A few years ago, I had the opportunity to expose some ideas about the logic of the infinite; on the use of the infinite in Mathematics, on the use made of it since Cantor; I explained why I did not regard as legitimate certain modes of reasoning which various eminent mathematicians had thought they could use. I naturally attracted green replies; these mathematicians did not believe they had been wrong, they thought they had the right to do what they had done. The discussion dragged on, not that new arguments were constantly coming up, but because we were always turning in the same circle, each repeating what he had just said, without appearing to have heard what the opponent had said. Every moment I was sent a new demonstration of the disputed principle, to put myself, it was said, safe from all objection; but this demonstration was always the same, barely made up. We have therefore arrived at no conclusion; if I said that I was surprised, I would give a sad idea of my psychological penetration.
In these circumstances, should I repeat once again the same arguments, to which I might perhaps give a new form, but to which I could not change anything in substance, since it seems to me that we have not even tried to refute them. It seems to me preferable to inquire what may be the origin of this difference of mentality which gives rise to such differences of opinion. I have just said that these irreducible divergences did not surprise me, that I had foreseen them from the first hour, but that does not exempt us from seeking the explanation; we can foresee a fact, as a result of repeated experiments, and yet be very embarrassed to explain it.
Let us seek, therefore, to study the psychology of the two opposing schools, from a purely objective point of view, as if we ourselves were placed outside these schools, as if we were describing a war between two anthills; we will first observe that there are two opposite tendencies among mathematicians in the way of considering the infinite. For some, the infinite derives from the finite, there is an infinite because there is an infinity of possible finite things; for others the infinite pre-exists the finite, the finite is obtained by cutting a small piece from infinity.
A theorem must be verifiable, but as we are finite ourselves, we can only operate on finite objects; even when the notion of infinity plays a part in the statement of the theorem, it must be in the verification that it is no longer a question; otherwise, this verification would be impossible. I will take as examples theorems like these: the sequence of prime numbers is unlimited, the series Σ1/n2 is convergent, and so on. ; each of them can result in equalities or inequalities in which only finite numbers appear. These theorems participate in the infinite, not because one of the possible verifications participates in it itself, but because the possible verifications are in infinite number.
In stating the theorem, I affirm that all these verifications would succeed; Of course, we do not do them all; there are some that I call possible because they would only require a finite time, but they would be practically impossible because they would require years of work. It is enough for me to conceive someone rich enough and crazy enough to tempt her by paying a sufficient number of auxiliaries. The proof of the theorem is precisely intended to render this madness useless.
Does a theorem that contains no verifiable conclusion have any meaning? or, more generally, does any theorem have any meaning beyond the verifications it contains? This is where mathematicians differ. Those of the first school, those whom I will call the Pragmatists (since they must be given a name) answer no, and when they are given a theorem without giving them a way to verify it, they only see the porridge for cats. They only want to consider objects that can be defined in a finite number of words; when in a reasoning we speak to them of an object A satisfying certain conditions, they imply an object that satisfies these conditions whatever the words that will be used to complete the definition provided that these words are in finite number.
Those of the other school, whom I shall call Cantorians to shorten, do not wish to admit this; a man, however talkative, will never pronounce in his life more than a billion words; and then are we to exclude from Science objects whose definition contains a billion and a word? and if we do not exclude them, why should we exclude those which can only be defined by an infinity of words, since the construction of some is like that of others beyond the reach of humanity?
This argument, of course, leaves the Pragmatists cold; no matter how talkative a man may be, humanity will be more talkative still, and as we do not know how long it will last, we can not limit the scope of its investigations in advance; we only know that this field will always remain limited; and even if we could fix the date of its disappearance, there are other stars which could resume the unfinished work on Earth; the Pragmatists would not have any repugnance to imagine a humanity much more talkative than ours, but still retaining something human; they refuse to reason on the hypothesis of I do not know what divinity infinitely talkative and capable of thinking an infinity of words in a finite time. On the contrary, others think that objects exist in a sort of department store, independent of all humanity or any divinity that could speak or think about it; that in this store we can not make our choice, that without doubt we have not enough appetite or enough money to buy everything; but that the inventory of the store is independent of the resources of the buyers. And from this initial misunderstanding result all kinds of discrepancies of detail.
