................................43 2.5. (...) Moivrean Variables......................................................................................49 2.6. Humean Variables........................................................................................53 2.7. Event Trees for Stochastic Processes...........................................................54 2.8. Timing in Event Trees.................................................................................56 2.9. Intersecting Event Trees...............................................................................60 2.10. Notes on the Literature...............................................................................61 Chapter 3. Probability Trees...........................................................................................63 3.1. Some Types of Probability Trees.................................................................64 ii 3.2. Axioms for the Probabilities of Moivrean Events.......................................68 3.3. Zero Probabilities....................................................................................... ..70 3.4. A Sample-Space Analysis of the Event-Tree Axioms.................................72 3.5. Probabilities and Expected Values for Variables.........................................74 3.6. Martingales......................................................................................... .........79 3.7. The Expectation of a Variable in a Cut........................................................83 3.8. Conditional Expected Value and Conditional Expectation.........................87 Chapter 4. The Meaning of Probability...........................................................................91 4.1. The Interpretation of Expected Value..........................................................92 4.2. The Interpretation of Expectation................................................................95 4.3. The Long Run..............................................................................................98 4.4. Changes in Belief.........................................................................................101 4.5. The Empirical Validation of Probability......................................................106 4.6. The Diversity of Uses of Probability...........................................................108 4.7. Notes on the Literature.................................................................................110 Chapter 5. Independent Events.......................................................................................113 5.1. Independence........................................................................................ .......114 5.2. Weak Independence.....................................................................................118 5.3. The Principle of the Common Cause...........................................................121 5.4. Conditional Independence............................................................................128 5.5. Notes on the Literature.................................................................................133 Chapter 6. Events Tracking Events.................................................................................135 6.1. Tracking............................................................................................ ...........137 6.2. Tracking and Conditional Independence.....................................................142 6.3. Stochastic Subsequence...............................................................................143 6.4. Singular Diagrams for Stochastic Subsequence...........................................147 6.5. Conjunctive and Interactive Forks...............................................................149 Chapter 7. Events as Signs of Events..............................................................................153 iii 7.1. Sign................................................................................................ ..............154 7.2. Weak Sign................................................................................................ ....159 7.3. The Ethics of Causal Talk............................................................................160 7.4. Screening Off...............................................................................................16 2 Chapter 8. Independent Variables...................................................................................167 8.1. Unconditional Independence........................................................................170 8.2. Conditional Independence............................................................................175 8.3. Independence for Partitions.........................................................................177 8.4. Independence for Families of Variables......................................................182 8.5. Individual Properties of the Independence Relations...................................186 Chapter 9. Variables Tracking Variables........................................................................189 9.1. Tracking and Conditional Independence: A Summary...............................190 9.2. Strong Tracking............................................................................................ 192 9.3. Strong Tracking and Conditional Independence..........................................198 9.4. Stochastic Subsequence...............................................................................201 9.5. Functional Dependence................................................................................203 9.6. Tracking in Mean.........................................................................................204 9.7. Linear Tracking............................................................................................ 