This page was last updated on October 29th, 2011
r i c h a r d T. W. a r t h u r

p r o f e s s o r . o f . p h i l o s o p h y, . m c m a s t e r . u n i v e r s i t y

 

a r t i c l e s... o n l i n e

o t h e r

1 7 t h . c e n t u r y . p h i l o s o p h y

The remarkable fecundity of Leibniz's work on infinite series: a review article on 2 Akademie volumes of Leibniz's writings, VII, 3: 1672-76: Differenzen, Folen Reihen, and III, 5: Mathematischer, naturwissenschaftlicher und technischer Briefwechsel .

Beeckman, Descartes and the force of motion: Journal for the History of Philosophy, 45,1, 1-28. In this reassessment of Descartes' debt to his mentor Isaac Beeckman, I argue that they share the same basic conception of motion: the force of a body's motion‚ understood as the force of persisting in that motion, shorn of any connotations of internal cause‚ is conserved through God's direct action, is proportional to the speed and magnitude of the body, and is gained or lost only through collisions. I contend that this constitutes a fully coherent ontology of motion, original with Beeckman and consistent with his atomism. Without acknowledging his debt to Beeckman, and despite his rejection of the latter's atomism, Descartes adopted the same basic conception; whereupon, and notwithstanding his own profoundly original contributions to the theory of motion, it becomes the bedrock of Descartes' own work in natural philosophy.

 The Enigma of Leibniz's Atomism: Oxford Studies in Early Modern Philosophy. Volume 1, 2003, 183-227.

Animal Generation and Substance in Sennert and Leibniz: to appear as a chapter in The Problem of Animal Generation in Modern Philosophy, ed. Justin Smith. Cambridge: Cambridge University Press, 2005.

Leibniz and the Zenonists: a reply to Paolo Rossi: This is an English translation of my first Italian publication: "Lo zenonismo come fonte delle monadi di Leibniz: una risposta a Paolo Rossi", a reply to Professor Rossi on the role of the Zenonists in the genesis of Leibniz's thought, Rivista di storia della filosofia (n. 2, 2003, 335-340). Rossi gave a rejoinder in the same issue (341-349).

Newton's Proof of the Vector Addition of Motive Forces: forthcoming in Infinitesimals, ed. William Harper and Wayne C. Myrvold, 2005?

 The transcendentality of π (pi) and Leibniz's philosophy of mathematics: Proceedings of the Canadian Society for History and Philosophy of Mathematics, 12, 13-19, 1999. Here I show that in an unpublished paper of 1676 (A VI iii N69) Leibniz conjectured that (pi) cannot be expressed even as the irrational root "of an equation of any degree", thus anticipating Legendre's famous conjecture of the transcendentality of by some 118 years.
 

. . t h e . i n f i n i t e ... ... ... ... ... ... ... ... ... ..back to top| back to home page

Leibniz and Cantor on the Actual Infinite: This constitutes the gist of a dialogue I am preparing in which Leibniz and Cantor debate the nature of the infinite. Although the paper is rough, the basic argument is discernible: i) Leibniz's syncategorematic actual infinite is a consistent third alternative to the Cantorian actual infinite and the Aristotelian potential infinite; ii) it is appropriate to his conception of the actual infinite division of matter as not involving infinite number; whereas iii) Cantor's actual infinite is not appropriate to such infinite division, since one cannot get to an infinitieth part by recursively dividing.

'A Complete Denial of the Continuous?' Leibniz's Law of Continuity: Noting the status of the Law of Continuity as one of Leibniz’s most cherished axioms, Bertrand Russell charged that his philosophy nevertheless amounted to “a complete denial of the continuous”. Georg Cantor made a similar accusation of inconsistency about Leibniz’s philosophy of the actual infinite. But I argue that neither doctrine is inconsistent when the subtleties of Leibniz’s syncategorematic interpretation are properly taken into account. Leibniz rejects the existence of infinite wholes: an infinite aggregate of actual things forms only a fictitious whole. Analogously, infinitesimals are only fictitious parts, this time of ideal wholes. That is, just as an actual infinity of terms can be understood syncategorematically as more terms than can be assigned a number, without there being any infinite numbers, so too the infinitely small can be given a syncategorematic interpretation by means of the Law of Continuity, without there existing any actual infinitesimals. By examining Leibniz’s justification of infinitesimals in his calculus, I argue that the syncategorematic interpretation is also applicable to series of changes, and thus exonerates Leibniz from Russell’s criticism: on this interpretation all naturally occurring transitions are continuous in that the difference between neighbouring states is smaller than any assignable. This means not that there exists a least difference, but that for any assignable finite difference, there exists a smaller one. Thus there is a true continuous transition, even though the states themselves and all assignable differences between them are actually discrete.

