And on the other hand what means the phrase 'very complex'? I have

already given one solution, but there are others. Complex causes we have

said produce a blend more and more intimate, but after how long a time

will this blend satisfy us? When will it have accumulated sufficient

complexity? When shall we have sufficiently shuffled the cards? If we

mix two powders, one blue, the other white, there comes a moment when

the tint of the mixture seems to us uniform because of the feebleness of

our senses; it will be uniform for the presbyte, forced to gaze from

afar, before it will be so for the myope. And when it has become uniform

for all eyes, we still could push back the limit by the use of

instruments. There is no chance for any man ever to discern the infinite

variety which, if the kinetic theory is true, hides under the uniform

appearance of a gas. And yet if we accept Gouy's ideas on the Brownian

movement, does not the microscope seem on the point of showing us

something analogous?

This new criterion is therefore relative like the first; and if it

retains an objective character, it is because all men have approximately

the same senses, the power of their instruments is limited, and besides

they use them only exceptionally.

IX

It is just the same in the moral sciences and particularly in history.

The historian is obliged to make a choice among the events of the epoch

he studies; he recounts only those which seem to him the most important.

He therefore contents himself with relating the most momentous events of

the sixteenth century, for example, as likewise the most remarkable

facts of the seventeenth century. If the first suffice to explain the

second, we say these conform to the laws of history. But if a great

event of the seventeenth century should have for cause a small fact of

the sixteenth century which no history reports, which all the world has

neglected, then we say this event is due to chance. This word has

therefore the same sense as in the physical sciences; it means that

slight causes have produced great effects.

The greatest bit of chance is the birth of a great man. It is only by

chance that meeting of two germinal cells, of different sex, containing

precisely, each on its side, the mysterious elements whose mutual

reaction must produce the genius. One will agree that these elements

must be rare and that their meeting is still more rare. How slight a

thing it would have required to deflect from its route the carrying

spermatozoon. It would have sufficed to deflect it a tenth of a

millimeter and Napoleon would not have been born and the destinies of a

continent would have been changed. No example can better make us

understand the veritable characteristics of chance.

One more word about the paradoxes brought out by the application of the

calculus of probabilities to the moral sciences. It has been proven that

no Chamber of Deputies will ever fail to contain a member of the

opposition, or at least such an event would be so improbable that we

might without fear wager the contrary, and bet a million against a sou.

Condorcet has striven to calculate how many jurors it would require to

make a judicial error practically impossible. If we had used the results

of this calculation, we should certainly have been exposed to the same

disappointments as in betting, on the faith of the calculus, that the

opposition would never be without a representative.

The laws of chance do not apply to these questions. If justice be not

always meted out to accord with the best reasons, it uses less than we

think the method of Bridoye. This is perhaps to be regretted, for then

the system of Condorcet would shield us from judicial errors.

What is the meaning of this? We are tempted to attribute facts of this

nature to chance because their causes are obscure; but this is not true

chance. The causes are unknown to us, it is true, and they are even

complex; but they are not sufficiently so, since they conserve

something. We have seen that this it is which distinguishes causes 'too

simple.' When men are brought together they no longer decide at random

and independently one of another; they influence one another. Multiplex

causes come into action. They worry men, dragging them to right or left,

but one thing there is they can not destroy, this is their Panurge

flock-of-sheep habits. And this is an invariant.

X

Difficulties are indeed involved in the application of the calculus of

probabilities to the exact sciences. Why are the decimals of a table of

logarithms, why are those of the number [pi] distributed in accordance

with the laws of chance? Elsewhere I have already studied the question

in so far as it concerns logarithms, and there it is easy. It is clear

that a slight difference of argument will give a slight difference of

logarithm, but a great difference in the sixth decimal of the logarithm.

Always we find again the same criterion.

But as for the number [pi], that presents more difficulties, and I have

at the moment nothing worth while to say.

There would be many other questions to resolve, had I wished to attack

them before solving that which I more specially set myself. When we

reach a simple result, when we find for example a round number, we say

that such a result can not be due to chance, and we seek, for its

explanation, a non-fortuitous cause. And in fact there is only a very

slight probability that among 10,000 numbers chance will give a round

number; for example, the number 10,000. This has only one chance in

10,000. But there is only one chance in 10,000 for the occurrence of any

other one number; and yet this result will not astonish us, nor will it

be hard for us to attribute it to chance; and that simply because it

will be less striking.

Is this a simple illusion of ours, or are there cases where this way of

thinking is legitimate? We must hope so, else were all science

impossible. When we wish to check a hypothesis, what do we do? We can

not verify all its consequences, since they would be infinite in number;

we content ourselves with verifying certain ones and if we succeed we

declare the hypothesis confirmed, because so much success could not be

due to chance. And this is always at bottom the same reasoning.

I can not completely justify it here, since it would take too much time;

but I may at least say that we find ourselves confronted by two

hypotheses, either a simple cause or that aggregate of complex causes we

call chance. We find it natural to suppose that the first should produce

a simple result, and then, if we find that simple result, the round

number for example, it seems more likely to us to be attributable to the

simple cause which must give it almost certainly, than to chance which

could only give it once in 10,000 times. It will not be the same if we

find a result which is not simple; chance, it is true, will not give

this more than once in 10,000 times; but neither has the simple cause

any more chance of producing it.

BOOK II

MATHEMATICAL REASONING

CHAPTER I

THE RELATIVITY OF SPACE

I

It is impossible to represent to oneself empty space; all our efforts to

imagine a pure space, whence should be excluded the changing images of

material objects, can result only in a representation where vividly

colored surfaces, for example, are replaced by lines of faint

coloration, and we can not go to the very end in this way without all

vanishing and terminating in nothingness. Thence comes the irreducible

relativity of space.

Whoever speaks of absolute space uses a meaningless phrase. This is a

truth long proclaimed by all who have reflected upon the matter, but

which we are too often led to forget.

I am at a determinate point in Paris, place du Panthéon for instance,

and I say: I shall come back _here_ to-morrow. If I be asked: Do you

mean you will return to the same point of space, I shall be tempted to

answer: yes; and yet I shall be wrong, since by to-morrow the earth will

have journeyed hence, carrying with it the place du Panthéon, which will

have traveled over more than two million kilometers. And if I tried to

speak more precisely, I should gain nothing, since our globe has run

over these two million kilometers in its motion with relation to the

sun, while the sun in its turn is displaced with reference to the Milky

Way, while the Milky Way itself is doubtless in motion without our being

able to perceive its velocity. So that we are completely ignorant, and

always shall be, of how much the place du Panthéon is displaced in a

day.

In sum, I meant to say: To-morrow I shall see again the dome and the

pediment of the Panthéon, and if there were no Panthéon my phrase would

be meaningless and space would vanish.

This is one of the most commonplace forms of the principle of the

relativity of space; but there is another, upon which Delbeuf has

particularly insisted. Suppose that in the night all the dimensions of

the universe become a thousand times greater: the world will have

remained _similar_ to itself, giving to the word _similitude_ the same

meaning as in Euclid, Book VI. Only what was a meter long will measure

thenceforth a kilometer, what was a millimeter long will become a meter.

The bed whereon I lie and my body itself will be enlarged in the same

proportion.

When I awake to-morrow morning, what sensation shall I feel in presence

of such an astounding transformation? Well, I shall perceive nothing at

all. The most precise measurements will be incapable of revealing to me

anything of this immense convulsion, since the measures I use will have

varied precisely in the same proportion as the objects I seek to

measure. In reality, this convulsion exists only for those who reason as

if space were absolute. If I for a moment have reasoned as they do, it

is the better to bring out that their way of seeing implies

contradiction. In fact it would be better to say that, space being

relative, nothing at all has happened, which is why we have perceived

nothing.

Has one the right, therefore, to say he knows the distance between two

points? No, since this distance could undergo enormous variations

without our being able to perceive them, provided the other distances

have varied in the same proportion. We have just seen that when I say: I

shall be here to-morrow, this does not mean: To-morrow I shall be at the

same point of space where I am to-day, but rather: To-morrow I shall be

at the same distance from the Panthéon as to-day. And we see that this

statement is no longer sufficient and that I should say: To-morrow and

to-day my distance from the Panthéon will be equal to the same number of

times the height of my body.

But this is not all; I have supposed the dimensions of the world to

vary, but that at least the world remained always similar to itself. We

might go much further, and one of the most astonishing theories of

modern physics furnishes us the occasion.

According to Lorentz and Fitzgerald, all the bodies borne along in the

motion of the earth undergo a deformation.

This deformation is, in reality, very slight, since all dimensions

parallel to the movement of the earth diminish by a hundred millionth,

while the dimensions perpendicular to this movement are unchanged. But

it matters little that it is slight, that it exists suffices for the

conclusion I am about to draw. And besides, I have said it was slight,

but in reality I know nothing about it; I have myself been victim of the

tenacious illusion which makes us believe we conceive an absolute space;

I have thought of the motion of the earth in its elliptic orbit around

the sun, and I have allowed thirty kilometers as its velocity. But its

real velocity (I mean, this time, not its absolute velocity, which is

meaningless, but its velocity with relation to the ether), I do not know

that, and have no means of knowing it: it is perhaps, 10, 100 times

greater, and then the deformation will be 100, 10,000 times more.

Can we show this deformation? Evidently not; here is a cube with edge

one meter; in consequence of the earth's displacement it is deformed,

one of its edges, that parallel to the motion, becomes smaller, the

others do not change. If I wish to assure myself of it by aid of a meter

measure, I shall measure first one of the edges perpendicular to the

motion and shall find that my standard meter fits this edge exactly; and

in fact neither of these two lengths is changed, since both are

perpendicular to the motion. Then I wish to measure the other edge, that

parallel to the motion; to do this I displace my meter and turn it so as

to apply it to the edge. But the meter, having changed orientation and

become parallel to the motion, has undergone, in its turn, the

deformation, so that though the edge be not a meter long, it will fit

exactly, I shall find out nothing.

You ask then of what use is the hypothesis of Lorentz and of Fitzgerald

if no experiment can permit of its verification? It is my exposition

that has been incomplete; I have spoken only of measurements that can be

made with a meter; but we can also measure a length by the time it takes

light to traverse it, on condition we suppose the velocity of light

constant and independent of direction. Lorentz could have accounted for

the facts by supposing the velocity of light greater in the direction

of the earth's motion than in the perpendicular direction. He preferred

to suppose that the velocity is the same in these different directions

but that the bodies are smaller in the one than in the other. If the

wave surfaces of light had undergone the same deformations as the

material bodies we should never have perceived the Lorentz-Fitzgerald

deformation.

In either case, it is not a question of absolute magnitude, but of the

measure of this magnitude by means of some instrument; this instrument

may be a meter, or the path traversed by light; it is only the relation

of the magnitude to the instrument that we measure; and if this relation

is altered, we have no way of knowing whether it is the magnitude or the

instrument which has changed.

