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Henri Poincaré, The evolution of laws (2)

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Henri PoincaréBut, it will be said, could it not be that the application of the preceding process would lead to a contradiction, or, if one wishes, that our differential equations admit no solution? Since the hypothesis of the immutability of laws, posited at the beginning of all our reasoning, would lead to an absurd consequence, we would have shown by absurdum that they have evolved, while being forever powerless to know in what sense.

Since our process is reversible, what we have just said applies in the future, and there seem to be cases where we could say that before that date the world must perish or change its laws; if for example the calculation shows us that at this date, one of the quantities we have to consider must become infinite, or take a physically impossible value. To perish, or to change its laws, is almost the same thing; a world that no longer has the laws of ours, it would no longer be our world, it would be another.

Is it possible that the study of the present world and its laws will lead us to formulas exposed to similar contradictions? Laws are obtained by experience; if they teach us that the state A of Sunday causes the state B of Monday, it is because we observed the two states A and B; therefore none of these two states is physically impossible. If we continue the process, and if we conclude by passing every day from one day to the next, from state A to state B, then from state B to state C, then from state C to state D, etc., that is all these states are physically possible; because if the state D for example was not, one could never have done experiment proving that the state C generates after one day the state D. However far the deductions are pushed, we will never reach a physically impossible state, that is, a contradiction. If one of our formulas was not exempt, it would be that we would have exceeded the experience, it would have been extrapolated. Suppose, for example, that it has been observed that in such and such a circumstance the temperature of a body drops by one degree per day; if it is currently 20° for example, we will conclude that in 300 days it will be280°; and this will be absurd, physically impossible, since the absolute zero is at -273°. What to say? Had it been observed that the temperature passed in a day from -279° to -280°? No, no doubt, since these two temperatures are unobservable. It had been seen, for example, that the law was true at about 0° and 20°, and it had been wrongly concluded that it should be true even to -273° and even beyond; an illegitimate extrapolation was made. But there is an infinity of ways to extrapolate an empirical formula, and among them one can always choose one that excludes physically impossible states.

We know the laws only imperfectly; experience only limits our choice, and among all the laws it allows us to choose, we will always find those which do not expose us to a contradiction of the kind of those which we have just spoken of and which could oblige us to conclude against immutability. This means of demonstrating such a development still escapes us, whether it is to demonstrate that the laws will change, or that they have changed.

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