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Contents
In an article I published in the Revue générale des
sciences, t. ii., p. 774, on the subject of non-Euclidean
geometry, I wrote the following sentences:
Beings with a mind made like ours, with the same senses that we
have, but without any prior education, could receive from a
properly selected world impressions such that they would be led
to construct a geometry different from Euclid's and to localize
the phenomena of that exterior world in a non-Euclidean space or
even in a four-dimensional space.
For us, educated in this present world, if we were suddenly
transported into that new world , we would have no difficulty in
relating its phenomena to our Euclidean space.
A man who devoted his life to it could perhaps succeed in
picturing to himself a fourth dimension. (1)
I did not follow this up with any further clarification and it
must have astonished several readers; it seems to me therefore
that it is necessary to develop my thought and that I owe some
explanations to the public.
Geometric space and representative space.
We often say that the images of external objects are localized in
space, and even that only on that condition can they be formed.
We also say that this space, which thus serves as a frame prepared
in advance for our sensations and representations, is identical
with that of the geometricians and that it possesses all the same
properties.
To all the good people who [think that way], the sentence cited
above must indeed have seemed quite extraordinary. But we ought
to see whether they might not be under some illusion that a
thorough analysis might dispel.
First, what are, strictly speaking, the properties of space? I
mean the space that is the object of geometry and which I call geometric
space. Here are some of the more essential:
. . . . .
Solid bodies and geometry. Among the
objects that surround us, there are some that frequently undergo
displacements susceptible to correction by a correlative movement
of our own bodies; they are the solid bodies.
Other objects, whose form is variable, only exceptionally undergo
similar displacements (change of position without change of
form). When a body moves and in so doing changes its form,
we can no longer by appropriate movements bring our sense organs
back into the same relative situation with respect to that
body; we can no longer, consequently, re-establish the original
set of impressions.
It is only later on and after a series of new experiences that we
learn to separate bodies of variable form into smaller elements
such that each of them moves in approximate accordance with the
laws of motion of solid bodies. We thus distinguish
"deformations" from other changes of state; in these
deformations each element undergoes a simple change of position,
which can be corrected, but the modification of the ensemble is
more profound and can no longer be corrected by a correlative
movement.
Such an idea is already very complex and could only have appeared
relatively late. It could moreover not have arisen if the
observation of solid bodies bad not already taught us to
distinguish changes of position.
If, therefore, there were no solid bodies in nature, there
would be no geometry.
Another remark also merits a moment of attention. Let us imagine
a solid body first occupying position a and passing
afterward to position b; in its first position, it would
cause us to receive a set of impressions A, and in its
second position a set of impressions B. Now let there be a
second solid body having qualities entirely different from the
first, for example a different color. Now let us suppose that it
passes from position a, where we receive the set of
impressions A', to position b, where we receive the
set of impressions B'.
In general set A will have nothing in common with set A'
, nor set B with set B'. The passage from set A
to set B and that from set A" to set B'
are therefore two changes which in themselves have in general
nothing in common.
And yet we consider both these changes as displacements, and what
is more, we consider them as the same displacement. How can that
be?
It is simply because we can correct both by the same correlative
movement of our body.
It is therefore the correlative movement which constitutes the only
link between two phenomena that it would not otherwise have
occurred to us to compare.
On the other hand, our bodies, thanks to the number of their
joints and muscles, can go through a host of different movements;
but not all are capable of "correcting" a modification
of external objects; those only are capable of it in which our
whole body, or at the very least, all the sense organs involved,
move en bloc, that is to say, like a solid body
without varying their relative positions.
In summary:
We have thus defined, thanks to this reciprocity, a particular class of phenomena that we call displacements. It is the laws of these phenomena that are the object of geometry.
The law of homogeneity. The first of
these laws is that of homogeneity. Let us suppose that, by an
external change a, we were to pass from a set of
impressions A to set B, then that this change were
corrected by a voluntary correlative movement b such that
we were brought back to set A. Let us suppose now that
another external change a' caused us again to pass from
set A to set B.
Experience teaches us that this change a' is, like a,
susceptible to correction by a voluntary movement b',
and that this voluntary movement b' corresponds to the
same muscular sensations as the movement b which corrected
a. It is this fact that we normally have in mind when we say that
space is homogeneous and isotropic.
We can also say that a movement that occurs once can be repeated
a second time, a third time, and so on, without its properties
varying.
Those readers who know the article I wrote in this journal on the
nature of mathematical reasoning will perhaps remember the
importance that I attribute to the possibility of indefinitely
repeating a single operation.