Take for example Zermelo’s theorem, according to which space is capable of being transformed into a well-ordered whole; Cantorians will be seduced by the rigor, real or apparent, of the demonstration; the Pragmatists will answer him: You say that you can transform space into a well-ordered whole; well ! turn it. – It would be too long. – So show us at least that someone who has enough time and patience could do the transformation. – No, we cannot because the number of operations to be done is infinite, it is even greater than Alefzero. – Can you show how one could express in a finite number of words the law that would order space? – No – and the Pragmatists conclude that the theorem is meaningless, or false, or at least undemonstrated.
Pragmatists place themselves in terms of extension and Cantorians in terms of understanding. When it comes to a finite collection, this distinction can only interest theorists of formal logic; but it appears to us much deeper as far as infinite collections are concerned. If we take the point of view of extension, a collection is constituted by the successive addition of new members; we can, by combining ancient objects, construct new objects, then with them even more new objects, and if the collection is infinite, it is because there is no reason to stop.
From the point of view of understanding, on the contrary, we start from the collection where pre-existing objects are, which appear to us at first as indistinct, but we end up recognizing some of them because we stick labels on them. and we put them in drawers; but the objects are older than the labels, and the collection would still exist if there were no preservatives to classify it.
For Cantorians the notion of cardinal number does not involve any mystery. Two collections have the same cardinal number when they can be stored in the same drawers; nothing is easier since the two collections pre-exist, and one can also look as pre-existing a collection of drawers independent of the curators charged with storing the objects there. For the Pragmatists, it is not the same; the collection does not pre-exist, it is enriched every day: new objects are constantly added to them that could not have been defined without relying on the notion of the objects already previously classified and on the way in which they are classified . At each new acquisition, the curator may be forced to upset its drawers to find a way to fit it: we will never know if two collections can be stored in the same drawers, since we can always fear that it is necessary to disturb.
For example, Pragmatists only admit objects that can be defined in a finite number of words; the possible definitions, being expressible by sentences, can always be numbered with ordinary numbers from one to infinity. At this count there would be only one infinite cardinal number possible, the number Alefzero; why do we say that the power of the continuum is not that of integers? Yes, given all the points of space that we know how to define with words in finite numbers, we know how to imagine a law, expressible itself by a finite number of words, which makes them correspond to the sequence of integers; but let us now consider sentences in which the notion of this law of correspondence appears; just now they had no sense since this law was not yet invented, and they could not be used to define points of space; now they have acquired a meaning, they will enable us to define new points of space; but these new points will no longer find a place in the classification adopted, which will compel us to upset it. And that is what we mean, according to the Pragmatists, when we say that the power of the continuum is not that of integers. We mean that it is impossible to establish between these two sets a law of correspondence which is sheltered from this sort of upheaval; instead we can do it for example when it comes to a line and a plane.
And then the Pragmatists are not certain that any set has, properly speaking, a cardinal number; or that, given two sets, one can always know if they have the same power, or if one has a greater power than the other. They come to doubt the existence of Alef-unu.
Another source of divergence comes from the way of conceiving the definition. There are several kinds of definitions; the direct definition that can be done either by genus proximum et differentiam specificam or by construction.
Incidentally, there are incomplete definitions in that they define not an individual, but an entire genus; they are legitimate, and they are even the most frequently used; but according to the Pragmatists, we must imply the group of individuals who satisfy the definition and who could be defined in a finite number of words; for the Cantorians this restriction is artificial and meaningless.
If there were only direct definitions, the powerlessness of pure logic can not be disputed; we could then in any proposition replace each of the terms by its definition; when we have completed this substitution, or the proposition would not be reduced to an identity and then it would not be susceptible of a purely logical demonstration; or it would be reduced to an identity and then it would be a tautology more or less cleverly disguised.
But we still have another kind of definitions, definitions by postulates; generally we will know that the object to be defined belongs to a genus, but when it is a question of stating the specific difference, it will not be stated directly, but by means of a “postulate” to which the object defined will have to satisfy. Thus mathematicians can define an amount x by an explicit equation x = f(y), or by an implicit equation F(x,y) = 0.