But what I wish to bring out is, that in this deformation the world has

not remained similar to itself; squares have become rectangles, circles

ellipses, spheres ellipsoids. And yet we have no way of knowing whether

this deformation be real.

Evidently one could go much further: in place of the Lorentz-Fitzgerald

deformation, whose laws are particularly simple, we could imagine any

deformation whatsoever. Bodies could be deformed according to any laws,

as complicated as we might wish, we never should notice it provided all

bodies without exception were deformed according to the same laws. In

saying, all bodies without exception, I include of course our own body

and the light rays emanating from different objects.

If we look at the world in one of those mirrors of complicated shape

which deform objects in a bizarre way, the mutual relations of the

different parts of this world would not be altered; if, in fact two real

objects touch, their images likewise seem to touch. Of course when we

look in such a mirror we see indeed the deformation, but this is because

the real world subsists alongside of its deformed image; and then even

were this real world hidden from us, something there is could not be

hidden, ourself; we could not cease to see, or at least to feel, our

body and our limbs which have not been deformed and which continue to

serve us as instruments of measure.

But if we imagine our body itself deformed in the same way as if seen

in the mirror, these instruments of measure in their turn will fail us

and the deformation will no longer be ascertainable.

Consider in the same way two worlds images of one another; to each

object _P_ of the world _A_ corresponds in the world _B_ an object _P'_,

its image; the coordinates of this image _P'_ are determinate functions

of those of the object _P_; moreover these functions may be any

whatsoever; I only suppose them chosen once for all. Between the

position of _P_ and that of _P'_ there is a constant relation; what this

relation is, matters not; enough that it be constant.

Well, these two worlds will be indistinguishable one from the other. I

mean the first will be for its inhabitants what the second is for its.

And so it will be as long as the two worlds remain strangers to each

other. Suppose we lived in world _A_, we shall have constructed our

science and in particular our geometry; during this time the inhabitants

of world _B_ will have constructed a science, and as their world is the

image of ours, their geometry will also be the image of ours or, better,

it will be the same. But if for us some day a window is opened upon

world _B_, how we shall pity them: "Poor things," we shall say, "they

think they have made a geometry, but what they call so is only a

grotesque image of ours; their straights are all twisted, their circles

are humped, their spheres have capricious inequalities." And we shall

never suspect they say the same of us, and one never will know who is

right.

We see in how broad a sense should be understood the relativity of

space; space is in reality amorphous and the things which are therein

alone give it a form. What then should be thought of that direct

intuition we should have of the straight or of distance? So little have

we intuition of distance in itself that in the night, as we have said, a

distance might become a thousand times greater without our being able to

perceive it, if all other distances had undergone the same alteration.

And even in a night the world _B_ might be substituted for the world _A_

without our having any way of knowing it, and then the straight lines of

yesterday would have ceased to be straight and we should never notice.

One part of space is not by itself and in the absolute sense of the word

equal to another part of space; because if so it is for us, it would not

be for the dwellers in world _B_; and these have just as much right to

reject our opinion as we to condemn theirs.

I have elsewhere shown what are the consequences of these facts from the

viewpoint of the idea we should form of non-Euclidean geometry and other

analogous geometries; to that I do not care to return; and to-day I

shall take a somewhat different point of view.

II

If this intuition of distance, of direction, of the straight line, if

this direct intuition of space in a word does not exist, whence comes

our belief that we have it? If this is only an illusion, why is this

illusion so tenacious? It is proper to examine into this. We have said

there is no direct intuition of size and we can only arrive at the

relation of this magnitude to our instruments of measure. We should

therefore not have been able to construct space if we had not had an

instrument to measure it; well, this instrument to which we relate

everything, which we use instinctively, it is our own body. It is in

relation to our body that we place exterior objects, and the only

spatial relations of these objects that we can represent are their

relations to our body. It is our body which serves us, so to speak, as

system of axes of coordinates.

For example, at an instant [alpha], the presence of the object _A_ is

revealed to me by the sense of sight; at another instant, [beta], the

presence of another object, _B_, is revealed to me by another sense,

that of hearing or of touch, for instance. I judge that this object _B_

occupies the same place as the object _A_. What does that mean? First

that does not signify that these two objects occupy, at two different

moments, the same point of an absolute space, which even if it existed

would escape our cognition, since, between the instants [alpha] and

[beta], the solar system has moved and we can not know its displacement.

That means these two objects occupy the same relative position with

reference to our body.

But even this, what does it mean? The impressions that have come to us

from these objects have followed paths absolutely different, the optic

nerve for the object _A_, the acoustic nerve for the object _B_.

They have nothing in common from the qualitative point of view. The

representations we are able to make of these two objects are absolutely

heterogeneous, irreducible one to the other. Only I know that to reach

the object _A_ I have just to extend the right arm in a certain way;

even when I abstain from doing it, I represent to myself the muscular

sensations and other analogous sensations which would accompany this

extension, and this representation is associated with that of the

object _A_.

Now, I likewise know I can reach the object _B_ by extending my right

arm in the same manner, an extension accompanied by the same train of

muscular sensations. And when I say these two objects occupy the same

place, I mean nothing more.

I also know I could have reached the object _A_ by another appropriate

motion of the left arm and I represent to myself the muscular sensations

which would have accompanied this movement; and by this same motion of

the left arm, accompanied by the same sensations, I likewise could have

reached the object _B_.

And that is very important, since thus I can defend myself against

dangers menacing me from the object _A_ or the object _B_. With each of

the blows we can be hit, nature has associated one or more parries which

permit of our guarding ourselves. The same parry may respond to several

strokes; and so it is, for instance, that the same motion of the right

arm would have allowed us to guard at the instant [alpha] against the

object _A_ and at the instant [beta] against the object _B_. Just so,

the same stroke can be parried in several ways, and we have said, for

instance, the object _A_ could be reached indifferently either by a

certain movement of the right arm or by a certain movement of the left

arm.

All these parries have nothing in common except warding off the same

blow, and this it is, and nothing else, which is meant when we say they

are movements terminating at the same point of space. Just so, these

objects, of which we say they occupy the same point of space, have

nothing in common, except that the same parry guards against them.

Or, if you choose, imagine innumerable telegraph wires, some

centripetal, others centrifugal. The centripetal wires warn us of

accidents happening without; the centrifugal wires carry the reparation.

Connections are so established that when a centripetal wire is traversed

by a current this acts on a relay and so starts a current in one of the

centrifugal wires, and things are so arranged that several centripetal

wires may act on the same centrifugal wire if the same remedy suits

several ills, and that a centripetal wire may agitate different

centrifugal wires, either simultaneously or in lieu one of the other

when the same ill may be cured by several remedies.

It is this complex system of associations, it is this table of

distribution, so to speak, which is all our geometry or, if you wish,

all in our geometry that is instinctive. What we call our intuition of

the straight line or of distance is the consciousness we have of these

associations and of their imperious character.

And it is easy to understand whence comes this imperious character

itself. An association will seem to us by so much the more

indestructible as it is more ancient. But these associations are not,

for the most part, conquests of the individual, since their trace is

seen in the new-born babe: they are conquests of the race. Natural

selection had to bring about these conquests by so much the more quickly

as they were the more necessary.

On this account, those of which we speak must have been of the earliest

in date, since without them the defense of the organism would have been

impossible. From the time when the cellules were no longer merely

juxtaposed, but were called upon to give mutual aid, it was needful that

a mechanism organize analogous to what we have described, so that this

aid miss not its way, but forestall the peril.

When a frog is decapitated, and a drop of acid is placed on a point of

its skin, it seeks to wipe off the acid with the nearest foot, and, if

this foot be amputated, it sweeps it off with the foot of the opposite

side. There we have the double parry of which I have just spoken,

allowing the combating of an ill by a second remedy, if the first fails.

And it is this multiplicity of parries, and the resulting coordination,

which is space.

We see to what depths of the unconscious we must descend to find the

first traces of these spatial associations, since only the inferior

parts of the nervous system are involved. Why be astonished then at the

resistance we oppose to every attempt made to dissociate what so long

has been associated? Now, it is just this resistance that we call the

evidence for the geometric truths; this evidence is nothing but the

repugnance we feel toward breaking with very old habits which have

always proved good.

III

The space so created is only a little space extending no farther than my

arm can reach; the intervention of the memory is necessary to push back

its limits. There are points which will remain out of my reach, whatever

effort I make to stretch forth my hand; if I were fastened to the ground

like a hydra polyp, for instance, which can only extend its tentacles,

all these points would be outside of space, since the sensations we

could experience from the action of bodies there situated, would be

associated with the idea of no movement allowing us to reach them, of no

appropriate parry. These sensations would not seem to us to have any

spatial character and we should not seek to localize them.

But we are not fixed to the ground like the lower animals; we can, if

the enemy be too far away, advance toward him first and extend the hand

when we are sufficiently near. This is still a parry, but a parry at

long range. On the other hand, it is a complex parry, and into the

representation we make of it enter the representation of the muscular

sensations caused by the movements of the legs, that of the muscular

sensations caused by the final movement of the arm, that of the

sensations of the semicircular canals, etc. We must, besides, represent

to ourselves, not a complex of simultaneous sensations, but a complex of

successive sensations, following each other in a determinate order, and

this is why I have just said the intervention of memory was necessary.

Notice moreover that, to reach the same point, I may approach nearer the

mark to be attained, so as to have to stretch my arm less. What more? It

is not one, it is a thousand parries I can oppose to the same danger.

All these parries are made of sensations which may have nothing in

common and yet we regard them as defining the same point of space, since

they may respond to the same danger and are all associated with the

notion of this danger. It is the potentiality of warding off the same

stroke which makes the unity of these different parries, as it is the

possibility of being parried in the same way which makes the unity of

the strokes so different in kind, which may menace us from the same

point of space. It is this double unity which makes the individuality of

each point of space, and, in the notion of point, there is nothing else.

The space before considered, which might be called _restricted space_,

was referred to coordinate axes bound to my body; these axes were fixed,

since my body did not move and only my members were displaced. What are

the axes to which we naturally refer the _extended space_? that is to

say the new space just defined. We define a point by the sequence of

movements to be made to reach it, starting from a certain initial

position of the body. The axes are therefore fixed to this initial

position of the body.

But the position I call initial may be arbitrarily chosen among all the

positions my body has successively occupied; if the memory more or less

unconscious of these successive positions is necessary for the genesis

of the notion of space, this memory may go back more or less far into

the past. Thence results in the definition itself of space a certain

indetermination, and it is precisely this indetermination which

constitutes its relativity.

There is no absolute space, there is only space relative to a certain

initial position of the body. For a conscious being fixed to the ground

like the lower animals, and consequently knowing only restricted space,

space would still be relative (since it would have reference to his

body), but this being would not be conscious of this relativity, because

the axes of reference for this restricted space would be unchanging!

