dimensions. What is meant when we say that a mathematical continuum or
that a physical continuum has two or three dimensions?
First we must introduce the notion of cut, studying first physical
continua. We have seen what characterizes the physical continuum. Each
of the elements of this continuum consists of a manifold of impressions;
and it may happen either that an element can not be discriminated from
another element of the same continuum, if this new element corresponds
to a manifold of impressions not sufficiently different, or, on the
contrary, that the discrimination is possible; finally it may happen
that two elements indistinguishable from a third may, nevertheless, be
distinguished one from the other.
That postulated, if _A_ and _B_ are two distinguishable elements of a
continuum _C_, a series of elements may be found, E_{1}, E_{2}, ...,
E_{_n_}, all belonging to this same continuum _C_ and such that
each of them is indistinguishable from the preceding, that E_{1} is
indistinguishable from _A_, and E_{_n_} indistinguishable from _B_.
Therefore we can go from _A_ to _B_ by a continuous route and without
quitting _C_. If this condition is fulfilled for any two elements _A_
and _B_ of the continuum _C_, we may say that this continuum _C_ is all
in one piece. Now let us distinguish certain of the elements of _C_
which may either be all distinguishable from one another, or themselves
form one or several continua. The assemblage of the elements thus chosen
arbitrarily among all those of _C_ will form what I shall call the _cut_
or the _cuts_.
Take on _C_ any two elements _A_ and _B_. Either we can also find a
series of elements E_{1}, E_{2}, ..., E_{_n_}, such: (1) that they all
belong to _C_; (2) that each of them is indistinguishable from the
following, E_{1} indistinguishable from _A_ and E_{_n_} from _B_; (3)
_and besides that none of the elements _E_ is indistinguishable from any
element of the cut_. Or else, on the contrary, in each of the series
E_{1}, E_{2}, ..., E_{_n_} satisfying the first two conditions, there
will be an element _E_ indistinguishable from one of the elements of the
cut. In the first case we can go from _A_ to _B_ by a continuous route
without quitting _C_ and _without meeting the cuts_; in the second case
that is impossible.
If then for any two elements _A_ and _B_ of the continuum _C_, it is
always the first case which presents itself, we shall say that _C_
remains all in one piece despite the cuts.
Thus, if we choose the cuts in a certain way, otherwise arbitrary, it
may happen either that the continuum remains all in one piece or that it
does not remain all in one piece; in this latter hypothesis we shall
then say that it is _divided_ by the cuts.
It will be noticed that all these definitions are constructed in setting
out solely from this very simple fact, that two manifolds of impressions
sometimes can be discriminated, sometimes can not be. That postulated,
if, to _divide_ a continuum, it suffices to consider as cuts a certain
number of elements all distinguishable from one another, we say that
this continuum _is of one dimension_; if, on the contrary, to divide a
continuum, it is necessary to consider as cuts a system of elements
themselves forming one or several continua, we shall say that this
continuum is _of several dimensions_.
If to divide a continuum _C_, cuts forming one or several continua of
one dimension suffice, we shall say that _C_ is a continuum _of two
dimensions_; if cuts suffice which form one or several continua of two
dimensions at most, we shall say that _C_ is a continuum _of three
dimensions_; and so on.
To justify this definition it is proper to see whether it is in this way
that geometers introduce the notion of three dimensions at the beginning
of their works. Now, what do we see? Usually they begin by defining
surfaces as the boundaries of solids or pieces of space, lines as the
boundaries of surfaces, points as the boundaries of lines, and they
affirm that the same procedure can not be pushed further.
This is just the idea given above: to divide space, cuts that are called
surfaces are necessary; to divide surfaces, cuts that are called lines
are necessary; to divide lines, cuts that are called points are
necessary; we can go no further, the point can not be divided, so the
point is not a continuum. Then lines which can be divided by cuts which
are not continua will be continua of one dimension; surfaces which can
be divided by continuous cuts of one dimension will be continua of two
dimensions; finally, space which can be divided by continuous cuts of
two dimensions will be a continuum of three dimensions.
Thus the definition I have just given does not differ essentially from
the usual definitions; I have only endeavored to give it a form
applicable not to the mathematical continuum, but to the physical
continuum, which alone is susceptible of representation, and yet to
retain all its precision. Moreover, we see that this definition applies
not alone to space; that in all which falls under our senses we find the
characteristics of the physical continuum, which would allow of the same
classification; that it would be easy to find there examples of continua
of four, of five, dimensions, in the sense of the preceding definition;
such examples occur of themselves to the mind.
I should explain finally, if I had the time, that this science, of which
I spoke above and to which Riemann gave the name of analysis situs,
teaches us to make distinctions among continua of the same number of
dimensions and that the classification of these continua rests also on
the consideration of cuts.
From this notion has arisen that of the mathematical continuum of
several dimensions in the same way that the physical continuum of one
dimension engendered the mathematical continuum of one dimension. The
formula
_A_ > _C_, _A_ = _B_, _B_ = _C_,
which summed up the data of crude experience, implied an intolerable
contradiction. To get free from it, it was necessary to introduce a new
notion while still respecting the essential characteristics of the
physical continuum of several dimensions. The mathematical continuum of
one dimension admitted of a scale whose divisions, infinite in number,
corresponded to the different values, commensurable or not, of one same
magnitude. To have the mathematical continuum of _n_ dimensions, it will
suffice to take _n_ like scales whose divisions correspond to different
values of _n_ independent magnitudes called coordinates. We thus shall
have an image of the physical continuum of _n_ dimensions, and this
image will be as faithful as it can be after the determination not to
allow the contradiction of which I spoke above.
4. _The Notion of Point_
It seems now that the question we put to ourselves at the start is
answered. When we say that space has three dimensions, it will be said,
we mean that the manifold of points of space satisfies the definition we
have just given of the physical continuum of three dimensions. To be
content with that would be to suppose that we know what is the manifold
of points of space, or even one point of space.
Now that is not as simple as one might think. Every one believes he
knows what a point is, and it is just because we know it too well that
we think there is no need of defining it. Surely we can not be required
to know how to define it, because in going back from definition to
definition a time must come when we must stop. But at what moment should
we stop?
We shall stop first when we reach an object which falls under our senses
or that we can represent to ourselves; definition then will become
useless; we do not define the sheep to a child; we say to him: _See_ the
sheep.
So, then, we should ask ourselves if it is possible to represent to
ourselves a point of space. Those who answer yes do not reflect that
they represent to themselves in reality a white spot made with the chalk
on a blackboard or a black spot made with a pen on white paper, and that
they can represent to themselves only an object or rather the
impressions that this object made on their senses.
When they try to represent to themselves a point, they represent the
impressions that very little objects made them feel. It is needless to
add that two different objects, though both very little, may produce
extremely different impressions, but I shall not dwell on this
difficulty, which would still require some discussion.
But it is not a question of that; it does not suffice to represent _one_
point, it is necessary to represent _a certain_ point and to have the
means of distinguishing it from an _other_ point. And in fact, that we
may be able to apply to a continuum the rule I have above expounded and
by which one may recognize the number of its dimensions, we must rely
upon the fact that two elements of this continuum sometimes can and
sometimes can not be distinguished. It is necessary therefore that we
should in certain cases know how to represent to ourselves _a specific_
element and to distinguish it from an _other_ element.
The question is to know whether the point that I represented to myself
an hour ago is the same as this that I now represent to myself, or
whether it is a different point. In other words, how do we know whether
the point occupied by the object _A_ at the instant [alpha] is the same
as the point occupied by the object _B_ at the instant [beta], or still
better, what this means?
I am seated in my room; an object is placed on my table; during a second
I do not move, no one touches the object. I am tempted to say that the
point _A_ which this object occupied at the beginning of this second is
identical with the point _B_ which it occupies at its end. Not at all;
from the point _A_ to the point _B_ is 30 kilometers, because the object
has been carried along in the motion of the earth. We can not know
whether an object, be it large or small, has not changed its absolute
position in space, and not only can we not affirm it, but this
affirmation has no meaning and in any case can not correspond to any
representation.
But then we may ask ourselves if the relative position of an object with
regard to other objects has changed or not, and first whether the
relative position of this object with regard to our body has changed. If
the impressions this object makes upon us have not changed, we shall be
inclined to judge that neither has this relative position changed; if
they have changed, we shall judge that this object has changed either in
state or in relative position. It remains to decide which of the _two_.
I have explained in 'Science and Hypothesis' how we have been led to
distinguish the changes of position. Moreover, I shall return to that
further on. We come to know, therefore, whether the relative position of
an object with regard to our body has or has not remained the same.
If now we see that two objects have retained their relative position
with regard to our body, we conclude that the relative position of these
two objects with regard to one another has not changed; but we reach
this conclusion only by indirect reasoning. The only thing that we know
directly is the relative position of the objects with regard to our
body. _A fortiori_ it is only by indirect reasoning that we think we
know (and, moreover, this belief is delusive) whether the absolute
position of the object has changed.
In a word, the system of coordinate axes to which we naturally refer all
exterior objects is a system of axes invariably bound to our body, and
carried around with us.
It is impossible to represent to oneself absolute space; when I try to
represent to myself simultaneously objects and myself in motion in
absolute space, in reality I represent to myself my own self motionless
and seeing move around me different objects and a man that is exterior
to me, but that I convene to call me.
Will the difficulty be solved if we agree to refer everything to these
axes bound to our body? Shall we know then what is a point thus defined
by its relative position with regard to ourselves? Many persons will
answer yes and will say that they 'localize' exterior objects.
What does this mean? To localize an object simply means to represent to
oneself the movements that would be necessary to reach it. I will
explain myself. It is not a question of representing the movements
themselves in space, but solely of representing to oneself the muscular
sensations which accompany these movements and which do not presuppose
the preexistence of the notion of space.
If we suppose two different objects which successively occupy the same
relative position with regard to ourselves, the impressions that these
two objects make upon us will be very different; if we localize them at
the same point, this is simply because it is necessary to make the same
movements to reach them; apart from that, one can not just see what they
could have in common.
But, given an object, we can conceive many different series of movements
which equally enable us to reach it. If then we represent to ourselves a
point by representing to ourselves the series of muscular sensations
which accompany the movements which enable us to reach this point, there
will be many ways entirely different of representing to oneself the same
point. If one is not satisfied with this solution, but wishes, for
instance, to bring in the visual sensations along with the muscular
sensations, there will be one or two more ways of representing to
oneself this same point and the difficulty will only be increased. In
any case the following question comes up: Why do we think that all these
representations so different from one another still represent the same
point?
Another remark: I have just said that it is to our own body that we
naturally refer exterior objects; that we carry about everywhere with us
a system of axes to which we refer all the points of space and that this
system of axes seems to be invariably bound to our body. It should be
noticed that rigorously we could not speak of axes invariably bound to
the body unless the different parts of this body were themselves
invariably bound to one another. As this is not the case, we ought,
before referring exterior objects to these fictitious axes, to suppose
our body brought back to the initial attitude.
5. _The Notion of Displacement_
I have shown in 'Science and Hypothesis' the preponderant rôle played by
the movements of our body in the genesis of the notion of space. For a
being completely immovable there would be neither space nor geometry; in
vain would exterior objects be displaced about him, the variations which
these displacements would make in his impressions would not be
attributed by this being to changes of position, but to simple changes
of state; this being would have no means of distinguishing these two
sorts of changes, and this distinction, fundamental for us, would have
no meaning for him.
The movements that we impress upon our members have as effect the
varying of the impressions produced on our senses by external objects;
other causes may likewise make them vary; but we are led to distinguish
the changes produced by our own motions and we easily discriminate them
for two reasons: (1) because they are voluntary; (2) because they are
accompanied by muscular sensations.
