Various Remarks.—We can now discuss several important questions:
1º Is the creative power of the mind exhausted by the creation of the mathematical continuum?
No: the works of Du Bois-Reymond demonstrate it in a striking way.
We know that mathematicians distinguish between infinitesimals of different orders and that those of the second order are infinitesimal, not only in an absolute way, but also in relation to those of the first order. It is not difficult to imagine infinitesimals of fractional or even of irrational order, and thus we find again that scale of the mathematical continuum which has been dealt with in the preceding pages.
Further, there are infinitesimals which are infinitely small in relation to those of the first order, and, on the contrary, infinitely great in relation to those of order 1 + ε, and that however small ε may be. Here, then, are new terms intercalated in our series, and if I may be permitted to revert to the phraseology lately employed which is very convenient though not consecrated by usage, I shall say that thus has been created a sort of continuum of the third order.
It would be easy to go further, but that would be idle; one would only be imagining symbols without possible application, and no one will think of doing that. The continuum of the third order, to which the consideration of the different orders of infinitesimals leads, is itself not useful enough to have won citizenship, and geometers regard it only as a mere curiosity. The mind uses its creative faculty only when experience requires it.
2º Once in possession of the concept of the mathematical continuum, is one safe from contradictions analogous to those which gave birth to it?
No, and I will give an example.
One must be very wise not to regard it as evident that every curve has a tangent; and in fact if we picture this curve and a straight as two narrow bands we can always so dispose them that they have a part in common without crossing. If we imagine then the breadth of these two bands to diminish indefinitely, this common part will always subsist and, at the limit, so to speak, the two lines will have a point in common without crossing, that is to say, they will be tangent.
The geometer who reasons in this way, consciously or not, is only doing what we have done above to prove two lines which cut have a point in common, and his intuition might seem just as legitimate.
It would deceive him however. We can demonstrate that there are curves which have no tangent, if such a curve is defined as an analytic continuum of the second order.
Without doubt some artifice analogous to those we have discussed above would have sufficed to remove the contradiction; but, as this is met with only in very exceptional cases, it has received no further attention.
Instead of seeking to reconcile intuition with analysis, we have been content to sacrifice one of the two, and as analysis must remain impeccable, we have decided against intuition.
The Physical Continuum of Several Dimensions.—We have discussed above the physical continuum as derived from the immediate data of our senses, or, if you wish, from the rough results of Fechner's experiments; I have shown that these results are summed up in the contradictory formulas
A=B,B=C,A Let us now see how this notion has been generalized and how
from it has come the concept of many-dimensional continua. Consider any two aggregates of sensations. Either we can
discriminate them one from another, or we can not, just as in
Fechner's experiments a weight of 10 grams can be distinguished
from a weight of 12 grams, but not from a weight of 11 grams.
This is all that is required to construct the continuum of several
dimensions. Let us call one of these aggregates of sensations anelement.
That will be something analogous to thepointof the mathematicians;
it will not be altogether the same thing however.
We can not say our element is without extension, since we can
not distinguish it from neighboring elements and it is thus
surrounded by a sort of haze. If the astronomical comparison
may be allowed, our 'elements' would be like nebulae, whereas
the mathematical points would be like stars. That being granted, a system of elements will form acontinuumif we can pass from any one of them to any other, by a
series of consecutive elements such that each is indistinguishable
from the preceding. Thislinearseries is to thelineof the
mathematician what an isolatedelementwas to the point. Before going farther, I must explain what is meant by acut. Consider a continuumCand remove from it certain of its
elements which for an instant we shall regard as no longer belonging
to this continuum. The aggregate of the elements so
removed will be called a cut. It may happen that, thanks to this
cut,Cmay besubdividedinto several distinct continua, the aggregate
of the remaining elements ceasing to form a unique continuum. There will then be onCtwo elements,AandB, that must be
regarded as belonging to two distinct continua, and this will be
recognized because it will be impossible to find a linear series
of consecutive elements ofC, each of these elements indistinguishable
from the preceding, the first beingAand the lastB,without one of the elements of this series being indistinguishable
from one of the elements of the cut. On the contrary, it may happen that the cut made is insufficient
to subdivide the continuumC. To classify the physical
continua, we will examine precisely what are the cuts which must
be made to subdivide them. If a physical continuumCcan be subdivided by a cut reducing
to a finite number of elements all distinguishable from one
another (and consequently forming neither a continuum, nor
several continua), we shall sayCis aone-dimensionalcontinuum. If, on the contrary,Ccan be subdivided only by cuts which
are themselves continua, we shall sayChas several dimensions.
