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Henri Poincaré: The evolution of laws (1)

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Henri PoincaréMr. Boutroux, in his work on the contingency of the laws of nature, wondered if natural laws are not likely to change, if while the world is changing continuously, the laws themselves, that is to say to say the rules according to which this evolution takes place, will be the only ones exempt from any variation. Such a conception has no chance of ever being adopted by scholars; in the sense that they would understand it, they could not adhere to it without denying the legitimacy and the very possibility of science. But the philosopher retains the right to ask himself the question, to consider the various solutions it entails, to examine the consequences of it, and to seek to reconcile them with the legitimate demands of scientists. I would like to consider some of the aspects that the problem may take; I shall thus be led not to conclusions proper, but to various reflections which may not be without interest. If, on the way, I allow myself to speak a little longer about certain related questions, please forgive myself.


Let’s first look at the mathematician’s point of view. Let us admit for a moment that the physical laws have undergone variations in the course of ages, and ask ourselves if we should have a means of perceiving them. Let us not forget at first that the few centuries during which humanity lived and thought were preceded by incomparably longer periods in which man did not yet live; they will undoubtedly be followed by other periods when our species will have disappeared. If one wants to believe in an evolution of the laws, it can not be, without contradiction, other than very slow, so that, during the few years that one thought, the laws of the nature could undergo only insignificant changes. If they have evolved in the past, then we must understand the geological past. Were the laws of former times those of today, the laws of tomorrow will still be the same? When we ask such a question, what meaning should we attach to words once, today and tomorrow? Today are the times whose history has preserved the memory; in the past, millions of years preceded history, and ichthyosaurs lived quietly without philosophizing; tomorrow, it is the millions of years that will come next, where the Earth will be cooled and where the man will have no more eyes to see or brain to think.

That put, what is a law? It is a constant link between the antecedent and the consequent, between the current state of the world and its immediate later state. Knowing the present state of every part of the universe, the ideal scientist who knew all the laws of nature would have fixed rules to deduce from them the state that these same parties will have the next day; it is conceivable that this process can be continued indefinitely. From Monday’s state of the world, one will deduce that of Tuesday; knowing that of Tuesday, it will be deduced by the same methods that of Wednesday; And so on. But that’s not all ; if there is a constant link between the state of Monday and that of Tuesday, we can deduce the second from the first, but we can do the opposite, that is to say that if we know the state of Tuesday, we can conclude that of Monday; from the state of Monday we will conclude likewise with that of Sunday, and so on; we can go back in time and we can go down. With the present and the laws, one can guess the future, but one can also guess the past. The process is essentially reversible.

Since we place ourselves here from the point of view of the mathematician, it is advisable to give this conception all the precision which it implies if we use mathematical language for it. We shall then say that the set of laws is equivalent to a system of differential equations which link the velocities of variations of the various elements of the universe to the present values ​​of these elements.

Such a system comprises, as we know, an infinity of solutions; but if we give ourselves the initial values ​​of all the elements, that is to say their values ​​at the moment t = 0, (the one that in the ordinary language we call the present) the solution is entirely determined, so that we can compute the values ​​of all the elements at any time, that we suppose t > 0, which corresponds to the future, that we suppose t < 0, which corresponds to the past. What is important to remember is that the way of concluding from present to past is not different from how to conclude from the present to the future.

What means do we have then to know the geological past, that is to say the history of the times when the laws could have formerly varied? This past could not be directly observed and we only know it by the traces it has left in the present, we only know it by the present, and we can only deduce it from the process that we have just described, and which would also allow us to deduce the future. But is this process able to reveal changes in the laws? Obviously not; we can not precisely apply it except by supposing that the laws have not changed; we know only the state of Monday, for example, and the rules which bind the state of Sunday to that of Monday; the application of these rules will then make us know the state of the Sunday; but if we wish to go further and deduce the state of Saturday, we must of necessity admit that the same rules which enabled us to go back up from Monday to Sunday, were still valid between Sunday and Saturday. Without this, the only conclusion we would be allowed is that it is impossible to know what happened on Saturday. If then the immutability of the laws appears in the premises of all our reasonings, we can not to not find it in our conclusions.

A Leverrier, knowing the current orbits of the planets, calculates, using Newton’s law, what will have become these orbits in 10,000 years. However he directs his calculations, he will never be able to find that Newton’s law will be false in a few thousand years. He could, by simply changing the sign of time in his formulas, calculate what these orbits were 10,000 years ago; but it is certain in advance not to find that Newton’s law has not always been true.

In short, we can know nothing of the past except on the condition of admitting that the laws have not changed; if we admit it, the question of the evolution of the laws does not arise; if we do not admit it, the question is insoluble, as are all those that relate to the past.

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