Definition of mass and force and the experimental support of the classic laws of mechanics

Henri Poincaré, one of the greatest mathematicians and rationalist of the 20th century, describes us in his little book, La science et l'hypothèse, how not only it appears at first difficult to define precisely what is mass and what is a force, but it is even impossible and one has to satisfy oneself only to axioms, that is to definitions which are not provable by experience.

Newton's famous law states that the force F is the product of the mass m and the acceleration a:

F = m . a.

According to Poincaré, all the great physicists have defined mass and force differently. For one physicist, mass is the density times the volume but for another one the density itself should be defined as the ratio of the mass to the volume. For another one, mass is the ratio of the force to the acceleration while for another one force is defined as mass times the acceleration. We are turning around and around.

Poincaré explains that for a definition to be useful and scientific, it needs to enable you to perform measurements:

"Quand on dit que la force est la cause d'un mouvement, on fait de la métaphysique, et cette définition, si on devait s'en contenter, serait absolument stérile. Pour qu'une définition puisse servir à quelque chose, il faut qu'elle nous apprenne à mesurer la force; cela suffit d'ailleurs, il n'est nullement nécessaire qu'elle nous apprenne ce que c'est que la force en soi, ni si elle est la cause ou l'effet d'un mouvement."

Poincaré goes on and explains that our experience gives us some ideas on how to measure a force in the case of an isolated system: we introduce the notion of the equality between action and reaction, we deduce that the centre of gravity of the isolated system has a rectilinear and uniform movement, etc. But because there is no perfectly isolated system, all of our deductions cannot be proven exactly by experience. The results of the experience will be close to our prediction but it will not be exact. Without surprise, because we know that, besides the entire Universe, there is no perfectly isolated system.

Thus, did we achieve anything? We have invented some principles using our experience but these principles cannot strictly be proven by experience. Our sole remedy, according to Poincaré is to take these principles as axioms, which would be true and provable by experience if we have a perfectly isolated system:

"Les principles de la dynamique nous apparaissaient d'abord comme des vérités expérimentales; mais nous avons été obligés de nous en servir comme définitions. C'est par définition que la force est égale au produit de la masse par l'accélération; voilà un principe qui est désormais hors de l'atteinte d'aucune expérience ultérieure. C'est de même par définition que l'action est égale à la réaction."

Thus, "[le principe de l'égalité de l'action et de la réaction] ne devrait être plus regardé comme une loi expérimentale, mais comme une définition." Furthermore, masses are only coefficients that have been introduced in the calculation: "les masses sont des coefficients qu'il est commode d'introduire dans les calculs." Indeed, Poincaré states, we could have chosen different values for the mass without contradicting the fundamental principles. The calculation would have been harder to perform, that's all.

One can wonder if all of this is useful? The answer is yes because although we cannot prove exactly the principles and axioms we have stated, the results of our prediction match almost perfectly the results of experience because in reality, many systems are almost isolated. This success should be enough -but is it?- to prevent existentialists, irrationalists and (cultural) relativists to state that science proves that no knowledge is attainable. Almost perfect knowledge is attainable, that is what Poincaré tells us. The fact that our knowledge is imperfect should in the same time reconcile everyone: that imperfection is maybe that little freedom of will, that little irrationality that little je ne sais quoi which is so important to fuel our imagination and creation. But maybe I already went too far...

(all quotations are from Chapter 6 La mécanique classique in La science et l'hypothèse)