If cuts which are continua of one dimension suffice, we
shall sayChas two dimensions; if cuts of two dimensions suffice,
we shall sayChas three dimensions, and so on. Thus is defined the notion of the physical continuum of several
dimensions, thanks to this very simple fact that two aggregates
of sensations are distinguishable or indistinguishable. The Mathematical Continuum of Several Dimensions.—Thence
the notion of the mathematical continuum ofndimensions
has sprung quite naturally by a process very like that we
discussed at the beginning of this chapter. A point of such a
continuum, you know, appears to us as defined by a system of
n distinct magnitudes called its coordinates. These magnitudes need not always be measurable; there is,
for instance, a branch of geometry independent of the measurement
of these magnitudes, in which it is only a question of knowing,
for example, whether on a curveABC, the pointBis between
the pointsAandC, and not of knowing whether the arcABis equal to the arcBCor twice as great. This is what is
calledAnalysis Situs. This is a whole body of doctrine which has attracted theattention of the greatest geometers and where we see flow one
from another a series of remarkable theorems. What distinguishes
these theorems from those of ordinary geometry is that
they are purely qualitative and that they would remain true if
the figures were copied by a draughtsman so awkward as to
grossly distort the proportions and replace straights by strokes
more or less curved. Through the wish to introduce measure next into the continuum
just defined this continuum becomes space, and geometry is
born. But the discussion of this is reserved for Part Second. Every conclusion supposes premises; these premises themselves
either are self-evident and need no demonstration, or can be
established only by relying upon other propositions, and since
we can not go back thus to infinity, every deductive science, and
in particular geometry, must rest on a certain number of undemonstrable
axioms. All treatises on geometry begin, therefore,
by the enunciation of these axioms. But among these there is a
distinction to be made: Some, for example, 'Things which are
equal to the same thing are equal to one another,' are not propositions
of geometry, but propositions of analysis. I regard them
as analytic judgmentsa priori, and shall not concern myself with
them. But I must lay stress upon other axioms which are peculiar to
geometry. Most treatises enunciate three of these explicitly: 1º Through two points can pass only one straight; 2º The straight line is the shortest path from one point to
another; 3º Through a given point there is not more than one parallel
to a given straight. Although generally a proof of the second of these axioms is
omitted, it would be possible to deduce it from the other two and
from those, much more numerous, which are implicitly admitted
without enunciating them, as I shall explain further on. It was long sought in vain to demonstrate likewise the third
axiom, known asEuclid's Postulate. What vast effort has been
wasted in this chimeric hope is truly unimaginable. Finally, inthe first quarter of the nineteenth century, and almost at the
same time, a Hungarian and a Russian, Bolyai and Lobachevski,
established irrefutably that this demonstration is impossible; they
have almost rid us of inventors of geometries 'sans postulatum';
since then the Académie des Sciences receives only about one or
two new demonstrations a year. The question was not exhausted; it soon made a great
stride by the publication of Riemann's celebrated memoir entitled:Ueber die Hypothesen welche der Geometrie zu Grunde
liegen. This paper has inspired most of the recent works of which
I shall speak further on, and among which it is proper to cite
those of Beltrami and of Helmholtz. The Bolyai-Lobachevski Geometry.—If it were possible to
deduce Euclid's postulate from the other axioms, it is evident
that in denying the postulate and admitting the other axioms, we
should be led to contradictory consequences; it would therefore
be impossible to base on such premises a coherent geometry. Now this is precisely what Lobachevski did. He assumes at the start that:Through a given point can be
drawn two parallels to a given straight. And he retains besides all Euclid's other axioms. From these
hypotheses he deduces a series of theorems among which it is
impossible to find any contradiction, and he constructs a
geometry whose faultless logic is inferior in nothing to that of
the Euclidean geometry. The theorems are, of course, very different from those to which
we are accustomed, and they can not fail to be at first a little
disconcerting. Thus the sum of the angles of a triangle is always less than
two right angles, and the difference between this sum and two
right angles is proportional to the surface of the triangle. It is impossible to construct a figure similar to a given figure
but of different dimensions. If we divide a circumference intonequal parts, and draw
tangents at the points of division, thesentangents will form a
polygon if the radius of the circle is small enough; but if this
radius is sufficiently great they will not meet. It is useless to multiply these examples; Lobachevski'spropositions have no relation to those of Euclid, but they are not less
logically bound one to another. Riemann's Geometry.—Imagine a world uniquely peopled
by beings of no thickness (height); and suppose these 'infinitely
flat' animals are all in the same plane and can not get out. Admit
besides that this world is sufficiently far from others to be
free from their influence. While we are making hypotheses, it
costs us no more to endow these beings with reason and believe
them capable of creating a geometry. In that case, they will certainly
attribute to space only two dimensions. But suppose now that these imaginary animals, while remaining
without thickness, have the form of a spherical, and not of a
plane, figure, and are all on the same sphere without power to get
off. What geometry will they construct? First it is clear they
will attribute to space only two dimensions; what will play for
them the rôle of the straight line will be the shortest path from
one point to another on the sphere, that is to say, an arc of a great
circle; in a word, their geometry will be the spherical geometry. What they will call space will be this sphere on which they
must stay, and on which happen all the phenomena they can
know. Their space will therefore beunboundedsince on a
sphere one can always go forward without ever being stopped,
and yet it will befinite; one can never find the end of it, but one
can make a tour of it. Well, Riemann's geometry is spherical geometry extended to
three dimensions. To construct it, the German mathematician
had to throw overboard, not only Euclid's postulate, but also the
first axiom:Only one straight can pass through two points. On a sphere, through two given points we can drawin generalonly one great circle (which, as we have just seen, would play the
rôle of the straight for our imaginary beings); but there is an
exception: if the two given points are diametrically opposite, an
infinity of great circles can be drawn through them. In the same way, in Riemann's geometry (at least in one of
its forms), through two points will pass in general only a single
straight; but there are exceptional cases where through two
points an infinity of straights can pass. There is a sort of opposition between Riemann's geometry and
that of Lobachevski. Thus the sum of the angles of a triangle is: Equal to two right angles in Euclid's geometry; Less than two right angles in that of Lobachevski; Greater than two right angles in that of Riemann. The number of straights through a given point that can be
drawn coplanar to a given straight, but nowhere meeting it, is
equal: To one in Euclid's geometry; To zero in that of Riemann; To infinity in that of Lobachevski. Add that Riemann's space is finite, although unbounded, in
the sense given above to these two words. The Surfaces of Constant Curvature.—One objection still
remained possible. The theorems of Lobachevski and of Riemann
present no contradiction; but however numerous the consequences
these two geometers have drawn from their hypotheses,
they must have stopped before exhausting them, since their number
would be infinite; who can say then that if they had pushed
their deductions farther they would not have eventually reached
some contradiction? This difficulty does not exist for Riemann's geometry, provided
it is limited to two dimensions; in fact, as we have seen,
two-dimensional Riemannian geometry does not differ from spherical
geometry, which is only a branch of ordinary geometry, and
consequently is beyond all discussion. Beltrami, in correlating likewise Lobachevski's two-dimensional
geometry with a branch of ordinary geometry, has equally
refuted the objection so far as it is concerned. Here is how he accomplished it. Consider any figure on a
surface. Imagine this figure traced on a flexible and inextensible
canvas applied over this surface in such a way that when the
canvas is displaced and deformed, the various lines of this figure
can change their form without changing their length. In general,
this flexible and inextensible figure can not be displaced
without leaving the surface; but there are certain particular surfacesfor which such a movement would be possible; these are the
surfaces of constant curvature. If we resume the comparison made above and imagine beings
without thickness living on one of these surfaces, they will regard
as possible the motion of a figure all of whose lines remain constant
in length. On the contrary, such a movement would appear
absurd to animals without thickness living on a surface of variable
curvature. These surfaces of constant curvature are of two sorts: Some
are ofpositive curvature, and can be deformed so as to be applied
over a sphere. The geometry of these surfaces reduces itself
therefore to the spherical geometry, which is that of Riemann. The others are ofnegative curvature. Beltrami has shown
that the geometry of these surfaces is none other than that of
Lobachevski. The two-dimensional geometries of Riemann and
Lobachevski are thus correlated to the Euclidean geometry. Interpretation of Non-Euclidean Geometries.—So vanishes
the objection so far as two-dimensional geometries are concerned. It would be easy to extend Beltrami's reasoning to three-dimensional
geometries. The minds that space of four dimensions
does not repel will see no difficulty in it, but they are few.