Doubtless the rock to which this being would be fettered would not be

motionless, since it would be carried along in the movement of our

planet; for us consequently these axes would change at each instant; but

for him they would be changeless. We have the faculty of referring our

extended space now to the position _A_ of our body, considered as

initial, again to the position _B_, which it had some moments afterward,

and which we are free to regard in its turn as initial; we make

therefore at each instant unconscious transformations of coordinates.

This faculty would be lacking in our imaginary being, and from not

having traveled, he would think space absolute. At every instant, his

system of axes would be imposed upon him; this system would have to

change greatly in reality, but for him it would be always the same,

since it would be always the _only_ system. Quite otherwise is it with

us, who at each instant have many systems between which we may choose at

will, on condition of going back by memory more or less far into the

past.

This is not all; restricted space would not be homogeneous; the

different points of this space could not be regarded as equivalent,

since some could be reached only at the cost of the greatest efforts,

while others could be easily attained. On the contrary, our extended

space seems to us homogeneous, and we say all its points are equivalent.

What does that mean?

If we start from a certain place _A_, we can, from this position, make

certain movements, _M_, characterized by a certain complex of muscular

sensations. But, starting from another position, _B_, we make movements

_M'_ characterized by the same muscular sensations. Let _a_, then, be

the situation of a certain point of the body, the end of the index

finger of the right hand for example, in the initial position _A_, and

_b_ the situation of this same index when, starting from this position

_A_, we have made the motions _M_. Afterwards, let _a'_ be the situation

of this index in the position _B_, and _b'_ its situation when, starting

from the position _B_, we have made the motions _M'_.

Well, I am accustomed to say that the points of space _a_ and _b_ are

related to each other just as the points _a'_ and _b'_, and this simply

means that the two series of movements _M_ and _M'_ are accompanied by

the same muscular sensations. And as I am conscious that, in passing

from the position _A_ to the position _B_, my body has remained capable

of the same movements, I know there is a point of space related to the

point _a'_ just as any point _b_ is to the point _a_, so that the two

points _a_ and _a'_ are equivalent. This is what is called the

homogeneity of space. And, at the same time, this is why space is

relative, since its properties remain the same whether it be referred to

the axes _A_ or to the axes _B_. So that the relativity of space and its

homogeneity are one sole and same thing.

Now, if I wish to pass to the great space, which no longer serves only

for me, but where I may lodge the universe, I get there by an act of

imagination. I imagine how a giant would feel who could reach the

planets in a few steps; or, if you choose, what I myself should feel in

presence of a miniature world where these planets were replaced by

little balls, while on one of these little balls moved a liliputian I

should call myself. But this act of imagination would be impossible for

me had I not previously constructed my restricted space and my extended

space for my own use.

IV

Why now have all these spaces three dimensions? Go back to the "table of

distribution" of which we have spoken. We have on the one side the list

of the different possible dangers; designate them by _A1_, _A2_, etc.;

and, on the other side, the list of the different remedies which I shall

call in the same way _B1_, _B2_, etc. We have then connections between

the contact studs or push buttons of the first list and those of the

second, so that when, for instance, the announcer of danger _A3_

functions, it will put or may put in action the relay corresponding to

the parry _B4_.

As I have spoken above of centripetal or centrifugal wires, I fear lest

one see in all this, not a simple comparison, but a description of the

nervous system. Such is not my thought, and that for several reasons:

first I should not permit myself to put forth an opinion on the

structure of the nervous system which I do not know, while those who

have studied it speak only circumspectly; again because, despite my

incompetence, I well know this scheme would be too simplistic; and

finally because on my list of parries, some would figure very complex,

which might even, in the case of extended space, as we have seen above,

consist of many steps followed by a movement of the arm. It is not a

question then of physical connection between two real conductors but of

psychologic association between two series of sensations.

If _A1_ and _A2_ for instance are both associated with the parry _B1_,

and if _A1_ is likewise associated with the parry _B2_, it will

generally happen that _A2_ and _B2_ will also themselves be associated.

If this fundamental law were not generally true, there would exist only

an immense confusion and there would be nothing resembling a conception

of space or a geometry. How in fact have we defined a point of space. We

have done it in two ways: it is on the one hand the aggregate of

announcers _A_ in connection with the same parry _B_; it is on the other

hand the aggregate of parries _B_ in connection with the same announcer

_A_. If our law was not true, we should say _A1_ and _A2_ correspond to

the same point since they are both in connection with _B1_; but we

should likewise say they do not correspond to the same point, since _A1_

would be in connection with _B2_ and the same would not be true of _A2_.

This would be a contradiction.

But, from another side, if the law were rigorously and always true,

space would be very different from what it is. We should have categories

strongly contrasted between which would be portioned out on the one hand

the announcers _A_, on the other hand the parries _B_; these categories

would be excessively numerous, but they would be entirely separated one

from another. Space would be composed of points very numerous, but

discrete; it would be _discontinuous_. There would be no reason for

ranging these points in one order rather than another, nor consequently

for attributing to space three dimensions.

But it is not so; permit me to resume for a moment the language of those

who already know geometry; this is quite proper since this is the

language best understood by those I wish to make understand me.

When I desire to parry the stroke, I seek to attain the point whence

comes this blow, but it suffices that I approach quite near. Then the

parry _B1_ may answer for _A1_ and for _A2_, if the point which

corresponds to _B1_ is sufficiently near both to that corresponding to

_A1_ and to that corresponding to _A2_. But it may happen that the point

corresponding to another parry _B2_ may be sufficiently near to the

point corresponding to A1 and not sufficiently near the point

corresponding to _A2_; so that the parry _B2_ may answer for _A1_

without answering for _A2_. For one who does not yet know geometry, this

translates itself simply by a derogation of the law stated above. And

then things will happen thus:

Two parries _B1_ and _B2_ will be associated with the same warning _A1_

and with a large number of warnings which we shall range in the same

category as _A1_ and which we shall make correspond to the same point of

space. But we may find warnings _A2_ which will be associated with _B2_

without being associated with _B1_, and which in compensation will be

associated with _B3_, which _B3_ was not associated with _A1_, and so

forth, so that we may write the series

_B1_, _A1_, _B2_, _A2_, _B3_, _A3_, _B4_, _A4_,

where each term is associated with the following and the preceding, but

not with the terms several places away.

Needless to add that each of the terms of these series is not isolated,

but forms part of a very numerous category of other warnings or of other

parries which have the same connections as it, and which may be regarded

as belonging to the same point of space.

The fundamental law, though admitting of exceptions, remains therefore

almost always true. Only, in consequence of these exceptions, these

categories, in place of being entirely separated, encroach partially one

upon another and mutually penetrate in a certain measure, so that space

becomes continuous.

On the other hand, the order in which these categories are to be ranged

is no longer arbitrary, and if we refer to the preceding series, we see

it is necessary to put _B2_ between _A1_ and _A2_ and consequently

between _B1_ and _B3_ and that we could not for instance put it between

_B3_ and _B4_.

There is therefore an order in which are naturally arranged our

categories which correspond to the points of space, and experience

teaches us that this order presents itself under the form of a table

of triple entry, and this is why space has three dimensions.

V

So the characteristic property of space, that of having three

dimensions, is only a property of our table of distribution, an internal

property of the human intelligence, so to speak. It would suffice to

destroy certain of these connections, that is to say of the associations

of ideas to give a different table of distribution, and that might be

enough for space to acquire a fourth dimension.

Some persons will be astonished at such a result. The external world,

they will think, should count for something. If the number of dimensions

comes from the way we are made, there might be thinking beings living in

our world, but who might be made differently from us and who would

believe space has more or less than three dimensions. Has not M. de Cyon

said that the Japanese mice, having only two pair of semicircular

canals, believe that space is two-dimensional? And then this thinking

being, if he is capable of constructing a physics, would he not make a

physics of two or of four dimensions, and which in a sense would still

be the same as ours, since it would be the description of the same world

in another language?

It seems in fact that it would be possible to translate our physics into

the language of geometry of four dimensions; to attempt this translation

would be to take great pains for little profit, and I shall confine

myself to citing the mechanics of Hertz where we have something

analogous. However, it seems that the translation would always be less

simple than the text, and that it would always have the air of a

translation, that the language of three dimensions seems the better

fitted to the description of our world, although this description can be

rigorously made in another idiom. Besides, our table of distribution was

not made at random. There is connection between the warning _A1_ and the

parry _B1_, this is an internal property of our intelligence; but why

this connection? It is because the parry _B1_ affords means effectively

to guard against the danger _A1_; and this is a fact exterior to us,

this is a property of the exterior world. Our table of distribution is

therefore only the translation of an aggregate of exterior facts; if it

has three dimensions, this is because it has adapted itself to a world

having certain properties; and the chief of these properties is that

there exist natural solids whose displacements follow sensibly the laws

we call laws of motion of rigid solids. If therefore the language of

three dimensions is that which permits us most easily to describe our

world, we should not be astonished; this language is copied from our

table of distribution; and it is in order to be able to live in this

world that this table has been established.

I have said we could conceive, living in our world, thinking beings

whose table of distribution would be four-dimensional and who

consequently would think in hyperspace. It is not certain however that

such beings, admitting they were born there, could live there and defend

themselves against the thousand dangers by which they would there be

assailed.

VI

A few remarks to end with. There is a striking contrast between the

roughness of this primitive geometry, reducible to what I call a table

of distribution, and the infinite precision of the geometers' geometry.

And yet this is born of that; but not of that alone; it must be made

fecund by the faculty we have of constructing mathematical concepts,

such as that of group, for instance; it was needful to seek among the

pure concepts that which best adapts itself to this rough space whose

genesis I have sought to explain and which is common to us and the

higher animals.

The evidence for certain geometric postulates, we have said, is only our

repugnance to renouncing very old habits. But these postulates are

infinitely precise, while these habits have something about them

essentially pliant. When we wish to think, we need postulates infinitely

precise, since this is the only way to avoid contradiction; but among

all the possible systems of postulates, there are some we dislike to

choose because they are not sufficiently in accord with our habits;

however pliant, however elastic they may be, these have a limit of

elasticity.

We see that if geometry is not an experimental science, it is a science

born apropos of experience; that we have created the space it studies,

but adapting it to the world wherein we live. We have selected the most

convenient space, but experience has guided our choice; as this choice

has been unconscious, we think it has been imposed upon us; some say

experience imposes it, others that we are born with our space ready

made; we see from the preceding considerations, what in these two

opinions is the part of truth, what of error.

In this progressive education whose outcome has been the construction of

space, it is very difficult to determine what is the part of the

individual, what the part of the race. How far could one of us,

transported from birth to an entirely different world, where were

dominant, for instance, bodies moving in conformity to the laws of

motion of non-Euclidean solids, renounce the ancestral space to build a

space completely new?

