Newton vs. Leibniz: Intransparency vs. Inconsistency

Since Antiquity it is known that theories claiming to decribe the "real world" have to deal with paradoxicalphenomena of plurality and motion. These paradoxes received their canonical form in the arguments ofZeno, and they seem hard, if not impossible, to overcome, except by what Poincaré so aptly calls "un aveud'impuissance": schemes to hide or avoid them, basically by the introduction of silent additionalhypotheses like a priori conceptions of space and time, or refutations that turn out to be themselvescircular, e.g. all part/whole arguments based on induction. Another option is to assign to paradoxesexplicitly a place in the description, in the hope that they will stay where they are and not swarm outlike viruses until the body of knowledge collapses. This is the case of atomism (in the proper sense),infinitesimal calculus, paraconsistent logic and the like. Except maybe for the case of paraconsistentlogic, all these approaches can be subsumed under a general strategy dating back to Antiquity, whichdivides the "world" into a layer of stability and one of change, connected by a relation of causality. Theselayers are again characterised by two kinds of infinity, the actual and the potential infinite, each withits proper mode of generation: (simultaneous) division and (stepwise) addition [Phys. 204a 6]. Thisstrategy we shall call "classical metaphysics". In Early Modernity its level of operation shifts fromlogic and metaphysics (in the traditional sense) to the foundations of mathematics and natural science. Itremains visible in the separate treatment of statics/cinematics (identity) and dynamics (causality) in thenew mechanical theory. Within this framework one can make sense of Meyerson's definition of causalityas that which grants identity through time (so that mere "lawfulness" is not enough), and use itto precisely distinguish the metaphysical differences between supposedly equivalent mechanical theorieslike Newtonian and Leibnizian mechanics. Their different approaches with respect to infinitesimal calculusare closely linked to these metaphysical differences. Meyerson's insight moreover helps to expose wherein both theories the original paradoxes hide, and how they are kept under control. In Newton's case theyare relegated to an immovable and omnipresent God who bridges the gap between actual and potential,between absolute and relative, by means of a law F = ma in which the rôle of causation is completelyobscured in the equally obscure relation between inertial and gravitational mass. We shall show that thisrelation nevertheless justifies the enigmatic "hypothesis I" (in the second edition of the Principia), "Thatthe centre of the system of the world is immovable". With Leibniz God grants perseverance and internalconsistency to everything in an infinity of interconnected worlds, actual and potential, great and small. They are all modally interconnected due to Leibniz's principle of the best of possible worlds. On thisbasis Leibniz's principle of the equivalence of hypotheses, the idea that this frame of reference is as goodas any other when describing the universe, rests. The cinematics that goes with it is evidently relativisticand its dynamics based on a conservation principle, so that there is no need to blur the notion of causality,quite the contrary. But Leibniz's system does not escape from inconsistency either, for his God, in orderto keep an eye on all these worlds, has to be able to take on all these mutually interrelated, possibleperspectives simultaneously. Specker however showed in a famous argument that this is inconsistent.Curiously enough, the Kochen-Specker theorem is a result related to inconsistencies in quantum theory, sothat one can only wonder at the kind of link that could exist between Leibnizian and quantum mechanics.

Publication year:2014

Keywords:history and philosophy of science