Let us now examine the pretended demonstrations of the principle of induction and in particular those of Whitehead and of Burali-Forti.
We shall speak of Whitehead"s first, and take advantage of certain new terms happily introduced by Russell in his recent memoir. Call _recurrent cla.s.s_ every cla.s.s containing zero, and containing _n_ + 1 if it contains _n_. Call _inductive number_ every number which is a part of _all_ the recurrent cla.s.ses. Upon what condition will this latter definition, which plays an essential role in Whitehead"s proof, be "predicative" and consequently acceptable?
In accordance with what has been said, it is necessary to understand by _all_ the recurrent cla.s.ses, all those in whose definition the notion of inductive number does not enter. Else we fall again upon the vicious circle which has engendered the antinomies.
Now _Whitehead has not taken this precaution_. Whitehead"s reasoning is therefore fallacious; it is the same which led to the antinomies. It was illegitimate when it gave false results; it remains illegitimate when by chance it leads to a true result.
A definition containing a vicious circle defines nothing. It is of no use to say, we are sure, whatever meaning we may give to our definition, zero at least belongs to the cla.s.s of inductive numbers; it is not a question of knowing whether this cla.s.s is void, but whether it can be rigorously deliminated. A "non-predicative" cla.s.s is not an empty cla.s.s, it is a cla.s.s whose boundary is undetermined. Needless to add that this particular objection leaves in force the general objections applicable to all the demonstrations.
IX
Burali-Forti has given another demonstration.[16] But he is obliged to a.s.sume two postulates: First, there always exists at least one infinite cla.s.s. The second is thus expressed:
u[epsilon]K(K - [iota][Lambda]) [inverted c] u <>
The first postulate is not more evident than the principle to be proved.
The second not only is not evident, but it is false, as Whitehead has shown; as moreover any recruit would see at the first glance, if the axiom had been stated in intelligible language, since it means that the number of combinations which can be formed with several objects is less than the number of these objects.
[16] In his article "Le cla.s.si finite," _Atti di Torino_, Vol. x.x.xII.
X
_Zermelo"s a.s.sumption_
A famous demonstration by Zermelo rests upon the following a.s.sumption: In any aggregate (or the same in each aggregate of an a.s.semblage of aggregates) we can always choose _at random_ an element (even if this a.s.semblage of aggregates should contain an infinity of aggregates). This a.s.sumption had been applied a thousand times without being stated, but, once stated, it aroused doubts. Some mathematicians, for instance M. Borel, resolutely reject it; others admire it. Let us see what, according to his last article, Russell thinks of it. He does not speak out, but his reflections are very suggestive.
And first a picturesque example: Suppose we have as many pairs of shoes as there are whole numbers, and so that we can number _the pairs_ from one to infinity, how many shoes shall we have? Will the number of shoes be equal to the number of pairs? Yes, if in each pair the right shoe is distinguishable from the left; it will in fact suffice to give the number 2_n_ - 1 to the right shoe of the _n_th pair, and the number 2_n_ to the left shoe of the _n_th pair. No, if the right shoe is just like the left, because a similar operation would become impossible--unless we admit Zermelo"s a.s.sumption, since then we could choose _at random_ in each pair the shoe to be regarded as the right.
XI
_Conclusions_
A demonstration truly founded upon the principles of a.n.a.lytic logic will be composed of a series of propositions. Some, serving as premises, will be ident.i.ties or definitions; the others will be deduced from the premises step by step. But though the bond between each proposition and the following is immediately evident, it will not at first sight appear how we get from the first to the last, which we may be tempted to regard as a new truth. But if we replace successively the different expressions therein by their definition and if this operation be carried as far as possible, there will finally remain only ident.i.ties, so that all will reduce to an immense tautology. Logic therefore remains sterile unless made fruitful by intuition.
This I wrote long ago; logistic professes the contrary and thinks it has proved it by actually proving new truths. By what mechanism? Why in applying to their reasonings the procedure just described--namely, replacing the terms defined by their definitions--do we not see them dissolve into ident.i.ties like ordinary reasonings? It is because this procedure is not applicable to them. And why? Because their definitions are not predicative and present this sort of hidden vicious circle which I have pointed out above; non-predicative definitions can not be subst.i.tuted for the terms defined. Under these conditions _logistic is not sterile, it engenders antinomies_.
It is the belief in the existence of the actual infinite which has given birth to those non-predicative definitions. Let me explain. In these definitions the word "all" figures, as is seen in the examples cited above. The word "all" has a very precise meaning when it is a question of a finite number of objects; to have another one, when the objects are infinite in number, would require there being an actual (given complete) infinity. Otherwise _all_ these objects could not be conceived as postulated anteriorly to their definition, and then if the definition of a notion _N_ depends upon _all_ the objects _A_, it may be infected with a vicious circle, if among the objects _A_ are some indefinable without the intervention of the notion _N_ itself.
The rules of formal logic express simply the properties of all possible cla.s.sifications. But for them to be applicable it is necessary that these cla.s.sifications be immutable and that we have no need to modify them in the course of the reasoning. If we have to cla.s.sify only a finite number of objects, it is easy to keep our cla.s.sifications without change. If the objects are _indefinite_ in number, that is to say if one is constantly exposed to seeing new and unforeseen objects arise, it may happen that the appearance of a new object may require the cla.s.sification to be modified, and thus it is we are exposed to antinomies. _There is no actual (given complete) infinity._ The Cantorians have forgotten this, and they have fallen into contradiction.