The part of the race seems indeed preponderant; yet if to it we owe

rough space, the soft space I have spoken of, the space of the higher

animals, is it not to the unconscious experience of the individual we

owe the infinitely precise space of the geometer? This is a question not

easy to solve. Yet we cite a fact showing that the space our ancestors

have bequeathed us still retains a certain plasticity. Some hunters

learn to shoot fish under water, though the image of these fish be

turned up by refraction. Besides they do it instinctively: they

therefore have learned to modify their old instinct of direction; or, if

you choose, to substitute for the association _A1_, _B1_, another

association _A1_, _B2_, because experience showed them the first would

not work.

CHAPTER II

MATHEMATICAL DEFINITIONS AND TEACHING

1. I should speak here of general definitions in mathematics; at least

that is the title, but it will be impossible to confine myself to the

subject as strictly as the rule of unity of action would require; I

shall not be able to treat it without touching upon a few other related

questions, and if thus I am forced from time to time to walk on the

bordering flower-beds on the right or left, I pray you bear with me.

What is a good definition? For the philosopher or the scientist it is a

definition which applies to all the objects defined, and only those; it

is the one satisfying the rules of logic. But in teaching it is not

that; a good definition is one understood by the scholars.

How does it happen that so many refuse to understand mathematics? Is

that not something of a paradox? Lo and behold! a science appealing only

to the fundamental principles of logic, to the principle of

contradiction, for instance, to that which is the skeleton, so to speak,

of our intelligence, to that of which we can not divest ourselves

without ceasing to think, and there are people who find it obscure! and

they are even in the majority! That they are incapable of inventing may

pass, but that they do not understand the demonstrations shown them,

that they remain blind when we show them a light which seems to us

flashing pure flame, this it is which is altogether prodigious.

And yet there is no need of a wide experience with examinations to know

that these blind men are in no wise exceptional beings. This is a

problem not easy to solve, but which should engage the attention of all

those wishing to devote themselves to teaching.

What is it, to understand? Has this word the same meaning for all the

world? To understand the demonstration of a theorem, is that to examine

successively each of the syllogisms composing it and to ascertain its

correctness, its conformity to the rules of the game? Likewise, to

understand a definition, is this merely to recognize that one already

knows the meaning of all the terms employed and to ascertain that it

implies no contradiction?

For some, yes; when they have done this, they will say: I understand.

For the majority, no. Almost all are much more exacting; they wish to

know not merely whether all the syllogisms of a demonstration are

correct, but why they link together in this order rather than another.

In so far as to them they seem engendered by caprice and not by an

intelligence always conscious of the end to be attained, they do not

believe they understand.

Doubtless they are not themselves just conscious of what they crave and

they could not formulate their desire, but if they do not get

satisfaction, they vaguely feel that something is lacking. Then what

happens? In the beginning they still perceive the proofs one puts under

their eyes; but as these are connected only by too slender a thread to

those which precede and those which follow, they pass without leaving

any trace in their head; they are soon forgotten; a moment bright, they

quickly vanish in night eternal. When they are farther on, they will no

longer see even this ephemeral light, since the theorems lean one upon

another and those they would need are forgotten; thus it is they become

incapable of understanding mathematics.

This is not always the fault of their teacher; often their mind, which

needs to perceive the guiding thread, is too lazy to seek and find it.

But to come to their aid, we first must know just what hinders them.

Others will always ask of what use is it; they will not have understood

if they do not find about them, in practise or in nature, the

justification of such and such a mathematical concept. Under each word

they wish to put a sensible image; the definition must evoke this image,

so that at each stage of the demonstration they may see it transform and

evolve. Only upon this condition do they comprehend and retain. Often

these deceive themselves; they do not listen to the reasoning, they look

at the figures; they think they have understood and they have only seen.

2. How many different tendencies! Must we combat them? Must we use them?

And if we wish to combat them, which should be favored? Must we show

those content with the pure logic that they have seen only one side of

the matter? Or need we say to those not so cheaply satisfied that what

they demand is not necessary?

In other words, should we constrain the young people to change the

nature of their minds? Such an attempt would be vain; we do not possess

the philosopher's stone which would enable us to transmute one into

another the metals confided to us; all we can do is to work with them,

adapting ourselves to their properties.

Many children are incapable of becoming mathematicians, to whom however

it is necessary to teach mathematics; and the mathematicians themselves

are not all cast in the same mold. To read their works suffices to

distinguish among them two sorts of minds, the logicians like

Weierstrass for example, the intuitives like Riemann. There is the same

difference among our students. The one sort prefer to treat their

problems 'by analysis' as they say, the others 'by geometry.'

It is useless to seek to change anything of that, and besides would it

be desirable? It is well that there are logicians and that there are

intuitives; who would dare say whether he preferred that Weierstrass had

never written or that there never had been a Riemann? We must therefore

resign ourselves to the diversity of minds, or better we must rejoice in

it.

3. Since the word understand has many meanings, the definitions which

will be best understood by some will not be best suited to others. We

have those which seek to produce an image, and those where we confine

ourselves to combining empty forms, perfectly intelligible, but purely

intelligible, which abstraction has deprived of all matter.

I know not whether it be necessary to cite examples. Let us cite them,

anyhow, and first the definition of fractions will furnish us an extreme

case. In the primary schools, to define a fraction, one cuts up an apple

or a pie; it is cut up mentally of course and not in reality, because I

do not suppose the budget of the primary instruction allows of such

prodigality. At the Normal School, on the other hand, or at the college,

it is said: a fraction is the combination of two whole numbers separated

by a horizontal bar; we define by conventions the operations to which

these symbols may be submitted; it is proved that the rules of these

operations are the same as in calculating with whole numbers, and we

ascertain finally that multiplying the fraction, according to these

rules, by the denominator gives the numerator. This is all very well

because we are addressing young people long familiarized with the notion

of fractions through having cut up apples or other objects, and whose

mind, matured by a hard mathematical education, has come little by

little to desire a purely logical definition. But the débutant to whom

one should try to give it, how dumfounded!

Such also are the definitions found in a book justly admired and greatly

honored, the _Foundations of Geometry_ by Hilbert. See in fact how he

begins: _We think three systems of_ THINGS _which we shall call points,

straights and planes_. What are these 'things'?

We know not, nor need we know; it would even be a pity to seek to know;

all we have the right to know of them is what the assumptions tell us;

this for example: _Two distinct points always determine a straight_,

which is followed by this remark: _in place of determine, we may say the

two points are on the straight, or the straight goes through these two

points or joins the two points_.

Thus 'to be on a straight' is simply defined as synonymous with

'determine a straight.' Behold a book of which I think much good, but

which I should not recommend to a school boy. Yet I could do so without

fear, he would not read much of it. I have taken extreme examples and no

teacher would dream of going that far. But even stopping short of such

models, does he not already expose himself to the same danger?

Suppose we are in a class; the professor dictates: the circle is the

locus of points of the plane equidistant from an interior point called

the center. The good scholar writes this phrase in his note-book; the

bad scholar draws faces; but neither understands; then the professor

takes the chalk and draws a circle on the board. "Ah!" think the

scholars, "why did he not say at once: a circle is a ring, we should

have understood." Doubtless the professor is right. The scholars'

definition would have been of no avail, since it could serve for no

demonstration, since besides it would not give them the salutary habit

of analyzing their conceptions. But one should show them that they do

not comprehend what they think they know, lead them to be conscious of

the roughness of their primitive conception, and of themselves to wish

it purified and made precise.

4. I shall return to these examples; I only wished to show you the two

opposed conceptions; they are in violent contrast. This contrast the

history of science explains. If we read a book written fifty years ago,

most of the reasoning we find there seems lacking in rigor. Then it was

assumed a continuous function can change sign only by vanishing; to-day

we prove it. It was assumed the ordinary rules of calculation are

applicable to incommensurable numbers; to-day we prove it. Many other

things were assumed which sometimes were false.

We trusted to intuition; but intuition can not give rigor, nor even

certainty; we see this more and more. It tells us for instance that

every curve has a tangent, that is to say that every continuous function

has a derivative, and that is false. And as we sought certainty, we had

to make less and less the part of intuition.

What has made necessary this evolution? We have not been slow to

perceive that rigor could not be established in the reasonings, if it

were not first put into the definitions.

The objects occupying mathematicians were long ill defined; we thought

we knew them because we represented them with the senses or the

imagination; but we had of them only a rough image and not a precise

concept upon which reasoning could take hold. It is there that the

logicians would have done well to direct their efforts.

So for the incommensurable number, the vague idea of continuity, which

we owe to intuition, has resolved itself into a complicated system of

inequalities bearing on whole numbers. Thus have finally vanished all

those difficulties which frightened our fathers when they reflected upon

the foundations of the infinitesimal calculus. To-day only whole numbers

are left in analysis, or systems finite or infinite of whole numbers,

bound by a plexus of equalities and inequalities. Mathematics we say is

arithmetized.

5. But do you think mathematics has attained absolute rigor without

making any sacrifice? Not at all; what it has gained in rigor it has

lost in objectivity. It is by separating itself from reality that it has

acquired this perfect purity. We may freely run over its whole domain,

formerly bristling with obstacles, but these obstacles have not

disappeared. They have only been moved to the frontier, and it would be

necessary to vanquish them anew if we wished to break over this frontier

to enter the realm of the practical.

We had a vague notion, formed of incongruous elements, some _a priori_,

others coming from experiences more or less digested; we thought we

knew, by intuition, its principal properties. To-day we reject the

empiric elements, retaining only the _a priori_; one of the properties

serves as definition and all the others are deduced from it by rigorous

reasoning. This is all very well, but it remains to be proved that this

property, which has become a definition, pertains to the real objects

which experience had made known to us and whence we drew our vague

intuitive notion. To prove that, it would be necessary to appeal to

experience, or to make an effort of intuition, and if we could not prove

it, our theorems would be perfectly rigorous, but perfectly useless.

Logic sometimes makes monsters. Since half a century we have seen arise

a crowd of bizarre functions which seem to try to resemble as little as

possible the honest functions which serve some purpose. No longer

continuity, or perhaps continuity, but no derivatives, etc. Nay more,

from the logical point of view, it is these strange functions which are

the most general, those one meets without seeking no longer appear

except as particular case. There remains for them only a very small

corner.

Heretofore when a new function was invented, it was for some practical

end; to-day they are invented expressly to put at fault the reasonings

of our fathers, and one never will get from them anything more than

that.

If logic were the sole guide of the teacher, it would be necessary to

begin with the most general functions, that is to say with the most

bizarre. It is the beginner that would have to be set grappling with

this teratologic museum. If you do not do it, the logicians might say,

you will achieve rigor only by stages.