207 9.8. Tracking by Partitions..................................................................................210 9.9. Tracking by Families of Variables...............................................................212 Chapter 10. Variables as Signs of Variables...................................................................215 10.1. Sign................................................................................................ ............219 10.2. Linear Sign................................................................................................ .222 10.3. Scored Sign................................................................................................ 225 10.4. Families of Variables.................................................................................227 Chapter 11. AnTheory of Event Trees.............................................................229 11.1. Event Trees as Sets of Sets........................................................................230 11.2. Event Trees as Partially Ordered Sets........................................................232 iv 11.3. Regular Event Trees...................................................................................240 11.4. The Resolution of Moivrean Variables......................................................244 11.5. Humean Events and Variables...................................................................246 Chapter 12. Martingale Trees..........................................................................................247 12.1. Examples of Decision Trees......................................................................249 12.2. The Meaning of Probability in a Decision Tree.........................................253 12.3. Martingales......................................................................................... .......257 12.4. The Structure of Martingale Trees.............................................................261 12.5. Probability and Causality...........................................................................265 12.6. Lower and Upper Probability.....................................................................269 12.7. The Law of Large Numbers.......................................................................272 12.8. Notes on the Literature...............................................................................274 Chapter 13. Refining............................................................................................ ...........275 13.1. Examples of Refinement............................................................................277 13.2. A Constructive Definition of Finite Refinement.......................................281 13.3. Axioms for Refinement..............................................................................282 13.4. Lifting Moivrean Events and Variables.....................................................288 13.5. Refining Martingale Trees.........................................................................288 13.6. Grounding........................................................................................... .......294 Chapter 14. Principles of Causal Conjecture..................................................................299 14.1. The Diversity of Causal Explanation.........................................................302 14.2. The Mean Effect of the Happening of a Moivrean Event..........................305 14.3. The Effect of a Humean Variable..............................................................311 14.4. Attribution and Generality.........................................................................316 14.5. The Statistical Measurement of the Effect of a Cause...............................319 14.6. Measurement by Experiment.....................................................................320 14.7. Using Our Knowledge of How Things Work............................................322 v 14.8. Sampling Error...........................................................................................329 14.9. The Sampling Frame..................................................................................329 14.10. Notes on the Literature.............................................................................330 Chapter 15. Causal Models.............................................................................................3 31 15.1. The Causal Interpretation of Statistical Prediction....................................333 15.2. Generalizing to a Family of Exogenous Variables....................................337 15.3 Some Joint Causal Diagrams......................................................................339 15.4. Causal Path Diagrams................................................................................342 15.5. Causal Relevance Diagrams.......................................................................346 15.6. The Meaning of Latent Variables..............................................................352 15.7. Notes on the Literature...............................................................................357 Chapter 16. Representing Probability Trees...................................................................359 16.1. Three Graphical Representations...............................................................361 16.2. Skeletal Simplifications.............................................................................368 16.3. Martingale Trees in Type Theory...............................................................371 Appendix A. Huygens’s Probability Trees.....................................................................379 Huygens’s Manuscript in Translation..................................................................380 Appendix B. Some Elements of Graph Theory..............................................................385 B1. Undirected Graphs........................................................................................385 B2. Directed Graphs............................................................................................38 6 Appendix C. Some Elements of Order Theory...............................................................393 C1. Partial and Quasi Orderings.........................................................................393 C2. Singular and Joint Diagrams for Binary Relations.......................................394 C3. Lattices............................................................................................ .............395 C4. The Lattice of Partitions of a Set..................................................................396 Appendix D. The Sample-Space Framework for Probability.........................................399 D1. Probability Measures....................................................................................399 D2. Variables........................................................................................... ...........400 vi D3. Families of Variables...................................................................................401 D4. Expected Value............................................................................................402 D5. The Law of Large Numbers.........................................................................405 D6. Conditional Probability................................................................................406 D7. Conditional Expected Value........................................................................407 Appendix E. Prediction in Probability Spaces................................................................409 E1. Conditional Distribution...............................................................................411 E2. Regression on a Single Variable...................................................................412 E3. Regression on a Partition or a Family of Variables......................................415 E4. Linear Regression on a Single Variable.......................................................418 E5. Linear Regression on a Family of Variables................................................422 Appendix F. Sample-Space Concepts of Independence.................................................425 F1. Overview............................................................................................ ...........426 F2. Independence Proper.....................................................................................432 F3. Unpredictability in Mean..............................................................................434 F4. Simple Uncorrelatedness..............................................................................437 F5. Mixed Uncorrelatedness...............................................................................438 F6. Partial Uncorrelatedness...............................................................................440 F7. Independence for Partitions..........................................................................442 F8. Independence for Families of Variables.......................................................445 F9. The Basic Role of Uncorrelatedness.............................................................448 F10. Dawid’s Axioms.........................................................................................449 Appendix G. Prediction Diagrams..................................................................................453 G1. Path Diagrams............................................................................................ ..454 G2. Generalized Path Diagrams..........................................................................462 G3. Relevance Diagrams.....................................................................................466 G4. Bubbled Relevance Diagrams......................................................................475 Appendix H. Abstract Stochastic Processes...................................................................479 vii H1. Probability Conditionals and Probability Distributions...............................477 H2. Abstract Stochastic Processes......................................................................479 H3. Embedding Variables and Processes in a Sample Space.............................480 References.......................................................................................... ..............................491. (shrink)

This item is a chapter from a book in progress, entitled "True Motion". Leibniz’s mechanics was, as we shall see, a theory of elastic collisions, not formulated like Huygens’ in terms of rules explicitly covering every possible combination of relative masses and velocities, but in terms of three conservation principles, including (effectively) the conservation of momentum and kinetic energy. That is, he proposed what we now call (ironically enough) ‘Newtonian’ (or ‘classical’) elastic collision theory. While such a theory is, for (...) instance, vital to the foundations of the kinetic theory of gases, it is not applicable to systems – like gravitational systems – in which fields of force are present. Thus, Leibniz’s mechanical principles never led to developments of the order of Newton’s in the Principia (additionally, he hamstrung their application by embedding them in a baroque philosophical system). (shrink)

The way Leibniz applied his philosophy to mathematics has been the subject of longstanding debates. A key piece of evidence is his letter to Masson on bodies. We offer an interpretation of this often misunderstood text, dealing with the status of infinite divisibility in nature, rather than in mathematics. In line with this distinction, we offer a reading of the fictionality of infinitesimals. The letter has been claimed to support a reading of infinitesimals according to which they are logical fictions, (...) contradictory in their definition, and thus absolutely impossible. The advocates of such a reading have lumped infinitesimals with infinite wholes, which are rejected by Leibniz as contradicting the part–whole principle. Far from supporting this reading, the letter is arguably consistent with the view that infinitesimals, as inassignable quantities, are mentis fictiones, i.e., (well-founded) fictions usable in mathematics, but possibly contrary to the Leibnizian principle of the harmony of things and not necessarily idealizing anything in rerum natura. Unlike infinite wholes, infinitesimals—as well as imaginary roots and other well-founded fictions—may involve accidental (as opposed to absolute) impossibilities, in accordance with the Leibnizian theories of knowledge and modality. (shrink)

Leibniz is well known for his formulation of the infinitesimal calculus. Nevertheless, the nature and logic of his discovery are seldom questioned: does it belong more to mathematics or metaphysics, and how is it connected to his physics? This book, composed of fourteen essays, investigates the nature and foundation of the calculus, its relationship to the physics of force and principle of continuity, and its overall method and metaphysics. The Leibnizian calculus is presented in its origin and context together (...) with its main contributors: Archimedes, Cavalieri, Wallis, Hobbes, Pascal, Huygens, Bernoulli, and Nieuwentijt. Many of us know and probably have used the Leibnizian formula: to .. (shrink)

This slight volume contains three short essays by the author: “The Idea of Nature In Contemporary Physics”, “Atomic Physics and Causal Law”, and “Classical Education”. Much more than half of the book is given over to a selection of brief readings from Kepler, Galileo, Newton, Huygens, D’Alembert, De la Mettrie, Ostwald, Hertz, and a short historical review by de Broglie of the evolution of quantum mechanics. These readings are meant to illustrate the author’s overall theme which appears to be this: (...) the development of modern science has been marked by a progressive change in the concept of nature. The mediaevals regarded Nature as God’s handiwork: “it would have been thought senseless to ask questions about the material world without reference to God”, Between Galileo and Newton a quite new outlook arose which encouraged the belief that an accurate and exhaustive account of Nature could be made in mathematical terms which made reference only to matter and motion. This was the mechanist view of Nature, which led logically to materialism. But modern physics has shown the inadequacy of this view. “The object of research is no longer nature itself, but man’s investigation of nature”. “Nature itself cannot be known”, only “man’s relationship with nature”. The uncertainty principle proves that an “objective” picture of Nature is impossible. The observer and the observed system cannot be clearly separated, and thus we must give up the “Cartesian” distinction between a world of “objective processes in space and time and a mind in which these processes are mirrored”. Thus, we no longer regard “nature” as God’s handiwork nor as an objective space-time-matter manifold but either as the “footprint of man” upon an unknowable substratum, or else as “the transparent clarity of a mathematics that no longer describes the behaviour of elementary particles but only our knowledge of this behaviour”. There is no “philosophy of nature” properly speaking; our knowledge of “nature”, such as it is, is derived from a philosophical consideration of science. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Infinitesimal Differences: Controversies between Leibniz and his ContemporariesFrançoise Monnoyeur-BroitmanUrsula Goldenbaum and Douglas Jesseph, editors. Infinitesimal Differences: Controversies between Leibniz and his Contemporaries. Berlin-New York: Walter de Gruyter, 2008. Pp. vi + 327. Cloth, $109.00.Leibniz is well known for his formulation of the infinitesimal calculus. Nevertheless, the nature and logic of his discovery are seldom questioned: does it belong more to mathematics or metaphysics, and how is it connected (...) to his physics? This book, composed of fourteen essays, investigates the nature and foundation of the calculus, its relationship to the physics of force and principle of continuity, and its overall method and metaphysics. The Leibnizian calculus is presented in its origin and context together with its main contributors: Archimedes, Cavalieri, Wallis, Hobbes, Pascal, Huygens, Bernoulli, and Nieuwentijt.Many of us know and probably have used the Leibnizian formula: to calculate the area under a curve; this book considers the origin, nature, method, and metaphysics of this formula. The most fascinating question discussed in it is the fiction introduced to define the infinitesimal calculation: in his mature period, Leibniz himself referred to his calculus as “a well-founded fiction.” We learn that he gradually gave up the idea of an actual infinite in favor of describing infinitely small quantities. As a result, the quantities became fictive and finite or indefinite instead of real and infinite, and the process of calculation shifted towards mathematics rather than metaphysics.In the first essay, Richard Arthur compares the calculus of Leibniz and Newton and demonstrates how it was based for both on the development of continuously varying quantity. He also shows how the understanding of this quantity derived from the Archimedean axiom that excludes the existence of actual infinite quantity. Ursula Goldenbaum next demonstrates how Hobbes, with his conatus considered as point and instant, had a direct impact on the Leibnizian conception of continuity; to confirm her interpretation, she includes at the end of her article the “Marginalia or Leibniz comments” on Hobbes. Samuel Levey establishes a connection between the notion of the infinitesimal as a “well-founded fiction” and the law of continuity, arguing that the law of continuity, understood as an extension of Archimedes’ principle, gives a finite foundation to the infinitesimal calculation. O. Bradley Bassler discusses the idea of the differential in terms of infinitely small quantities and the use of proportions between infinitely small and finite quantities in order to measure [End Page 527] the infinitely small; here, differentials are also presented as fictions. Fritz Nagel discusses the methodological sophistication of the calculus and its opponents. His main questions concern the different grades of infinity, quantities regarded as zero, and the infinitesimal as a fiction that can be calculated. Douglas Jesseph considers the reality of the infinitesimal and the “well-founded fiction,” showing how Leibniz methodically constructs infinitesimal quantities without reality by returning to Hobbes’ notion of conatus. Hobbes is, for Jesseph, the initiator of these infinitely small quantities without existence.A second group of essays examines the connection between algebra and geometry in the calculus. Philip Beeley demonstrates how John Wallis, on the path of Cavalieri’s method of the indivisibles, found a way to transform geometric problems into sums of arithmetic sequences. Siegmund Probst expresses the different phases in the discovery of the Leibnizian method. He mentions that Leibniz coined the term ‘infinitesimal’ in 1673, and points out that he did not need the method of indivisibles to find a curve from its given arc, but only a series expansion using differences of higher order. Emily Grosholz looks at how, thanks to his algebraic and geometrical notations of curves, Leibniz could formulate the equation of his infinitesimal calculus. According to Grosholz, Leibniz exploits the ambiguity of notations in order to express the geometrical and dynamical aspect of the calculus. Herbert Breger underlines the novelty and abstract nature of Leibniz’s new method of calculation, showing how the theory of infinitesimals became an algebraic calculus independent of geometry.A final group of articles studies the relationship between the calculus and Leibniz’s physics. François Duchesneau shows how the differential and integral calculus provided a model for the integrative laws of Leibniz’s... (shrink)

This is a truly outstanding contribution to contemporary scholarship on Leibniz’s methodology and natural philosophy. By providing the original edition of Leibnizian manuscripts dating back to the period 1676-1680, in particular the De corporum concursu, Michel Fichant has confirmed what some previous scholars had surmised—that Leibniz had substituted his new principle of conservation of vis viva for the Cartesian principle of conservation of quantity of motion measured by mv shortly after his return from France, and quite a few (...) years before the publication of the Brevis demonstratio erroris memorabilis Cartesii in the Acta Eruditorum. Fichant has conclusively shown that the turning point rightly occurs in the midst of Leibniz’s writing the De corporum concursu in January-February 1678. At the time, Leibniz was groping for a coherent system of physical and mathematical reasons to ground the mechanical laws of motion beyond the relativistic formulas which Huygens, Wallis, and Wren had provided in response to the enquiry launched by the Royal Society in 1668. (shrink)

I address a question recently raised by Simon Saunders concerning the relationship between the space-time structure of Newton-Cartan theory and that of what I will call “Maxwell-Huygens space-time.” This discussion will also clarify a connection between Saunders’s work and a recent paper by Eleanor Knox.