The definition by postulate has value only when the existence of the definite object has been demonstrated; in mathematical language, this means that the postulate does not imply contradiction; we have no right to neglect this condition; we must either admit the absence of contradiction as an intuitive truth, as an axiom, by a kind of act of faith; but then we must realize what we are doing and know that we have extended the list of indemonstrable axioms; or it is necessary to construct a demonstration in rule, either by the example, or by the use of reasoning by recurrence. It is not that this demonstration is less necessary when it comes to a direct definition, but it is generally easier.
Some Pragmatists will be more demanding: for them to look at a definition as legitimate, it will not be enough for them not to lead to contradictions in terms, they will still have to have a meaning, from their particular point of view that I sought to define above.
Be that as it may, will the logic remain sterile after the introduction of the definitions by postulates? We can no longer, given a proposition, replace a term by its definition; all we can do is eliminate this term between the proposition and the postulate that serves as its definition. If this operation, made from what might be called the rules of logical elimination, does not lead us to an identity, it is because the proposition is indemonstrable by pure logic; if it leads to an identity, it is because the proposition is only a tautology. We have nothing to change to our earlier conclusions.
But there is a third kind of definition, which is the origin of a new misunderstanding between Pragmatists and Cantorians. These are still definitions by postulate, but the postulate here is a relation between the object to be defined and all the individuals of a genus whose object to be defined is supposed to be part of (or which are supposed to be part of beings that can only be defined by the object to be defined). This is what happens if we ask the following two assumptions:
X (object to be defined) has such relation with all the individuals of the genus G.
X is part of the genus G.
or the following three postulates:
X has such a relation with all individuals of the genus G.
Y has such relation with X.
Y is part of G.
For Pragmatists such a definition implies a vicious circle. We cannot define X without knowing all the individuals of the genus G, and therefore without knowing X which is one of these individuals. Cantorians do not admit that; the genus G is given to us, therefore we know all the individuals, the definition is only intended to discern among these individuals the one who has with all his comrades the stated relationship. No, answer their opponents, knowledge of the genre does not make you know all its individuals, it only gives you the opportunity to build them all, or rather to build as many as you want. They will exist only after they have been constructed, that is, after they have been defined; X exists only by its definition which makes sense only if we know in advance all the individuals of G and in particular X. It would be pointless to say, they add, that it is not a vicious circle to define X by its relation to X, that this relation is in fact a postulate that can be used to define X; for it would be necessary to establish beforehand that this postulate does not imply contradiction, but it is not usual what one does in this kind of definitions. We first prove that whatever the genus G, of which all the individuals are supposed to be known, there exists a being X which has with this kind the relation in question; that is to say, the existence of this being does not entail contradiction; it remains to show that there is no contradiction between the existence of this being and the hypothesis that this being is itself part of the genre.
The debate could continue for a long time; but the point I would like to stress is that, if such definitions were admitted, the logic would no longer be sterile, and the proof is that we built in this way a host of arguments intended to demonstrate proposals that were not tautologies because there are people who wonder if they are not false. And then we admire the power that a word can have. This is an object from which nothing could have been drawn until he was baptized; it was enough to give him a name so that he would do wonders. How is it possible ? It is because by giving it a name, we implicitly asserted that the object existed (that is, was pure from all contradiction) and that it was entirely determined. Now, that we do not know anything about what the Pragmatists claim. What is the mechanism that makes the demonstration fruitful? it is very simple, we deny the proposition to be demonstrated and we show that we are in contradiction with the existence of the object X; and this is only legitimate if one is certain of this existence, and secondly, if one knows that the object is entirely determined. And indeed if X is deduced from the genus G by the definition, that if we then complete the genus G by adding the object X and other individuals of the same kind who can derive from it; if we call G’ the genus thus completed and X’ which would be deduced from G’ by the definition in the same way that X has been deduced from G, we need to be sure that X’ is identical to X. If it were not so, and by denying the proposition to be proved, one would be led to two contradictory statements.
φ1(X) = 0, φ2(X) = 0
how do we know that it is the same X that appears in both? If X appeared in one and X’ in the other, both proposals would be written
φ1(X) = 0, φ2(X’) = 0
and would no longer be contradictory in general.
Why do Pragmatists make this objection? This is because the genus G only appears to them as a collection likely to grow indefinitely, as new individuals are constructed with the proper characters; this is how G can never be set ne varietur, as Cantorians do, and we are not sure that, by new annexations, it will not become G’.