It is from this repetition that mathematical reasoning draws its
strength; it is thanks to the law of homogeneity which it has
taken from geometric facts.
To be complete, we should annex to the law of homogeneity a host
of other analogous laws into the details of which I do not wish
to enter, but which mathematicians subsume into a single word by
saying that displacements form a "group".
The number of dimensions. I feel more
difficulty in explaining my thought concerning the origin of the
notion of point and the number of dimensions: it is markedly
different from opinions generally accepted and it is not easy to
state it in ordinary language.
We understand displacements in terms of the passage from a set of
impressions A to a different set b; but among these
displacements we distinguish some such that the initial set A
and the final set B conserve certain common qualities. I
do not wish to go into further detail nor to seek to determine in
exactly what these common qualities consist.
I am satisfied to note that we are led to distinguish certain
special displacements such that we may say they leave fixed one
of the points of space.
That is the origin of the idea of point.
The set of all displacements constitutes what we call a group;
the set of those displacements that leave fixed a point of space
constitutes a partial or subgroup.
It is in the relation of this group and subgroup that we must
seek the explanation of the fact that space has three dimensions.
The group total is of the order 6, that is to say that any such
displacement can be considered as a combination of six elementary
and independent movements.
The subgroup is of the order 3, that is to say that any
displacement belonging to this subgroup, or, in other words, any
displacement which leaves fixed a point of space, can be
considered a combination of three elementary and independent
movements.
The difference 6 - 3 represents the number of dimensions.
The Non-Euclidean world. If geometric
space were a frame imposed on each of our sensations,
considered individually, it would be impossible to represent an
image stripped of that frame, and we could change nothing in our
geometry.
But that is not the case; geometry is only the résumé of the
laws in accordance with which images succeed one another. Nothing
prevents our imagining a series of representations, in every way
similar to our ordinary representations, but succeeding one
another according to laws different from those to which we are
accustomed.
We can therefore conceive that beings whose education took place
in a milieu where these laws were inoperative could have a
geometry very different from ours.
Let us imagine, for example, a world enclosed in a great sphere
and subject to the following laws:
The temperature is not uniform; it is maximal in the center, and diminishes in proportion as we move away, reducing to absolute zero when we get to the sphere in which the world is enclosed.
I will further specify the precise law according to which the
temperature varies. Let R be the radius of the limiting
sphere; let r be the distance from a given point to the
center of this sphere. The absolute temperature will be
proportional to R²- r².
I will suppose, in addition, that in this world, all bodies have
the same coefficient of expansion such that the length of any
rule will be proportional to the absolute temperature.
Finally I will suppose that an object transported from one point
to another whose temperature is different, immediately assumes
calorific equilibrium in its new environment.
Nothing in these hypotheses is contradictory or unimaginable.
A mobile object will then become smaller and smaller as it
approaches the limiting sphere.
First let us observe that although this world is limited from the
point of view of our habitual geometry, it will appear infinite
to its inhabitants.
Indeed, when they wish to approach the limiting sphere, they get
colder and become smaller and smaller. The steps they take are
therefore also smaller and smaller, so that they can never reach
the limiting sphere.
If, for us, geometry is only the study of the laws by which
invariable solids move, for these imaginary beings, it would be a
study of the laws by which solids move deformed by those
differences of temperature I have just mentioned.
True, in our world natural solids also submit to variations of
form and volume due to beating and cooling. But we neglect these
variations in laying the foundation of geometry, for in addition
to their being very weak, they are irregular and consequently
seem to us to be accidental.
In this hypothetical world, this would not be the case, -and
these variations would follow regular and simple laws.
Moreover, the diverse solid particles making up the bodies of the
inhabitants would also undergo the same variations ,of form and
volume.
I will add one further hypothesis. I will assume that light
crosses diversely refracting media in such a way that the index
of refraction is inversely proportional to R² - r².
It is easy to see that in these conditions, light rays would be
not rectilinear but circular.
To justify the foregoing, it remains to be shown that certain
changes occurring in the position of exterior objects could be corrected
by correlative movements of the sensory beings who inhabit
this imaginary world; and in such a way as to restore the
original group of impressions felt by these sensory beings.
Let us suppose that an object moves, while changing shape, not as
an invariable solid, but as a solid undergoing unequal dilations
in exact accord with the law of temperature that I proposed
above. Permit me, in the interest of succinct language, to call
such a movement a non-Euclidean displacement. . .
If a sensory being were in the area, his impressions would be
modified by the displacement of the object, but he could
re-establish them by making suitable movements. It suffices that
finally the object and the sensory being, considered as forming a
single body, should have undergone one of those particular
displacements that I have just called non-Euclidean.
