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.

IV

_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 embarra.s.sing 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 a.s.sumes 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.

V

_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-Konig 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-Konig antinomy which is of a slightly different nature. The Richard antinomy[15] 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 a.s.semblage from 1 to infinity. Suppose the numbering effected, and define a number _N_ as follows: If the _n_th decimal of the _n_th number of the a.s.semblage _E_ is

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

the _n_th decimal of _N_ shall be:

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

[15] _Revue generale des sciences_, June 30, 1905.

As we see, _N_ is not equal to the _n_th number of _E_, and as _n_ is arbitrary, _N_ does not appertain to _E_ and yet _N_ should belong to this a.s.semblage 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 English words; and by definition this number should not be capable of definition by a like phrase.

VI

_Zigzag Theory and No-cla.s.s Theory_

What is Mr. Russell's att.i.tude in presence of these contradictions?

After having a.n.a.lyzed those of which we have just spoken, and cited still others, after having given them a form recalling Epimenides, he does not hesitate to conclude: "A propositional function of one variable does not always determine a cla.s.s." A propositional function (that is to say a definition) does not always determine a cla.s.s. A 'propositional function' or 'norm' may be 'non-predicative.' And this does not mean that these non-predicative propositions determine an empty cla.s.s, a null cla.s.s; this does not mean that there is no value of x satisfying the definition and capable of being one of the elements of the cla.s.s. The elements exist, but they have no right to unite in a syndicate to form a cla.s.s.

But this is only the beginning and it is needful to know how to recognize whether a definition is or is not predicative. To solve this problem Russell hesitates between three theories which he calls

A. The zigzag theory;

B. The theory of limitation of size;

C. The no-cla.s.s theory.

According to the zigzag theory "definitions (propositional functions) determine a cla.s.s when they are very simple and cease to do so only when they are complicated and obscure." Who, now, is to decide whether a definition may be regarded as simple enough to be acceptable? To this question there is no answer, if it be not the loyal avowal of a complete inability: "The rules which enable us to recognize whether these definitions are predicative would be extremely complicated and can not commend themselves by any plausible reason. This is a fault which might be remedied by greater ingenuity or by using distinctions not yet pointed out. But hitherto in seeking these rules, I have not been able to find any other directing principle than the absence of contradiction."

This theory therefore remains very obscure; in this night a single light--the word zigzag. What Russell calls the 'zigzaginess' is doubtless the particular characteristic which distinguishes the argument of Epimenides.

According to the theory of limitation of size, a cla.s.s would cease to have the right to exist if it were too extended. Perhaps it might be infinite, but it should not be too much so. But we always meet again the same difficulty; at what precise moment does it begin to be too much so? Of course this difficulty is not solved and Russell pa.s.ses on to the third theory.

In the no-cla.s.ses theory it is forbidden to speak the word 'cla.s.s' and this word must be replaced by various periphrases. What a change for logistic which talks only of cla.s.ses and cla.s.ses of cla.s.ses! It becomes necessary to remake the whole of logistic. Imagine how a page of logistic would look upon suppressing all the propositions where it is a question of cla.s.s. There would only be some scattered survivors in the midst of a blank page. _Apparent rari nantes in gurgite vasto._

Be that as it may, we see how Russell hesitates and the modifications to which he submits the fundamental principles he has. .h.i.therto adopted.

Criteria are needed to decide whether a definition is too complex or too extended, and these criteria can only be justified by an appeal to intuition.

It is toward the no-cla.s.ses theory that Russell finally inclines. Be that as it may, logistic is to be remade and it is not clear how much of it can be saved. Needless to add that Cantorism and logistic are alone under consideration; real mathematics, that which is good for something, may continue to develop in accordance with its own principles without bothering about the storms which rage outside it, and go on step by step with its usual conquests which are final and which it never has to abandon.

VII

_The True Solution_

What choice ought we to make among these different theories? It seems to me that the solution is contained in a letter of M. Richard of which I have spoken above, to be found in the _Revue generale des sciences_ of June 30, 1905. After having set forth the antinomy we have called Richard's antinomy, he gives its explanation. Recall what has already been said of this antinomy. _E_ is the aggregate of _all_ the numbers definable by a finite number of words, _without introducing the notion of the aggregate E itself_. Else the definition of _E_ would contain a vicious circle; we must not define _E_ by the aggregate _E_ itself.

Now we have defined _N_ with a finite number of words, it is true, but with the aid of the notion of the aggregate _E_. And this is why _N_ is not part of _E_. In the example selected by M. Richard, the conclusion presents itself with complete evidence and the evidence will appear still stronger on consulting the text of the letter itself. But the same explanation holds good for the other antinomies, as is easily verified.

Thus _the definitions which should be regarded as not predicative are those which contain a vicious circle_. And the preceding examples sufficiently show what I mean by that. Is it this which Russell calls the 'zigzaginess'? I put the question without answering it.

VIII

_The Demonstrations of the Principle of Induction_