So we naturally divide the changes that our impressions may undergo into
two categories to which perhaps I have given an inappropriate
designation: (1) the internal changes, which are voluntary and
accompanied by muscular sensations; (2) the external changes, having the
opposite characteristics.
We then observe that among the external changes are some which can be
corrected, thanks to an internal change which brings everything back to
the primitive state; others can not be corrected in this way (it is thus
that, when an exterior object is displaced, we may then by changing our
own position replace ourselves as regards this object in the same
relative position as before, so as to reestablish the original aggregate
of impressions; if this object was not displaced, but changed its state,
that is impossible). Thence comes a new distinction among external
changes: those which may be so corrected we call changes of position;
and the others, changes of state.
Think, for example, of a sphere with one hemisphere blue and the other
red; it first presents to us the blue hemisphere, then it so revolves as
to present the red hemisphere. Now think of a spherical vase containing
a blue liquid which becomes red in consequence of a chemical reaction.
In both cases the sensation of red has replaced that of blue; our senses
have experienced the same impressions which have succeeded each other in
the same order, and yet these two changes are regarded by us as very
different; the first is a displacement, the second a change of state.
Why? Because in the first case it is sufficient for me to go around the
sphere to place myself opposite the blue hemisphere and reestablish the
original blue sensation.
Still more; if the two hemispheres, in place of being red and blue, had
been yellow and green, how should I have interpreted the revolution of
the sphere? Before, the red succeeded the blue, now the green succeeds
the yellow; and yet I say that the two spheres have undergone the same
revolution, that each has turned about its axis; yet I can not say that
the green is to yellow as the red is to blue; how then am I led to
decide that the two spheres have undergone the _same_ displacement?
Evidently because, in one case as in the other, I am able to reestablish
the original sensation by going around the sphere, by making the same
movements, and I know that I have made the same movements because I have
felt the same muscular sensations; to know it, I do not need, therefore,
to know geometry in advance and to represent to myself the movements of
my body in geometric space.
Another example: An object is displaced before my eye; its image was
first formed at the center of the retina; then it is formed at the
border; the old sensation was carried to me by a nerve fiber ending at
the center of the retina; the new sensation is carried to me by
_another_ nerve fiber starting from the border of the retina; these two
sensations are qualitatively different; otherwise, how could I
distinguish them?
Why then am I led to decide that these two sensations, qualitatively
different, represent the same image, which has been displaced? It is
because I _can follow the object with the eye_ and by a displacement of
the eye, voluntary and accompanied by muscular sensations, bring back
the image to the center of the retina and reestablish the primitive
sensation.
I suppose that the image of a red object has gone from the center _A_ to
the border _B_ of the retina, then that the image of a blue object goes
in its turn from the center _A_ to the border _B_ of the retina; I shall
decide that these two objects have undergone the _same_ displacement.
Why? Because in both cases I shall have been able to reestablish the
primitive sensation, and that to do it I shall have had to execute the
_same_ movement of the eye, and I shall know that my eye has executed
the same movement because I shall have felt the _same_ muscular
sensations.
If I could not move my eye, should I have any reason to suppose that the
sensation of red at the center of the retina is to the sensation of red
at the border of the retina as that of blue at the center is to that of
blue at the border? I should only have four sensations qualitatively
different, and if I were asked if they are connected by the proportion I
have just stated, the question would seem to me ridiculous, just as if I
were asked if there is an analogous proportion between an auditory
sensation, a tactile sensation and an olfactory sensation.
Let us now consider the internal changes, that is, those which are
produced by the voluntary movements of our body and which are
accompanied by muscular changes. They give rise to the two following
observations, analogous to those we have just made on the subject of
external changes.
1. I may suppose that my body has moved from one point to another, but
that the same _attitude_ is retained; all the parts of the body have
therefore retained or resumed the same _relative_ situation, although
their absolute situation in space may have varied. I may suppose that
not only has the position of my body changed, but that its attitude is
no longer the same, that, for instance, my arms which before were folded
are now stretched out.
I should therefore distinguish the simple changes of position without
change of attitude, and the changes of attitude. Both would appear to me
under form of muscular sensations. How then am I led to distinguish
them? It is that the first may serve to correct an external change, and
that the others can not, or at least can only give an imperfect
correction.
This fact I proceed to explain as I would explain it to some one who
already knew geometry, but it need not thence be concluded that it is
necessary already to know geometry to make this distinction; before
knowing geometry I ascertain the fact (experimentally, so to speak),
without being able to explain it. But merely to make the distinction
between the two kinds of change, I do not need to _explain_ the fact, it
suffices me _to ascertain_ it.
However that may be, the explanation is easy. Suppose that an exterior
object is displaced; if we wish the different parts of our body to
resume with regard to this object their initial relative position, it is
necessary that these different parts should have resumed likewise their
initial relative position with regard to one another. Only the internal
changes which satisfy this latter condition will be capable of
correcting the external change produced by the displacement of that
object. If, therefore, the relative position of my eye with regard to my
finger has changed, I shall still be able to replace the eye in its
initial relative situation with regard to the object and reestablish
thus the primitive visual sensations, but then the relative position of
the finger with regard to the object will have changed and the tactile
sensations will not be reestablished.
2. We ascertain likewise that the same external change may be corrected
by two internal changes corresponding to different muscular sensations.
Here again I can ascertain this without knowing geometry; and I have no
need of anything else; but I proceed to give the explanation of the
fact, employing geometrical language. To go from the position _A_ to the
position _B_ I may take several routes. To the first of these routes
will correspond a series _S_ of muscular sensations; to a second route
will correspond another series _S''_, of muscular sensations which
generally will be completely different, since other muscles will be
used.
How am I led to regard these two series _S_ and _S''_ as corresponding
to the same displacement _AB_? It is because these two series are
capable of correcting the same external change. Apart from that, they
have nothing in common.
Let us now consider two external changes: [alpha] and [beta], which
shall be, for instance, the rotation of a sphere half blue, half red,
and that of a sphere half yellow, half green; these two changes have
nothing in common, since the one is for us the passing of blue into red
and the other the passing of yellow into green. Consider, on the other
hand, two series of internal changes _S_ and _S''_; like the others,
they will have nothing in common. And yet I say that [alpha] and [beta]
correspond to the same displacement, and that _S_ and _S''_ correspond
also to the same displacement. Why? Simply because _S_ can correct
[alpha] as well as [beta] and because [alpha] can be corrected by _S''_
as well as by _S_. And then a question suggests itself:
If I have ascertained that _S_ corrects [alpha] and [beta] and that
_S''_ corrects [alpha], am I certain that _S''_ likewise corrects
[beta]? Experiment alone can teach us whether this law is verified. If
it were not verified, at least approximately, there would be no
geometry, there would be no space, because we should have no more
interest in classifying the internal and external changes as I have just
done, and, for instance, in distinguishing changes of state from changes
of position.
It is interesting to see what has been the rôle of experience in all
this. It has shown me that a certain law is approximately verified. It
has not told me _how_ space is, and that it satisfies the condition in
question. I knew, in fact, before all experience, that space satisfied
this condition or that it would not be; nor have I any right to say that
experience told me that geometry is possible; I very well see that
geometry is possible, since it does not imply contradiction; experience
only tells me that geometry is useful.
6. _Visual Space_
Although motor impressions have had, as I have just explained, an
altogether preponderant influence in the genesis of the notion of space,
which never would have taken birth without them, it will not be without
interest to examine also the rôle of visual impressions and to
investigate how many dimensions 'visual space' has, and for that purpose
to apply to these impressions the definition of § 3.
A first difficulty presents itself: consider a red color sensation
affecting a certain point of the retina; and on the other hand a blue
color sensation affecting the same point of the retina. It is necessary
that we have some means of recognizing that these two sensations,
qualitatively different, have something in common. Now, according to the
considerations expounded in the preceding paragraph, we have been able
to recognize this only by the movements of the eye and the observations
to which they have given rise. If the eye were immovable, or if we were
unconscious of its movements, we should not have been able to recognize
that these two sensations, of different quality, had something in
common; we should not have been able to disengage from them what gives
them a geometric character. The visual sensations, without the muscular
sensations, would have nothing geometric, so that it may be said there
is no pure visual space.
To do away with this difficulty, consider only sensations of the same
nature, red sensations, for instance, differing one from another only as
regards the point of the retina that they affect. It is clear that I
have no reason for making such an arbitrary choice among all the
possible visual sensations, for the purpose of uniting in the same class
all the sensations of the same color, whatever may be the point of the
retina affected. I should never have dreamt of it, had I not before
learned, by the means we have just seen, to distinguish changes of state
from changes of position, that is, if my eye were immovable. Two
sensations of the same color affecting two different parts of the retina
would have appeared to me as qualitatively distinct, just as two
sensations of different color.
In restricting myself to red sensations, I therefore impose upon myself
an artificial limitation and I neglect systematically one whole side of
the question; but it is only by this artifice that I am able to analyze
visual space without mingling any motor sensation.
Imagine a line traced on the retina and dividing in two its surface; and
set apart the red sensations affecting a point of this line, or those
differing from them too little to be distinguished from them. The
aggregate of these sensations will form a sort of cut that I shall call
_C_, and it is clear that this cut suffices to divide the manifold of
possible red sensations, and that if I take two red sensations affecting
two points situated on one side and the other of the line, I can not
pass from one of these sensations to the other in a continuous way
without passing at a certain moment through a sensation belonging to the
cut.
If, therefore, the cut has _n_ dimensions, the total manifold of my red
sensations, or if you wish, the whole visual space, will have _n_ + 1.
Now, I distinguish the red sensations affecting a point of the cut _C_.
The assemblage of these sensations will form a new cut _C'_. It is clear
that this will divide the cut _C_, always giving to the word divide the
same meaning.
If, therefore, the cut _C'_ has _n_ dimensions, the cut _C_ will have
_n_ + 1 and the whole of visual space _n_ + 2.
If all the red sensations affecting the same point of the retina were
regarded as identical, the cut _C'_ reducing to a single element would
have 0 dimensions, and visual space would have 2.
And yet most often it is said that the eye gives us the sense of a third
dimension, and enables us in a certain measure to recognize the distance
of objects. When we seek to analyze this feeling, we ascertain that it
reduces either to the consciousness of the convergence of the eyes, or
to that of the effort of accommodation which the ciliary muscle makes to
focus the image.
Two red sensations affecting the same point of the retina will therefore
be regarded as identical only if they are accompanied by the same
sensation of convergence and also by the same sensation of effort of
accommodation or at least by sensations of convergence and accommodation
so slightly different as to be indistinguishable.
On this account the cut _C'_ is itself a continuum and the cut _C_ has
more than one dimension.
But it happens precisely that experience teaches us that when two visual
sensations are accompanied by the same sensation of convergence, they
are likewise accompanied by the same sensation of accommodation. If then
we form a new cut _C''_ with all those of the sensations of the cut
_C'_, which are accompanied by a certain sensation of convergence, in
accordance with the preceding law they will all be indistinguishable and
may be regarded as identical. Therefore _C''_ will not be a continuum
and will have 0 dimension; and as _C''_ divides _C'_ it will thence
result that _C'_ has one, _C_ two and _the whole visual space three
dimensions_.
But would it be the same if experience had taught us the contrary and if
a certain sensation of convergence were not always accompanied by the
same sensation of accommodation? In this case two sensations affecting
the same point of the retina and accompanied by the same sense of
convergence, two sensations which consequently would both appertain to
the cut _C''_, could nevertheless be distinguished since they would be
accompanied by two different sensations of accommodation. Therefore
_C''_ would be in its turn a continuum and would have one dimension (at
least); then _C'_ would have two, _C_ three and _the whole visual space
would have four dimensions_.
