The Birth of an alternative Mechanics
Antonino Drago
Dept. of Physical Sciences, Univ. "Federico II", Napoli
I interpreted Leibniz' program for a reform of dynamics according to the following five points:
i) Instead to consider the whole Leibniz' production on mechanics, I focussed the attention upon his last writings only, representing a mature attitude on the subject: Essay of dynamique (1692), Specimen Dynamicum (1695) and Essay de dynamique sur les lois du mouvement (1698).(1)
ii) Against a widespread prejudice of an idealistic metaphysics pervading Leibniz' thinking, I followed MacDonald Ross' analysis; ...
Abstract
I interpreted Leibniz' program for a reform of dynamics according to the following five points:
i) Instead to consider the whole Leibniz' production on mechanics, I focussed the attention upon his last writings only, representing a mature attitude on the subject: Essay of dynamique (1692), Specimen Dynamicum (1695) and Essay de dynamique sur les lois du mouvement (1698).(1)
ii) Against a widespread prejudice of an idealistic metaphysics pervading Leibniz' thinking, I followed MacDonald Ross' analysis;(2) in Leibniz there are two "radically distinct conceptions of metaphysics... the one of the science of general principles and concepts [in my opinion, the foundations of science]; the other as the science of immaterial reality...; and the two conceptions fused together."
iii) I reconstructed Leibniz' mechanics (3) according to the following sequence of basic features, principles and laws: 1) The logicophilosophical framework for the whole theory is constituted by the principle of sufficient reason ("Nothing is without reason"); 2) the theory organisation is aimed to find out a new scientific method for solving an universal problem, i.e. the laws for the impact of bodies, a phenomenon by which one can reconstruct all physical phenomena; 3) nonclassical logic holds true, i.e. the logical law   A = A may fail (in present text, underlined words emphasise instances of these failures); 4) the theory is composed by contingent statements (whose contrary statements do not imply contradiction); 5) dynamics is put as first, rather than statics; 6) as first physical principle is put the impossibility of a motion without an end: 7) "our minds look for conservation": invariants (symmetries); 8) the principle of conservation of kinetic energy; 9) relativity of space, time and motion; composition and decomposition of motions; 10) elastic bodies (no hard body); 11) the principle of actionreaction; 12) indifference of a body to rest or to move; inertia as an immanent property of bodies; 13) a first, consistent sketch of a theory of the impact of all kinds of body; 14) a rough definition of work; 15) Torricelli's principle (i.e., a first historical version of the principle of virtual velocities, stated by J. Bernoulli in 1717, one year after Leibniz' death). In conclusion, Leibniz' program looking for an at all different theory of mechanics, was achieved in a great part.
iv) I discovered that, against a widespread prejudice on Leibniz' program as a blind alley  mainly since it does not include Newton's f=ma , it was substantiated in 1783 by a Leibniz' follower, Lazare Carnot; he achieved a new formulation of mechanics, based upon a new equation generalising the principle of virtual velocities.(4) I showed that this principle cannot be drawn from Newtonian principles since it is a version of the impossibility of a perpetual motion, and more in general, an application of the principle of sufficient reason to constraints.(5) In fact, L. Carnot's theory presents no axiomsprinciples, no f=ma, but methodological "hypotheses" only; in particular, L. Carnot, by adhering to both the principle of sufficient reason and experimental data, gave a new version of inertia principle. From L. Carnot's new equation the solution of the impact of bodies, i.e. the classical conservation laws, are obtained as the invariants of a first mathematical technique of symmetry, bounded to a plain algebraicvectorial calculation. Unfortunately, L. Carnot's mechanics was subsequently considered as no more than the starting point for "technical physics", without any "theoretical value".
v) According to my interpretation of the foundations of a physical theory  capable to reconstruct Leibniz' Scientia Generalis,(6)  Newton's mechanics is characterised by means of two basic choices, i.e. the use of actual infinity (at least in his calculus) and the apodictic organisation of a theory;(7) whereas, Leibniz' theory  as well as L. Carnot' theory  is characterised by the opposite choices, i.e. the exclusion of any "being of reason"  say, infinitesimals  from the contingent statements of physics, and an organisation which is modelled according to the principle of sufficient reason; in theoretical physics this principle allows methodological principles only  rather than axiomprinciples , all stated through double negated sentences. All that proves the radical change started by Leibniz' reform of dynamics.
As a general result, contrarily to a deeply rooted and widespread paradigm, the foundations of mechanics result to be pluralistic in nature. The historical development of theoretical mechanics present two main lines, diverging just owing to the contrast between Newton and Leibniz. In the subsequent develpment of theoretical physics, thermodynamics' basic notions are no more at odd with those of theoretical mechanics, provided that one refers them to L. Carnot's mechanics.(8) Moreover, electromagnetic theory is not a priori included in  or in agreement with  Newton's mechanics.(9) In 20th Century special relativity discarded Newton's absolute space, absolute time and f=ma as a principle, since it is not invariant. Rather, a modern version of Leibniz' theory as relativistic mechanics may be obtained via L. Carnot's equation(10).
I conclude that, contrarely to a widespread and rooted prejudice, Leibniz was the last philosopher capable to master the whole science of his time, so deeply to found anew the whole hard science of his time; in fact, he anticipated modern mathematical logic, he invented infinitesimal analysis and he started a valid and fruitful "dynamics reform".
Info
Drago A.: "The Birth of an alternative Mechanics from Leibniz' Principle of sufficient reason", in Proceedings LeibnizKongress, Berlin, 1014 September 2001, in press
