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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
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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unpublished paper of 1676 (A VI iii N69) Leibniz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 ¹ (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 ¹ by some 118 years.
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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.
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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.
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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).
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