Will it then be said that it is experience which teaches us that space
has three dimensions, since it is in setting out from an experimental
law that we have come to attribute three to it? But we have therein
performed, so to speak, only an experiment in physiology; and as also it
would suffice to fit over the eyes glasses of suitable construction to
put an end to the accord between the feelings of convergence and of
accommodation, are we to say that putting on spectacles is enough to
make space have four dimensions and that the optician who constructed
them has given one more dimension to space? Evidently not; all we can
say is that experience has taught us that it is convenient to attribute
three dimensions to space.
But visual space is only one part of space, and in even the notion of
this space there is something artificial, as I have explained at the
beginning. The real space is motor space and this it is that we shall
examine in the following chapter.
CHAPTER IV
SPACE AND ITS THREE DIMENSIONS
1. _The Group of Displacements_
Let us sum up briefly the results obtained. We proposed to investigate
what was meant in saying that space has three dimensions and we have
asked first what is a physical continuum and when it may be said to have
_n_ dimensions. If we consider different systems of impressions and
compare them with one another, we often recognize that two of these
systems of impressions are indistinguishable (which is ordinarily
expressed in saying that they are too close to one another, and that our
senses are too crude, for us to distinguish them) and we ascertain
besides that two of these systems can sometimes be discriminated from
one another though indistinguishable from a third system. In that case
we say the manifold of these systems of impressions forms a physical
continuum _C_. And each of these systems is called an _element_ of the
continuum _C_.
How many dimensions has this continuum? Take first two elements _A_ and
_B_ of _C_, and suppose there exists a series [Sigma] of elements, all
belonging to the continuum _C_, of such a sort that _A_ and _B_ are the
two extreme terms of this series and that each term of the series is
indistinguishable from the preceding. If such a series [Sigma] can be
found, we say that _A_ and _B_ are joined to one another; and if any two
elements of _C_ are joined to one another, we say that _C_ is all of one
piece.
Now take on the continuum _C_ a certain number of elements in a way
altogether arbitrary. The aggregate of these elements will be called a
_cut_. Among the various series [Sigma] which join _A_ to _B_, we shall
distinguish those of which an element is indistinguishable from one of
the elements of the cut (we shall say that these are they which _cut_
the cut) and those of which _all_ the elements are distinguishable from
all those of the cut. If _all_ the series [Sigma] which join _A_ to _B_
cut the cut, we shall say that _A_ and _B_ are _separated_ by the cut,
and that the cut _divides_ _C_. If we can not find on _C_ two elements
which are separated by the cut, we shall say that the cut _does not
divide_ _C_.
These definitions laid down, if the continuum _C_ can be divided by cuts
which do not themselves form a continuum, this continuum _C_ has only
one dimension; in the contrary case it has several. If a cut forming a
continuum of 1 dimension suffices to divide _C_, _C_ will have 2
dimensions; if a cut forming a continuum of 2 dimensions suffices, _C_
will have 3 dimensions, etc. Thanks to these definitions, we can always
recognize how many dimensions any physical continuum has. It only
remains to find a physical continuum which is, so to speak, equivalent
to space, of such a sort that to every point of space corresponds an
element of this continuum, and that to points of space very near one
another correspond indistinguishable elements. Space will have then as
many dimensions as this continuum.
The intermediation of this physical continuum, capable of
representation, is indispensable; because we can not represent space to
ourselves, and that for a multitude of reasons. Space is a mathematical
continuum, it is infinite, and we can represent to ourselves only
physical continua and finite objects. The different elements of space,
which we call points, are all alike, and, to apply our definition, it is
necessary that we know how to distinguish the elements from one another,
at least if they are not too close. Finally absolute space is nonsense,
and it is necessary for us to begin by referring space to a system of
axes invariably bound to our body (which we must always suppose put back
in the initial attitude).
Then I have sought to form with our visual sensations a physical
continuum equivalent to space; that certainly is easy and this example
is particularly appropriate for the discussion of the number of
dimensions; this discussion has enabled us to see in what measure it is
allowable to say that 'visual space' has three dimensions. Only this
solution is incomplete and artificial. I have explained why, and it is
not on visual space but on motor space that it is necessary to bring our
efforts to bear. I have then recalled what is the origin of the
distinction we make between changes of position and changes of state.
Among the changes which occur in our impressions, we distinguish, first
the _internal_ changes, voluntary and accompanied by muscular
sensations, and the _external_ changes, having opposite characteristics.
We ascertain that it may happen that an external change may be
_corrected_ by an internal change which reestablishes the primitive
sensations. The external changes, capable of being corrected by an
internal change are called _changes of position_, those not capable of
it are called _changes of state_. The internal changes capable of
correcting an external change are called _displacements of the whole
body_; the others are called _changes of attitude_.
Now let [alpha] and [beta] be two external changes, [alpha]' and [beta]'
two internal changes. Suppose that a may be corrected either by [alpha]'
or by [beta]', and that [alpha]' can correct either [alpha] or [beta];
experience tells us then that [beta]' can likewise correct [beta]. In
this case we say that [alpha] and [beta] correspond to the _same_
displacement and also that [alpha]' and [beta]' correspond to the _same_
displacement. That postulated, we can imagine a physical continuum which
we shall call _the continuum or group of displacements_ and which we
shall define in the following manner. The elements of this continuum
shall be the internal changes capable of correcting an external change.
Two of these internal changes [alpha]' and [beta]' shall be regarded as
indistinguishable: (1) if they are so naturally, that is, if they are
too close to one another; (2) if [alpha]' is capable of correcting the
same external change as a third internal change naturally
indistinguishable from [beta]'. In this second case, they will be, so to
speak, indistinguishable by convention, I mean by agreeing to disregard
circumstances which might distinguish them.
Our continuum is now entirely defined, since we know its elements and
have fixed under what conditions they may be regarded as
indistinguishable. We thus have all that is necessary to apply our
definition and determine how many dimensions this continuum has. We
shall recognize that it has _six_. The continuum of displacements is,
therefore, not equivalent to space, since the number of dimensions is
not the same; it is only related to space. Now how do we know that this
continuum of displacements has six dimensions? We know it _by
experience_.
It would be easy to describe the experiments by which we could arrive
at this result. It would be seen that in this continuum cuts can be made
which divide it and which are continua; that these cuts themselves can
be divided by other cuts of the second order which yet are continua, and
that this would stop only after cuts of the sixth order which would no
longer be continua. From our definitions that would mean that the group
of displacements has six dimensions.
That would be easy, I have said, but that would be rather long; and
would it not be a little superficial? This group of displacements, we
have seen, is related to space, and space could be deduced from it, but
it is not equivalent to space, since it has not the same number of
dimensions; and when we shall have shown how the notion of this
continuum can be formed and how that of space may be deduced from it, it
might always be asked why space of three dimensions is much more
familiar to us than this continuum of six dimensions, and consequently
doubted whether it was by this detour that the notion of space was
formed in the human mind.
2. _Identity of Two Points_
What is a point? How do we know whether two points of space are
identical or different? Or, in other words, when I say: The object _A_
occupied at the instant [alpha] the point which the object _B_ occupies
at the instant [beta], what does that mean?
Such is the problem we set ourselves in the preceding chapter, §4. As I
have explained it, it is not a question of comparing the positions of
the objects _A_ and _B_ in absolute space; the question then would
manifestly have no meaning. It is a question of comparing the positions
of these two objects with regard to axes invariably bound to my body,
supposing always this body replaced in the same attitude.
I suppose that between the instants [alpha] and [beta] I have moved
neither my body nor my eye, as I know from my muscular sense. Nor have I
moved either my head, my arm or my hand. I ascertain that at the instant
[alpha] impressions that I attributed to the object _A_ were transmitted
to me, some by one of the fibers of my optic nerve, the others by one of
the sensitive tactile nerves of my finger; I ascertain that at the
instant [beta] other impressions which I attribute to the object _B_ are
transmitted to me, some by this same fiber of the optic nerve, the
others by this same tactile nerve.
Here I must pause for an explanation; how am I told that this impression
which I attribute to _A_, and that which I attribute to _B_, impressions
which are qualitatively different, are transmitted to me by the same
nerve? Must we suppose, to take for example the visual sensations, that
_A_ produces two simultaneous sensations, a sensation purely luminous
_a_ and a colored sensation _a'_, that _B_ produces in the same way
simultaneously a luminous sensation _b_ and a colored sensation _b'_,
that if these different sensations are transmitted to me by the same
retinal fiber, _a_ is identical with _b_, but that in general the
colored sensations _a'_ and _b'_ produced by different bodies are
different? In that case it would be the identity of the sensation _a_
which accompanies _a'_ with the sensation _b_ which accompanies _b'_,
which would tell that all these sensations are transmitted to me by the
same fiber.
However it may be with this hypothesis and although I am led to prefer
to it others considerably more complicated, it is certain that we are
told in some way that there is something in common between these
sensations _a_ + _a'_ and _b_ +_b'_, without which we should have no
means of recognizing that the object _B_ has taken the place of the
object _A_.
Therefore I do not further insist and I recall the hypothesis I have
just made: I suppose that I have ascertained that the impressions which
I attribute to _B_ are transmitted to me at the instant [beta] by the
same fibers, optic as well as tactile, which, at the instant [alpha],
had transmitted to me the impressions that I attributed to _A_. If it is
so, we shall not hesitate to declare that the point occupied by _B_ at
the instant [beta] is identical with the point occupied by _A_ at the
instant [alpha].
I have just enunciated two conditions for these points being identical;
one is relative to sight, the other to touch. Let us consider them
separately. The first is necessary, but is not sufficient. The second is
at once necessary and sufficient. A person knowing geometry could easily
explain this in the following manner: Let _O_ be the point of the retina
where is formed at the instant [alpha] the image of the body _A_; let
_M_ be the point of space occupied at the instant [alpha] by this body
_A_; let _M'_ be the point of space occupied at the instant [beta] by
the body _B_. For this body _B_ to form its image in _O_, it is not
necessary that the points _M_ and _M'_ coincide; since vision acts at a
distance, it suffices for the three points _O_ _M_ _M'_ to be in a
straight line. This condition that the two objects form their image on
_O_ is therefore necessary, but not sufficient for the points _M_ and
_M'_ to coincide. Let now _P_ be the point occupied by my finger and
where it remains, since it does not budge. As touch does not act at a
distance, if the body _A_ touches my finger at the instant [alpha], it
is because _M_ and _P_ coincide; if _B_ touches my finger at the instant
[beta], it is because _M'_ and _P_ coincide. Therefore _M_ and _M'_
coincide. Thus this condition that if _A_ touches my finger at the
instant [alpha], _B_ touches it at the instant [beta], is at once
necessary and sufficient for _M_ and _M'_ to coincide.
But we who, as yet, do not know geometry can not reason thus; all that
we can do is to ascertain experimentally that the first condition
relative to sight may be fulfilled without the second, which is relative
to touch, but that the second can not be fulfilled without the first.
Suppose experience had taught us the contrary, as might well be; this
hypothesis contains nothing absurd. Suppose, therefore, that we had
ascertained experimentally that the condition relative to touch may be
fulfilled without that of sight being fulfilled and that, on the
contrary, that of sight can not be fulfilled without that of touch being
also. It is clear that if this were so we should conclude that it is
touch which may be exercised at a distance, and that sight does not
operate at a distance.
But this is not all; up to this time I have supposed that to determine
the place of an object I have made use only of my eye and a single
finger; but I could just as well have employed other means, for example,
all my other fingers.