I prefer therefore to proceed otherwise. Consider a certain plane, which I shall call the fundamental
plane, and construct a sort of dictionary, by making correspond
each to each a double series of terms written in two columns, just
as correspond in the ordinary dictionaries the words of two languages
whose significance is the same: Space: Portion of space situated above the fundamental plane. Plane: Sphere cutting the fundamental plane orthogonally. Straight: Circle cutting the fundamental plane orthogonally. Sphere: Sphere. Circle: Circle. Angle: Angle. Distance between two points: Logarithm of the cross ratio of
these two points and the intersections of the fundamental plane
with a circle passing through these two points and cutting it
orthogonally. Etc., Etc. Now take Lobachevski's theorems and translate them with
the aid of this dictionary as we translate a German text with the
aid of a German-English dictionary.We shall thus obtain theorems
of the ordinary geometry.For example, that theorem of
Lobachevski: 'the sum of the angles of a triangle is less than two
right angles' is translated thus: "If a curvilinear triangle has
for sides circle-arcs which prolonged would cut orthogonally the
fundamental plane, the sum of the angles of this curvilinear triangle
will be less than two right angles." Thus, however far the
consequences of Lobachevski's hypotheses are pushed, they will
never lead to a contradiction. In fact, if two of Lobachevski's
theorems were contradictory, it would be the same with the translations
of these two theorems, made by the aid of our dictionary,
but these translations are theorems of ordinary geometry and no
one doubts that the ordinary geometry is free from contradiction.
Whence comes this certainty and is it justified? That is a question
I can not treat here because it would require to be enlarged
upon, but which is very interesting and I think not insoluble. Nothing remains then of the objection above formulated.
This is not all. Lobachevski's geometry, susceptible of a concrete
interpretation, ceases to be a vain logical exercise and is capable
of applications; I have not the time to speak here of these applications,
nor of the aid that Klein and I have gotten from them
for the integration of linear differential equations. This interpretation moreover is not unique, and several dictionaries
analogous to the preceding could be constructed, which
would enable us by a simple 'translation' to transform Lobachevski's
theorems into theorems of ordinary geometry. The Implicit Axioms.—Are the axioms explicitly enunciated
in our treatises the sole foundations of geometry? We may be
assured of the contrary by noticing that after they are successively
abandoned there are still left over some propositions common
to the theories of Euclid, Lobachevski and Riemann. These
propositions must rest on premises the geometers admit without
enunciation. It is interesting to try to disentangle them from
the classic demonstrations. Stuart Mill has claimed that every definition contains anaxiom, because in defining one affirms implicitly the existence
of the object defined. This is going much too far; it is rare that
in mathematics a definition is given without its being followed by
the demonstration of the existence of the object defined, and
when this is dispensed with it is generally because the reader
can easily supply it. It must not be forgotten that the word
existence has not the same sense when it refers to a mathematical
entity and when it is a question of a material object. A mathematical
entity exists, provided its definition implies no contradiction,
either in itself, or with the propositions already admitted. But if Stuart Mill's observation can not be applied to all
definitions, it is none the less just for some of them. The plane
is sometimes defined as follows: The plane is a surface such that the straight which joins any
two of its points is wholly on this surface. This definition manifestly hides a new axiom; it is true we
might change it, and that would be preferable, but then we
should have to enunciate the axiom explicitly. Other definitions would suggest reflections not less important. Such, for example, is that of the equality of two figures; two
figures are equal when they can be superposed; to superpose
them one must be displaced until it coincides with the other; but
how shall it be displaced? If we should ask this, no doubt we
should be told that it must be done without altering the shape
and as a rigid solid. The vicious circle would then be evident. In fact this definition defines nothing; it would have no meaning
for a being living in a world where there were only fluids.
If it seems clear to us, that is because we are used to the properties
of natural solids which do not differ much from those of the
ideal solids, all of whose dimensions are invariable. Yet, imperfect as it may be, this definition implies an axiom. The possibility of the motion of a rigid figure is not a self-evident
truth, or at least it is so only in the fashion of Euclid's
postulate and not as an analytic judgmenta prioriwould be. Moreover, in studying the definitions and the demonstrations
of geometry, we see that one is obliged to admit without proof
not only the possibility of this motion, but some of its properties
besides. This is at once seen from the definition of the straight line.
