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Henri Poincaré, The evolution of laws (3)

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Henri PoincaréAt this point, we can oppose an argument of fact. “You say that in seeking to ascend, thanks to the knowledge of the laws, from the present to the past, one will never meet with a contradiction, and yet the scientists have met with it, from which it does not seem that one can escape as easily as you think. That they are only apparent, that we may preserve the hope of lifting them, I grant you; but according to your reasoning, even an apparent contradiction should be impossible.”

Let’s quote an example immediately. If we calculate from the laws of thermodynamics, the time from which the sun has been able to send us its heat, we find about 50 million years; this time can not suffice for geologists; not only has the evolution of organized forms not been able to take place so rapidly – this is a point on which we could discuss – but the deposition of layers where we find remains of plants or animals which do not were able to live without sun, required a number of years perhaps ten times greater.

What has made the contradiction possible is that the reasoning on which the geological evidence is based differs very much from that of the mathematician. Observing identical effects, we conclude to the identity of the causes, and for example by retracing the fossil remains of animals belonging to a family currently alive, we conclude that at the time when the layer containing these fossils, the conditions without which the animals of this family could not live, were all realized at once.

At first sight, it is the same thing that the mathematician did, whose point of view we had adopted in the preceding paragraphs; he, too, concluded that since the laws had not changed, identical effects could only have been produced by identical causes. There is one essential difference, however. Let us consider the state of the world, at a given instant, and at another previous instant; the state of the world, or even that of a very small part of the world, is something extremely complex and depends on a very large number of elements. I suppose, to simplify the exposition, only two elements, so that two data suffice to define this state. At the next moment, these data will be for example A and B; at the previous moment A’ and B’.

The formula of the mathematician, constructed with all the observed laws, teaches him that the state AB can only have been generated by the previous state A’B’; but if he knows only one of the data, A for example, without knowing if it is accompanied by the other data B, its formula does not allow him any conclusion. At most, if the phenomena A and A’ appear to be related to each other, but relatively independent of B and B’, conclude from A to A’; in any case, it will not deduce the double circumstance A’ and B’ from the unique circumstance A. The geologist, on the other hand, observing the effect A alone, will conclude that it could only be produced by the combination of the causes A’ and B’ who often give birth to it before our eyes; for in many cases this effect A is so special that another combination of causes leading to the same effect would be absolutely improbable.

If two organisms are identical or merely analogous, this analogy can not be due to chance, and we can affirm that they lived under such conditions; finding the debris, we shall be sure not only that it has pre-existed a germ analogous to that from which we see similar beings coming out, but that the external temperature was not too high for this germ to develop. Otherwise these debris could only be a ludus naturæ, as was believed in the seventeenth century; and it is useless to say that such a conclusion absolutely shocks reason. The existence of organized debris is, moreover, only an extreme case more striking than the others, and without leaving the mineral world, we could have cited examples of the same kind.

The geologist can therefore conclude where the mathematician would be powerless. But we see that it is no longer guaranteed against contradiction as was the mathematician. If from a single circumstance, he concludes to multiple previous circumstances; if the extent of the conclusion is somewhat greater than that of the premises, it is possible that what one infers from one observation is at odds with what one will learn from another. Each isolated fact becomes, so to speak, a center of irradiation: from each of them the mathematician deduced a unique fact; the geologist deduces from it multiple facts; from the luminous point which is given to it, it makes a brilliant disc more or less extended; two bright spots will then give him two disks that can encroach on each other, hence the possibility of a conflict. For example if he finds in a layer of molluscs that can not live below 20°, he will conclude that the seas of this time were hot; but if afterwards one of his colleagues discovered in the same stratum of other animals that a temperature of more than 5° would be killed, he would conclude that these seas were cold.

There may be reason to hope that the observations will not contradict each other in fact, or that the contradictions will not be irreducible, but we are no longer guaranteed, as it were, against the risk of a contradiction by the very rules of formal logic. And then we may ask ourselves whether, in reasoning like geologists, we will not someday fall into some absurd consequence, so that we will have to conclude that the laws are mutable.

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