I suppose that my first finger receives at the instant [alpha] a tactile
impression which I attribute to the object _A_. I make a series of
movements, corresponding to a series _S_ of muscular sensations. After
these movements, at the instant [alpha]', my _second_ finger receives a
tactile impression that I attribute likewise to _A_. Afterward, at the
instant [beta], without my having budged, as my muscular sense tells me,
this same second finger transmits to me anew a tactile impression which
I attribute this time to the object _B_; I then make a series of
movements, corresponding to a series _S'_ of muscular sensations. I know
that this series _S'_ is the inverse of the series _S_ and corresponds
to contrary movements. I know this because many previous experiences
have shown me that if I made successively the two series of movements
corresponding to _S_ and to _S'_, the primitive impressions would be
reestablished, in other words, that the two series mutually compensate.
That settled, should I expect that at the instant [beta]', when the
second series of movements is ended, my _first finger_ would feel a
tactile impression attributable to the object _B_?
To answer this question, those already knowing geometry would reason as
follows: There are chances that the object _A_ has not budged, between
the instants [alpha] and [alpha]', nor the object _B_ between the
instants [beta] and [beta]'; assume this. At the instant [alpha], the
object _A_ occupied a certain point _M_ of space. Now at this instant it
touched my first finger, and _as touch does not operate at a distance_,
my first finger was likewise at the point _M_. I afterward made the
series _S_ of movements and at the end of this series, at the instant
[alpha]', I ascertained that the object _A_ touched my second finger. I
thence conclude that this second finger was then at _M_, that is, that
the movements _S_ had the result of bringing the second finger to the
place of the first. At the instant [beta] the object _B_ has come in
contact with my second finger: as I have not budged, this second finger
has remained at _M_; therefore the object _B_ has come to _M_; by
hypothesis it does not budge up to the instant [beta]'. But between the
instants [beta] and [beta]' I have made the movements _S'_; as these
movements are the inverse of the movements _S_, they must have for
effect bringing the first finger in the place of the second. At the
instant [beta]' this first finger will, therefore, be at _M_; and as the
object _B_ is likewise at _M_, this object _B_ will touch my first
finger. To the question put, the answer should therefore be yes.
We who do not yet know geometry can not reason thus; but we ascertain
that this anticipation is ordinarily realized; and we can always explain
the exceptions by saying that the object _A_ has moved between the
instants [alpha] and [alpha]', or the object _B_ between the instants
[beta] and [beta]'.
But could not experience have given a contrary result? Would this
contrary result have been absurd in itself? Evidently not. What should
we have done then if experience had given this contrary result? Would
all geometry thus have become impossible? Not the least in the world. We
should have contented ourselves with concluding _that touch can operate
at a distance_.
When I say, touch does not operate at a distance, but sight operates at
a distance, this assertion has only one meaning, which is as follows: To
recognize whether _B_ occupies at the instant [beta] the point occupied
by _A_ at the instant [alpha], I can use a multitude of different
criteria. In one my eye intervenes, in another my first finger, in
another my second finger, etc. Well, it is sufficient for the criterion
relative to one of my fingers to be satisfied in order that all the
others should be satisfied, but it is not sufficient that the criterion
relative to the eye should be. This is the sense of my assertion. I
content myself with affirming an experimental fact which is ordinarily
verified.
At the end of the preceding chapter we analyzed visual space; we saw
that to engender this space it is necessary to bring in the retinal
sensations, the sensation of convergence and the sensation of
accommodation; that if these last two were not always in accord, visual
space would have four dimensions in place of three; we also saw that if
we brought in only the retinal sensations, we should obtain 'simple
visual space,' of only two dimensions. On the other hand, consider
tactile space, limiting ourselves to the sensations of a single finger,
that is in sum to the assemblage of positions this finger can occupy.
This tactile space that we shall analyze in the following section and
which consequently I ask permission not to consider further for the
moment, this tactile space, I say, has three dimensions. Why has space
properly so called as many dimensions as tactile space and more than
simple visual space? It is because touch does not operate at a distance,
while vision does operate at a distance. These two assertions have the
same meaning and we have just seen what this is.
Now I return to a point over which I passed rapidly in order not to
interrupt the discussion. How do we know that the impressions made on
our retina by _A_ at the instant [alpha] and _B_ at the instant [beta]
are transmitted by the same retinal fiber, although these impressions
are qualitatively different? I have suggested a simple hypothesis, while
adding that other hypotheses, decidedly more complex, would seem to me
more probably true. Here then are these hypotheses, of which I have
already said a word. How do we know that the impressions produced by the
red object A at the instant [alpha], and by the blue object _B_ at the
instant [beta], if these two objects have been imaged on the same point
of the retina, have something in common? The simple hypothesis above
made may be rejected and we may suppose that these two impressions,
qualitatively different, are transmitted by two different though
contiguous nervous fibers. What means have I then of knowing that these
fibers are contiguous? It is probable that we should have none, if the
eye were immovable. It is the movements of the eye which have told us
that there is the same relation between the sensation of blue at the
point _A_ and the sensation of blue at the point _B_ of the retina as
between the sensation of red at the point _A_ and the sensation of red
at the point _B_. They have shown us, in fact, that the same movements,
corresponding to the same muscular sensations, carry us from the first
to the second, or from the third to the fourth. I do not emphasize these
considerations, which belong, as one sees, to the question of local
signs raised by Lotze.
3. _Tactile Space_
Thus I know how to recognize the identity of two points, the point
occupied by _A_ at the instant [alpha] and the point occupied by _B_ at
the instant [beta], but only _on one condition_, namely, that I have not
budged between the instants [alpha] and [beta]. That does not suffice
for our object. Suppose, therefore, that I have moved in any manner in
the interval between these two instants, how shall I know whether the
point occupied by _A_ at the instant [alpha] is identical with the point
occupied by _B_ at the instant [beta]? I suppose that at the instant
[alpha], the object _A_ was in contact with my first finger and that in
the same way, at the instant [beta], the object _B_ touches this first
finger; but at the same time my muscular sense has told me that in the
interval my body has moved. I have considered above two series of
muscular sensations _S_ and _S'_, and I have said it sometimes happens
that we are led to consider two such series _S_ and _S'_ as inverse one
of the other, because we have often observed that when these two series
succeed one another our primitive impressions are reestablished.
If then my muscular sense tells me that I have moved between the two
instants [alpha] and [beta], but so as to feel successively the two
series of muscular sensations _S_ and _S'_ that I consider inverses, I
shall still conclude, just as if I had not budged, that the points
occupied by _A_ at the instant [alpha] and by _B_ at the instant [beta]
are identical, if I ascertain that my first finger touches _A_ at the
instant [alpha], and _B_ at the instant [beta].
This solution is not yet completely satisfactory, as one will see. Let
us see, in fact, how many dimensions it would make us attribute to
space. I wish to compare the two points occupied by _A_ and _B_ at the
instants [alpha] and [beta], or (what amounts to the same thing since I
suppose that my finger touches _A_ at the instant [alpha] and _B_ at the
instant [beta]) I wish to compare the two points occupied by my finger
at the two instants [alpha] and [beta]. The sole means I use for this
comparison is the series [Sigma] of muscular sensations which have
accompanied the movements of my body between these two instants. The
different imaginable series [Sigma] form evidently a physical continuum
of which the number of dimensions is very great. Let us agree, as I have
done, not to consider as distinct the two series [Sigma] and [Sigma] +
_S_ + _S'_, when _S_ and _S'_ are inverses one of the other in the sense
above given to this word; in spite of this agreement, the aggregate of
distinct series [Sigma] will still form a physical continuum and the
number of dimensions will be less but still very great.
To each of these series [Sigma] corresponds a point of space; to two
series [Sigma] and [Sigma]' thus correspond two points _M_ and _M'_. The
means we have hitherto used enable us to recognize that _M_ and _M'_ are
not distinct in two cases: (1) if [Sigma] is identical with [Sigma]';
(2) if [Sigma]' = [Sigma] + _S_ + _S'_, _S_ and _S'_ being inverses one
of the other. If in all the other cases we should regard _M_ and _M'_ as
distinct, the manifold of points would have as many dimensions as the
aggregate of distinct series [Sigma], that is, much more than three.
For those who already know geometry, the following explanation would be
easily comprehensible. Among the imaginable series of muscular
sensations, there are those which correspond to series of movements
where the finger does not budge. I say that if one does not consider as
distinct the series [Sigma] and [Sigma] + [sigma], where the series
[sigma] corresponds to movements where the finger does not budge, the
aggregate of series will constitute a continuum of three dimensions, but
that if one regards as distinct two series [Sigma] and [Sigma]' unless
[Sigma]' = [Sigma] + _S_ + _S'_, _S_ and _S'_ being inverses, the
aggregate of series will constitute a continuum of more than three
dimensions.
In fact, let there be in space a surface _A_, on this surface a line
_B_, on this line a point _M_. Let C_{0} be the aggregate of all series
[Sigma]. Let C_{1} be the aggregate of all the series [Sigma], such that
at the end of corresponding movements the finger is found upon the
surface _A_, and C_{2} or C_{3} the aggregate of series [Sigma] such
that at the end the finger is found on _B_, or at _M_. It is clear,
first that C_{1} will constitute a cut which will divide C_{0}, that
C_{2} will be a cut which will divide C_{1}, and C_{3} a cut which will
divide C_2. Thence it results, in accordance with our definitions, that
if C_{3} is a continuum of _n_ dimensions, C_{0} will be a physical
continuum of _n_ + 3 dimensions.
Therefore, let [Sigma] and [Sigma]' = [Sigma] + [sigma] be two series
forming part of C_{3}; for both, at the end of the movements, the finger
is found at _M_; thence results that at the beginning and at the end of
the series [sigma] the finger is at the same point _M_. This series
[sigma] is therefore one of those which correspond to movements where
the finger does not budge. If [Sigma] and [Sigma] + [sigma] are not
regarded as distinct, all the series of C_{3} blend into one; therefore
C_{3} will have 0 dimension, and C_{0} will have 3, as I wished to
prove. If, on the contrary, I do not regard [Sigma] and [Sigma] +
[sigma] as blending (unless [sigma] = _S_ + _S'_, _S_ and _S'_ being
inverses), it is clear that C_{3} will contain a great number of series
of distinct sensations; because, without the finger budging, the body
may take a multitude of different attitudes. Then C_{3} will form a
continuum and C_{0} will have more than three dimensions, and this also
I wished to prove.
We who do not yet know geometry can not reason in this way; we can only
verify. But then a question arises; how, before knowing geometry, have
we been led to distinguish from the others these series [sigma] where
the finger does not budge? It is, in fact, only after having made this
distinction that we could be led to regard [Sigma] and [Sigma] + [sigma]
as identical, and it is on this condition alone, as we have just seen,
that we can arrive at space of three dimensions.
We are led to distinguish the series [sigma], because it often happens
that when we have executed the movements which correspond to these
series [sigma] of muscular sensations, the tactile sensations which are
transmitted to us by the nerve of the finger that we have called the
first finger, persist and are not altered by these movements. Experience
alone tells us that and it alone could tell us.
If we have distinguished the series of muscular sensations _S_ + _S'_
formed by the union of two inverse series, it is because they preserve
the totality of our impressions; if now we distinguish the series
[sigma], it is because they preserve _certain_ of our impressions. (When
I say that a series of muscular sensations _S_ 'preserves' one of our
impressions _A_, I mean that we ascertain that if we feel the impression
_A_, then the muscular sensations _S_, we _still_ feel the impression
_A_ _after_ these sensations _S_.)
I have said above it often happens that the series [sigma] do not alter
the tactile impressions felt by our first finger; I said _often_, I did
not say _always_. This it is that we express in our ordinary language by
saying that the tactile impressions would not be altered if the finger
has not moved, _on the condition_ that _neither has_ the object _A_,
which was in contact with this finger, moved. Before knowing geometry,
we could not give this explanation; all we could do is to ascertain that
the impression often persists, but not always.