I have endeavored to explain as clearly and as impartially as I could what the differences between the two schools of mathematicians consist of; and it seems to me that we already perceive the true cause; scholars of both schools have opposite mental tendencies; those whom I have called the Pragmatists are idealists, the Cantorians are realists.
There is one thing that will confirm us in this way of seeing things. We see that the Cantorians (that I spend this term convenient although I want to speak here not mathematicians who follow the path opened by Cantor, or maybe even philosophers who claim to him, but those who have the same tendencies in an independent way), that the Cantorians, I say, constantly speak of epistemology, that is, science of science; and it is understood that this epistemology is entirely independent of psychology; that is to say, it must teach us what the sciences would be if there were no scientists; that we must study the sciences, no doubt by supposing that there are no scientists, but at least not supposing that there are any. So not only is Nature a reality independent of the physicist who might be tempted to study it, but physics itself is also a reality that would subsist if there were no physicists. This is realism.
And why do Pragmatists refuse to admit objects that could not be defined by a finite number of words? It is because they consider that an object exists only when it is thought, and that it is impossible to conceive an object thought independently of a thinking subject. This is idealism. And as a thinking subject is a man, or something that resembles man, that it is therefore a finite being, the infinite can have no other meaning than the possibility of creating so many objects finished as we want.
And then we can make a curious remark. Realists are usually placed in the physical point of view; it is the material objects, or the individual souls, or what they call the substances, of which they affirm the independent existence. The world for them existed before the creation of man, even before that of living beings; it would still exist even if there were no God or any thinking subject. This is the point of view of common sense, and it is only through reflection that we may be led to abandon it. Proponents of physical realism are generally finitists; in the question of Kantian antinomies, they hold for theses; they believe that the world is limited. Such is, for example, the view of Mr. Evellin. On the contrary, idealists do not have the same repugnance and are ready to subscribe to the antitheses.
But Cantorians are realistic, even with regard to mathematical entities; these entities appear to them to have an independent existence; the geometer does not create them, he discovers them. These objects then exist, so to speak, without existing, since they are reduced to pure essences; but as, by nature, these objects are infinitely numerous, the partisans of mathematical realism are much more infinitists than the idealists; their infinity is no longer a becoming, since it pre-exists the spirit that discovers it; whether they admit it or deny it, they must believe in the infinite present.
Here we recognize the theory of Plato’s ideas; and it may seem strange to see Plato ranked among the realists; Nothing, however, is more opposed to contemporary idealism than Platonism, although this doctrine is also far remote from physical realism.
I have never known a more realistic mathematician, in the Platonic sense, than Hermit, and yet I must admit that I have not met any more refractory to Cantorism. There is an apparent contradiction here, especially since he kept saying: I am anticantorist because I am realistic. He criticized Cantor for creating objects, instead of just discovering them. No doubt, because of his religious convictions, he considered it a kind of impiety to want to penetrate into a field that only God can embrace and not to wait for him to reveal one by one the mysteries. He compared the mathematical sciences to the natural sciences. A naturalist who would have sought to divine the secret of God, instead of consulting the experience, would have seemed to him not only presumptuous but disrespectful to the divine majesty; the Cantorians seemed to him wanting to do the same in mathematics. And that’s why, realistic in theory, he was idealistic in practice. There is a reality to know and it is external to us and independent of us; but all that we know of it depends on us, and is nothing more than a becoming, a sort of stratification of successive conquests. The rest is real but eternally unknowable.
Hermite’s case is isolated, and I do not dwell on it any further. There have always been opposing trends in philosophy, and it does not seem that these tendencies are about to be reconciled. It is undoubtedly because there are different souls and that to these souls we can not change anything. So there is no hope of seeing the agreement between the Pragmatists and the Cantorians. Men do not get along because they do not speak the same language and there are languages that can not be learned.
And yet in mathematics they are accustomed to agree; but it is precisely because of what I have called verifications; they judge in the last resort and before them everyone bows down. But where these verifications are lacking, mathematicians are no more advanced than mere philosophers. When it comes to knowing if a theorem can have meaning without being verifiable, who will be able to judge since by definition one is forbidden to check? One would have no more resource than to drive his adversary to a contradiction. But the experiment was done and it did not succeed.
Many antinomies have been reported, and disagreement has subsisted, no one has been convinced; of a contradiction, one can always get out by a push; I mean by a distinguo.