Although from the point of view of our habitual geometry bodies
are deformed in this displacement and their diverse parts are no
longer in the same relative position, nevertheless we shall see
that the impressions of that sensory being have again become the
same.
Indeed, although the mutual distances of the diverse parts could
and did vary, nevertheless the parts originally in contact have
returned into contact. Therefore the tactile impressions have not
changed.
Furthermore, taking into account the hypothesis set forth above
as to the refraction and curve of light rays, the visual
impressions will also have remained the same.
These imaginary beings will thus be led like us to classify the
phenomena they witness, and to distinguish among them those
"changes of position" that are susceptible to
correction by a voluntary correlative movement.
If they were to found a geometry, it would not be like ours, the
study of the movements of our invariable solids, it would be that
of the changes of position that they will thus have
distinguished, and which are none other than the
"non-Euclidean displacements"; it will be
non-Euclidean geometry.
Thus, beings like us, educated in such a world, would not have
the same geometry as we.
The four-dimensional World. In the same
way as a non-Euclidean world, we can represent a world having
four dimensions.
The sense of sight, even with a single eye, joined to muscular
sensations relative to movements of the eyeball could suffice to
our knowing a three-dimensional world.
The images of exterior objects come and paint themselves on the
retina, which is a two-dimensional tabula; these are
perspectives.
But, since these objects are mobile, and since the same is true
of our eye, we see successively different perspectives of the
same body, taken from several different points of view. We notice
at the same time that the passage from one perspective to another
is often accompanied by muscular sensations.
If the passage from perspective A to perspective B
and that from perspective A' to perspective B' are
accompanied by the same muscular sensations, we consider them to
be operations of the same kind.
Studying afterward the laws according to which these operations
combine, we recognize that they form a group having the same
structure as that of the movements of invariable solids.
Now, we have seen that it is from the properties of this group
that we have deduced the notions of geometric and of three
dimensions.
Thus we understand how the idea of a three-dimensional space was
able to arise from the spectacle of these perspectives, even
though each of them has only two dimensions, because they
succeed one another in accordance with certain laws.
In the same way that we can make on a plane the perspective of a
figure having three dimensions, we can make that of a
four-dimensional figure on a tabula with three (or two)
dimensions. It is only a game for the geometrician.
We can even take from a single figure several perspectives from
several different viewpoints.
We can easily represent these perspectives since they have only
three dimensions.
Let us imagine that the diverse perspectives of a single object
succeed one another; that the passage from one to the other is
accompanied by muscular sensations.
We will naturally consider two of these passages as two
operations of the same kind when they are associated with the
same muscular sensations.
Then nothing prevents our imagining that these operations might
combine following any law that we wish, for example in such a way
as to form a group having the same structure as that of an
invariable four-dimensional solid.
In all this there is nothing we cannot represent and nevertheless
these sensations are precisely those that a being furnished with
a two-dimensional retina would feel if be could move in
four-dimensional space.
It is in this sense that we may say that we can represent the
fourth dimension.
Conclusions. We say that experience
plays an indispensable role in the genesis of geometry; but it
would be an error to conclude that geometry is an experimental
science, even in part.
If it were experimental it would be only approximate and
provisional. And what a crude approximation!
Geometry would only be the study of the movements of solids; but
in reality it is not concerned with natural solids, it has as its
object certain ideal solids, absolutely invariable, which are
only a simplified and very distant image of the natural ones.
The notion of ideal bodies is drawn entirely from our minds and
experience is only an occasion that invites us to construct such
a notion.
The object of geometry is the study of a particular
"group"; but the general concept of group pre-exists in
our minds, at least potentially. It is imposed on us, not as a
form of sensibility, but as a form of understanding.
Still, among all possible groups, we must choose that one which
will be, so to speak, the standard to which we will refer
natural phenomena.
Experience guides us in this choice which it does not impose on
us; it makes us recognize not what is the truest geometry, but
what is the most convenient. It will be noticed that I have been
able to describe the fantastic worlds that I imagined above without
ceasing to use the language of ordinary geometry.
Indeed, we should not have to change anything if we were to be
transported to such a world.
Beings who were educated there would no doubt find it more
convenient to create a geometry different from ours, better
adapted to their impressions. As for us, in the face of the same
impressions, it is certain that we would find it more convenient
not to change our habits.
(1) This selection first appeared as an article, "L'Espace et la géométrie," in Revue de métaphysique et de morale, 1895 t.iii, pp. 631-46. It was especially translated for this volume by William Ryding of the Department of French, Columbia University.
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