6. Yes, perhaps, but we can not make so cheap of reality, and I mean not

only the reality of the sensible world, which however has its worth,

since it is to combat against it that nine tenths of your students ask

of you weapons. There is a reality more subtile, which makes the very

life of the mathematical beings, and which is quite other than logic.

Our body is formed of cells, and the cells of atoms; are these cells and

these atoms then all the reality of the human body? The way these cells

are arranged, whence results the unity of the individual, is it not also

a reality and much more interesting?

A naturalist who never had studied the elephant except in the

microscope, would he think he knew the animal adequately? It is the same

in mathematics. When the logician shall have broken up each

demonstration into a multitude of elementary operations, all correct, he

still will not possess the whole reality; this I know not what which

makes the unity of the demonstration will completely escape him.

In the edifices built up by our masters, of what use to admire the work

of the mason if we can not comprehend the plan of the architect? Now

pure logic can not give us this appreciation of the total effect; this

we must ask of intuition.

Take for instance the idea of continuous function. This is at first only

a sensible image, a mark traced by the chalk on the blackboard. Little

by little it is refined; we use it to construct a complicated system of

inequalities, which reproduces all the features of the primitive image;

when all is done, we have _removed the centering_, as after the

construction of an arch; this rough representation, support thenceforth

useless, has disappeared and there remains only the edifice itself,

irreproachable in the eyes of the logician. And yet, if the professor

did not recall the primitive image, if he did not restore momentarily

the _centering_, how could the student divine by what caprice all these

inequalities have been scaffolded in this fashion one upon another? The

definition would be logically correct, but it would not show him the

veritable reality.

7. So back we must return; doubtless it is hard for a master to teach

what does not entirely satisfy him; but the satisfaction of the master

is not the unique object of teaching; we should first give attention to

what the mind of the pupil is and to what we wish it to become.

Zoologists maintain that the embryonic development of an animal

recapitulates in brief the whole history of its ancestors throughout

geologic time. It seems it is the same in the development of minds. The

teacher should make the child go over the path his fathers trod; more

rapidly, but without skipping stations. For this reason, the history of

science should be our first guide.

Our fathers thought they knew what a fraction was, or continuity, or the

area of a curved surface; we have found they did not know it. Just so

our scholars think they know it when they begin the serious study of

mathematics. If without warning I tell them: "No, you do not know it;

what you think you understand, you do not understand; I must prove to

you what seems to you evident," and if in the demonstration I support

myself upon premises which to them seem less evident than the

conclusion, what shall the unfortunates think? They will think that the

science of mathematics is only an arbitrary mass of useless subtilities;

either they will be disgusted with it, or they will play it as a game

and will reach a state of mind like that of the Greek sophists.

Later, on the contrary, when the mind of the scholar, familiarized with

mathematical reasoning, has been matured by this long frequentation, the

doubts will arise of themselves and then your demonstration will be

welcome. It will awaken new doubts, and the questions will arise

successively to the child, as they arose successively to our fathers,

until perfect rigor alone can satisfy him. To doubt everything does not

suffice, one must know why he doubts.

8. The principal aim of mathematical teaching is to develop certain

faculties of the mind, and among them intuition is not the least

precious. It is through it that the mathematical world remains in

contact with the real world, and if pure mathematics could do without

it, it would always be necessary to have recourse to it to fill up the

chasm which separates the symbol from reality. The practician will

always have need of it, and for one pure geometer there should be a

hundred practicians.

The engineer should receive a complete mathematical education, but for

what should it serve him?

To see the different aspects of things and see them quickly; he has no

time to hunt mice. It is necessary that, in the complex physical objects

presented to him, he should promptly recognize the point where the

mathematical tools we have put in his hands can take hold. How could he

do it if we should leave between instruments and objects the deep chasm

hollowed out by the logicians?

9. Besides the engineers, other scholars, less numerous, are in their

turn to become teachers; they therefore must go to the very bottom; a

knowledge deep and rigorous of the first principles is for them before

all indispensable. But this is no reason not to cultivate in them

intuition; for they would get a false idea of the science if they never

looked at it except from a single side, and besides they could not

develop in their students a quality they did not themselves possess.

For the pure geometer himself, this faculty is necessary; it is by logic

one demonstrates, by intuition one invents. To know how to criticize is

good, to know how to create is better. You know how to recognize if a

combination is correct; what a predicament if you have not the art of

choosing among all the possible combinations. Logic tells us that on

such and such a way we are sure not to meet any obstacle; it does not

say which way leads to the end. For that it is necessary to see the end

from afar, and the faculty which teaches us to see is intuition. Without

it the geometer would be like a writer who should be versed in grammar

but had no ideas. Now how could this faculty develop if, as soon as it

showed itself, we chase it away and proscribe it, if we learn to set it

at naught before knowing the good of it.

And here permit a parenthesis to insist upon the importance of written

exercises. Written compositions are perhaps not sufficiently emphasized

in certain examinations, at the polytechnic school, for instance. I am

told they would close the door against very good scholars who have

mastered the course, thoroughly understanding it, and who nevertheless

are incapable of making the slightest application. I have just said the

word understand has several meanings: such students only understand in

the first way, and we have seen that suffices neither to make an

engineer nor a geometer. Well, since choice must be made, I prefer those

who understand completely.

10. But is the art of sound reasoning not also a precious thing, which

the professor of mathematics ought before all to cultivate? I take good

care not to forget that. It should occupy our attention and from the

very beginning. I should be distressed to see geometry degenerate into I

know not what tachymetry of low grade and I by no means subscribe to the

extreme doctrines of certain German Oberlehrer. But there are occasions

enough to exercise the scholars in correct reasoning in the parts of

mathematics where the inconveniences I have pointed out do not present

themselves. There are long chains of theorems where absolute logic has

reigned from the very first and, so to speak, quite naturally, where the

first geometers have given us models we should constantly imitate and

admire.

It is in the exposition of first principles that it is necessary to

avoid too much subtility; there it would be most discouraging and

moreover useless. We can not prove everything and we can not define

everything; and it will always be necessary to borrow from intuition;

what does it matter whether it be done a little sooner or a little

later, provided that in using correctly premises it has furnished us, we

learn to reason soundly.

11. Is it possible to fulfill so many opposing conditions? Is this

possible in particular when it is a question of giving a definition? How

find a concise statement satisfying at once the uncompromising rules of

logic, our desire to grasp the place of the new notion in the totality

of the science, our need of thinking with images? Usually it will not be

found, and this is why it is not enough to state a definition; it must

be prepared for and justified.

What does that mean? You know it has often been said: every definition

implies an assumption, since it affirms the existence of the object

defined. The definition then will not be justified, from the purely

logical point of view, until one shall have _proved_ that it involves no

contradiction, neither in the terms, nor with the verities previously

admitted.

But this is not enough; the definition is stated to us as a convention;

but most minds will revolt if we wish to impose it upon them as an

_arbitrary_ convention. They will be satisfied only when you have

answered numerous questions.

Usually mathematical definitions, as M. Liard has shown, are veritable

constructions built up wholly of more simple notions. But why assemble

these elements in this way when a thousand other combinations were

possible?

Is it by caprice? If not, why had this combination more right to exist

than all the others? To what need does it respond? How was it foreseen

that it would play an important rôle in the development of the science,

that it would abridge our reasonings and our calculations? Is there in

nature some familiar object which is so to speak the rough and vague

image of it?

This is not all; if you answer all these questions in a satisfactory

manner, we shall see indeed that the new-born had the right to be

baptized; but neither is the choice of a name arbitrary; it is needful

to explain by what analogies one has been guided and that if analogous

names have been given to different things, these things at least differ

only in material and are allied in form; that their properties are

analogous and so to say parallel.

At this cost we may satisfy all inclinations. If the statement is

correct enough to please the logician, the justification will satisfy

the intuitive. But there is still a better procedure; wherever possible,

the justification should precede the statement and prepare for it; one

should be led on to the general statement by the study of some

particular examples.

Still another thing: each of the parts of the statement of a definition

has as aim to distinguish the thing to be defined from a class of other

neighboring objects. The definition will be understood only when you

have shown, not merely the object defined, but the neighboring objects

from which it is proper to distinguish it, when you have given a grasp

of the difference and when you have added explicitly: this is why in

stating the definition I have said this or that.

But it is time to leave generalities and examine how the somewhat

abstract principles I have expounded may be applied in arithmetic,

geometry, analysis and mechanics.

ARITHMETIC

12. The whole number is not to be defined; in return, one ordinarily

defines the operations upon whole numbers; I believe the scholars learn

these definitions by heart and attach no meaning to them. For that there

are two reasons: first they are made to learn them too soon, when their

mind as yet feels no need of them; then these definitions are not

satisfactory from the logical point of view. A good definition for

addition is not to be found just simply because we must stop and can not

define everything. It is not defining addition to say it consists in

adding. All that can be done is to start from a certain number of

concrete examples and say: the operation we have performed is called

addition.

For subtraction it is quite otherwise; it may be logically defined as

the operation inverse to addition; but should we begin in that way? Here

also start with examples, show on these examples the reciprocity of the

two operations; thus the definition will be prepared for and justified.

Just so again for multiplication; take a particular problem; show that

it may be solved by adding several equal numbers; then show that we

reach the result more quickly by a multiplication, an operation the

scholars already know how to do by routine and out of that the logical

definition will issue naturally.

Division is defined as the operation inverse to multiplication; but

begin by an example taken from the familiar notion of partition and show

on this example that multiplication reproduces the dividend.

There still remain the operations on fractions. The only difficulty is

for multiplication. It is best to expound first the theory of

proportion; from it alone can come a logical definition; but to make

acceptable the definitions met at the beginning of this theory, it is

necessary to prepare for them by numerous examples taken from classic

problems of the rule of three, taking pains to introduce fractional

data.

Neither should we fear to familiarize the scholars with the notion of

proportion by geometric images, either by appealing to what they

remember if they have already studied geometry, or in having recourse to

direct intuition, if they have not studied it, which besides will

prepare them to study it. Finally I shall add that after defining

multiplication of fractions, it is needful to justify this definition by

showing that it is commutative, associative and distributive, and

calling to the attention of the auditors that this is established to

justify the definition.

One sees what a rôle geometric images play in all this; and this rôle is

justified by the philosophy and the history of the science. If

arithmetic had remained free from all admixture of geometry, it would

have known only the whole number; it is to adapt itself to the needs of

geometry that it invented anything else.

GEOMETRY

In geometry we meet forthwith the notion of the straight line. Can the

straight line be defined? The well-known definition, the shortest path

from one point to another, scarcely satisfies me. I should start simply

with the _ruler_ and show at first to the scholar how one may verify a

ruler by turning; this verification is the true definition of the

straight line; the straight line is an axis of rotation. Next he should

be shown how to verify the ruler by sliding and he would have one of the

most important properties of the straight line.