It is true that Cantorism has been of service, but this was when applied to a real problem whose terms were precisely defined, and then we could advance without fear.
Logistic also forgot it, like the Cantorians, and encountered the same difficulties. But the question is to know whether they went this way by accident or whether it was a necessity for them. For me, the question is not doubtful; belief in an actual infinity is essential in the Russell logic. It is just this which distinguishes it from the Hilbert logic.
Hilbert takes the view-point of extension, precisely in order to avoid the Cantorian antinomies. Russell takes the view-point of comprehension.
Consequently for him the genus is anterior to the species, and the _summum genus_ is anterior to all. That would not be inconvenient if the _summum genus_ was finite; but if it is infinite, it is necessary to postulate the infinite, that is to say to regard the infinite as actual (given complete). And we have not only infinite cla.s.ses; when we pa.s.s from the genus to the species in restricting the concept by new conditions, these conditions are still infinite in number. Because they express generally that the envisaged object presents such or such a relation with all the objects of an infinite cla.s.s.
But that is ancient history. Russell has perceived the peril and takes counsel. He is about to change everything, and, what is easily understood, he is preparing not only to introduce new principles which shall allow of operations formerly forbidden, but he is preparing to forbid operations he formerly thought legitimate. Not content to adore what he burned, he is about to burn what he adored, which is more serious. He does not add a new wing to the building, he saps its foundation.
The old logistic is dead, so much so that already the zigzag theory and the no-cla.s.ses theory are disputing over the succession. To judge of the new, we shall await its coming.
BOOK III
THE NEW MECHANICS
CHAPTER I
MECHANICS AND RADIUM
I
_Introduction_
The general principles of Dynamics, which have, since Newton, served as foundation for physical science, and which appeared immovable, are they on the point of being abandoned or at least profoundly modified? This is what many people have been asking themselves for some years. According to them, the discovery of radium has overturned the scientific dogmas we believed the most solid: on the one hand, the impossibility of the trans.m.u.tation of metals; on the other hand, the fundamental postulates of mechanics.
Perhaps one is too hasty in considering these novelties as finally established, and breaking our idols of yesterday; perhaps it would be proper, before taking sides, to await experiments more numerous and more convincing. None the less is it necessary, from to-day, to know the new doctrines and the arguments, already very weighty, upon which they rest.
In few words let us first recall in what those principles consist:
_A._ The motion of a material point isolated and apart from all exterior force is straight and uniform; this is the principle of inertia: without force no acceleration;
_B._ The acceleration of a moving point has the same direction as the resultant of all the forces to which it is subjected; it is equal to the quotient of this resultant by a coefficient called _ma.s.s_ of the moving point.
The ma.s.s of a moving point, so defined, is a constant; it does not depend upon the velocity acquired by this point; it is the same whether the force, being parallel to this velocity, tends only to accelerate or to r.e.t.a.r.d the motion of the point, or whether, on the contrary, being perpendicular to this velocity, it tends to make this motion deviate toward the right, or the left, that is to say to _curve_ the trajectory;
_C._ All the forces affecting a material point come from the action of other material points; they depend only upon the _relative_ positions and velocities of these different material points.
Combining the two principles _B_ and _C_, we reach the _principle of relative motion_, in virtue of which the laws of the motion of a system are the same whether we refer this system to fixed axes, or to moving axes animated by a straight and uniform motion of translation, so that it is impossible to distinguish absolute motion from a relative motion with reference to such moving axes;
_D._ If a material point _A_ acts upon another material point _B_, the body _B_ reacts upon _A_, and these two actions are two equal and directly opposite forces. This is _the principle of the equality of action and reaction_, or, more briefly, the _principle of reaction_.
Astronomic observations and the most ordinary physical phenomena seem to have given of these principles a confirmation complete, constant and very precise. This is true, it is now said, but it is because we have never operated with any but very small velocities; Mercury, for example, the fastest of the planets, goes scarcely 100 kilometers a second. Would this planet act the same if it went a thousand times faster? We see there is yet no need to worry; whatever may be the progress of automobilism, it will be long before we must give up applying to our machines the cla.s.sic principles of dynamics.
How then have we come to make actual speeds a thousand times greater than that of Mercury, equal, for instance, to a tenth or a third of the velocity of light, or approaching still more closely to that velocity?
It is by aid of the cathode rays and the rays from radium.
We know that radium emits three kinds of rays, designated by the three Greek letters [alpha], [beta], [gamma]; in what follows, unless the contrary be expressly stated, it will always be a question of the [beta]
rays, which are a.n.a.logous to the cathode rays.
After the discovery of the cathode rays two theories appeared. Crookes attributed the phenomena to a veritable molecular bombardment; Hertz, to special undulations of the ether. This was a renewal of the debate which divided physicists a century ago about light; Crookes took up the emission theory, abandoned for light; Hertz held to the undulatory theory. The facts seem to decide in favor of Crookes.