An interesting passage from Poincare's "Science and Hypothesis".

Question

I've found Poincare's non-technical writing "Science and Hypothesis" a piece of very pleasant bedtime reading material, yet it contains quite a lot that I don't already know. For example, in the chapter "Energy and Thermodynamics", he writes

The energetic theory has the following advantages over the classical(comment: in the context, "classical" seems to refer to Newtonian formulation of mechanics). First, it is less incomplete--that is to say, the principles of the conservation of energy and of Hamilton(least action) teach us more than the fundamental principles of the classical theory, and exclude certain motions which do not occur in nature and which would be compatible with the classical theory. Second, it frees us from the hypothesis of atoms, which it was almost impossible to avoid with the classical theory.

On the first point, what is Poincare possible referring to when he says "and exclude certain motions which do not occur in nature and which would be compatible with the classical theory."? I've hold the belief that Newtonian and Hamiltonian/Lagrangian formulations of mechanics are equivalent in all systems of interest.

Regarding the second point, what does "hypothesis of atoms" mean given the context? I've thought about two possibilities: 1. Concept of "material point"; 2. Concept of indivisible constituents of matter. Which one is more appropriate in the context? And why would it be inevitable in Newtonian formulation? 

Comments

I did not read this paper, but I think this passage concerns Thermodynamics solely. At that time there were arguments between physicists on whether Thermodynamics could be "derived" from Classical Mechanics or not.

The main drawback of the "mechanical" approach, in my humble opinion, is in neglecting the radiation subsystem, which is not localised and often leads to the irreversibility of the mechanical subsystem evolution.

Answer 1

Your "in all systems of interest" is very nearly the same as "in all systems that occur in nature", which you might find an acceptable answer to your first question. Hamiltonian/Lagrangian systems are a subset of all Newtonian systems. Newtonian systems that have nonconservative force laws are of interest, but Poincaré is arguing in a fundamentalist vein with all degrees of freedom explicitly included.

In the early 20th Century, chemists were using the atomic hypothesis freely, but there was still no robust proof of the existence of atoms as opposed to there being a system of useful empirical rules. Precise measurements and the atomic explanation of Brownian motion was one of the tipping points, but at least until that time philosophers of an empirical bent denied the existence of atoms. There were some significant holdouts until the 1910's. Whether the chemist's atoms were point-like or indivisible was not especially of practical importance.

Energetics is more than just Hamiltonian/Lagrangian physics; it is also a part of a thermodynamic approach to nature, without taking statistical mechanics as an explanation for the thermodynamics. There can be bulk thermodynamic properties such as temperature and the associated heats without requiring an explanation, whereas Newtonian mechanics, not having a fundamental concept of heat, can only(?) explain heat as a statistical property of large numbers of very small objects that (mostly, because chemistry) don't break apart.

I suggest you try the Stanford Encyclopedia of Philosophy article http://plato.stanford.edu/entries/atomism-modern/, perhaps particularly Section 5.4 (SEP is generally a pretty good online resource).

Comments

Thank you, it's late here, I'll read them tomorrow.

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Your comment on 2nd point, i.e. energetics/thermodynamics vs. Newtonian mechanics, makes sense to me.

On the 1st point, are you basically saying that, Poincare is suggesting no non-conservative system shall exist on the fundamental level, since they shall all be expressible in least-action-principle language? I have reservation on your speculation. If Poincare took it as a possibility--as shown by his reluctance to talk about atomic hypothesis--that heat might be ontologically different with, but equally fundamental as, mechanical energy, then it seems plausible to speculate non-conservative system exists at a fundamental level, and he knew well that irreversible thermodynamic process couldn't be well expressed by a least action principle, as he mentioned later in the same chapter. It would seem contradictory if that's what Poincare meant on the 1st point.

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@JiaJiyang, I think Poincaré is constructing and trying to refine a conventionalist approach to physics, within which what is or is not in a class of mathematical model is determined by convention — but I think he doesn't talk enough about how what is or is not in a given class of model is also determined by what is sufficient to model nature or by other theoretical virtues of the overall system, such as simplicity, tractability, ... (not that anyone has ever managed to tie this down very well). Talk about what is fundamental is as delicate a business for conventionalism as it is for anyone, but Poincaré does it anyway.

I take Poincaré to be saying that if we're constructing Newtonian models it is better to use the smaller subset of conservative models in preference to including non-conservative models if we can, a distinction that is captured by using a Hamiltonian/Lagrangian (although this pretty clearly forces Poincaré into my understanding of his broader position as I see it). Then, however, the complexity of using conservative Newtonian mechanics can seem greater than the complexity of energetics, which has to be set against the former perhaps seeming more explanatory.

Poincaré is a genius, and I tried to be a conventionalist for a while, but the influence of conventionalism on our ideas of scientific method a hundred years later is somewhat indirect, perhaps mostly through logical positivism. I recommend you read Duhem as well if you haven't already.

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Thanks, I'll get a copy of Duhem if I have a chance.

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By the way, is your "conventionalism" the same as "nominalism" as mentioned in the Poincare's book?

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I believe yes.