Many defective definitions have been given, but the true one is
that which is implied in all the demonstrations where the straight
line enters: "It may happen that the motion of a rigid figure is such that
all the points of a line belonging to this figure remain motionless
while all the points situated outside of this line move. Such a
line will be called a straight line." We have designedly, in this
enunciation, separated the definition from the axiom it implies. Many demonstrations, such as those of the cases of the equality
of triangles, of the possibility of dropping a perpendicular from
a point to a straight, presume propositions which are not enunciated,
for they require the admission that it is possible to transport
a figure in a certain way in space. The Fourth Geometry.—Among these implicit axioms, there
is one which seems to me to merit some attention, because when
it is abandoned a fourth geometry can be constructed as coherent
as those of Euclid, Lobachevski and Riemann. To prove that a perpendicular may always be erected at a
pointAto a straightAB, we consider a straightACmovable
around the pointAand initially coincident with the fixed
straightAB; and we make it turn about the pointAuntil it
comes into the prolongation ofAB. Thus two propositions are presupposed: First, that such a rotation
is possible, and next that it may be continued until the
two straights come into the prolongation one of the other. If the first point is admitted and the second rejected, we are
led to a series of theorems even stranger than those of Lobachevski
and Riemann, but equally exempt from contradiction. I shall cite only one of these theorems and that not the most
singular:A real straight may be perpendicular to itself. Lie's Theorem.—The number of axioms implicitly introduced
in the classic demonstrations is greater than necessary, and
it would be interesting to reduce it to a minimum. It may first
be asked whether this reduction is possible, whether the number
of necessary axioms and that of imaginable geometries are not
infinite. A theorem of Sophus Lie dominates this whole discussion. It
may be thus enunciated: Suppose the following premises are admitted: 1º Space hasndimensions; 2º The motion of a rigid figure is possible; 3º It requirespconditions to determine the position of this
figure in space. The number of geometries compatible with these premises will
be limited. I may even add that ifnis given, a superior limit can be
assigned top. If therefore the possibility of motion is admitted, there can
be invented only a finite (and even a rather small) number of
three-dimensional geometries. Riemann's Geometries.—Yet this result seems contradicted
by Riemann, for this savant constructs an infinity of different
geometries, and that to which his name is ordinarily given is only
a particular case. All depends, he says, on how the length of a curve is defined.
Now, there is an infinity of ways of defining this length, and each
of them may be the starting point of a new geometry. That is perfectly true, but most of these definitions are incompatible
with the motion of a rigid figure, which in the theorem
of Lie is supposed possible. These geometries of Riemann, in
many ways so interesting, could never therefore be other than
purely analytic and would not lend themselves to demonstrations
analogous to those of Euclid. On the Nature of Axioms.—Most mathematicians regard
Lobachevski's geometry only as a mere logical curiosity; some of
them, however, have gone farther. Since several geometries are
possible, is it certain ours is the true one? Experience no doubt
teaches us that the sum of the angles of a triangle is equal to two
right angles; but this is because the triangles we deal with are
too little; the difference, according to Lobachevski, is proportional
to the surface of the triangle; will it not perhaps become
sensible when we shall operate on larger triangles or when our
measurements shall become more precise? The Euclidean geometry
would thus be only a provisional geometry. To discuss this opinion, we should first ask ourselves what
is the nature of the geometric axioms. Are they synthetica priorijudgments, as Kant said? They would then impose themselves upon us with such force
that we could not conceive the contrary proposition, nor build
upon it a theoretic edifice. There would be no non-Euclidean
geometry. To be convinced of it take a veritable synthetica priorijudgment, the following, for instance, of which we have seen
the preponderant rôle in the first chapter: If a theorem is true for the number 1, and if it has been proved
that it is true of n + 1 provided it is true of n, it will be true of
all the positive whole numbers. Then try to escape from that and, denying this proposition,
try to found a false arithmetic analogous to non-Euclidean
geometry—it can not be done; one would even be tempted at first
blush to regard these judgments as analytic. Moreover, resuming our fiction of animals without thickness,
we can hardly admit that these beings, if their minds are like
ours, would adopt the Euclidean geometry which would be contradicted
by all their experience. Should we therefore conclude that the axioms of geometry are
experimental verities? But we do not experiment on ideal
straights or circles; it can only be done on material objects. On
what then could be based experiments which should serve as
foundation for geometry? The answer is easy. We have seen above that we constantly reason as if the geometric
figures behaved like solids. What geometry would borrow
from experience would therefore be the properties of these
bodies. The properties of light and its rectilinear propagation
have also given rise to some of the propositions of geometry,
and in particular those of projective geometry, so that from this
point of view one would be tempted to say that metric geometry
is the study of solids, and projective, that of light. But a difficulty remains, and it is insurmountable. If geometry
were an experimental science, it would not be an exact
science, it would be subject to a continual revision. Nay, it
would from this very day be convicted of error, since we know
that there is no rigorously rigid solid. Theaxioms of geometry therefore are neither synthetica priorijudgments nor experimental facts. They areconventions; our choice among all possible conventions
isguidedby experimental facts; but it remainsfreeand is
limited only by the necessity of avoiding all contradiction. Thus
