1

The logicians have attempted to answer the preceding considerations. For that, a transformation of logistic was necessary, and Russell in particular has modified on certain points his original views. Without entering into the details of the debate, I should like to return to the two questions to my mind most important: Have the rules of logistic demonstrated their fruitfulness and infallibility? Is it true they afford means of proving the principle of complete induction without any appeal to intuition?
2
The Infallibility of Logistic

On the question of fertility, it seems M. Couturat has na?ve illusions. Logistic, according to him, lends invention ‘stilts and wings,’ and on the next page: ”Ten years ago, Peano published the first edition of his Formulaire.” How is that, ten years of wings and not to have flown!

I have the highest esteem for Peano, who has done very pretty things (for instance his ‘space-filling curve,’ a phrase now discarded); but after all he has not gone further nor higher nor quicker than the majority of wingless mathematicians, and would have done just as well with his legs.

On the contrary I see in logistic only shackles for the inventor. It is no aid to conciseness — far from it, and if twenty-seven equations were necessary to establish that 1 is a number, how many would be needed to prove a real theorem? If we distinguish, with Whitehead, the individual x, the class of which the only member is x and which shall be called ιx, then the class of which the only member is the class of which the only member is x and which shall be called μx, do you think these distinctions, useful as they may be, go far to quicken our pace?

Logistic forces us to say all that is ordinarily left to be understood; it makes us advance step by step; this is perhaps surer but not quicker.

It is not wings you logisticians give us, but leading-strings. And then we have the right to require that these leading-strings prevent our falling. This will be their only excuse. When a bond does not bear much interest, it should at least be an investment for a father of a family.

Should your rules be followed blindly? Yes, else only intuition could enable us to distinguish among them; but then they must be infallible; for only in an infallible authority can one have a blind confidence. This, therefore, is for you a necessity. Infallible you shall be, or not at all.

You have no right to say to us: “It is true we make mistakes, but so do you.” For us to blunder is a misfortune, a very great misfortune; for you it is death.

Nor may you ask: Does the infallibility of arithmetic prevent errors in addition? The rules of calculation are infallible, and yet we see those blunder who do not apply these rules; but in checking their calculation it is at once seen where they went wrong. Here it is not at all the case; the logicians have applied their rules, and they have fallen into contradiction; and so true is this, that they are preparing to change these rules and to “sacrifice the notion of class.” Why change them if they were infallible?

“We are not obliged,” you say, “to solve hic et nunc all possible problems.” Oh, we do not ask so much of you. If, in face of a problem, you would give no solution, we should have nothing to say; but on the contrary you give us two of them and those contradictory, and consequently at least one false; this it is which is failure.

Russell seeks to reconcile these contradictions, which can only be done, according to him, “by restricting or even sacrificing the notion of class.” And M. Couturat, discovering the success of his attempt, adds: “If the logicians succeed where others have failed, M. Poincaré will remember this phrase, and give the honor of the solution to logistic.”

But no! Logistic exists, it has its code which has already had four editions; or rather this code is logistic itself. Is Mr. Russell preparing to show that one at least of the two contradictory reasonings has transgressed the code? Not at all; he is preparing to change these laws and to abrogate a certain number of them. If he succeeds, I shall give the honor of it to Russell’s intuition and not to the Peanian logistic which he will have destroyed.
3
The Liberty of Contradiction

I made two principal objections to the definition of whole number adopted in logistic. What says M. Couturat to the first of these objections?

What does the word exist mean in mathematics? It means, I said, to be free from contradiction. This M. Couturat contests. “Logical existence,” says he, “is quite another thing from the absence of contradiction. It consists in the fact that a class is not empty.” To say: a‘s exist, is, by definition, to affirm that the class a is not null.

And doubtless to affirm that the class a is not null, is, by definition, to affirm that a‘s exist. But one of the two affirmations is as denuded of meaning as the other, if they do not both signify, either that one may see or touch a‘s which is the meaning physicists or naturalists give them, or that one may conceive an a without being drawn into contradictions, which is the meaning given them by logicians and mathematicians.

For M. Couturat, “it is not non-contradiction that proves existence, but it is existence that proves non-contradiction.” To establish the existence of a class, it is necessary therefore to establish, by an example, that there is an individual belonging to this class: “But, it will be said, how is the existence of this individual proved? Must not this existence be established, in order that the existence of the class of which it is a part may be deduced? Well, no; however paradoxical may appear the assertion, we never demonstrate the existence of an individual. Individuals, just because they are individuals, are always considered as existent. . . . We never have to express that an individual exists, absolutely speaking, but only that it exists in a class.” M. Couturat finds his own assertion paradoxical, and he will certainly not be the only one. Yet it must have a meaning. It doubtless means that the existence of an individual, alone in the world, and of which nothing is affirmed, can not involve contradiction; in so far as it is all alone it evidently will not embarrass any one. Well, so let it be; we shall admit the existence of the individual, ‘absolutely speaking,’ but nothing more. It remains to prove the existence of the individual ‘in a class,’ and for that it will always be necessary to prove that the affirmation, “Such an individual belongs to such a class,” is neither contradictory in itself, nor to the other postulates adopted.