But that the impression often continues is enough to make the series
[sigma] appear remarkable to us, to lead us to put in the same class the
series [Sigma] and [Sigma] + [sigma], and hence not regard them as
distinct. Under these conditions we have seen that they will engender a
physical continuum of three dimensions.
Behold then a space of three dimensions engendered by my first finger.
Each of my fingers will create one like it. It remains to consider how
we are led to regard them as identical with visual space, as identical
with geometric space.
But one reflection before going further; according to the foregoing, we
know the points of space, or more generally the final situation of our
body, only by the series of muscular sensations revealing to us the
movements which have carried us from a certain initial situation to this
final situation. But it is clear that this final situation will depend,
on the one hand, upon these movements and, _on the other hand, upon the
initial situation_ from which we set out. Now these movements are
revealed to us by our muscular sensations; but nothing tells us the
initial situation; nothing can distinguish it for us from all the other
possible situations. This puts well in evidence the essential relativity
of space.
4. _Identity of the Different Spaces_
We are therefore led to compare the two continua _C_ and _C'_
engendered, for instance, one by my first finger _D_, the other by my
second finger _D'_. These two physical continua both have three
dimensions. To each element of the continuum _C_, or, if you prefer, to
each point of the first tactile space, corresponds a series of muscular
sensations [Sigma], which carry me from a certain initial situation to a
certain final situation.[8] Moreover, the same point of this first space
will correspond to [Sigma] and [Sigma] + [sigma], if [sigma] is a series
of which we know that it does not make the finger _D_ move.
[8] In place of saying that we refer space to axes rigidly bound to
our body, perhaps it would be better to say, in conformity to
what precedes, that we refer it to axes rigidly bound to the
initial situation of our body.
Similarly to each element of the continuum _C'_, or to each point of the
second tactile space, corresponds a series of sensations [Sigma]', and
the same point will correspond to [Sigma]' and to [Sigma]' + [sigma]',
if [sigma]' is a series which does not make the finger _D'_ move.
What makes us distinguish the various series designated [sigma] from
those called [sigma]' is that the first do not alter the tactile
impressions felt by the finger _D_ and the second preserve those the
finger _D'_ feels.
Now see what we ascertain: in the beginning my finger _D'_ feels a
sensation _A'_; I make movements which produce muscular sensations _S_;
my finger _D_ feels the impression _A_; I make movements which produce a
series of sensations [sigma]; my finger _D_ continues to feel the
impression _A_, since this is the characteristic property of the series
[sigma]; I then make movements which produce the series _S'_ of muscular
sensations, _inverse_ to _S_ in the sense above given to this word. I
ascertain then that my finger _D'_ feels anew the impression _A'_. (It
is of course understood that _S_ has been suitably chosen.)
This means that the series _S_ + [sigma] + _S'_, preserving the tactile
impressions of the finger _D'_, is one of the series I have called
[sigma]'. Inversely, if one takes any series [sigma]', _S'_ + [sigma]' +
_S_ will be one of the series that we call [sigma]'.
Thus if _S_ is suitably chosen, _S_ + [sigma] + _S'_ will be a series
[sigma]', and by making [sigma] vary in all possible ways, we shall
obtain all the possible series [sigma]'.
Not yet knowing geometry, we limit ourselves to verifying all that, but
here is how those who know geometry would explain the fact. In the
beginning my finger _D'_ is at the point _M_, in contact with the object
_a_, which makes it feel the impression _A'_. I make the movements
corresponding to the series _S_; I have said that this series should be
suitably chosen, I should so make this choice that these movements carry
the finger _D_ to the point originally occupied by the finger _D'_, that
is, to the point _M_; this finger _D_ will thus be in contact with the
object _a_, which will make it feel the impression _A_.
I then make the movements corresponding to the series [sigma]; in these
movements, by hypothesis, the position of the finger _D_ does not
change, this finger therefore remains in contact with the object a and
continues to feel the impression _A_. Finally I make the movements
corresponding to the series _S'_. As _S'_ is inverse to _S_, these
movements carry the finger _D'_ to the point previously occupied by the
finger _D_, that is, to the point _M_. If, as may be supposed, the
object _a_ has not budged, this finger _D'_ will be in contact with this
object and will feel anew the impression _A'_.... _Q.E.D._
Let us see the consequences. I consider a series of muscular sensations
[Sigma]. To this series will correspond a point _M_ of the first tactile
space. Now take again the two series _S_ and _S'_, inverses of one
another, of which we have just spoken. To the series _S_ + [Sigma] +
_S'_ will correspond a point _N_ of the second tactile space, since to
any series of muscular sensations corresponds, as we have said, a
point, whether in the first space or in the second.
I am going to consider the two points _N_ and _M_, thus defined, as
corresponding. What authorizes me so to do? For this correspondence to
be admissible, it is necessary that if two points _M_ and _M'_,
corresponding in the first space to two series [Sigma] and [Sigma]', are
identical, so also are the two corresponding points of the second space
_N_ and _N'_, that is, the two points which correspond to the two series
_S_ + [Sigma] + _S'_ and _S_ + [Sigma]' + _S'_. Now we shall see that
this condition is fulfilled.
First a remark. As _S_ and _S'_ are inverses of one another, we shall
have _S_ + _S'_ = 0, and consequently _S_ + _S'_ + [Sigma] = [Sigma] +
_S_ + _S'_ = [Sigma], or again [Sigma] + _S_ + _S'_ + [Sigma]' = [Sigma]
+ [Sigma]'; but it does not follow that we have _S_ + [Sigma] + _S'_ =
[Sigma]; because, though we have used the addition sign to represent the
succession of our sensations, it is clear that the order of this
succession is not indifferent: we can not, therefore, as in ordinary
addition, invert the order of the terms; to use abridged language, our
operations are associative, but not commutative.
That fixed, in order that [Sigma] and [Sigma]' should correspond to the
same point _M_ = _M'_ of the first space, it is necessary and sufficient
for us to have [Sigma]' = [Sigma] + [sigma]. We shall then have: _S_ +
[Sigma]' + _S'_ = _S_ + [Sigma] + [sigma] + _S'_ = _S_ + [Sigma] + _S'_
+ _S_ + [sigma] + _S'_.
But we have just ascertained that _S_ + [sigma] + _S'_ was one of the
series [sigma]'. We shall therefore have: _S_ + [Sigma]' + _S'_ = _S_ +
[Sigma] + _S'_ + [sigma]', which means that the series _S_ + [Sigma]' +
_S'_ and _S_ + [Sigma] + _S'_ correspond to the same point _N_ = _N'_ of
the second space. Q.E.D.
Our two spaces therefore correspond point for point; they can be
'transformed' one into the other; they are isomorphic. How are we led to
conclude thence that they are identical?
Consider the two series [sigma] and _S_ + [sigma] + _S'_ = [sigma]'. I
have said that often, but not always, the series [sigma] preserves the
tactile impression _A_ felt by the finger _D_; and similarly it often
happens, but not always, that the series [sigma]' preserves the tactile
impression _A'_ felt by the finger _D'_. Now I ascertain that it happens
_very often_ (that is, much more often than what I have just called
'often') that when the series [sigma] has preserved the impression _A_
of the finger _D_, the series [sigma]' preserves at the same time the
impression _A'_ of the finger _D'_; and, inversely, that if the first
impression is altered, the second is likewise. That happens _very
often_, but not always.
We interpret this experimental fact by saying that the unknown object
_a_ which gives the impression _A_ to the finger _D_ is identical with
the unknown object _a'_ which gives the impression _A'_ to the finger
_D'_. And in fact when the first object moves, which the disappearance
of the impression _A_ tells us, the second likewise moves, since the
impression _A'_ disappears likewise. When the first object remains
motionless, the second remains motionless. If these two objects are
identical, as the first is at the point _M_ of the first space and the
second at the point _N_ of the second space, these two points are
identical. This is how we are led to regard these two spaces as
identical; or better, this is what we mean when we say that they are
identical.
What we have just said of the identity of the two tactile spaces makes
unnecessary our discussing the question of the identity of tactile space
and visual space, which could be treated in the same way.
5. _Space and Empiricism_
It seems that I am about to be led to conclusions in conformity with
empiristic ideas. I have, in fact, sought to put in evidence the rôle of
experience and to analyze the experimental facts which intervene in the
genesis of space of three dimensions. But whatever may be the importance
of these facts, there is one thing we must not forget and to which
besides I have more than once called attention. These experimental facts
are often verified but not always. That evidently does not mean that
space has often three dimensions, but not always.
I know well that it is easy to save oneself and that, if the facts do
not verify, it will be easily explained by saying that the exterior
objects have moved. If experience succeeds, we say that it teaches us
about space; if it does not succeed, we hie to exterior objects which we
accuse of having moved; in other words, if it does not succeed, it is
given a fillip.
These fillips are legitimate; I do not refuse to admit them; but they
suffice to tell us that the properties of space are not experimental
truths, properly so called. If we had wished to verify other laws, we
could have succeeded also, by giving other analogous fillips. Should we
not always have been able to justify these fillips by the same reasons?
One could at most have said to us: 'Your fillips are doubtless
legitimate, but you abuse them; why move the exterior objects so often?'
To sum up, experience does not prove to us that space has three
dimensions; it only proves to us that it is convenient to attribute
three to it, because thus the number of fillips is reduced to a minimum.
I will add that experience brings us into contact only with
representative space, which is a physical continuum, never with
geometric space, which is a mathematical continuum. At the very most it
would appear to tell us that it is convenient to give to geometric space
three dimensions, so that it may have as many as representative space.
The empiric question may be put under another form. Is it impossible to
conceive physical phenomena, the mechanical phenomena, for example,
otherwise than in space of three dimensions? We should thus have an
objective experimental proof, so to speak, independent of our
physiology, of our modes of representation.
But it is not so; I shall not here discuss the question completely, I
shall confine myself to recalling the striking example given us by the
mechanics of Hertz. You know that the great physicist did not believe in
the existence of forces, properly so called; he supposed that visible
material points are subjected to certain invisible bonds which join them
to other invisible points and that it is the effect of these invisible
bonds that we attribute to forces.
But that is only a part of his ideas. Suppose a system formed of n
material points, visible or not; that will give in all 3_n_ coordinates;
let us regard them as the coordinates of a _single_ point in space of
3_n_ dimensions. This single point would be constrained to remain upon a
surface (of any number of dimensions < 3_n_) in virtue of the bonds of
which we have just spoken; to go on this surface from one point to
another, it would always take the shortest way; this would be the
single principle which would sum up all mechanics.
Whatever should be thought of this hypothesis, whether we be allured by
its simplicity, or repelled by its artificial character, the simple fact
that Hertz was able to conceive it, and to regard it as more convenient
than our habitual hypotheses, suffices to prove that our ordinary ideas,
and, in particular, the three dimensions of space, are in no wise
imposed upon mechanics with an invincible force.
6. _Mind and Space_
Experience, therefore, has played only a single rôle, it has served as
occasion. But this rôle was none the less very important; and I have
thought it necessary to give it prominence. This rôle would have been
useless if there existed an _a priori_ form imposing itself upon our
sensitivity, and which was space of three dimensions.
