First, we have seen that a definition is acceptable only on condition that it implies no contradiction. We have shown likewise that for the first definition this demonstration is impossible; on the other hand, we have just recalled that for the second Hilbert has given a complete proof.

As to the third, evidently it implies no contradiction. Does this mean that the definition guarantees, as it should, the existence of the object defined? We are here no longer in the mathematical sciences, but in the physical, and the word existence has no longer the same meaning.

It no longer signifies absence of contradiction; it means objective existence.

You already see a first reason for the distinction I made between the three cases; there is a second. In the applications we have to make of these three concepts, do they present themselves to us as defined by these three postulates?

The possible applications of the principle of induction are innumerable; take, for example, one of those we have expounded above, and where it is sought to prove that an aggregate of a.s.sumptions can lead to no contradiction. For this we consider one of the series of syllogisms we may go on with in starting from these a.s.sumptions as premises. When we have finished the _n_th syllogism, we see we can make still another and this is the _n_ + 1th. Thus the number _n_ serves to count a series of successive operations; it is a number obtainable by successive additions. This therefore is a number from which we may go back to unity by _successive subtractions_. Evidently we could not do this if we had _n_ = _n_ - 1, since then by subtraction we should always obtain again the same number. So the way we have been led to consider this number _n_ implies a definition of the finite whole number and this definition is the following: A finite whole number is that which can be obtained by successive additions; it is such that _n_ is not equal to _n_ - 1.

That granted, what do we do? We show that if there has been no contradiction up to the _n_th syllogism, no more will there be up to the _n_ + 1th, and we conclude there never will be. You say: I have the right to draw this conclusion, since the whole numbers are by definition those for which a like reasoning is legitimate. But that implies another definition of the whole number, which is as follows: A whole number is that on which we may reason by recurrence. In the particular case it is that of which we may say that, if the absence of contradiction up to the time of a syllogism of which the number is an integer carries with it the absence of contradiction up to the time of the syllogism whose number is the following integer, we need fear no contradiction for any of the syllogisms whose number is an integer.

The two definitions are not identical; they are doubtless equivalent, but only in virtue of a synthetic judgment _a priori_; we can not pa.s.s from one to the other by a purely logical procedure. Consequently we have no right to adopt the second, after having introduced the whole number by a way that presupposes the first.

On the other hand, what happens with regard to the straight line? I have already explained this so often that I hesitate to repeat it again, and shall confine myself to a brief recapitulation of my thought. We have not, as in the preceding case, two equivalent definitions logically irreducible one to the other. We have only one expressible in words.

Will it be said there is another which we feel without being able to word it, since we have the intuition of the straight line or since we represent to ourselves the straight line? First of all, we can not represent it to ourselves in geometric s.p.a.ce, but only in representative s.p.a.ce, and then we can represent to ourselves just as well the objects which possess the other properties of the straight line, save that of satisfying Euclid's postulate. These objects are 'the non-Euclidean straights,' which from a certain point of view are not ent.i.ties void of sense, but circles (true circles of true s.p.a.ce) orthogonal to a certain sphere. If, among these objects equally capable of representation, it is the first (the Euclidean straights) which we call straights, and not the latter (the non-Euclidean straights), this is properly by definition.

And arriving finally at the third example, the definition of phosphorus, we see the true definition would be: Phosphorus is the bit of matter I see in yonder flask.

XII

And since I am on this subject, still another word. Of the phosphorus example I said: "This proposition is a real verifiable physical law, because it means that all bodies having all the other properties of phosphorus, save its point of fusion, melt like it at 44." And it was answered: "No, this law is not verifiable, because if it were shown that two bodies resembling phosphorus melt one at 44 and the other at 50, it might always be said that doubtless, besides the point of fusion, there is some other unknown property by which they differ."

That was not quite what I meant to say. I should have written, "All bodies possessing such and such properties finite in number (to wit, the properties of phosphorus stated in the books on chemistry, the fusion-point excepted) melt at 44."

And the better to make evident the difference between the case of the straight and that of phosphorus, one more remark. The straight has in nature many images more or less imperfect, of which the chief are the light rays and the rotation axis of the solid. Suppose we find the ray of light does not satisfy Euclid's postulate (for example by showing that a star has a negative parallax), what shall we do? Shall we conclude that the straight being by definition the trajectory of light does not satisfy the postulate, or, on the other hand, that the straight by definition satisfying the postulate, the ray of light is not straight?

a.s.suredly we are free to adopt the one or the other definition and consequently the one or the other conclusion; but to adopt the first would be stupid, because the ray of light probably satisfies only imperfectly not merely Euclid's postulate, but the other properties of the straight line, so that if it deviates from the Euclidean straight, it deviates no less from the rotation axis of solids which is another imperfect image of the straight line; while finally it is doubtless subject to change, so that such a line which yesterday was straight will cease to be straight to-morrow if some physical circ.u.mstance has changed.

Suppose now we find that phosphorus does not melt at 44, but at 43.9.

Shall we conclude that phosphorus being by definition that which melts at 44, this body that we did call phosphorus is not true phosphorus, or, on the other hand, that phosphorous melts at 43.9? Here again we are free to adopt the one or the other definition and consequently the one or the other conclusion; but to adopt the first would be stupid because we can not be changing the name of a substance every time we determine a new decimal of its fusion-point.

XIII

To sum up, Russell and Hilbert have each made a vigorous effort; they have each written a work full of original views, profound and often well warranted. These two works give us much to think about and we have much to learn from them. Among their results, some, many even, are solid and destined to live.

But to say that they have finally settled the debate between Kant and Leibnitz and ruined the Kantian theory of mathematics is evidently incorrect. I do not know whether they really believed they had done it, but if they believed so, they deceived themselves.

CHAPTER V

THE LATEST EFFORTS OF THE LOGISTICIANS

I

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?

II

_The Infallibility of Logistic_

On the question of fertility, it seems M. Couturat has nave 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 's.p.a.ce-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 cla.s.s of which the only member is _x_ and which shall be called [iota]_x_, then the cla.s.s of which the only member is the cla.s.s of which the only member is _x_ and which shall be called [mu]_x_, 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 cla.s.s." 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 cla.s.s." And M. Couturat, discovering the success of his attempt, adds: "If the logicians succeed where others have failed, M. Poincare 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.

III

_The Liberty of Contradiction_

I made two princ.i.p.al 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 cla.s.s is not empty." To say: _a_'s exist, is, by definition, to affirm that the cla.s.s _a_ is not null.

And doubtless to affirm that the cla.s.s _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 cla.s.s, it is necessary therefore to establish, by an _example_, that there is an individual belonging to this cla.s.s: "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 cla.s.s of which it is a part may be deduced? Well, no; however paradoxical may appear the a.s.sertion, 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 cla.s.s." M.

Couturat finds his own a.s.sertion 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 embarra.s.s 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 cla.s.s,'

and for that it will always be necessary to prove that the affirmation, "Such an individual belongs to such a cla.s.s," 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 a.s.sent 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.