it is that the postulates can remainrigorouslytrue even though
the experimental laws which have determined their adoption are
only approximative. In other words,the axioms of geometry(I do not speak of
those of arithmetic)are merely disguised definitions. Then what are we to think of that question: Is the Euclidean
geometry true? It has no meaning. As well ask whether the metric system is true and the old
measures false; whether Cartesian coordinates are true and polar
coordinates false. One geometry can not be more true than another;
it can only bemore convenient. Now, Euclidean geometry is, and will remain, the most convenient: 1º Because it is the simplest; and it is so not only in consequence
of our mental habits, or of I know not what direct intuition
that we may have of Euclidean space; it is the simplest in
itself, just as a polynomial of the first degree is simpler than one
of the second; the formulas of spherical trigonometry are more
complicated than those of plane trigonometry, and they would
still appear so to an analyst ignorant of their geometric signification. 2º Because it accords sufficiently well with the properties of
natural solids, those bodies which our hands and our eyes compare
and with which we make our instruments of measure. Let us begin by a little paradox. Beings with minds like ours, and having the same senses as
we, but without previous education, would receive from a suitably
chosen external world impressions such that they would be led
to construct a geometry other than that of Euclid and to localize
the phenomena of that external world in a non-Euclidean space,
or even in a space of four dimensions. As for us, whose education has been accomplished by our
actual world, if we were suddenly transported into this new
world, we should have no difficulty in referring its phenomena to
our Euclidean space. Conversely, if these beings were transported
into our environment, they would be led to relate our
phenomena to non-Euclidean space. Nay more; with a little effort we likewise could do it. A
person who should devote his existence to it might perhaps attain
to a realization of the fourth dimension. Geometric Space and Perceptual Space.—It is often said
the images of external objects are localized in space, even that
they can not be formed except on this condition. It is also said
that this space, which serves thus as a ready preparedframefor
our sensations and our representations, is identical with that of
the geometers, of which it possesses all the properties. To all the good minds who think thus, the preceding statement
must have appeared quite extraordinary. But let us see
whether they are not subject to an illusion that a more profound
analysis would dissipate. What, first of all, are the properties of space, properly so
called? I mean of that space which is the object of geometry
and which I shall callgeometric space. The following are some of the most essential: 1º It is continuous; 2º It is infinite; 3º It has three dimensions; 4º It is homogeneous, that is to say, all its points are identical
one with another; 5º It is isotropic, that is to say, all the straights which pass
through the same point are identical one with another. Compare it now to the frame of our representations and our
sensations, which I may callperceptual space. Visual Space.—Consider first a purely visual impression, due
to an image formed on the bottom of the retina. A cursory analysis shows us this image as continuous, but as
possessing only two dimensions; this already distinguishes from
geometric space what we may callpure visual space. Besides, this image is enclosed in a limited frame. Finally, there is another difference not less important:this
pure visual space is not homogeneous. All the points of the
retina, aside from the images which may there be formed, do not
play the same rôle. The yellow spot can in no way be regarded
as identical with a point on the border of the retina. In fact, not
only does the same object produce there much more vivid impressions,
but in everylimitedframe the point occupying the
center of the frame will never appear as equivalent to a point
near one of the borders. No doubt a more profound analysis would show us that this
continuity of visual space and its two dimensions are only an
illusion; it would separate it therefore still more from geometric
space, but we shall not dwell on this remark. Sight, however, enables us to judge of distances and consequently
to perceive a third dimension. But every one knows
that this perception of the third dimension reduces itself to the
sensation of the effort at accommodation it is necessary to make,
and to that of the convergence which must be given to the two
eyes, to perceive an object distinctly. These are muscular sensations altogether different from the
visual sensations which have given us the notion of the first two
dimensions. The third dimension therefore will not appear to
us as playing the same rôle as the other two. What may be
calledcomplete visual spaceis therefore not an isotropic space. It has, it is true, precisely three dimensions, which means that
the elements of our visual sensations (those at least which combine
to form the notion of extension) will be completely defined
when three of them are known; to use the language of
mathematics, they will be functions of three independent
variables. But examine the matter a little more closely. The third
dimension is revealed to us in two different ways: by the effort
of accommodation and by the convergence of the eyes. No doubt these two indications are always concordant, there
is a constant relation between them, or, in mathematical terms,
the two variables which measure these two muscular sensations
do not appear to us as independent; or again, to avoid an appeal
to mathematical notions already rather refined, we may go back
to the language of the preceding chapter and enunciate the same
fact as follows: If two sensations of convergence,AandB, are
indistinguishable, the two sensations of accommodation,A´andB´, which respectively accompany them, will be equally indistinguishable. But here we have, so to speak, an experimental fact;a priorinothing prevents our supposing the contrary, and if the contrary
takes place, if these two muscular sensations vary independently
of one another, we shall have to take account of one more independent
variable, and 'complete visual space' will appear to us
as a physical continuum of four dimensions. We have here even, I will add, a fact ofexternalexperience.
