Leibniz and the post-Copernican universe. Koyré revisited

This paper employs the revised conception of Leibniz emerging from recent research to reassess critically the ‘radical spiritual revolution’ which, according to Alexandre Koyré’s landmark book, From the closed world to the infinite universe (1957) was precipitated in the seventeenth century by the revolutions in physics, astronomy, and cosmology. While conceding that the cosmological revolution necessitated a reassessment of the place of value-concepts within cosmology, it argues that this reassessment did not entail a spiritual revolution of the kind assumed by Koyré, in which ‘value-concepts, such as perfection, harmony, meaning and aim’ were shed from the conception of the structure of the universe altogether. On the contrary, thanks to his pioneering intuition of the distinction between physical and metaphysical levels of explanation, Leibniz saw with great clarity that a scientific explanation of the universe which rejected the ‘closed world’ typical of Aristotelian cosmology did not necessarily require the abandonment of key metaphysical doctrines underlying the Aristotelian conception of the universe. Indeed the canon of value-concepts mentioned by Koyré—meaning, aim, perfection and harmony—reads like a list of the most important concepts underlying the Leibnizian conception of the metaphysical structure of the universe. Moreover, Leibniz’s universe, far from being a universe without God—because, as Clarke insinuated, it does not need intervention from God—is a universe which in its deepest ontological fabric is interwoven with the presence of God.     

Leibniz on the laws of nature and the best deductive system

Many philosophers who do not analyze laws of nature as the axioms and theorems of the best deductive systems nevertheless believe that membership in those systems is evidence for being a law. This raises the question, “If the best systems analysis fails, what explains the fact that being a member of the best systems is evidence for being a law?” In this essay I answer this question on behalf of Leibniz. I argue that although Leibniz’s philosophy of laws is inconsistent with the best systems analysis, his philosophy of nature’s perfection enables him to explain why membership in the best systems is evidence for being a law of nature.     

Leibniz on the requisites of an exact arithmetical quadrature

In this paper we will try to explain how Leibniz justified the idea of an exact arithmetical quadrature. We will do this by comparing Leibniz's exposition with that of John Wallis. In short, we will show that the idea of exactitude in matters of quadratures relies on two fundamental requisites that, according to Leibniz, the infinite series have, namely, that of regularity and that of completeness. In the first part of this paper, we will go deeper into three main features of Leibniz's method, that is: it is an infinitesimal method, it looks for an arithmetical quadrature and it proposes a result that is not approximate, but exact. After that, we will deal with the requisite of the regularity of the series, pointing out that, unlike the inductive method proposed by Wallis, Leibniz propounded some sort of intellectual recognition of what is invariant in the series. Finally, we will consider the requisite of completeness of the series. We will see that, although both Wallis and Leibniz introduced the supposition of completeness, the German thinker went beyond the English mathematician, since he recognized that it is not necessary to look for a number for the quadrature of the circle, given that we have a series that is equal to the area of that curvilinear figure.     

The celestial mechanics of Leibniz in the light of Newtonian criticism

By the middle of the nineteenth century science was developing into a profession demanding advanced training and devotion to research. American institutions, however, were still better suited to an earlier stage of popular science. Many of the difficulties and frustrations for would-be scientists created by the time lag in institutional change are illustrated in the career of Cleveland Abbe. In the fifteen years between 1856 and 1871 his attempts to become an astronomer touched upon many significant aspects of American science as a profession, including the American observatory movement, the creation of graduate education, government support for science, and the tension between the joint goals of the increase and the diffusion of knowledge.     

Unbekannte Studien von Leibniz zur Eliminations- und Explikationstheorie

Ohne Zusammenfassung Vorgelegt von C. Truesdell     

The Work of Tschirnhaus, La Hire and Leibniz on Catacaustics and the Birth of the Envelopes of Lines in the 17th Century

The aim of this paper is to examine the work of Tschirnhaus, La Hire and Leibniz on the theory of caustics, a subject whose history is closely linked to geometrical optics. The curves in question were examined by the most eminent mathematicians of the 17th century such as Huygens, Barrow and Newton and were subsequently studied analytically from the time of Tschirnhaus until the 19th century.Leibniz was interested in caustics and the subject probably inspired him in his discovery of the concept of envelopes of lines.     

Explaining Leibniz equivalence as difference of non-inertial appearances: Dis-solution of the Hole Argument and physical individuation of point-events

“The last remnant of physical objectivity of space–time” is disclosed in the case of a continuous family of spatially non-compact models of general relativity (GR). The physical individuation of point-events is furnished by the autonomous degrees of freedom of the gravitational field (viz., the Dirac observables) which represent—as it were—the ontic part of the metric field. The physical role of the epistemic part (viz. the gauge variables) is likewise clarified as embodying the unavoidable non-inertial aspects of GR. At the end the philosophical import of the Hole Argument is substantially weakened and in fact the Argument itself dissolved, while a specific four-dimensional holistic and structuralist view of space–time (called point-structuralism) emerges, including elements common to the tradition of both substantivalism and relationism. The observables of our models undergo real temporal change: this gives new evidence to the fact that statements like the frozen-time character of evolution, as other ontological claims about GR, are model dependent.     

Kant’s third law of mechanics: The long shadow of Leibniz

This paper examines the origin, range and meaning of the Principle of Action and Reaction in Kant’s mechanics. On the received view, it is a version of Newton’s Third Law. I argue that Kant meant his principle as foundation for a Leibnizian mechanics. To find a ‘Newtonian’ law of action and reaction, we must look to Kant’s ‘dynamics,’ or theory of matter.     

Neuron numbers in hypothalamic nuclei of young, middle-aged and aged male rats

Morphologic analysis of nine hypothalamic areas revealed significant decreases in the number of neurons per unit area in the ventral medial and arcuate nuclei. These data suggest that altered neuron numbers in the VMW and perhaps the ARC may participate in the well documented reductions in endocrine and neuroendocrine function observed in aging rats.     

Neuron numbers in hypothalamic nuclei in young,middle-aged and aged male rats

Summary Morphologic analysis of nine hypothalamic areas revealed significant decreases in the number of neurons per unit area in the ventral medial and arcuate nuclei. These data suggest that altered neuron numbers in the VMW and perhaps the ARC may participate in the well documented reductions in endocrine and neuroendocrine function observed in aging rats.     

Adrenal responses to hypoxia and hypercapnia in the young calf

Summary Quantitative analysis of adrenal responses to moderate hypoxia and hypercapnia in the conscious calf shows that the sensitivity of the adrenal cortical response far exceeds that of the adrenal medulla.Acknowledgment. This work has been supported by grants from the Agricultural Research Council, the Medical Research Council and the Wellcome Trust. We are also indebted to Dr D. M. Burley (CIBA) for his continued support.