As to this other property of being the shortest path from one point to

another, it is a theorem which can be demonstrated apodictically, but

the demonstration is too delicate to find a place in secondary teaching.

It will be worth more to show that a ruler previously verified fits on a

stretched thread. In presence of difficulties like these one need not

dread to multiply assumptions, justifying them by rough experiments.

It is needful to grant these assumptions, and if one admits a few more

of them than is strictly necessary, the evil is not very great; the

essential thing is to learn to reason soundly on the assumptions

admitted. Uncle Sarcey, who loved to repeat, often said that at the

theater the spectator accepts willingly all the postulates imposed upon

him at the beginning, but the curtain once raised, he becomes

uncompromising on the logic. Well, it is just the same in mathematics.

For the circle, we may start with the compasses; the scholars will

recognize at the first glance the curve traced; then make them observe

that the distance of the two points of the instrument remains constant,

that one of these points is fixed and the other movable, and so we shall

be led naturally to the logical definition.

The definition of the plane implies an axiom and this need not be

hidden. Take a drawing board and show that a moving ruler may be kept

constantly in complete contact with this plane and yet retain three

degrees of freedom. Compare with the cylinder and the cone, surfaces on

which an applied straight retains only two degrees of freedom; next take

three drawing boards; show first that they will glide while remaining

applied to one another and this with three degrees of freedom; and

finally to distinguish the plane from the sphere, show that two of these

boards which fit a third will fit each other.

Perhaps you are surprised at this incessant employment of moving things;

this is not a rough artifice; it is much more philosophic than one would

at first think. What is geometry for the philosopher? It is the study of

a group. And what group? That of the motions of solid bodies. How define

this group then without moving some solids?

Should we retain the classic definition of parallels and say parallels

are two coplanar straights which do not meet, however far they be

prolonged? No, since this definition is negative, since it is

unverifiable by experiment, and consequently can not be regarded as an

immediate datum of intuition. No, above all because it is wholly strange

to the notion of group, to the consideration of the motion of solid

bodies which is, as I have said, the true source of geometry. Would it

not be better to define first the rectilinear translation of an

invariable figure, as a motion wherein all the points of this figure

have rectilinear trajectories; to show that such a translation is

possible by making a square glide on a ruler?

From this experimental ascertainment, set up as an assumption, it would

be easy to derive the notion of parallel and Euclid's postulate itself.

MECHANICS

I need not return to the definition of velocity, or acceleration, or

other kinematic notions; they may be advantageously connected with that

of the derivative.

I shall insist, on the other hand, upon the dynamic notions of force and

mass.

I am struck by one thing: how very far the young people who have

received a high-school education are from applying to the real world the

mechanical laws they have been taught. It is not only that they are

incapable of it; they do not even think of it. For them the world of

science and the world of reality are separated by an impervious

partition wall.

If we try to analyze the state of mind of our scholars, this will

astonish us less. What is for them the real definition of force? Not

that which they recite, but that which, crouching in a nook of their

mind, from there directs it wholly. Here is the definition: forces are

arrows with which one makes parallelograms. These arrows are imaginary

things which have nothing to do with anything existing in nature. This

would not happen if they had been shown forces in reality before

representing them by arrows.

How shall we define force?

I think I have elsewhere sufficiently shown there is no good logical

definition. There is the anthropomorphic definition, the sensation of

muscular effort; this is really too rough and nothing useful can be

drawn from it.

Here is how we should go: first, to make known the genus force, we must

show one after the other all the species of this genus; they are very

numerous and very different; there is the pressure of fluids on the

insides of the vases wherein they are contained; the tension of threads;

the elasticity of a spring; the gravity working on all the molecules of

a body; friction; the normal mutual action and reaction of two solids in

contact.

This is only a qualitative definition; it is necessary to learn to

measure force. For that begin by showing that one force may be replaced

by another without destroying equilibrium; we may find the first example

of this substitution in the balance and Borda's double weighing.

Then show that a weight may be replaced, not only by another weight,

but by force of a different nature; for instance, Prony's brake permits

replacing weight by friction.

From all this arises the notion of the equivalence of two forces.

The direction of a force must be defined. If a force _F_ is equivalent

to another force _F'_ applied to the body considered by means of a

stretched string, so that _F_ may be replaced by _F'_ without affecting

the equilibrium, then the point of attachment of the string will be by

definition the point of application of the force _F'_, and that of the

equivalent force _F_; the direction of the string will be the direction

of the force _F'_ and that of the equivalent force _F_.

From that, pass to the comparison of the magnitude of forces. If a force

can replace two others with the same direction, it equals their sum;

show for example that a weight of 20 grams may replace two 10-gram

weights.

Is this enough? Not yet. We now know how to compare the intensity of two

forces which have the same direction and same point of application; we

must learn to do it when the directions are different. For that, imagine

a string stretched by a weight and passing over a pulley; we shall say

that the tensor of the two legs of the string is the same and equal to

the tension weight.

This definition of ours enables us to compare the tensions of the two

pieces of our string, and, using the preceding definitions, to compare

any two forces having the same direction as these two pieces. It should

be justified by showing that the tension of the last piece of the string

remains the same for the same tensor weight, whatever be the number and

the disposition of the reflecting pulleys. It has still to be completed

by showing this is only true if the pulleys are frictionless.

Once master of these definitions, it is to be shown that the point of

application, the direction and the intensity suffice to determine a

force; that two forces for which these three elements are the same are

_always_ equivalent and may _always_ be replaced by one another, whether

in equilibrium or in movement, and this whatever be the other forces

acting.

It must be shown that two concurrent forces may always be replaced by a

unique resultant; and that _this resultant remains the same_, whether

the body be at rest or in motion and whatever be the other forces

applied to it.

Finally it must be shown that forces thus defined satisfy the principle

of the equality of action and reaction.

Experiment it is, and experiment alone, which can teach us all that. It

will suffice to cite certain common experiments, which the scholars make

daily without suspecting it, and to perform before them a few

experiments, simple and well chosen.

It is after having passed through all these meanders that one may

represent forces by arrows, and I should even wish that in the

development of the reasonings return were made from time to time from

the symbol to the reality. For instance it would not be difficult to

illustrate the parallelogram of forces by aid of an apparatus formed of

three strings, passing over pulleys, stretched by weights and in

equilibrium while pulling on the same point.

Knowing force, it is easy to define mass; this time the definition

should be borrowed from dynamics; there is no way of doing otherwise,

since the end to be attained is to give understanding of the distinction

between mass and weight. Here again, the definition should be led up to

by experiments; there is in fact a machine which seems made expressly to

show what mass is, Atwood's machine; recall also the laws of the fall of

bodies, that the acceleration of gravity is the same for heavy as for

light bodies, and that it varies with the latitude, etc.

Now, if you tell me that all the methods I extol have long been applied

in the schools, I shall rejoice over it more than be surprised at it. I

know that on the whole our mathematical teaching is good. I do not wish

it overturned; that would even distress me. I only desire betterments

slowly progressive. This teaching should not be subjected to brusque

oscillations under the capricious blast of ephemeral fads. In such

tempests its high educative value would soon founder. A good and sound

logic should continue to be its basis. The definition by example is

always necessary, but it should prepare the way for the logical

definition, it should not replace it; it should at least make this

wished for, in the cases where the true logical definition can be

advantageously given only in advanced teaching.

Understand that what I have here said does not imply giving up what I

have written elsewhere. I have often had occasion to criticize certain

definitions I extol to-day. These criticisms hold good completely. These

definitions can only be provisory. But it is by way of them that we must

pass.

CHAPTER III

MATHEMATICS AND LOGIC

INTRODUCTION

Can mathematics be reduced to logic without having to appeal to

principles peculiar to mathematics? There is a whole school, abounding

in ardor and full of faith, striving to prove it. They have their own

special language, which is without words, using only signs. This

language is understood only by the initiates, so that commoners are

disposed to bow to the trenchant affirmations of the adepts. It is

perhaps not unprofitable to examine these affirmations somewhat closely,

to see if they justify the peremptory tone with which they are

presented.

But to make clear the nature of the question it is necessary to enter

upon certain historical details and in particular to recall the

character of the works of Cantor.

Since long ago the notion of infinity had been introduced into

mathematics; but this infinite was what philosophers call a _becoming_.

The mathematical infinite was only a quantity capable of increasing

beyond all limit: it was a variable quantity of which it could not be

said that it _had passed_ all limits, but only that it _could pass_

them.

Cantor has undertaken to introduce into mathematics an _actual

infinite_, that is to say a quantity which not only is capable of

passing all limits, but which is regarded as having already passed them.

He has set himself questions like these: Are there more points in space

than whole numbers? Are there more points in space than points in a

plane? etc.

And then the number of whole numbers, that of the points of space, etc.,

constitutes what he calls a _transfinite cardinal number_, that is to

say a cardinal number greater than all the ordinary cardinal numbers.

And he has occupied himself in comparing these transfinite cardinal

numbers. In arranging in a proper order the elements of an aggregate

containing an infinity of them, he has also imagined what he calls

transfinite ordinal numbers upon which I shall not dwell.

Many mathematicians followed his lead and set a series of questions of

the sort. They so familiarized themselves with transfinite numbers that

they have come to make the theory of finite numbers depend upon that of

Cantor's cardinal numbers. In their eyes, to teach arithmetic in a way

truly logical, one should begin by establishing the general properties

of transfinite cardinal numbers, then distinguish among them a very

small class, that of the ordinary whole numbers. Thanks to this détour,

one might succeed in proving all the propositions relative to this

little class (that is to say all our arithmetic and our algebra) without

using any principle foreign to logic. This method is evidently contrary

to all sane psychology; it is certainly not in this way that the human

mind proceeded in constructing mathematics; so its authors do not dream,

I think, of introducing it into secondary teaching. But is it at least

logic, or, better, is it correct? It may be doubted.

The geometers who have employed it are however very numerous. They have

accumulated formulas and they have thought to free themselves from what

was not pure logic by writing memoirs where the formulas no longer

alternate with explanatory discourse as in the books of ordinary

mathematics, but where this discourse has completely disappeared.

Unfortunately they have reached contradictory results, what are called

the _cantorian antinomies_, to which we shall have occasion to return.

These contradictions have not discouraged them and they have tried to

modify their rules so as to make those disappear which had already shown

themselves, without being sure, for all that, that new ones would not

manifest themselves.

It is time to administer justice on these exaggerations. I do not hope

to convince them; for they have lived too long in this atmosphere.

Besides, when one of their demonstrations has been refuted, we are sure

to see it resurrected with insignificant alterations, and some of them

have already risen several times from their ashes. Such long ago was the

Lernæan hydra with its famous heads which always grew again. Hercules

got through, since his hydra had only nine heads, or eleven; but here

there are too many, some in England, some in Germany, in Italy, in

France, and he would have to give up the struggle. So I appeal only to

men of good judgment unprejudiced.

