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The Roots of Hertz’s Philosophy

After the time of Newton, actually long after Newton, there was an attempt to eliminate this situation from Mechanics, by replacing the action at a distance between particles by space constraints (Hertz, 2003). The Differential Geometry was, by the end of 19^th century, advanced enough in order to provide the mathematical tool for this enterprise, and Hertz set out to eliminate altogether the force from any mechanical considerations. His main argument, however, was not directed specifically towards the falsifiability of the Third Principle, in spite of the fact that he pointed out correctly that this axiom is a problem, but towards the fact that the force was simultaneously cause and effect, and this contradictory duality is not admissible for a concept in general, not only for the force:

“In our laws of motion force was a cause of motion and was present before the motion. Can we, without confusing our ideas, suddenly begin to speak of forces which arise through motion, which are the consequence of motion? Can we behave as if we had already asserted anything about forces of this new kind in our laws, as if by calling them forces we could invest them with the properties of forces? These questions must be clearly answered in the negative. The only possible explanation is that, properly speaking, centrifugal force is not a force at all. Its name, like the name vis viva, is accepted as a historic tradition; it is convenient to retain it, although we should rather apologize for its retention than endeavor to justify it. But what now becomes of the demand of the third law, which requires a force exerted by the inert stone upon the hand, and which can only be satisfied by an actual force, not a mere name?

…

The force spoken of in the definition and in the first two laws acts upon a body in one definite direction. The sense of the third law is that forces always connect two bodies, and are directed from the first to the second as well as from the second to the first. It seems to me that the conception of force assumed and created in us by the third law on one hand, and the first two laws on the other hand, are slightly different. This slight difference may be enough to produce the logical obscurity of which the consequences are manifest in the above examples” (Hertz, 2003, p. 6, our Italics)

It is quite obvious here that Hertz sensed very well that there is a difference between the First and Second Laws of Motion on one hand and the Third Law on the other: they do not refer to the same thing! The main point of Hertz is that while the first two Laws involve the force as a vector (”acts upon a body in one definite direction”), the Third Law involves two different space points, and we are under impression that he finds this situation ambiguous; anyway not in accordance with the definition of a vector (”directed from the first to the second as well as from the second to the first”). Fact is that in such a situation the Third Principle is false. The principle acts plainly when the vectors have the same application point. Even though this fact is not explicitly stated, judging by overall aspect of the Hertz’s treatise just cited, we think that the idea was in the background. This is almost explicitly proved by the fact that Hertz insists in defining some concepts that were not properly defined in Classical Mechanics up until the time of Hertz, or ever since for that matter.

     Indeed, Hertz appears to be the closest one in realizing that the guilty part here 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 concerned, first and foremost, 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 (Hertz, 2003). 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 fuelled 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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