Nothing prevents our supposing that a being with a mind like
ours, having the same sense organs that we have, may be placed
in a world where light would only reach him after having
traversed reflecting media of complicated form. The two indications
which serve us in judging distances would cease to be
connected by a constant relation. A being who should achieve
in such a world the education of his senses would no doubt
attribute four dimensions to complete visual space. Tactile Space and Motor Space.—'Tactile space' is still
more complicated than visual space and farther removed from
geometric space. It is superfluous to repeat for touch the discussion
I have given for sight. But apart from the data of sight and touch, there are other
sensations which contribute as much and more than they to the
genesis of the notion of space. These are known to every one;
they accompany all our movements, and are usually called muscular
sensations. The corresponding frame constitutes what may be calledmotor
space. Each muscle gives rise to a special sensation capable of augmenting
or of diminishing, so that the totality of our muscular
sensations will depend upon as many variables as we have
muscles. From this point of view,motor space would have as
many dimensions as we have muscles. I know it will be said that if the muscular sensations contribute
to form the notion of space, it is because we have the
sense of thedirectionof each movement and that it makes an
integrant part of the sensation. If this were so, if a muscular
sensation could not arise except accompanied by this geometric
sense of direction, geometric space would indeed be a form imposed
upon our sensibility. But I perceive nothing at all of this when I analyze my sensations. What I do see is that the sensations which correspond to movements
in the same direction are connected in my mind by a mereassociation of ideas. It is to this association that what we call
'the sense of direction' is reducible. This feeling therefore can
not be found in a single sensation. This association is extremely complex, for the contraction of
the same muscle may correspond, according to the position of the
limbs, to movements of very different direction. Besides, it is evidently acquired; it is, like all associations of
ideas, the result of ahabit; this habit itself results from very
numerousexperiences; without any doubt, if the education of our
senses had been accomplished in a different environment, where
we should have been subjected to different impressions, contrary
habits would have arisen and our muscular sensations
would have been associated according to other laws. Characteristics of Perceptual Space.—Thus perceptual
space, under its triple form, visual, tactile and motor, is essentially
different from geometric space. It is neither homogeneous, nor isotropic; one can not even say
that it has three dimensions. It is often said that we 'project' into geometric space the
objects of our external perception; that we 'localize' them. Has this a meaning, and if so what? Does it mean that werepresentto ourselves external objects in
geometric space? Our representations are only the reproduction of our sensations;
they can therefore be ranged only in the same frame as
these, that is to say, in perceptual space. It is as impossible for us to represent to ourselves external
bodies in geometric space, as it is for a painter to paint on a
plane canvas objects with their three dimensions. Perceptual space is only an image of geometric space, an
image altered in shape by a sort of perspective, and we can represent
to ourselves objects only by bringing them under the laws of
this perspective. Therefore we do notrepresentto ourselves external bodies in
geometric space, but wereasonon these bodies as if they were
situated in geometric space. When it is said then that we 'localize' such and such an object
at such and such a point of space, what does it mean? It simply means that we represent to ourselves the movements
it would be necessary to make to reach that object; and one may
not say that to represent to oneself these movements, it is necessary
to project the movements themselves in space and that the
notion of space must, consequently, pre-exist. When I say that we represent to ourselves these movements,
I mean only that we represent to ourselves the muscular sensations
which accompany them and which have no geometric character
whatever, which consequently do not at all imply the preexistence
of the notion of space. Change of State and Change of Position.—But, it will
be said, if the idea of geometric space is not imposed upon our
mind, and if, on the other hand, none of our sensations can
furnish it, how could it have come into existence? This is what we have now to examine, and it will take some
time, but I can summarize in a few words the attempt at explanation
that I am about to develop. None of our sensations, isolated, could have conducted us to
the idea of space; we are led to it only in studying the laws,
according to which these sensations succeed each other. We see first that our impressions are subject to change; but
among the changes we ascertain we are soon led to make a distinction. At one time we say that the objects which cause these impressions
have changed state, at another time that they have
changed position, that they have only been displaced. Whether an object changes its state or merely its position,
this is always translated for us in the same manner:by a modification
in an aggregate of impressions. How then could we have been led to distinguish between the
two? It is easy to account for. If there has only been a
change of position, we can restore the primitive aggregate of
impressions by making movements which replace us opposite the
mobile object in the samerelativesituation. We thuscorrectthe modification that happened and we reestablish the initial
state by an inverse modification. If it is a question of sight, for example, and if an object
changes its place before our eye, we can 'follow it with the
eye' and maintain its image on the same point of the retina by
appropriate movements of the eyeball. These movements we are conscious of because they are voluntary
and because they are accompanied by muscular sensations,
but that does not mean that we represent them to ourselves in
geometric space. So what characterizes change of position, what distinguishes
it from change of state, is that it can always be corrected in this
way. It may therefore happen that we pass from the totality of
impressionsAto the totalityBin two different ways: 1º Involuntarily and without experiencing muscular sensations;
this happens when it is the object which changes place; 2º Voluntarily and with muscular sensations; this happens
when the object is motionless, but we move so that the object has
relative motion with reference to us. If this be so, the passage from the totalityAto the totalityBis only a change of position. It follows from this that sight and touch could not have
given us the notion of space without the aid of the 'muscular
sense.' Not only could this notion not be derived from a single sensation
or evenfrom a series of sensations, but what is more, animmobilebeing could never have acquired it, since, not being
able tocorrectby his movements the effects of the changes of
position of exterior objects, he would have had no reason whatever
to distinguish them from changes of state. Just as little
could he have acquired it if his motions had not been voluntary
or were unaccompanied by any sensations. Conditions of Compensation.—How is a like compensation
possible, of such sort that two changes, otherwise independent of
each other, reciprocally correct each other? A mind already familiar with geometry would reason as follows:
Evidently, if there is to be compensation, the various
parts of the external object, on the one hand, and the various
sense organs, on the other hand, must be in the samerelativeposition after the double change. And, for that to be the case,
the various parts of the external object must likewise have
retained in reference to each other the same relative position,
and the same must be true of the various parts of our body in
regard to each other. In other words, the external object, in the first change, must
be displaced as is a rigid solid, and so must it be with the whole
of our body in the second change which corrects the first. Under these conditions, compensation may take place. But we who as yet know nothing of geometry, since for us the
notion of space is not yet formed, we can not reason thus, we
can not foreseea prioriwhether compensation is possible. But
experience teaches us that it sometimes happens, and it is from
this experimental fact that we start to distinguish changes of
state from changes of position. Solid Bodies and Geometry.—Among surrounding objects
there are some which frequently undergo displacements susceptible
of being thus corrected by a correlative movement of
our own body; these are thesolid bodies. The other objects,whose form is variable, only exceptionally undergo like displacements
(change of position without change of form). When a
body changes its placeand its shape, we can no longer, by appropriate
movements, bring back our sense-organs into the samerelativesituation with regard to this body; consequently we can
no longer reestablish the primitive totality of impressions. It is only later, and as a consequence of new experiences, that
we learn how to decompose the bodies of variable form into
smaller elements, such that each is displaced almost in accordance
with the same laws as solid bodies. Thus we distinguish
'deformations' from other changes of state; in these deformations,
each element undergoes a mere change of position, which
can be corrected, but the modification undergone by the aggregate
is more profound and is no longer susceptible of correction
by a correlative movement. Such a notion is already very complex and must have been
relatively late in appearing; moreover it could not have arisen if
the observation of solid bodies had not already taught us to distinguish
changes of position. Therefore, if there were no solid bodies in nature, there would
be no geometry. Another remark also deserves a moment's attention. Suppose
a solid body to occupy successively the positions α and β; in its
first position, it will produce on us the totality of impressionsA,
and in its second position the totality of impressionsB. Let
there be now a second solid body, having qualities entirely different
from the first, for example, a different color. Suppose it to
pass from the position α, where it gives us the totality of impressionsA´, to the position β, where it gives the totality of impressionsB´. In general, the totalityAwill have nothing in common with
the totalityA´, nor the totalityBwith the totalityB´. The transition
from the totalityAto the totalityBand that from the
totalityA´to the totalityB´are therefore two changes whichin
themselveshave in general nothing in common. And yet we regard these two changes both as displacements
and, furthermore, we consider them as thesamedisplacement.