Does this form exist, or, if you choose, can we represent to ourselves
space of more than three dimensions? And first what does this question
mean? In the true sense of the word, it is clear that we can not
represent to ourselves space of four, nor space of three, dimensions; we
can not first represent them to ourselves empty, and no more can we
represent to ourselves an object either in space of four, or in space of
three, dimensions: (1) Because these spaces are both infinite and we can
not represent to ourselves a figure _in_ space, that is, the part _in_
the whole, without representing the whole, and that is impossible,
because it is infinite; (2) because these spaces are both mathematical
continua, and we can represent to ourselves only the physical continuum;
(3) because these spaces are both homogeneous, and the frames in which
we enclose our sensations, being limited, can not be homogeneous.
Thus the question put can only be understood in one way; is it possible
to imagine that, the results of the experiences related above having
been different, we might have been led to attribute to space more than
three dimensions; to imagine, for instance, that the sensation of
accommodation might not be constantly in accord with the sensation of
convergence of the eyes; or indeed that the experiences of which we
have spoken in § 2, and of which we express the result by saying 'that
touch does not operate at a distance,' might have led us to an inverse
conclusion.
And then yes evidently that is possible; from the moment one imagines an
experience, one imagines just thereby the two contrary results it may
give. That is possible, but that is difficult, because we have to
overcome a multitude of associations of ideas, which are the fruit of a
long personal experience and of the still longer experience of the race.
Is it these associations (or at least those of them that we have
inherited from our ancestors), which constitute this _a priori_ form of
which it is said that we have pure intuition? Then I do not see why one
should declare it refractory to analysis and should deny me the right of
investigating its origin.
When it is said that our sensations are 'extended' only one thing can be
meant, that is that they are always associated with the idea of certain
muscular sensations, corresponding to the movements which enable us to
reach the object which causes them, which enable us, in other words, to
defend ourselves against it. And it is just because this association is
useful for the defense of the organism, that it is so old in the history
of the species and that it seems to us indestructible. Nevertheless, it
is only an association and we can conceive that it may be broken; so
that we may not say that sensation can not enter consciousness without
entering in space, but that in fact it does not enter consciousness
without entering in space, which means, without being entangled in this
association.
No more can I understand one's saying that the idea of time is logically
subsequent to space, since we can represent it to ourselves only under
the form of a straight line; as well say that time is logically
subsequent to the cultivation of the prairies, since it is usually
represented armed with a scythe. That one can not represent to himself
simultaneously the different parts of time, goes without saying, since
the essential character of these parts is precisely not to be
simultaneous. That does not mean that we have not the intuition of time.
So far as that goes, no more should we have that of space, because
neither can we represent it, in the proper sense of the word, for the
reasons I have mentioned. What we represent to ourselves under the name
of straight is a crude image which as ill resembles the geometric
straight as it does time itself.
Why has it been said that every attempt to give a fourth dimension to
space always carries this one back to one of the other three? It is easy
to understand. Consider our muscular sensations and the 'series' they
may form. In consequence of numerous experiences, the ideas of these
series are associated together in a very complex woof, our series are
_classed_. Allow me, for convenience of language, to express my thought
in a way altogether crude and even inexact by saying that our series of
muscular sensations are classed in three classes corresponding to the
three dimensions of space. Of course this classification is much more
complicated than that, but that will suffice to make my reasoning
understood. If I wish to imagine a fourth dimension, I shall suppose
another series of muscular sensations, making part of a fourth class.
But as _all_ my muscular sensations have already been classed in one of
the three pre-existent classes, I can only represent to myself a series
belonging to one of these three classes, so that my fourth dimension is
carried back to one of the other three.
What does that prove? This: that it would have been necessary first to
destroy the old classification and replace it by a new one in which the
series of muscular sensations should have been distributed into four
classes. The difficulty would have disappeared.
It is presented sometimes under a more striking form. Suppose I am
enclosed in a chamber between the six impassable boundaries formed by
the four walls, the floor and the ceiling; it will be impossible for me
to get out and to imagine my getting out. Pardon, can you not imagine
that the door opens, or that two of these walls separate? But of course,
you answer, one must suppose that these walls remain immovable. Yes, but
it is evident that I have the right to move; and then the walls that we
suppose absolutely at rest will be in motion with regard to me. Yes, but
such a relative motion can not be arbitrary; when objects are at rest,
their relative motion with regard to any axes is that of a rigid solid;
now, the apparent motions that you imagine are not in conformity with
the laws of motion of a rigid solid. Yes, but it is experience which has
taught us the laws of motion of a rigid solid; nothing would prevent our
_imagining_ them different. To sum up, for me to imagine that I get out
of my prison, I have only to imagine that the walls seem to open, when I
move.
I believe, therefore, that if by space is understood a mathematical
continuum of three dimensions, were it otherwise amorphous, it is the
mind which constructs it, but it does not construct it out of nothing;
it needs materials and models. These materials, like these models,
preexist within it. But there is not a single model which is imposed
upon it; it has _choice_; it may choose, for instance, between space of
four and space of three dimensions. What then is the rôle of experience?
It gives the indications following which the choice is made.
Another thing: whence does space get its quantitative character? It
comes from the rôle which the series of muscular sensations play in its
genesis. These are series which may _repeat themselves_, and it is from
their repetition that number comes; it is because they can repeat
themselves indefinitely that space is infinite. And finally we have
seen, at the end of section 3, that it is also because of this that
space is relative. So it is repetition which has given to space its
essential characteristics; now, repetition supposes time; this is enough
to tell that time is logically anterior to space.
7. _Rôle of the Semicircular Canals_
I have not hitherto spoken of the rôle of certain organs to which the
physiologists attribute with reason a capital importance, I mean the
semicircular canals. Numerous experiments have sufficiently shown that
these canals are necessary to our sense of orientation; but the
physiologists are not entirely in accord; two opposing theories have
been proposed, that of Mach-Delage and that of M. de Cyon.
M. de Cyon is a physiologist who has made his name illustrious by
important discoveries on the innervation of the heart; I can not,
however, agree with his ideas on the question before us. Not being a
physiologist, I hesitate to criticize the experiments he has directed
against the adverse theory of Mach-Delage; it seems to me, however, that
they are not convincing, because in many of them the _total_ pressure
was made to vary in one of the canals, while, physiologically, what
varies is the _difference_ between the pressures on the two extremities
of the canal; in others the organs were subjected to profound lesions,
which must alter their functions.
Besides, this is not important; the experiments, if they were
irreproachable, might be convincing against the old theory. They would
not be convincing _for_ the new theory. In fact, if I have rightly
understood the theory, my explaining it will be enough for one to
understand that it is impossible to conceive of an experiment confirming
it.
The three pairs of canals would have as sole function to tell us that
space has three dimensions. Japanese mice have only two pairs of canals;
they believe, it would seem, that space has only two dimensions, and
they manifest this opinion in the strangest way; they put themselves in
a circle, and, so ordered, they spin rapidly around. The lampreys,
having only one pair of canals, believe that space has only one
dimension, but their manifestations are less turbulent.
It is evident that such a theory is inadmissible. The sense-organs are
designed to tell us of _changes_ which happen in the exterior world. We
could not understand why the Creator should have given us organs
destined to cry without cease: Remember that space has three dimensions,
since the number of these three dimensions is not subject to change.
We must, therefore, come back to the theory of Mach-Delage. What the
nerves of the canals can tell us is the difference of pressure on the
two extremities of the same canal, and thereby: (1) the direction of the
vertical with regard to three axes rigidly bound to the head; (2) the
three components of the acceleration of translation of the center of
gravity of the head; (3) the centrifugal forces developed by the
rotation of the head; (4) the acceleration of the motion of rotation of
the head.
It follows from the experiments of M. Delage that it is this last
indication which is much the most important; doubtless because the
nerves are less sensible to the difference of pressure itself than to
the brusque variations of this difference. The first three indications
may thus be neglected.
Knowing the acceleration of the motion of rotation of the head at each
instant, we deduce from it, by an unconscious integration, the final
orientation of the head, referred to a certain initial orientation taken
as origin. The circular canals contribute, therefore, to inform us of
the movements that we have executed, and that on the same ground as the
muscular sensations. When, therefore, above we speak of the series _S_
or of the series [Sigma], we should say, not that these were series of
muscular sensations alone, but that they were series at the same time of
muscular sensations and of sensations due to the semicircular canals.
Apart from this addition, we should have nothing to change in what
precedes.
In the series _S_ and [Sigma], these sensations of the semicircular
canals evidently hold a very important place. Yet alone they would not
suffice, because they can tell us only of the movements of the head;
they tell us nothing of the relative movements of the body or of the
members in regard to the head. And more, it seems that they tell us only
of the rotations of the head and not of the translations it may
undergo.
PART II
THE PHYSICAL SCIENCES
CHAPTER V
ANALYSIS AND PHYSICS
I
You have doubtless often been asked of what good is mathematics and
whether these delicate constructions entirely mind-made are not
artificial and born of our caprice.
Among those who put this question I should make a distinction; practical
people ask of us only the means of money-making. These merit no reply;
rather would it be proper to ask of them what is the good of
accumulating so much wealth and whether, to get time to acquire it, we
are to neglect art and science, which alone give us souls capable of
enjoying it, 'and for life's sake to sacrifice all reasons for living.'
Besides, a science made solely in view of applications is impossible;
truths are fecund only if bound together. If we devote ourselves solely
to those truths whence we expect an immediate result, the intermediary
links are wanting and there will no longer be a chain.
The men most disdainful of theory get from it, without suspecting it,
their daily bread; deprived of this food, progress would quickly cease,
and we should soon congeal into the immobility of old China.
But enough of uncompromising practicians! Besides these, there are those
who are only interested in nature and who ask us if we can enable them
to know it better.
To answer these, we have only to show them the two monuments already
rough-hewn, Celestial Mechanics and Mathematical Physics.
They would doubtless concede that these structures are well worth the
trouble they have cost us. But this is not enough. Mathematics has a
triple aim. It must furnish an instrument for the study of nature. But
that is not all: it has a philosophic aim and, I dare maintain, an
esthetic aim. It must aid the philosopher to fathom the notions of
number, of space, of time. And above all, its adepts find therein
delights analogous to those given by painting and music. They admire the
delicate harmony of numbers and forms; they marvel when a new discovery
opens to them an unexpected perspective; and has not the joy they thus
feel the esthetic character, even though the senses take no part
therein? Only a privileged few are called to enjoy it fully, it is true,
but is not this the case for all the noblest arts?
This is why I do not hesitate to say that mathematics deserves to be
cultivated for its own sake, and the theories inapplicable to physics as
well as the others. Even if the physical aim and the esthetic aim were
not united, we ought not to sacrifice either.
But more: these two aims are inseparable and the best means of attaining
one is to aim at the other, or at least never to lose sight of it. This
is what I am about to try to demonstrate in setting forth the nature of
the relations between the pure science and its applications.
The mathematician should not be for the physicist a mere purveyor of
formulas; there should be between them a more intimate collaboration.
Mathematical physics and pure analysis are not merely adjacent powers,
maintaining good neighborly relations; they mutually interpenetrate and
their spirit is the same. This will be better understood when I have
shown what physics gets from mathematics and what mathematics, in
return, borrows from physics.
II
The physicist can not ask of the analyst to reveal to him a new truth;
the latter could at most only aid him to foresee it. It is a long time
since one still dreamt of forestalling experiment, or of constructing
the entire world on certain premature hypotheses. Since all those
constructions in which one yet took a naïve delight it is an age, to-day
only their ruins remain.
All laws are therefore deduced from experiment; but to enunciate them, a
special language is needful; ordinary language is too poor, it is
besides too vague, to express relations so delicate, so rich, and so
precise.
This therefore is one reason why the physicist can not do without
mathematics; it furnishes him the only language he can speak. And a
well-made language is no indifferent thing; not to go beyond physics,
the unknown man who invented the word _heat_ devoted many generations to
error. Heat has been treated as a substance, simply because it was
designated by a substantive, and it has been thought indestructible.
