## Kant’s Natural Philosophy and Generalized Mechanics

Abstract:

In this paper, I uncover in Kant’s Metaphysical Foundations of Natural Science a deep conflict: between his Dynamics and Mechanics. Specifically, his theory of matter entails continuous bodies, but his laws of mechanics can handle only discrete particles not continua. Kant’s laws of motion are too weak to be fully general, namely to yield equations of motion for all possible bodies, including continuous bodies. To remedy this defect, Kant would need an additional principle: the Torque Law, also called the Balance Law of Angular Momentum: viz. that the impressed torque equals the change in angular momentum. Euler discovered and advertised the Law widely in Germany from 1750 to 1776, and so Kant could—and should—have known about that Law.
As Kant never discusses the Torque Law explicitly, I examine two strategies to obtain it on his behalf, from premises he had or could have had. (A) One requires a strong version of Newton’s Third Law: namely, that the forces between any two particles are equal, opposite, and central, i.e. exerted along the straight line between the particles. (B) The other requires a proof that certain kinds of shear forces are equal (in modern terms, that the stress tensor is symmetric). Strategy (B) presupposes that matter is a physical continuum, just as Kant has it. So, he ought to pursue (B). However, Kant rules out, on a priori grounds, all shear forces. So, avenue (B) is closed to him in principle.
To pursue (A)—and so to ensure that his mechanics is truly general, as he claims—Kant needs the Strong Third Law. But, that law holds only for discrete particles, not continuous media. I argue that, if Kant had kept his early theory of matter, or physical monadology, the conflict I signaled above would not have arisen. Physical monads, i.e. mass points, do obey the Strong Third Law. Hence, they would allow Kant—or his interpreters—to derive the Torque Law from them. Moreover, physical monads cohere with Kant’s kinematics, or Phoronomy—whereas continuous matter is too strong for it. Lastly, physical monads can ground a mechanics of continua, which Kant desires. I conclude that physical monads alone—not continuous matter—would give Kant’s mature natural philosophy the unity and generality it seeks.
I explain, in §1, why and where Kant needs the Torque Law in his foundation for mechanics. And, I document briefly Euler’s discovery of the Torque Law, the other fundamental principle of Newtonian mechanics (next to Newton’s own F = Ma). In §2, I present two ways of proving the Torque Law from premises available to Kant. I show that Kant, rather inexplicably, denies himself the second way, and so prospect of grounding the Torque Law in continuous matter. Finally, in §3, I show that, if Kant had kept his physical monadology, it would have given him a basis for the Torque Law—and thus the philosophical basis for a unified, fully general theory of mechanics.