How can that be? It is simply because they can both be corrected by thesamecorrelative movement of our body. 'Correlative movement' therefore constitutes thesole connectionbetween two phenomena which otherwise we never should
have dreamt of likening. On the other hand, our body, thanks to the number of its
articulations and muscles, may make a multitude of different
movements; but all are not capable of 'correcting' a modification
of external objects; only those will be capable of it in which our
whole body, or at least all those of our sense-organs which come
into play, are displaced as a whole, that is, without their relative
positions varying, or in the fashion of a solid body. To summarize: 1º We are led at first to distinguish two categories of phenomena: Some, involuntary, unaccompanied by muscular sensations, are
attributed by us to external objects; these are external changes; Others, opposite in character and attributed by us to the
movements of our own body, are internal changes; 2º We notice that certain changes of each of these categories
may be corrected by a correlative change of the other category; 3º We distinguish among external changes those which have
thus a correlative in the other category; these we call displacements;
and just so among the internal changes, we distinguish
those which have a correlative in the first category. Thus are defined, thanks to this reciprocity, a particular class
of phenomena which we call displacements. The laws of these phenomena constitute the object of geometry. Law of Homogeneity.—The first of these laws is the law of
homogeneity. Suppose that, by an external change α, we pass from the totality
of impressionsAto the totalityB, then that this change
α is corrected by a correlative voluntary movement β, so that we
are brought back to the totalityA. Suppose now that another external change α´ makes us pass
anew from the totalityAto the totalityB. Experience teaches us that this change α´ is, like α, susceptible
of being corrected by a correlative voluntary movementβ´ and that this movement β´ corresponds to the same muscular
sensations as the movement β which corrected α. This fact is usually enunciated by saying thatspace is homogeneous
and isotropic. It may also be said that a movement which has once been produced
may be repeated a second and a third time, and so on,
without its properties varying. In the first chapter, where we discussed the nature of mathematical
reasoning, we saw the importance which must be
attributed to the possibility of repeating indefinitely the same
operation. It is from this repetition that mathematical reasoning gets its
power; it is, therefore, thanks to the law of homogeneity, that it
has a hold on the geometric facts. For completeness, to the law of homogeneity should be added
a multitude of other analogous laws, into the details of which I
do not wish to enter, but which mathematicians sum up in a word
by saying that displacements form 'a group.' The Non-Euclidean World.—If geometric space were a
frame imposed oneachof our representations, considered individually,
it would be impossible to represent to ourselves an
image stripped of this frame, and we could change nothing of
our geometry. But this is not the case; geometry is only the résumé of the
laws according to which these images succeed each other. Nothing
then prevents us from imagining a series of representations,
similar in all points to our ordinary representations, but succeeding
one another according to laws different from those to
which we are accustomed. We can conceive then that beings who received their education
in an environment where these laws were thus upset might
have a geometry very different from ours. Suppose, for example, a world enclosed in a great sphere and
subject to the following laws: The temperature is not uniform; it is greatest at the center,
and diminishes in proportion to the distance from the center, to
sink to absolute zero when the sphere is reached in which this
world is enclosed. To specify still more precisely the law in accordance with
which this temperature varies: LetRbe the radius of the limiting
sphere; letrbe the distance of the point considered from
the center of this sphere. The absolute temperature shall be
proportional toR2−r2. I shall further suppose that, in this world, all bodies have
the same coefficient of dilatation, so that the length of any rule
is proportional to its absolute temperature. Finally, I shall suppose that a body transported from one
point to another of different temperature is put immediately into
thermal equilibrium with its new environment. Nothing in these hypotheses is contradictory or unimaginable. A movable object will then become smaller and smaller in proportion
as it approaches the limit-sphere. Note first that, though this world is limited from the point
of view of our ordinary geometry, it will appear infinite to its
inhabitants. In fact, when these try to approach the limit-sphere, they cool
off and become smaller and smaller. Therefore the steps they
take are also smaller and smaller, so that they can never reach the
limiting sphere. If, for us, geometry is only the study of the laws according
to which rigid solids move, for these imaginary beings it will be
the study of the laws of motion of solidsdistorted by the differences
of temperaturejust spoken of. No doubt, in our world, natural solids likewise undergo variations
of form and volume due to warming or cooling. But we
neglect these variations in laying the foundations of geometry,
because, besides their being very slight, they are irregular and
consequently seem to us accidental. In our hypothetical world, this would no longer be the case,
and these variations would follow regular and very simple laws. Moreover, the various solid pieces of which the bodies of its
inhabitants would be composed would undergo the same variations
of form and volume. I will make still another hypothesis; I will suppose light
traverses media diversely refractive and such that the index ofrefraction is inversely proportional toR2−r2. It is easy to
see that, under these conditions, the rays of light would not be
rectilinear, but circular. To justify what precedes, it remains for me to show that
certain changes in the position of external objects can becorrectedby correlative movements of the sentient beings inhabiting
this imaginary world, and that in such a way as to restore the
primitive aggregate of impressions experienced by these sentient
beings. Suppose in fact that an object is displaced, undergoing deformation,
not as a rigid solid, but as a solid subjected to unequal
dilatations in exact conformity to the law of temperature above
supposed. Permit me for brevity to call such a movement anon-Euclidean displacement. If a sentient being happens to be in the neighborhood, his
impressions will be modified by the displacement of the object,
but he can reestablish them by moving in a suitable manner. It
suffices if finally the aggregate of the object and the sentient
being, considered as forming a single body, has undergone one of
those particular displacements I have just called non-Euclidean.
