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Introduction

Hertz’s philosophy gives us the possibility of an expression of the Third Principle of Classical Dynamics, in the most general scientific form, i.e. as a falsifiable proposition. It also gives us the possibility to “formalize”, so to speak, the Newtonian approach of Natural Philosophy, inasmuch as the Mathematics necessary for this task has already been written, in the form of exterior differential calculus. Here we present the general principle and some striking examples of a general interest, just in order to make the point.Hertz appears to be the closest one in realizing that, in the problem of ambiguity of the concept of force,  the guilty part is not the force or at least not the force alone. This is why he insisted upon the correct geometrical and physical definitions of the very points of application, therefore of the point of action, of the forces. This is actually the most one can do within the framework of the vector model for the force. Indeed, Hertz was actually concerned in axiomatically defining the notions of material particles and material points, two notions that we use freely today with the same connotation, but which nevertheless have to be carefully considered when it comes to describing the action of forces. In spite of his careful analysis Hertz had to sacrifice the force from among the fundamental concepts of the Mechanics. First, there was not, at that time, a possibility of algebraic formalization of the equilibrium of two forces acting in the same point. Secondly, the lack of logical characterization of the matter itself led to complications connected with the very concept of vector. One might even say that it is the concept of vector that fueled the extension of the Third Principle of Newtonian System in order to make it unfalsifiable. And this unfalsifiability is, for instance, the reason of introduction of the concealed qualities of the matter (masses, coordinates, etc.). As a matter of fact it is the very reason of upholding these concepts even today, in the form of the so-called “missing masses”. But the real reason of his revision of the fundamentals of Mechanics stays, we imagine, in the fact that Hertz realized that action at distance has always been identified with the force, and this fact is true only in special cases. And, as the action at distance is more general than the force, this last one has, naturally, to go! So it was, for Hertz anyway, and only for a moment, because the force has been reinstated right away.

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