“It is then,” continues M. Couturat, “arbitrary and misleading to maintain that a definition is valid only if we first prove it is not contradictory.” One could not claim in prouder and more energetic terms the liberty of contradiction. “In any case, the onus probandi rests upon those who believe that these principles are contradictory.” Postulates are presumed to be compatible until the contrary is proved, just as the accused person is presumed innocent. Needless to add that I do not assent to this claim. But, you say, the demonstration you require of us is impossible, and you can not ask us to jump over the moon. Pardon me; that is impossible for you, but not for us, who admit the principle of induction as a synthetic judgment a priori. And that would be necessary for you, as for us.

To demonstrate that a system of postulates implies no contradiction, it is necessary to apply the principle of complete induction; this mode of reasoning not only has nothing ‘bizarre’ about it, but it is the only correct one. It is not ‘unlikely’ that it has ever been employed; and it is not hard to find ‘examples and precedents’ of it. I have cited two such instances borrowed from Hilbert’s article. He is not the only one to have used it, and those who have not done so have been wrong. What I have blamed Hilbert for is not his having recourse to it (a born mathematician such as he could not fail to see a demonstration was necessary and this the only one possible), but his having recourse without recognizing the reasoning by recurrence.

4
The Second Objection

I pointed out a second error of logistic in Hilbert’s article. To-day Hilbert is excommunicated and M. Couturat no longer regards him as of the logistic cult; so he asks if I have found the same fault among the orthodox. No, I have not seen it in the pages I have read; I know not whether I should find it in the three hundred pages they have written which I have no desire to read.

Only, they must commit it the day they wish to make any application of mathematics. This science has not as sole object the eternal contemplation of its own navel; it has to do with nature and some day it will touch it. Then it will be necessary to shake off purely verbal definitions and to stop paying oneself with words.

To go back to the example of Hilbert: always the point at issue is reasoning by recurrence and the question of knowing whether a system of postulates is not contradictory. M. Couturat will doubtless say that then this does not touch him, but it perhaps will interest those who do not claim, as he does, the liberty of contradiction.

We wish to establish, as above, that we shall never encounter contradiction after any number of deductions whatever, provided this number be finite. For that, it is necessary to apply the principle of induction. Should we here understand by finite number every number to which by definition the principle of induction applies? Evidently not, else we should be led to most embarrassing consequences. To have the right to lay down a system of postulates, we must be sure they are not contradictory. This is a truth admitted by most scientists; I should have written by all before reading M. Couturat’s last article. But what does this signify? Does it mean that we must be sure of not meeting contradiction after a finite number of propositions, the finite number being by definition that which has all properties of recurrent nature, so that if one of these properties fails — if, for instance, we come upon a contradiction — we shall agree to say that the number in question is not finite? In other words, do we mean that we must be sure not to meet contradictions, on condition of agreeing to stop just when we are about to encounter one? To state such a proposition is enough to condemn it.

So, Hilbert’s reasoning not only assumes the principle of induction, but it supposes that this principle is given us not as a simple definition, but as a synthetic judgment a priori.

To sum up:

A demonstration is necessary.

The only demonstration possible is the proof by recurrence.

This is legitimate only if we admit the principle of induction and if we regard it not as a definition but as a synthetic judgment.
5
The Cantor Antinomies

Now to examine Russell’s new memoir. This memoir was written with the view to conquer the difficulties raised by those Cantor antinomies to which frequent allusion has already been made. Cantor thought he could construct a science of the infinite; others went on in the way he opened, but they soon ran foul of strange contradictions. These antinomies are already numerous, but the most celebrated are:

1. The Burali-Forti antinomy;

2. The Zermelo-K?nig antinomy;

3. The Richard antinomy.

Cantor proved that the ordinal numbers (the question is of transfinite ordinal numbers, a new notion introduced by him) can be ranged in a linear series; that is to say that of two unequal ordinals one is always less than the other. Burali-Forti proves the contrary; and in fact he says in substance that if one could range all the ordinals in a linear series, this series would define an ordinal greater than all the others; we could afterwards adjoin 1 and would obtain again an ordinal which would be still greater, and this is contradictory.

We shall return later to the Zermelo-K?nig antinomy which is of a slightly different nature. The Richard antinomy15 is as follows: Consider all the decimal numbers definable by a finite number of words; these decimal numbers form an aggregate E, and it is easy to see that this aggregate is countable, that is to say we can number the different decimal numbers of this assemblage from 1 to infinity. Suppose the numbering effected, and define a number N as follows: If the nth decimal of the nth number of the assemblage E is

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

the nth decimal of N shall be:

1, 2, 3, 4, 5, 6, 7, 8, 1, 1

As we see, N is not equal to the nth number of E, and as n is arbitrary, N does not appertain to E and yet N should belong to this assemblage since we have defined it with a finite number of words.

We shall later see that M. Richard has himself given with much sagacity the explanation of his paradox and that this extends, mutatis mutandis, to the other like paradoxes. Again, Russell cites another quite amusing paradox: What is the least whole number which can not be defined by a phrase composed of less than a hundred English words?

This number exists; and in fact the numbers capable of being defined by a like phrase are evidently finite in number since the words of the English language are not infinite in number. Therefore among them will be one less than all the others. And, on the other hand, this number does not exist, because its definition implies contradiction. This number, in fact, is defined by the phrase in italics which is composed of less than a hundred Engli............