I

In these latter years numerous works have been published on pure

mathematics and the philosophy of mathematics, trying to separate and

isolate the logical elements of mathematical reasoning. These works have

been analyzed and expounded very clearly by M. Couturat in a book

entitled: _The Principles of Mathematics_.

For M. Couturat, the new works, and in particular those of Russell and

Peano, have finally settled the controversy, so long pending between

Leibnitz and Kant. They have shown that there are no synthetic judgments

a priori (Kant's phrase to designate judgments which can neither be

demonstrated analytically, nor reduced to identities, nor established

experimentally), they have shown that mathematics is entirely reducible

to logic and that intuition here plays no rôle.

This is what M. Couturat has set forth in the work just cited; this he

says still more explicitly in his Kant jubilee discourse, so that I

heard my neighbor whisper: "I well see this is the centenary of Kant's

_death_."

Can we subscribe to this conclusive condemnation? I think not, and I

shall try to show why.

II

What strikes us first in the new mathematics is its purely formal

character: "We think," says Hilbert, "three sorts of _things_, which we

shall call points, straights and planes. We convene that a straight

shall be determined by two points, and that in place of saying this

straight is determined by these two points, we may say it passes through

these two points, or that these two points are situated on this

straight." What these _things_ are, not only we do not know, but we

should not seek to know. We have no need to, and one who never had seen

either point or straight or plane could geometrize as well as we. That

the phrase _to pass through_, or the phrase _to be situated upon_ may

arouse in us no image, the first is simply a synonym of to _be

determined_ and the second of _to determine_.

Thus, be it understood, to demonstrate a theorem, it is neither

necessary nor even advantageous to know what it means. The geometer

might be replaced by the _logic piano_ imagined by Stanley Jevons; or,

if you choose, a machine might be imagined where the assumptions were

put in at one end, while the theorems came out at the other, like the

legendary Chicago machine where the pigs go in alive and come out

transformed into hams and sausages. No more than these machines need the

mathematician know what he does.

I do not make this formal character of his geometry a reproach to

Hilbert. This is the way he should go, given the problem he set himself.

He wished to reduce to a minimum the number of the fundamental

assumptions of geometry and completely enumerate them; now, in

reasonings where our mind remains active, in those where intuition still

plays a part, in living reasonings, so to speak, it is difficult not to

introduce an assumption or a postulate which passes unperceived. It is

therefore only after having carried back all the geometric reasonings to

a form purely mechanical that he could be sure of having accomplished

his design and finished his work.

What Hilbert did for geometry, others have tried to do for arithmetic

and analysis. Even if they had entirely succeeded, would the Kantians be

finally condemned to silence? Perhaps not, for in reducing mathematical

thought to an empty form, it is certainly mutilated.

Even admitting it were established that all the theorems could be

deduced by procedures purely analytic, by simple logical combinations of

a finite number of assumptions, and that these assumptions are only

conventions; the philosopher would still have the right to investigate

the origins of these conventions, to see why they have been judged

preferable to the contrary conventions.

And then the logical correctness of the reasonings leading from the

assumptions to the theorems is not the only thing which should occupy

us. The rules of perfect logic, are they the whole of mathematics? As

well say the whole art of playing chess reduces to the rules of the

moves of the pieces. Among all the constructs which can be built up of

the materials furnished by logic, choice must be made; the true geometer

makes this choice judiciously because he is guided by a sure instinct,

or by some vague consciousness of I know not what more profound and more

hidden geometry, which alone gives value to the edifice constructed.

To seek the origin of this instinct, to study the laws of this deep

geometry, felt, not stated, would also be a fine employment for the

philosophers who do not want logic to be all. But it is not at this

point of view I wish to put myself, it is not thus I wish to consider

the question. The instinct mentioned is necessary for the inventor, but

it would seem at first we might do without it in studying the science

once created. Well, what I wish to investigate is if it be true that,

the principles of logic once admitted, one can, I do not say discover,

but demonstrate, all the mathematical verities without making a new

appeal to intuition.

III

I once said no to this question:[12] should our reply be modified by the

recent works? My saying no was because "the principle of complete

induction" seemed to me at once necessary to the mathematician and

irreducible to logic. The statement of this principle is: "If a property

be true of the number 1, and if we establish that it is true of _n_ + 1

provided it be of _n_, it will be true of all the whole numbers."

Therein I see the mathematical reasoning par excellence. I did not mean

to say, as has been supposed, that all mathematical reasonings can be

reduced to an application of this principle. Examining these reasonings

closely, we there should see applied many other analogous principles,

presenting the same essential characteristics. In this category of

principles, that of complete induction is only the simplest of all and

this is why I have chosen it as type.

[12] See _Science and Hypothesis_, chapter I.

The current name, principle of complete induction, is not justified.

This mode of reasoning is none the less a true mathematical induction

which differs from ordinary induction only by its certitude.

IV

DEFINITIONS AND ASSUMPTIONS

The existence of such principles is a difficulty for the uncompromising

logicians; how do they pretend to get out of it? The principle of

complete induction, they say, is not an assumption properly so called or

a synthetic judgment _a priori_; it is just simply the definition of

whole number. It is therefore a simple convention. To discuss this way

of looking at it, we must examine a little closely the relations between

definitions and assumptions.

Let us go back first to an article by M. Couturat on mathematical

definitions which appeared in _l'Enseignement mathématique_, a magazine

published by Gauthier-Villars and by Georg at Geneva. We shall see there

a distinction between the _direct definition and the definition by

postulates_.

"The definition by postulates," says M. Couturat, "applies not to a

single notion, but to a system of notions; it consists in enumerating

the fundamental relations which unite them and which enable us to

demonstrate all their other properties; these relations are postulates."

If previously have been defined all these notions but one, then this

last will be by definition the thing which verifies these postulates.

Thus certain indemonstrable assumptions of mathematics would be only

disguised definitions. This point of view is often legitimate; and I

have myself admitted it in regard for instance to Euclid's postulate.

The other assumptions of geometry do not suffice to completely define

distance; the distance then will be, by definition, among all the

magnitudes which satisfy these other assumptions, that which is such as

to make Euclid's postulate true.

Well the logicians suppose true for the principle of complete induction

what I admit for Euclid's postulate; they want to see in it only a

disguised definition.

But to give them this right, two conditions must be fulfilled. Stuart

Mill says every definition implies an assumption, that by which the

existence of the defined object is affirmed. According to that, it

would no longer be the assumption which might be a disguised definition,

it would on the contrary be the definition which would be a disguised

assumption. Stuart Mill meant the word existence in a material and

empirical sense; he meant to say that in defining the circle we affirm

there are round things in nature.

Under this form, his opinion is inadmissible. Mathematics is independent

of the existence of material objects; in mathematics the word exist can

have only one meaning, it means free from contradiction. Thus rectified,

Stuart Mill's thought becomes exact; in defining a thing, we affirm that

the definition implies no contradiction.

If therefore we have a system of postulates, and if we can demonstrate

that these postulates imply no contradiction, we shall have the right to

consider them as representing the definition of one of the notions

entering therein. If we can not demonstrate that, it must be admitted

without proof, and that then will be an assumption; so that, seeking the

definition under the postulate, we should find the assumption under the

definition.

Usually, to show that a definition implies no contradiction, we proceed

by _example_, we try to make an example of a thing satisfying the

definition. Take the case of a definition by postulates; we wish to

define a notion _A_, and we say that, by definition, an _A_ is anything

for which certain postulates are true. If we can prove directly that all

these postulates are true of a certain object _B_, the definition will

be justified; the object _B_ will be an _example_ of an _A_. We shall be

certain that the postulates are not contradictory, since there are cases

where they are all true at the same time.

But such a direct demonstration by example is not always possible.

To establish that the postulates imply no contradiction, it is then

necessary to consider all the propositions deducible from these

postulates considered as premises, and to show that, among these

propositions, no two are contradictory. If these propositions are finite

in number, a direct verification is possible. This case is infrequent

and uninteresting. If these propositions are infinite in number, this

direct verification can no longer be made; recourse must be had to

procedures where in general it is necessary to invoke just this

principle of complete induction which is precisely the thing to be

proved.

This is an explanation of one of the conditions the logicians should

satisfy, _and further on we shall see they have not done it_.

V

There is a second. When we give a definition, it is to use it.

We therefore shall find in the sequel of the exposition the word

defined; have we the right to affirm, of the thing represented by this

word, the postulate which has served for definition? Yes, evidently, if

the word has retained its meaning, if we do not attribute to it

implicitly a different meaning. Now this is what sometimes happens and

it is usually difficult to perceive it; it is needful to see how this

word comes into our discourse, and if the gate by which it has entered

does not imply in reality a definition other than that stated.

This difficulty presents itself in all the applications of mathematics.

The mathematical notion has been given a definition very refined and

very rigorous; and for the pure mathematician all doubt has disappeared;

but if one wishes to apply it to the physical sciences for instance, it

is no longer a question of this pure notion, but of a concrete object

which is often only a rough image of it. To say that this object

satisfies, at least approximately, the definition, is to state a new

truth, which experience alone can put beyond doubt, and which no longer

has the character of a conventional postulate.

But without going beyond pure mathematics, we also meet the same

difficulty.

You give a subtile definition of numbers; then, once this definition

given, you think no more of it; because, in reality, it is not it which

has taught you what number is; you long ago knew that, and when the word

number further on is found under your pen, you give it the same sense as

the first comer. To know what is this meaning and whether it is the same

in this phrase or that, it is needful to see how you have been led to

speak of number and to introduce this word into these two phrases. I

shall not for the moment dilate upon this point, because we shall have

occasion to return to it.

Thus consider a word of which we have given explicitly a definition _A_;

afterwards in the discourse we make a use of it which implicitly

supposes another definition _B_. It is possible that these two

definitions designate the same thing. But that this is so is a new truth

which must either be demonstrated or admitted as an independent

assumption.

_We shall see farther on that the logicians have not fulfilled the

second condition any better than the first._

VI

The definitions of number are very numerous and very different; I forego

the enumeration even of the names of their authors. We should not be

astonished that there are so many. If one among them was satisfactory,

no new one would be given. If each new philosopher occupying himself

with this question has thought he must invent another one, this was

because he was not satisfied with those of his predecessors, and he was

not satisfied with them because he thought he saw a petitio principii.

I have always felt, in reading the writings devoted to this problem, a

profound feeling of discomfort; I was always expecting to run against a

petitio principii, and when I did not immediately perceive it, I feared

I had overlooked it.

This is because it is impossible to give a definition without using a

sentence, and difficult to make a sentence without using a number word,

or at least the word several, or at least a word in the plural. And then

the declivity is slippery and at each instant there is risk of a fall

into petitio principii.

I shall devote my attention in what follows only to those of these

definitions where the petitio principii is most ably concealed.

