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Henri Poincaré, The logic of infinity: What a classification must be

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Can the ordinary rules of logic be applied without change, as soon as we consider collections taking an infinite number of objects? This is a question we did not ask at first, but we were led to examine when the mathematicians who made themselves a specialty of the study of the infinite suddenly clashed with certain contradictions at least apparent. Do these contradictions stem from the fact that the rules of logic have been incorrectly applied, or that they cease to be valid outside their proper domain, which is that of collections formed only of a finite number of objects ? I think that it will be useful to say a few words on this subject, and to give the readers an idea of ​​the debates to which this problem has given rise.

Formal logic is nothing but the study of the properties common to all classifications; it tells us that two soldiers belonging to the same regiment belong by that very fact to the same brigade, and consequently to the same division, and that is the whole theory of syllogism. What then is the condition for the rules of this logic to be valid? This is because the classification adopted is immutable. We learn that two soldiers belong to the same regiment, and we want to conclude that they belong to the same brigade; we have the right to do so provided that during the time we put into reasoning, one of the two men was not transferred from one regiment to another.

The antinomies that have been reported all come from the oblivion of this simple condition: we used a classification that was not immutable and could not be; we have taken the precaution of proclaiming it immutable; but this precaution was insufficient; it had to be made immutable and there are cases where this is not possible.

Allow me to quote an example cited by Mr. Russell. It was against me, moreover, that he invoked it. He wanted to prove that the difficulties did not come from the introduction of the present infinite, since they can occur even when we consider only finite numbers. I will come back to this point later, but that is not what it is for the moment and I choose this example because it is fun and it highlights the fact that I come to report.

What is the smallest integer that can not be defined by a sentence of less than one hundred French words? And first, does this number exist?

Yes, because with one hundred French words, one can only build a finite number of sentences, since the number of words in the French dictionary is limited. Of these phrases, there will be some that will have no meaning or that will not define any integer. But each of them can define at most only one integer. The number of integers that can be defined in this way is therefore limited; therefore, there are certainly integers that can not be; and among these integers, there is certainly one which is smaller than all the others.

No ; for if this whole existed, its existence would imply contradiction, since it would be defined by a sentence of less than one hundred French words, namely by the very phrase which affirms that it can not be.

This reasoning is based on a classification of whole numbers into two categories, those which can be defined by a sentence of less than one hundred French words and those which can not be. In asking the question, we implicitly claim that this classification is immutable and that we begin to reason only after having definitively established it. But that is not possible. The classification can only be definitive when we have reviewed all the sentences of less than a hundred words, we will have rejected those which have no meaning, and we will definitively fix the meaning of those who have one. But among these sentences, there are some that can only make sense after the classification is stopped, those are those where it is question of this classification itself. In summary, the classification of numbers can be stopped only after the sorting of the sentences is completed, and this sorting can not be completed until the classification is stopped, so that neither classification nor sorting will ever be possible to be completed.

These difficulties will meet much more often when it comes to infinite collections. Suppose we wish to classify the elements of one of these collections and that the principle of classification rests on some relation of the element to be classified with the whole collection. Can such a classification ever be conceived as arrested? There is no present infinity, and when we speak of an infinite collection, we mean a collection to which we can continually add new elements (similar to a subscription list that would never be closed in the future expectation of new subscribers). But the classification could only be stopped when this list is closed; whenever we add new elements to the collection, we modify this collection; we can thus modify the relation of this collection with the elements already classified; and since it is from this relation that these elements have been arranged in this or that drawer, it may happen that once this relation is modified, these elements are no longer in the right drawer and that we are obliged to move. As long as we have new elements to introduce, we must be afraid of having to start all his work again; but it will never happen that we have no new elements to introduce; the classification will never be stopped.

Hence a distinction between two kinds of classifications applicable to the elements of infinite collections; predicative classifications, which can not be overturned by the introduction of new elements; the non-predicative classifications where the introduction of the new elements requires to be constantly reworked.

Suppose for example that we classify the integers in two families according to their size. We can recognize whether a number is larger or smaller than 10 without having to consider the relations of this number to all other integers. When we have defined, I suppose, the first 100 numbers, we will know which of them are smaller and which are larger than 10; when we then introduce the number 101, or any of the following numbers, those of the first 100 integers that were smaller than 10 will remain smaller than 10, those that were larger will remain larger; the classification is predicative.

Imagine, on the contrary, that one wants to classify the points of space and that one distinguishes those which can be defined in a finite number of words and those which can not. Some of the possible sentences will allude to the entire collection, that is, to space, or parts of space. When we introduce new points into space, these sentences will change their meaning, they will no longer define the same point; or they will lose all sense; or they will acquire meaning when they did not have one before. And then points that were not definable will become susceptible to be defined; others who were, ceased to be. They will have to move from one category to another. The classification will not be predictive.

There are good minds who consider that the only objects that can be reasoned are those that can be defined in a finite number of words, and I would be worse off by not seeing them as good spirits, that I will soon be defending their opinion. We can thus find that the previous example is badly chosen, but it is easy to modify it.

To classify integers, or points of space, I will consider the sentence that defines each integer, or each point. As it can happen that the same number or the same point can be defined by several sentences, I will arrange these sentences in alphabetical order and I will choose the first of them. That being said, this sentence will end with a vowel or a consonant, and we could classify according to this criterion. But this classification would not be predicative; by the introduction of new integers, or new points, sentences that made no sense can acquire one. And then on the board of sentences which define an integer or a point already introduced, it will become necessary to add new sentences, which were previously meaningless, which have just acquired one, and which define precisely the same point. It may happen that these new sentences take the lead in alphabetical order, and that they end with a vowel, while the old sentences ended with a consonant. And then our whole or our point which had been provisionally arranged in one category, will have to be transferred in the other.

If, on the contrary, we classify the points of space according to the magnitude of their coordinates, if we agree to classify all those whose abscissa is smaller than 10, the introduction of new points will not change the classification; the points already introduced that met the condition will not stop responding after this introduction. The classification will be predicative.

What we have just said about classifications applies immediately to definitions. Any definition is indeed a classification. It separates the objects that satisfy the definition, and those that do not, and places them in two distinct classes. If it proceeds, as the School says, per proximum genus and differentiam specificam, it is obviously based on the subdivision of the genus into cash. A definition like any classification may or may not be predicative.

But here a difficulty presents itself. Let’s go back to the previous example. Integers belong to the class A or class B, depending on whether they are smaller or larger than 10.5. I defined some integers α β γ … I divided them between these two classes A and B. I define and introduce new integers. I said that the distribution was not changed and therefore the classification was predicative. But in order that the place of the number α in the classification is not modified, it is not enough that the frames of the classification have not changed, it is necessary that the number α remained the same, that is to say that its definition is predicative. So from a certain point of view, one should not say that a classification is predicative in an absolute way, but that it is predicative with respect to a mode of definition.

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