## Leibniz’s Dynamic Error

Abstract:

One of the key claims of Leibniz’s two decade long dynamics project (c. 1678-1700) was the calculation of force (vis) as mv2. This measure of force was developed not only from his understanding of his mentor Huygens’ work on motion and collision but also his interpretation (and generalization) of Galileo’s law of falling bodies. It would serve as Leibniz’s guide since the start of his dynamics project first in the refutation of Descartes’ laws of collision, then in his mature rejection of the mechanist view of physical causation and finally in providing a foundation for his unique synthesis of the “new sciences” and his idiosyncratic Neo-Aristotelianism.
However, since many of Leibniz’s demonstrations of this quantity mv2 resemble problems of the conservation of energy-work, it immediately strikes us that Leibniz leaves out ½ from his calculation of force. The quantity ½mv2 from the standard Newtonian expression cannot be something that Leibniz could missed in his own justification of such a quantity. The simple mechanics of integrating over E=mvdv seems to imply just this ½. Important commentators have attempted to explain away this “error” by reference to Leibniz’s practice in writing equations where constants are often ignored. The problem here is that this ½ is not merely a constant but a feature of the mathematics that underlies the entire methodology of the dynamics.
Following the work of I. Szabó, I argue that Leibniz’s error points to a more systematic problem. That is, Leibniz continued to think of force through static rather than dynamic means. This means above all that Leibniz’s theory of force is not drawn from the integration of E=mvdv but rather from static “moments”. This is indeed a major systematic limitation of Leibniz’s dynamics. With this error however, we are in a better position to reevaluate the gap between the intentions and results of Leibniz’s dynamics from the perspective of his methodology.
My claim is that the omission of the ½ from mv2 can only be understood in terms of the three architectonic principles guiding his dynamics: equipollence of effect and cause, equivalence of hypotheses and continuity. In turn, Leibniz’s conservation of force in nature should be detached from its possible entangling with the contemporaneous Newtonian project (and Newtonian force) and rather be understood on its own systematic terms regulating the relation between these three principles. This clarifies not only how Leibniz was responding to his scientific forebears like Galileo, Descartes and Huygens but also demonstrate its impact a few decades later in the development of classical mechanic’s “analytic turn”.
Most importantly this analysis of Leibniz’s error reveals the depth of Leibniz thinking, albeit incomplete, on the structural rather than mechanistic nature of physical causation. By means of statics, Leibniz develops a methodological foundation for physics that aims at the structural nature of physical causality rather than physical motion itself.