I explain and assess here Huygens’ concept of relative motion. I show that it allows him to ground most of the Law of Inertia, and also to explain rotation. Thereby his concept obviates the need for Newton’s absolute space. Thus his account is a powerful foundation for mechanics, though not without some tension.

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 61.3 (2000) 415-432 [Access article in PDF] Christiaan Huygens's Attitude toward Animals Nathaniel Wolloch The debate on the status of animals has interested people since ancient times. In the early modern era this debate reached one of its most historically important and sedulous stages, drawing the attention of some of the most famous minds in Europe. Curiously enough, the historiography of this debate (...) has failed to take cognizance of the part played in it by the famous Dutch scientist Christiaan Huygens (1629-95), renowned for his invention of the pendulum clock, founding of the wave theory of light, and discovery of the true shape of the rings of Saturn, to name only a few of his prolific and varied activities. 1 While Huygens's contribution does not occupy a historically central part in this debate, which did not usually directly engross his attention, it is possible, based on some of his writings and activities, to piece together a picture of his general attitude toward animals. Such a study, as presented here, reveals more than anecdotal importance, since it both entails and affords an enlightening discussion of the connection between philosophy, science, and the treatment of animals, a discussion, moreover, which is relevant to the understanding not only of the early modern debate on animals, but also to the current debate of this issue. [End Page 415]One of the most prevalent features of the development of the attitude toward animals has been the recurrence of anthropocentric arguments. Such arguments were exhibited both by theriophiles (lovers of animals) and anti-theriophiles. 2 A significant instance of this is the recurring argument that harming animals is not bad in itself but rather because it might lead to the harming or corruption of human beings. While such an approach in effect states that those who are kind to animals would also be kind to human beings, it also implies that cruelty toward animals is excusable if committed to advance good causes, such as science and medicine. 3 Such anthropocentric views ultimately derive from both classical culture and philosophy, and especially the notion, rooted in biblical cosmology, according to which all of nature was given to man for his use by God, i.e., man was given the stewardship of nature. 4Theriophilic views which tend to regard animals as sentient beings in their own right became relatively common in early modern times. The most influential of these theriophiles was Michel de Montaigne, known for his love and praise [End Page 416] of animals, 5 and among the many theriophiles influenced by his views one of the most important was his disciple Pierre Charron. 6However, the most influential early modern view of animals, which became the main reference-point for the debate on animals, was anti-theriophilic. This was the Cartesian theory of the "beast-machine," which viewed animals as mere automata, devoid of an immortal soul as possessed by human beings; they were regarded as devoid of mind, and their behavior was compared to that of a clock. Descartes himself seems to have been rather agnostic regarding this issue. He did not deny animals some capacity for feeling but claimed that it was more probable they were senseless automata than beings with souls like human beings. 7 However, while many theriophiles attacked the beast-machine theory, its anti-theriophilic implications were developed and radicalized by various seventeenth-century Cartesians beyond what Descartes himself probably intended. The most prominent example of this was the leading Cartesian Nicolas Malebranche. The French intellectual Bernard Le Bovier, sieur de Fontenelle, related how he saw Malebranche, a generally kind man, kick a pregnant bitch, and when confronted with his (Fontenelle's) alarm, regarded this act as inconsequential, claiming that she did not feel a thing. 8 As cruel as the beast-machine [End Page 417] theory may seem, however, one should remember that it tried to explain away the suffering of animals. It implied that had such suffering been proven to exist, it would have required moral consideration. The views of Spinoza... (shrink)