This is possible if it be supposed that the limbs of these beings
dilate according to the same law as the other bodies of the world
they inhabit. Although from the point of view of our ordinary geometry
there is a deformation of the bodies in this displacement and
their various parts are no longer in the same relative position,
nevertheless we shall see that the impressions of the sentient
being have once more become the same. In fact, though the mutual distances of the various parts may
have varied, yet the parts originally in contact are again in
contact. Therefore the tactile impressions have not changed. On the other hand, taking into account the hypothesis made
above in regard to the refraction and the curvature of the rays
of light, the visual impressions will also have remained the same. These imaginary beings will therefore like ourselves be led
to classify the phenomena they witness and to distinguish among
them the 'changes of position' susceptible of correction by a correlative
voluntary movement. If they construct a geometry, it will not be, as ours is, thestudy of the movements of our rigid solids; it will be the study
of the changes of position which they will thus have distinguished
and which are none other than the 'non-Euclidean displacements';it will be non-Euclidean geometry. Thus beings like ourselves, educated in such a world, would
not have the same geometry as ours. The World of Four Dimensions.—We can represent to ourselves
a four-dimensional world just as well as a non-Euclidean. The sense of sight, even with a single eye, together with the
muscular sensations relative to the movements of the eyeball,
would suffice to teach us space of three dimensions. The images of external objects are painted on the retina, which
is a two-dimensional canvas; they areperspectives. But, as eye and objects are movable, we see in succession various
perspectives of the same body, taken from different points
of view. At the same time, we find that the transition from one perspective
to another is often accompanied by muscular sensations. If the transition from the perspectiveAto the perspectiveB, and that from the perspectiveA´to the perspectiveB´are
accompanied by the same muscular sensations, we liken them one
to the other as operations of the same nature. Studying then the laws according to which these operations
combine, we recognize that they form a group, which has the
same structure as that of the movements of rigid solids. Now, we have seen that it is from the properties of this group
we have derived the notion of geometric space and that of three
dimensions. We understand thus how the idea of a space of three dimensions
could take birth from the pageant of these perspectives,
though each of them is of only two dimensions, sincethey follow
one another according to certain laws. Well, just as the perspective of a three-dimensional figure
can be made on a plane, we can make that of a four-dimensional
figure on a picture of three (or of two) dimensions. To a
geometer this is only child's play. We can even take of the same figure several perspectives from
several different points of view. We can easily represent to ourselves these perspectives, since
they are of only three dimensions. Imagine that the various perspectives of the same object succeed
one another, and that the transition from one to the other
is accompanied by muscular sensations. We shall of course consider two of these transitions as two
operations of the same nature when they are associated with the
same muscular sensations. Nothing then prevents us from imagining that these operations
combine according to any law we choose, for example, so as
to form a group with the same structure as that of the movements
of a rigid solid of four dimensions. Here there is nothing unpicturable, and yet these sensations
are precisely those which would be felt by a being possessed of
a two-dimensional retina who could move in space of four dimensions.
In this sense we may say the fourth dimension is
imaginable. Conclusions.—We see that experience plays an indispensable
rôle in the genesis of geometry; but it would be an error thence
to conclude that geometry is, even in part, an experimental
science. If it were experimental, it would be only approximative and
provisional. And what rough approximation! Geometry would be only the study of the movements of solids;
but in reality it is not occupied with natural solids, it has for
object certain ideal solids, absolutely rigid, which are only a
simplified and very remote image of natural solids. The notion of these ideal solids is drawn from all parts of our
mind, and experience is only an occasion which induces us to
bring it forth from them. The object of geometry is the study of a particular 'group';
but the general group concept pre-exists, at least potentially, in
our minds. It is imposed on us, not as form of our sense, but as
form of our understanding. Only, from among all the possible groups, that must be chosen
which will be, so to speak, thestandardto which we shall refer
natural phenomena. Experience guides us in this choice without forcing it uponus; it tells us not which is the truest geometry, but which is the
mostconvenient. Notice that I have been able to describe the fantastic worlds
above imaginedwithout ceasing to employ the language of ordinary
geometry. And, in fact, we should not have to change it if transported
thither. Beings educated there would doubtless find it more convenient
to create a geometry different from ours, and better adapted to
their impressions. As for us, in face of thesameimpressions, it
is certain we should find it more convenient not to change our
habits.
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