On the other hand, he who invented the word _electricity_ had the
unmerited good fortune to implicitly endow physics with a _new_ law,
that of the conservation of electricity, which, by a pure chance, has
been found exact, at least until now.
Well, to continue the simile, the writers who embellish a language, who
treat it as an object of art, make of it at the same time a more supple
instrument, more apt for rendering shades of thought.
We understand, then, how the analyst, who pursues a purely esthetic aim,
helps create, just by that, a language more fit to satisfy the
physicist.
But this is not all: law springs from experiment, but not immediately.
Experiment is individual, the law deduced from it is general; experiment
is only approximate, the law is precise, or at least pretends to be.
Experiment is made under conditions always complex, the enunciation of
the law eliminates these complications. This is what is called
'correcting the systematic errors.'
In a word, to get the law from experiment, it is necessary to
generalize; this is a necessity imposed upon the most circumspect
observer. But how generalize? Every particular truth may evidently be
extended in an infinity of ways. Among these thousand routes opening
before us, it is necessary to make a choice, at least provisional; in
this choice, what shall guide us?
It can only be analogy. But how vague is this word! Primitive man knew
only crude analogies, those which strike the senses, those of colors or
of sounds. He never would have dreamt of likening light to radiant
heat.
What has taught us to know the true, profound analogies, those the eyes
do not see but reason divines?
It is the mathematical spirit, which disdains matter to cling only to
pure form. This it is which has taught us to give the same name to
things differing only in material, to call by the same name, for
instance, the multiplication of quaternions and that of whole numbers.
If quaternions, of which I have just spoken, had not been so promptly
utilized by the English physicists, many persons would doubtless see in
them only a useless fancy, and yet, in teaching us to liken what
appearances separate, they would have already rendered us more apt to
penetrate the secrets of nature.
Such are the services the physicist should expect of analysis; but for
this science to be able to render them, it must be cultivated in the
broadest fashion without immediate expectation of utility--the
mathematician must have worked as artist.
What we ask of him is to help us to see, to discern our way in the
labyrinth which opens before us. Now, he sees best who stands highest.
Examples abound, and I limit myself to the most striking.
The first will show us how to change the language suffices to reveal
generalizations not before suspected.
When Newton's law has been substituted for Kepler's we still know only
elliptic motion. Now, in so far as concerns this motion, the two laws
differ only in form; we pass from one to the other by a simple
differentiation. And yet from Newton's law may be deduced by an
immediate generalization all the effects of perturbations and the whole
of celestial mechanics. If, on the other hand, Kepler's enunciation had
been retained, no one would ever have regarded the orbits of the
perturbed planets, those complicated curves of which no one has ever
written the equation, as the natural generalizations of the ellipse. The
progress of observations would only have served to create belief in
chaos.
The second example is equally deserving of consideration.
When Maxwell began his work, the laws of electro-dynamics admitted up to
his time accounted for all the known facts. It was not a new experiment
which came to invalidate them. But in looking at them under a new bias,
Maxwell saw that the equations became more symmetrical when a term was
added, and besides, this term was too small to produce effects
appreciable with the old methods.
You know that Maxwell's _a priori_ views awaited for twenty years an
experimental confirmation; or, if you prefer, Maxwell was twenty years
ahead of experiment. How was this triumph obtained?
It was because Maxwell was profoundly steeped in the sense of
mathematical symmetry; would he have been so, if others before him had
not studied this symmetry for its own beauty?
It was because Maxwell was accustomed to 'think in vectors,' and yet it
was through the theory of imaginaries (neomonics) that vectors were
introduced into analysis. And those who invented imaginaries hardly
suspected the advantage which would be obtained from them for the study
of the real world, of this the name given them is proof sufficient.
In a word, Maxwell was perhaps not an able analyst, but this ability
would have been for him only a useless and bothersome baggage. On the
other hand, he had in the highest degree the intimate sense of
mathematical analogies. Therefore it is that he made good mathematical
physics.
Maxwell's example teaches us still another thing.
How should the equations of mathematical physics be treated? Should we
simply deduce all the consequences and regard them as intangible
realities? Far from it; what they should teach us above all is what can
and what should be changed. It is thus that we get from them something
useful.
The third example goes to show us how we may perceive mathematical
analogies between phenomena which have physically no relation either
apparent or real, so that the laws of one of these phenomena aid us to
divine those of the other.
The very same equation, that of Laplace, is met in the theory of
Newtonian attraction, in that of the motion of liquids, in that of the
electric potential, in that of magnetism, in that of the propagation of
heat and in still many others. What is the result? These theories seem
images copied one from the other; they are mutually illuminating,
borrowing their language from each other; ask electricians if they do
not felicitate themselves on having invented the phrase flow of force,
suggested by hydrodynamics and the theory of heat.
Thus mathematical analogies not only may make us foresee physical
analogies, but besides do not cease to be useful when these latter fail.
To sum up, the aim of mathematical physics is not only to facilitate for
the physicist the numerical calculation of certain constants or the
integration of certain differential equations. It is besides, it is
above all, to reveal to him the hidden harmony of things in making him
see them in a new way.
Of all the parts of analysis, the most elevated, the purest, so to
speak, will be the most fruitful in the hands of those who know how to
use them.
III
Let us now see what analysis owes to physics.
It would be necessary to have completely forgotten the history of
science not to remember that the desire to understand nature has had on
the development of mathematics the most constant and happiest influence.
In the first place the physicist sets us problems whose solution he
expects of us. But in proposing them to us, he has largely paid us in
advance for the service we shall render him, if we solve them.
If I may be allowed to continue my comparison with the fine arts, the
pure mathematician who should forget the existence of the exterior world
would be like a painter who knew how to harmoniously combine colors and
forms, but who lacked models. His creative power would soon be
exhausted.
The combinations which numbers and symbols may form are an infinite
multitude. In this multitude how shall we choose those which are worthy
to fix our attention? Shall we let ourselves be guided solely by our
caprice? This caprice, which itself would besides soon tire, would
doubtless carry us very far apart and we should quickly cease to
understand each other.
But this is only the smaller side of the question. Physics will
doubtless prevent our straying, but it will also preserve us from a
danger much more formidable; it will prevent our ceaselessly going
around in the same circle.
History proves that physics has not only forced us to choose among
problems which came in a crowd; it has imposed upon us such as we should
without it never have dreamed of. However varied may be the imagination
of man, nature is still a thousand times richer. To follow her we must
take ways we have neglected, and these paths lead us often to summits
whence we discover new countries. What could be more useful!
It is with mathematical symbols as with physical realities; it is in
comparing the different aspects of things that we are able to comprehend
their inner harmony, which alone is beautiful and consequently worthy of
our efforts.
The first example I shall cite is so old we are tempted to forget it; it
is nevertheless the most important of all.
The sole natural object of mathematical thought is the whole number. It
is the external world which has imposed the continuum upon us, which we
doubtless have invented, but which it has forced us to invent. Without
it there would be no infinitesimal analysis; all mathematical science
would reduce itself to arithmetic or to the theory of substitutions.
On the contrary, we have devoted to the study of the continuum almost
all our time and all our strength. Who will regret it; who will think
that this time and this strength have been wasted? Analysis unfolds
before us infinite perspectives that arithmetic never suspects; it shows
us at a glance a majestic assemblage whose array is simple and
symmetric; on the contrary, in the theory of numbers, where reigns the
unforeseen, the view is, so to speak, arrested at every step.
Doubtless it will be said that outside of the whole number there is no
rigor, and consequently no mathematical truth; that the whole number
hides everywhere, and that we must strive to render transparent the
screens which cloak it, even if to do so we must resign ourselves to
interminable repetitions. Let us not be such purists and let us be
grateful to the continuum, which, if _all_ springs from the whole
number, was alone capable of making _so much_ proceed therefrom.
Need I also recall that M. Hermite obtained a surprising advantage from
the introduction of continuous variables into the theory of numbers?
Thus the whole number's own domain is itself invaded, and this invasion
has established order where disorder reigned.
See what we owe to the continuum and consequently to physical nature.
Fourier's series is a precious instrument of which analysis makes
continual use, it is by this means that it has been able to represent
discontinuous functions; Fourier invented it to solve a problem of
physics relative to the propagation of heat. If this problem had not
come up naturally, we should never have dared to give discontinuity its
rights; we should still long have regarded continuous functions as the
only true functions.
The notion of function has been thereby considerably extended and has
received from some logician-analysts an unforeseen development. These
analysts have thus adventured into regions where reigns the purest
abstraction and have gone as far away as possible from the real world.
Yet it is a problem of physics which has furnished them the occasion.
After Fourier's series, other analogous series have entered the domain
of analysis; they have entered by the same door; they have been imagined
in view of applications.
The theory of partial differential equations of the second order has an
analogous history. It has been developed chiefly by and for physics. But
it may take many forms, because such an equation does not suffice to
determine the unknown function, it is necessary to adjoin to it
complementary conditions which are called conditions at the limits;
whence many different problems.
If the analysts had abandoned themselves to their natural tendencies,
they would never have known but one, that which Madame Kovalevski has
treated in her celebrated memoir. But there are a multitude of others
which they would have ignored. Each of the theories of physics, that of
electricity, that of heat, presents us these equations under a new
aspect. It may, therefore, be said that without these theories we should
not know partial differential equations.
It is needless to multiply examples. I have given enough to be able to
conclude: when physicists ask of us the solution of a problem, it is not
a duty-service they impose upon us, it is on the contrary we who owe
them thanks.
IV
But this is not all; physics not only gives us the occasion to solve
problems; it aids us to find the means thereto, and that in two ways. It
makes us foresee the solution; it suggests arguments to us.
I have spoken above of Laplace's equation which is met in a multitude of
diverse physical theories. It is found again in geometry, in the theory
of conformal representation and in pure analysis, in that of
imaginaries.
In this way, in the study of functions of complex variables, the
analyst, alongside of the geometric image, which is his usual
instrument, finds many physical images which he may make use of with the
same success. Thanks to these images, he can see at a glance what pure
deduction would show him only successively. He masses thus the separate
elements of the solution, and by a sort of intuition divines before
being able to demonstrate.
To divine before demonstrating! Need I recall that thus have been made
all the important discoveries? How many are the truths that physical
analogies permit us to present and that we are not in condition to
establish by rigorous reasoning!
For example, mathematical physics introduces a great number of
developments in series. No one doubts that these developments converge;
but the mathematical certitude is lacking. These are so many conquests
assured for the investigators who shall come after us.
On the other hand, physics furnishes us not alone solutions; it
furnishes us besides, in a certain measure, arguments. It will suffice
to recall how Felix Klein, in a question relative to Riemann surfaces,
has had recourse to the properties of electric currents.
It is true, the arguments of this species are not rigorous, in the sense
the analyst attaches to this word. And here a question arises: How can a
demonstration not sufficiently rigorous for the analyst suffice for the
physicist? It seems there can not be two rigors, that rigor is or is
not, and that, where it is not there can not be deduction.
This apparent paradox will be better understood by recalling under what
conditions number is applied to natural phenomena. Whence come in
general the difficulties encountered in seeking rigor? We strike them
almost always in seeking to establish that some quantity tends to some
limit, or that some function is continuous, or that it has a derivative.
Now the numbers the physicist measures by experiment are never known
except approximately; and besides, any function always differs as little
as you choose from a discontinuous function, and at the same time it
differs as little as you choose from a continuous function. The
physicist may, therefore, at will suppose that the function studied is
continuous, or that it is discontinuous; that it has or has not a
derivative; and may do so without fear of ever being contradicted,
either by present experience or by any future experiment. We see that
with such liberty he makes sport of difficulties which stop the analyst.