From Actuals to Fictions: Four Phases in Leibniz’s Early Thought On Infinitesimals: In this paper I attempt to trace the development of Gottfried Leibniz’s early thought on the status of the actually infinitely small in relation to the continuum. I argue that before he arrived at his mature interpretation of infinitesimals as fictions, he had advocated their existence as actually existing entities in the continuum. From among his early attempts on the continuum problem I distinguish four distinct phases in his interpretation of infinitesimals: (i) (1669) the continuum consists of assignable points separated by unassignable gaps; (ii) (1670-71) the continuum is composed of an infinity of indivisible points, or parts smaller than any assignable, with no gaps between them; (iii) (1672-75) a continuous line is composed not of points but of infinitely many infinitesimal lines, each of which is divisible and proportional to a generating motion at an instant (conatus); (iv) (1676 onward) infinitesimals are fictitious entities, which may be used as compendia loquendi to abbreviate mathematical reasonings; they are justifiable in terms of finite quantities taken as arbitrarily small, in such a way that the resulting error is smaller than any pre-assigned margin. Thus according to this analysis Leibniz arrived at his interpretation of infinitesimals as fictions already in 1676, and not in the 1700's in response to the controversy between Nieuwentijt and Varignon, as is often believed.

. . p h i l o s o p h y . o f . t i m e                                             .back to top| back to home page

Minkowski Spacetime and the Dimensions of the Present: to appear in a volume on the Ontology of Spcetime, edited by Dennis Dieks. In Einstein-Minkowski spacetime, because of the relativity of simultaneity to the inertial frame chosen, there is no unique world-at-an-instant. Thus the classical view that there is a unique set of events existing now in a three dimensional space cannot be sustained. The two solutions most often advanced are (i) that the four-dimensional structure of events and processes is alone real, and that becoming present is not an objective part of reality; and (ii) that present existence is not an absolute notion, but is relative to inertial frame; the world-at-an-instant is a three dimensional, but relative, reality. According to a third view, advanced by Robb, Capek and Stein, (iii) what is present at a given spacetime point is, strictly speaking, constituted by that point alone. I argue here against the first of these views that the four-dimensional universe cannot be said to exist now, already, or indeed at any time at all; so that talk of its existence or reality as if that precludes the existence or reality of the present is a non sequitur. The second view assumes that in relativistic physics time lapse is measured by the time co-ordinate function; against this I maintain that it is in fact measured by the proper time, as I argue by reference to the Twin Paradox. The third view, although formally correct, is tarnished by its unrealistic assumption of point-events. This makes it susceptible to paradox, and also sets it at variance with our normal intuitions of the present. I argue that a defensible concept of the present is nonetheless obtainable when account is taken of the non-instantaneity of events, including that of conscious awareness, as (iv) that region of spacetime comprised between the forward lightcone of the beginning of a small interval of proper time t (e.g. that during which conscious experience is laid down) and the backward lightcone of the end of that interval. This gives a serviceable notion of what is present to a given event of short duration, as well as saving our intuition of the "reality" or robustness of present events.

. . p h i l o s o p h y . o f . p h y s i c s                                    .back to top| back to home page

On thought experiments as a priori science: International Studies in the Philosophy of Science, 13, 3, 215-229, 1999. Against Norton's claim that all thought experiments can be reduced to explicit arguments, I defend J. R. Brown's position that certain thought experiments yield a priori knowledge. They do this, I argue, not by allowing us to perceive "Platonic universals" (Brown), even though they may contain non-propositional components that are epistemically indispensable, but by helping to identify certain tacit presuppositions or "natural interpretations" (Feyerabend's term) that lead to a contradiction when the phenomenon is described in terms of them, and by suggesting a new natural interpretation in terms of which the phenomenon can be redescribed free of contradiction.

Can thought experiments be resolved by experiment? The case of 'Aristotle's Wheel': for Philosophical Thought Experiments, ed. Letitia Meynell, Jim Brown and
Melanie Frappier, a volume in the Routledge Philosophy of Science series, exp. publ. date 2012.

Virtual Processes and Quantum Tunnelling as Fictions: submitted for a special issue of Science & Education devoted to the appraisal of Mario Bunge's contribution to
philosophy, ed. David Blitz.

Review of What was Mechanical about Mechanics: The Concept of Force between Metaphysics and Mechanics from Newton to Lagrange, by J. Christiaan Boudri (Dordrecht/Boston/London: Kluwer Academic Publishers, 2002).

. . o t h e r

back to top| back to home page

Logic Book sample

Chapter 8 is a sample of a Logic textbook I am writing. This chapter is on reductio arguments, enormously important in the history of thought, but rather cursorily treated in most logic texts. Also here are the exercises and selected solutions to them.

Click to see further abstracts and work in progress.