VII

PASIGRAPHY

The symbolic language created by Peano plays a very grand rôle in these

new researches. It is capable of rendering some service, but I think M.

Couturat attaches to it an exaggerated importance which must astonish

Peano himself.

The essential element of this language is certain algebraic signs which

represent the different conjunctions: if, and, or, therefore. That these

signs may be convenient is possible; but that they are destined to

revolutionize all philosophy is a different matter. It is difficult to

admit that the word _if_ acquires, when written C, a virtue it had not

when written if. This invention of Peano was first called _pasigraphy_,

that is to say the art of writing a treatise on mathematics without

using a single word of ordinary language. This name defined its range

very exactly. Later, it was raised to a more eminent dignity by

conferring on it the title of _logistic_. This word is, it appears,

employed at the Military Academy, to designate the art of the

quartermaster of cavalry, the art of marching and cantoning troops; but

here no confusion need be feared, and it is at once seen that this new

name implies the design of revolutionizing logic.

We may see the new method at work in a mathematical memoir by

Burali-Forti, entitled: _Una Questione sui numeri transfiniti_, inserted

in Volume XI of the _Rendiconti del circolo matematico di Palermo_.

I begin by saying this memoir is very interesting, and my taking it here

as example is precisely because it is the most important of all those

written in the new language. Besides, the uninitiated may read it,

thanks to an Italian interlinear translation.

What constitutes the importance of this memoir is that it has given the

first example of those antinomies met in the study of transfinite

numbers and making since some years the despair of mathematicians. The

aim, says Burali-Forti, of this note is to show there may be two

transfinite numbers (ordinals), _a_ and _b_, such that _a_ is neither

equal to, greater than, nor less than _b_.

To reassure the reader, to comprehend the considerations which follow,

he has no need of knowing what a transfinite ordinal number is.

Now, Cantor had precisely proved that between two transfinite numbers as

between two finite, there can be no other relation than equality or

inequality in one sense or the other. But it is not of the substance of

this memoir that I wish to speak here; that would carry me much too far

from my subject; I only wish to consider the form, and just to ask if

this form makes it gain much in rigor and whether it thus compensates

for the efforts it imposes upon the writer and the reader.

First we see Burali-Forti define the number 1 as follows:

1 = [iota]T'{Ko[(n_](u, h)[epsilon](u[epsilon]Un)},

a definition eminently fitted to give an idea of the number 1 to persons

who had never heard speak of it.

I understand Peanian too ill to dare risk a critique, but still I fear

this definition contains a petitio principii, considering that I see the

figure 1 in the first member and Un in letters in the second.

However that may be, Burali-Forti starts from this definition and, after

a short calculation, reaches the equation:

(27) 1[epsilon]No,

which tells us that One is a number.

And since we are on these definitions of the first numbers, we recall

that M. Couturat has also defined 0 and 1.

What is zero? It is the number of elements of the null class. And what

is the null class? It is that containing no element.

To define zero by null, and null by no, is really to abuse the wealth of

language; so M. Couturat has introduced an improvement in his

definition, by writing:

0 = [iota][Lambda]:[phi]x = [Lambda]·[inverted c]·[Lambda]

= (x[epsilon][phi]x),

which means: zero is the number of things satisfying a condition never

satisfied.

But as never means _in no case_ I do not see that the progress is great.

I hasten to add that the definition M. Couturat gives of the number 1 is

more satisfactory.

One, says he in substance, is the number of elements in a class in which

any two elements are identical.

It is more satisfactory, I have said, in this sense that to define 1, he

does not use the word one; in compensation, he uses the word two. But I

fear, if asked what is two, M. Couturat would have to use the word one.

VIII

But to return to the memoir of Burali-Forti; I have said his conclusions

are in direct opposition to those of Cantor. Now, one day M. Hadamard

came to see me and the talk fell upon this antinomy.

"Burali-Forti's reasoning," I said, "does it not seem to you

irreproachable?" "No, and on the contrary I find nothing to object to in

that of Cantor. Besides, Burali-Forti had no right to speak of the

aggregate of _all_ the ordinal numbers."

"Pardon, he had the right, since he could always put

[Omega] = T'(No,[epsilon]>).

I should like to know who was to prevent him, and can it be said a thing

does not exist, when we have called it [Omega]?"

It was in vain, I could not convince him (which besides would have been

sad, since he was right). Was it merely because I do not speak the

Peanian with enough eloquence? Perhaps; but between ourselves I do not

think so.

Thus, despite all this pasigraphic apparatus, the question was not

solved. What does that prove? In so far as it is a question only of

proving one a number, pasigraphy suffices, but if a difficulty presents

itself, if there is an antinomy to solve, pasigraphy becomes impotent.

CHAPTER IV

THE NEW LOGICS

I

_The Russell Logic_

To justify its pretensions, logic had to change. We have seen new logics

arise of which the most interesting is that of Russell. It seems he has

nothing new to write about formal logic, as if Aristotle there had

touched bottom. But the domain Russell attributes to logic is infinitely

more extended than that of the classic logic, and he has put forth on

the subject views which are original and at times well warranted.

First, Russell subordinates the logic of classes to that of

propositions, while the logic of Aristotle was above all the logic of

classes and took as its point of departure the relation of subject to

predicate. The classic syllogism, "Socrates is a man," etc., gives place

to the hypothetical syllogism: "If _A_ is true, _B_ is true; now if _B_

is true, _C_ is true," etc. And this is, I think, a most happy idea,

because the classic syllogism is easy to carry back to the hypothetical

syllogism, while the inverse transformation is not without difficulty.

And then this is not all. Russell's logic of propositions is the study

of the laws of combination of the conjunctions _if_, _and_, _or_, and

the negation _not_.

In adding here two other conjunctions, _and_ and _or_, Russell opens to

logic a new field. The symbols _and_, _or_ follow the same laws as the

two signs × and +, that is to say the commutative associative and

distributive laws. Thus _and_ represents logical multiplication, while

_or_ represents logical addition. This also is very interesting.

Russell reaches the conclusion that any false proposition implies all

other propositions true or false. M. Couturat says this conclusion will

at first seem paradoxical. It is sufficient however to have corrected a

bad thesis in mathematics to recognize how right Russell is. The

candidate often is at great pains to get the first false equation; but

that once obtained, it is only sport then for him to accumulate the most

surprising results, some of which even may be true.

II

We see how much richer the new logic is than the classic logic; the

symbols are multiplied and allow of varied combinations _which are no

longer limited in number_. Has one the right to give this extension to

the meaning of the word _logic_? It would be useless to examine this

question and to seek with Russell a mere quarrel about words. Grant him

what he demands; but be not astonished if certain verities declared

irreducible to logic in the old sense of the word find themselves now

reducible to logic in the new sense--something very different.

A great number of new notions have been introduced, and these are not

simply combinations of the old. Russell knows this, and not only at the

beginning of the first chapter, 'The Logic of Propositions,' but at the

beginning of the second and third, 'The Logic of Classes' and 'The Logic

of Relations,' he introduces new words that he declares indefinable.

And this is not all; he likewise introduces principles he declares

indemonstrable. But these indemonstrable principles are appeals to

intuition, synthetic judgments _a priori_. We regard them as intuitive

when we meet them more or less explicitly enunciated in mathematical

treatises; have they changed character because the meaning of the word

logic has been enlarged and we now find them in a book entitled

_Treatise on Logic_? _They have not changed nature; they have only

changed place._

III

Could these principles be considered as disguised definitions? It would

then be necessary to have some way of proving that they imply no

contradiction. It would be necessary to establish that, however far one

followed the series of deductions, he would never be exposed to

contradicting himself.

We might attempt to reason as follows: We can verify that the

operations of the new logic applied to premises exempt from

contradiction can only give consequences equally exempt from

contradiction. If therefore after _n_ operations we have not met

contradiction, we shall not encounter it after _n_ + 1. Thus it is

impossible that there should be a moment when contradiction _begins_,

which shows we shall never meet it. Have we the right to reason in this

way? No, for this would be to make use of complete induction; and

_remember, we do not yet know the principle of complete induction_.

We therefore have not the right to regard these assumptions as disguised

definitions and only one resource remains for us, to admit a new act of

intuition for each of them. Moreover I believe this is indeed the

thought of Russell and M. Couturat.

Thus each of the nine indefinable notions and of the twenty

indemonstrable propositions (I believe if it were I that did the

counting, I should have found some more) which are the foundation of the

new logic, logic in the broad sense, presupposes a new and independent

act of our intuition and (why not say it?) a veritable synthetic

judgment _a priori_. On this point all seem agreed, but what Russell

claims, and _what seems to me doubtful, is that after these appeals to

intuition, that will be the end of it; we need make no others and can

build all mathematics without the intervention of any new element_.

IV

M. Couturat often repeats that this new logic is altogether independent

of the idea of number. I shall not amuse myself by counting how many

numeral adjectives his exposition contains, both cardinal and ordinal,

or indefinite adjectives such as several. We may cite, however, some

examples:

"The logical product of _two_ or _more_ propositions is....";

"All propositions are capable only of _two_ values, true and false";

"The relative product of _two_ relations is a relation";

"A relation exists between two terms," etc., etc.

Sometimes this inconvenience would not be unavoidable, but sometimes

also it is essential. A relation is incomprehensible without two terms;

it is impossible to have the intuition of the relation, without having

at the same time that of its two terms, and without noticing they are

two, because, if the relation is to be conceivable, it is necessary that

there be two and only two.

V

_Arithmetic_

I reach what M. Couturat calls the _ordinal theory_ which is the

foundation of arithmetic properly so called. M. Couturat begins by

stating Peano's five assumptions, which are independent, as has been

proved by Peano and Padoa.

1. Zero is an integer.

2. Zero is not the successor of any integer.

3. The successor of an integer is an integer. To this it would be proper

to add,

Every integer has a successor.

4. Two integers are equal if their successors are.

The fifth assumption is the principle of complete induction.

M. Couturat considers these assumptions as disguised definitions; they

constitute the definition by postulates of zero, of successor, and of

integer.

But we have seen that for a definition by postulates to be acceptable we

must be able to prove that it implies no contradiction.

Is this the case here? Not at all.

The demonstration can not be made _by example_. We can not take a part

of the integers, for instance the first three, and prove they satisfy

the definition.

If I take the series 0, 1, 2, I see it fulfils the assumptions 1, 2, 4

and 5; but to satisfy assumption 3 it still is necessary that 3 be an

integer, and consequently that the series 0, 1, 2, 3, fulfil the

assumptions; we might prove that it satisfies assumptions 1, 2, 4, 5,

but assumption 3 requires besides that 4 be an integer and that the

series 0, 1, 2, 3, 4 fulfil the assumptions, and so on.

It is therefore impossible to demonstrate the assumptions for certain

integers without proving them for all; we must give up proof by

example.