He may always reason as if all the functions which occur in his
calculations were entire polynomials.
Thus the sketch which suffices for physics is not the deduction which
analysis requires. It does not follow thence that one can not aid in
finding the other. So many physical sketches have already been
transformed into rigorous demonstrations that to-day this transformation
is easy. There would be plenty of examples did I not fear in citing them
to tire the reader.
I hope I have said enough to show that pure analysis and mathematical
physics may serve one another without making any sacrifice one to the
other, and that each of these two sciences should rejoice in all which
elevates its associate.
CHAPTER VI
ASTRONOMY
Governments and parliaments must find that astronomy is one of the
sciences which cost most dear: the least instrument costs hundreds of
thousands of dollars, the least observatory costs millions; each eclipse
carries with it supplementary appropriations. And all that for stars
which are so far away, which are complete strangers to our electoral
contests, and in all probability will never take any part in them. It
must be that our politicians have retained a remnant of idealism, a
vague instinct for what is grand; truly, I think they have been
calumniated; they should be encouraged and shown that this instinct does
not deceive them, that they are not dupes of that idealism.
We might indeed speak to them of navigation, of which no one can
underestimate the importance, and which has need of astronomy. But this
would be to take the question by its smaller side.
Astronomy is useful because it raises us above ourselves; it is useful
because it is grand; that is what we should say. It shows us how small
is man's body, how great his mind, since his intelligence can embrace
the whole of this dazzling immensity, where his body is only an obscure
point, and enjoy its silent harmony. Thus we attain the consciousness of
our power, and this is something which can not cost too dear, since this
consciousness makes us mightier.
But what I should wish before all to show is, to what point astronomy
has facilitated the work of the other sciences, more directly useful,
since it has given us a soul capable of comprehending nature.
Think how diminished humanity would be if, under heavens constantly
overclouded, as Jupiter's must be, it had forever remained ignorant of
the stars. Do you think that in such a world we should be what we are? I
know well that under this somber vault we should have been deprived of
the light of the sun, necessary to organisms like those which inhabit
the earth. But if you please, we shall assume that these clouds are
phosphorescent and emit a soft and constant light. Since we are making
hypotheses, another will cost no more. Well! I repeat my question: Do
you think that in such a world we should be what we are?
The stars send us not only that visible and gross light which strikes
our bodily eyes, but from them also comes to us a light far more subtle,
which illuminates our minds and whose effects I shall try to show you.
You know what man was on the earth some thousands of years ago, and what
he is to-day. Isolated amidst a nature where everything was a mystery to
him, terrified at each unexpected manifestation of incomprehensible
forces, he was incapable of seeing in the conduct of the universe
anything but caprice; he attributed all phenomena to the action of a
multitude of little genii, fantastic and exacting, and to act on the
world he sought to conciliate them by means analogous to those employed
to gain the good graces of a minister or a deputy. Even his failures did
not enlighten him, any more than to-day a beggar refused is discouraged
to the point of ceasing to beg.
To-day we no longer beg of nature; we command her, because we have
discovered certain of her secrets and shall discover others each day. We
command her in the name of laws she can not challenge, because they are
hers; these laws we do not madly ask her to change, we are the first to
submit to them. Nature can only be governed by obeying her.
What a change must our souls have undergone to pass from the one state
to the other! Does any one believe that, without the lessons of the
stars, under the heavens perpetually overclouded that I have just
supposed, they would have changed so quickly? Would the metamorphosis
have been possible, or at least would it not have been much slower?
And first of all, astronomy it is which taught that there are laws. The
Chaldeans, who were the first to observe the heavens with some
attention, saw that this multitude of luminous points is not a confused
crowd wandering at random, but rather a disciplined army. Doubtless the
rules of this discipline escaped them, but the harmonious spectacle of
the starry night sufficed to give them the impression of regularity,
and that was in itself already a great thing. Besides, these rules were
discerned by Hipparchus, Ptolemy, Copernicus, Kepler, one after another,
and finally, it is needless to recall that Newton it was who enunciated
the oldest, the most precise, the most simple, the most general of all
natural laws.
And then, taught by this example, we have seen our little terrestrial
world better and, under the apparent disorder, there also we have found
again the harmony that the study of the heavens had revealed to us. It
also is regular, it also obeys immutable laws, but they are more
complicated, in apparent conflict one with another, and an eye untrained
by other sights would have seen there only chaos and the reign of chance
or caprice. If we had not known the stars, some bold spirits might
perhaps have sought to foresee physical phenomena; but their failures
would have been frequent, and they would have excited only the derision
of the vulgar; do we not see, that even in our day the meteorologists
sometimes deceive themselves, and that certain persons are inclined to
laugh at them.
How often would the physicists, disheartened by so many checks, have
fallen into discouragement, if they had not had, to sustain their
confidence, the brilliant example of the success of the astronomers!
This success showed them that nature obeys laws; it only remained to
know what laws; for that they only needed patience, and they had the
right to demand that the sceptics should give them credit.
This is not all: astronomy has not only taught us that there are laws,
but that from these laws there is no escape, that with them there is no
possible compromise. How much time should we have needed to comprehend
that fact, if we had known only the terrestrial world, where each
elemental force would always seem to us in conflict with other forces?
Astronomy has taught us that the laws are infinitely precise, and that
if those we enunciate are approximative, it is because we do not know
them well. Aristotle, the most scientific mind of antiquity, still
accorded a part to accident, to chance, and seemed to think that the
laws of nature, at least here below, determine only the large features
of phenomena. How much has the ever-increasing precision of
astronomical predictions contributed to correct such an error, which
would have rendered nature unintelligible!
But are these laws not local, varying in different places, like those
which men make; does not that which is truth in one corner of the
universe, on our globe, for instance, or in our little solar system,
become error a little farther away? And then could it not be asked
whether laws depending on space do not also depend upon time, whether
they are not simple habitudes, transitory, therefore, and ephemeral?
Again it is astronomy that answers this question. Consider the double
stars; all describe conics; thus, as far as the telescope carries, it
does not reach the limits of the domain which obeys Newton's law.
Even the simplicity of this law is a lesson for us; how many complicated
phenomena are contained in the two lines of its enunciation; persons who
do not understand celestial mechanics may form some idea of it at least
from the size of the treatises devoted to this science; and then it may
be hoped that the complication of physical phenomena likewise hides from
us some simple cause still unknown.
It is therefore astronomy which has shown us what are the general
characteristics of natural laws; but among these characteristics there
is one, the most subtle and the most important of all, which I shall ask
leave to stress.
How was the order of the universe understood by the ancients; for
instance, by Pythagoras, Plato or Aristotle? It was either an immutable
type fixed once for all, or an ideal to which the world sought to
approach. Kepler himself still thought thus when, for instance, he
sought whether the distances of the planets from the sun had not some
relation to the five regular polyhedrons. This idea contained nothing
absurd, but it was sterile, since nature is not so made. Newton has
shown us that a law is only a necessary relation between the present
state of the world and its immediately subsequent state. All the other
laws since discovered are nothing else; they are in sum, differential
equations; but it is astronomy which furnished the first model for them,
without which we should doubtless long have erred.
Astronomy has also taught us to set at naught appearances. The day
Copernicus proved that what was thought the most stable was in motion,
that what was thought moving was fixed, he showed us how deceptive could
be the infantile reasonings which spring directly from the immediate
data of our senses. True, his ideas did not easily triumph, but since
this triumph there is no longer a prejudice so inveterate that we can
not shake it off. How can we estimate the value of the new weapon thus
won?
The ancients thought everything was made for man, and this illusion must
be very tenacious, since it must ever be combated. Yet it is necessary
to divest oneself of it; or else one will be only an eternal myope,
incapable of seeing the truth. To comprehend nature one must be able to
get out of self, so to speak, and to contemplate her from many different
points of view; otherwise we never shall know more than one side. Now,
to get out of self is what he who refers everything to himself can not
do. Who delivered us from this illusion? It was those who showed us that
the earth is only one of the smallest planets of the solar system, and
that the solar system itself is only an imperceptible point in the
infinite spaces of the stellar universe.
At the same time astronomy taught us not to be afraid of big numbers.
This was needful, not only for knowing the heavens, but to know the
earth itself; and was not so easy as it seems to us to-day. Let us try
to go back and picture to ourselves what a Greek would have thought if
told that red light vibrates four hundred millions of millions of times
per second. Without any doubt, such an assertion would have appeared to
him pure madness, and he never would have lowered himself to test it.
To-day a hypothesis will no longer appear absurd to us because it
obliges us to imagine objects much larger or smaller than those our
senses are capable of showing us, and we no longer comprehend those
scruples which arrested our predecessors and prevented them from
discovering certain truths simply because they were afraid of them. But
why? It is because we have seen the heavens enlarging and enlarging
without cease; because we know that the sun is 150 millions of
kilometers from the earth and that the distances of the nearest stars
are hundreds of thousands of times greater yet. Habituated to the
contemplation of the infinitely great, we have become apt to comprehend
the infinitely small. Thanks to the education it has received, our
imagination, like the eagle's eye that the sun does not dazzle, can look
truth in the face.
Was I wrong in saying that it is astronomy which has made us a soul
capable of comprehending nature; that under heavens always overcast and
starless, the earth itself would have been for us eternally
unintelligible; that we should there have seen only caprice and
disorder; and that, not knowing the world, we should never have been
able to subdue it? What science could have been more useful? And in thus
speaking I put myself at the point of view of those who only value
practical applications. Certainly, this point of view is not mine; as
for me, on the contrary, if I admire the conquests of industry, it is
above all because if they free us from material cares, they will one day
give to all the leisure to contemplate nature. I do not say: Science is
useful, because it teaches us to construct machines. I say: Machines are
useful, because in working for us, they will some day leave us more time
to make science. But finally it is worth remarking that between the two
points of view there is no antagonism, and that man having pursued a
disinterested aim, all else has been added unto him.
Auguste Comte has said somewhere, that it would be idle to seek to know
the composition of the sun, since this knowledge would be of no use to
sociology. How could he be so short-sighted? Have we not just seen that
it is by astronomy that, to speak his language, humanity has passed from
the theological to the positive state? He found an explanation for that
because it had happened. But how has he not understood that what
remained to do was not less considerable and would be not less
profitable? Physical astronomy, which he seems to condemn, has already
begun to bear fruit, and it will give us much more, for it only dates
from yesterday.
First was discovered the nature of the sun, what the founder of
positivism wished to deny us, and there bodies were found which exist on
the earth, but had here remained undiscovered; for example, helium, that
gas almost as light as hydrogen. That already contradicted Comte. But to
the spectroscope we owe a lesson precious in a quite different way; in
the most distant stars, it shows us the same substances. It might have
been asked whether the terrestrial elements were not due to some chance
which had brought together more tenuous atoms to construct of them the
more complex edifice that the chemists call atom; whether, in other
regions of the universe, other fortuitous meetings had not engendered
edifices entirely different. Now we know that this is not so, that the
laws of our chemistry are the general laws of nature, and that they owe
nothing to the chance which caused us to be born on the earth.
But, it will be said, astronomy has given to the other sciences all it
can give them, and now that the heavens have procured for us the
instruments which enable us to study terrestrial nature, they could
without danger veil themselves forever. After what we have just said, is
there still need to answer this objection? One could have reasoned the
same in Ptolemy's time; then also men thought they knew everything, and
they still had almost everything to learn.
The stars are majestic laboratories, gigantic crucibles, such as no
chemist could dream. There reign temperatures impossible for us to
realize. Their only defect is being a little far away; but the telescope
will soon bring them near to us, and then we shall see how matter acts
there. What good fortune for the physicist and the chemist!