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)

Some comments on entropy

Definition

Since I reviewed the basic teachings concerning entropy, I see entropy everywhere. The most helpful definition of entropy is the one regarding the state of order. The highest the order, the lowest the entropy. Thus, I declare myself an enemy of entropy, in the sense that I, we, always try to create some order, to put things in order: putting the plates away in the kitchen, entangling an electric wire, etc. Why is it actually easier to put things in disorder and to create entropy than the reverse? One useful explanation to comprehend this difference is to see that a state in order is an improbable state, while a state in disorder is a probable one: the electric wire is more likely to be tangled after so many years than to stay untangled, and there are many ways to put a mess in a room, but only one to put things at their places.

Entropy and time

There is some controversy concerning the relationship between entropy and time. The problem is that the fundamental laws of classical physics (from Newton) are reversible in time; that is whatever happens in one direction (toward to the future) can very well happen in the other direction (toward the past). Thus, as we see a drop of milk in a tea cup spreading and diffusing throughout the volume, we should see all the milk particles to come back and form the initial drop of milk. This is indeed possible according to Poincaré's recurrence theorem, although, because the state of the drop is very unlikely compared to all the states where the milk is spread, the probability that this happens is tiny (but in theory, it could happen!).

On the other hand, the second law of thermodynamics says that for a closed system, the entropy has to increase: the spreading of the drop of milk within the cup is a perfect example of entropy increase. Some, such as Prigogine, argues that entropy carries with itself the so-called arrow of time: because entropy increases, we can make the difference between past and future. But the question then remained, is the second law compatible to the reversible laws of classical physics? Roger Penrose, in his book The emperor's new mind, argues that the second law is not only compatible, but also, contrary to Prigogine's view, that the entropy does not carry the arrow of time with it. Whatever the direction, toward the past or the future, the entropy has to increase within a closed system, in particular within a system where there is no constraint on the entropy. In the case of the drop of milk, although toward the future there is no constraint and the entropy increases indeed, toward the past, there is the constraint of the initial conditions saying that the entropy is low at the beginning: thus, if you run the experiment backwards, there is the constraint that at the end, the entropy is lower than at the beginning. Because of this constraint, the second law does not apply as such and in consequence, the entropy does not itself carry the arrow of time. The arrow of time exists because our system started with a state of low entropy. The remaining question is thus, why and how did we start with a state of low entropy?

Source of low entropy

Both L. Botlzmann and R. Penrose describe the struggle for life as a struggle for low entropy, with the ultimate source of low entropy being the sun. L. Botlzmann writes

"The general struggle for existence of animate beings is therefore not a struggle for raw materials [...] nor energy [...], but a struggle for entropy, which becomes available through the transition of energy from the hot sun to the cold earth."

and R. Penrose says

"We do not need to gain energy from our environment because energy is conserved. But we are continually fighting against the second law of thermodynamics. Entropy is not conserved; it is increasing all the time. To keep ourselves alive, we need to keep lowering the entropy that is within ourselves."

Thus, how do we get this low entropy? The ultimate source of entropy is the sun, and plants are the organisms which are using directly this entropy source, transforming it into molecular structures, themselves ready to be eaten. We, humans, via the food web, are eating plants or animals who themselves eat plants, to get low entropy for our body.

In this aspect, I am then wondering if we can class the food web in terms of entropy content. The plants would get a source of low entropy S_i, some of it would be used such that the entropy content gained in the eaten plant would be actually larger, S_i < S_plant . This process would repeat in that the higher in the food web, the higher the entropy content. In that respect, I would conclude that 1) humans would be organisms with some of the highest entropy (the most disorder) and 2) we should all be vegetarians in order to efficiently get low entropy in our diet. Do you agree with these conclusions?

Why the sun?

R. Penrose also explains why the sun is a source of low entropy and I was very surprised to learn that the reason is nearly a geometrical one. The sun is a hot spot, a small disk of light compared to the entire sky. Because of this geometrical configuration, the energy we receive from the sun has a much lower entropy that the energy sent back to space by earth because this energy is sent into all directions. Thus, if I understand correctly, if the earth was surrounded by many suns so much so that they would cover the entire sky, there would not be any source of low entropy and life would be unable to exist? Of course, one still needs to explain why the sun, itself a compact star and thus a source of low entropy, exists but the explanation goes on with cosmological arguments that I understand much less.

References
Ludwig Boltzmann, The second law of thermodynamics, in Theoretical physics and philosophical problems
Jean Bricmont, Science of chaos or chaos in science?
Roger Penrose, The emperor's new mind