This rapid review of the various parts of mathematical physics has convinced us that all these problems, despite the extreme variety of boundary conditions and even differential equations, have, so to speak, a certain air of family which it is impossible to misunderstand. We must therefore expect to find them a very large number of common properties.
Unfortunately, the first of the properties common to all these problems is their extreme difficulty. Not only can they most often not be resolved completely, but it is only at the cost of the greatest efforts that their possibility can be rigorously demonstrated.
Is this demonstration necessary? Most physicists would make it cheap. Since experience does not allow us to doubt, for example, the possibility of electrical equilibrium, we cannot doubt, either, it seems, the possibility of the equations which express this equilibrium. We cannot be satisfied with this defeat; analysis must be able to suffice in itself and besides such a reasoning, if it does not apply perhaps to the problems that one encounters directly in physics, could not apply in the same way to a crowd of simpler problems, which arise on their own as soon as one tries to solve the first ones. Moreover, any rigorous demonstration of the possibility of a problem is always a solution to it; in the case that concerns us, this solution will generally be crude and quite unsuitable for numerical calculation; however it will always teach us something.
Now, if this demonstration is necessary, must we nevertheless subject ourselves to the same rigor as in a question of pure analysis? This would in many cases be quite unnecessary pedantry. The differential equations which physical phenomena obey have often only been established by loose reasoning; we only regard them as approximations; the experimental results, to which it is a question of comparing the consequences of the theory, are themselves approximate. Under these conditions, absolute rigor is of little value, and it often seems that there is no point in seeking it if one has to pay too much effort for it.
But then how will we recognize that a reasoning whose rigor is not absolute, is not a simple paralogism? When will we have the right to say that such a demonstration, insufficient for analysis, is rigorous enough for physics? The limit is very difficult to draw. Yet I will try to do so; I will endeavor to mark this boundary clearly and to explain why beyond that we are still in the domain of science, and beyond that in that of paralogism.
Nevertheless, whenever I can, I will aim for absolute rigor, and that for two reasons; in the first place, it is always hard for a surveyor to approach a problem without solving it completely; secondly, the equations which I will study are susceptible, not only of physical applications, but also of analytical applications. It is on the possibility of Dirichlet’s problem that Riemann based his magnificent theory of Abei’s functions. Since then other geometers have made important applications of this same principle to the most fundamental parts of pure analysis. Is it still permissible to settle for half-rigor? And who tells us that the other problems of mathematical physics will not one day, as the simplest of them has already been, come to play a considerable role in analysis?
Source: Henri Poincaré, Sur les Equations aux Dérivées Partielles de la Physique Mathématique, American Journal of Mathematics, Mar., 1890, Vol. 12, No. 3 (Mar., 1890), pp. 211-294. Translation by Nicolae Sfetcu. © 2023 Nicolae Sfetcu
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