The idea of space has given rise to so many theories that it is difficult
to restrain ourselves within the strict limits of psychology, and of
our particular subject. Whether or no this concept be innate, given _à
priori_ or derived from our cerebral constitution, we have here—setting
aside all question of origin—only to inquire by what ways and means we
attain full consciousness of it and determine it to be a fundamental
concept.
In order to follow its development we must necessarily set out from
experience; since space, like number or time, is perceived before it is
conceived. For the sake of clearness and precision, let us designate the
primitive concrete data, the result of perception, as _extension_, and
the concept, the result of abstraction, as _space_—properly so called.
I. At the outset what is given us by intuition is extension under a
concrete form. What first becomes known to us is not space but a limited
and determined extension—what the child can hold in its hand, reach by a
movement of its arms, later on the room which it crosses with uncertain
steps; it is a street, a square traversed, a journey made by carriage or
by train, the horizon which the eye embraces, the nebulæ vaguely seen in
the nocturnal sky, etc. All this is concrete and measurable, and can be
reduced to a measure, i. e., to a concrete extension such as the metre
and its fractions.
These different extensions, although given by the senses, and therefore
concrete, are already abstract; since they co-exist with other qualities
(resistance, color, cold, heat, etc.) from which a spontaneous analysis
separates them, in order to consider them individually. This analysis is
translated by the common terms, long, short, high, deep, near, far, to
the right, to the left, in front, behind, etc.
By a simplification which occurred much later (for it implies the
foundation of geometry) this somewhat confused and incoherent list is
replaced by a more rational analysis: height, breadth, depth, distance,
position. It marks the transition from the concrete-abstract to the
abstract period. It is in fact certain that before constituting itself
as a science founded upon reasoning, geometry traversed a semi-empirical
stage, it was born of practical needs—the necessity of measuring fields,
building houses, and the rest. Moreover certain great mathematicians
have by no means disdained to admit its relations with experience: Gauss
called it the “science of the eye,” and Sylvester declared “that most if
not all the chief ideas of modern mathematics originated in observation.”
Let us, without insisting further, recollect that extension is given us
by touch and sight. Touch is _par excellence_ the sense of extension:
thus geometry reduces the problems of equality or inequality to
superpositions, and all measure of extension is finally reducible to
tactile and muscular sensations. The terms touch and vision ought in
fact to be completely co-extensive, representing not merely a passive
impression upon the cutaneous surface, or the retina, but an active
reaction of the motor elements proper to the sensorial organs.
The term _acoustic_ space has recently come into use. Much work has been
done on the semi-circular canals, leaving no doubt as to the part they
play in the sense of bodily equilibrium;[98] some authors have even
localised a “space-sense” in them. Münsterberg relates from his personal
experience that while the vestibule and the cochlea receive excitations
whence result the purely qualitative sensations of sound (height,
intensity, etc.), the semi-circular canals receive others which depend
upon the _position_ of the source of the sound: these excitations would
produce reflexes, probably in the cerebellum, the purpose of which would
be to bring the head into the position best adapted for clear audition.
The synthesis of sounds, of the modifications perceived in the canals,
and of the aforesaid movements (or images of movement) would constitute
the elements of an acoustic space. Wundt, who opposes this theory, sees
nothing more in the semi-circular canals than internal tactile organs,
auxiliary to external touch.[99]
Leaving this hypothesis of acoustic space (which is by no means
well-established), we know from numerous observations that the different
modalities of tactile and visual extension, notably that of distance, are
only known with precision after much groping and long apprenticeship.[100]
Extension under all its aspects, whether perceived or imagined, presents
according to constitution, age, or circumstances, a character of
variability which is in complete contrast with the stability and fixity
of the concept of space. The conditions of this relativity have been
exposed at length by Herbert Spencer. “A creature without eyes cannot
have the same conception of space as one that has eyes; and it is the
same with the congenitally blind as compared with those who are in full
possession of their eyesight; and for the creature whose locomotion is
rapid and powerful as compared with the creature which moves slowly
and painfully.” Our bodily bulk and organic dimensions also affect
conceptions of space; distances which seemed great to the boy seem
moderate to the man, and buildings once thought imposing in height and
mass dwindle into insignificance. Without speaking of nervous subjects,
who illusively imagine their bodies enormously large or infinitely
small, there are also transient and momentary states of the organism
which considerably modify the consciousness of space; thus, De Quincy,
describing some of his opium dreams, says that “buildings and landscapes
were exhibited in proportions so vast as the bodily eye is not fitted to
receive. _Space swelled_, and was amplified to an extent of unutterable
infinity.”[101] “Deliberate analysis of their movements,” says Lotze,
“is so little practised by women that it can be asserted without fear of
error that such expressions as ‘to the right,’ ‘to the left,’ ‘forwards,’
‘backwards,’ etc., express in their language no mathematical relations
whatever, but merely certain particular feelings which they experience
when during work they perform movements in these directions.”[102] In
fine, the consciousness of concrete extension varies in quantity and
quality with the structure, position, age, and momentary condition of
the feeling subject.
II. Starting from these concrete data—extension as included in our
perceptions—how does the intellect arrive at the abstract notion of space?
The immense majority of men left to their own resources do not rise above
a confused notion, half-concrete, half-abstract, of the properties of
extension, and what Lotze says (_supra_) applies even better to their
total idea of space-relations. The fundamental conception in such minds
is simply the possibility of going very far in all directions, or of
placing a succession of bodies one behind the other. As to limit, this
operation remains vaguely undetermined. It is however translated into
current parlance, e. g., “bodies are in space,” and other analogues.
Space is conceived, or rather imagined, as an immense sphere which
encloses everything, as the receptacle of all extension. It contains
bodies, as a barrel holds wine. The primitive cosmologies which yet
demand a certain development of reflexion and of abstraction reveal the
nature of this conception to us when they speak of the circle of the
horizon, the vault of heaven, the firmament (a kind of _firm_ enclosure),
and other expressions which denote belief in an insurmountable limit.
This conception, which is wholly imaginative at bottom, is a fine example
of abstraction elevated into an entity, and the phantom thus created
becomes in its turn the source of idle or badly-stated problems such as
the following.
‘We have never,’ contends J. S. Mill, ‘perceived an object, or a portion
of space, without there being space beyond it, and from the moment
of our birth we have always perceived objects or parts of space. It
follows from this that the idea of an object or part of space must be
inseparably associated with the notion of a further space beyond it.
Each moment of our life tends to rivet this association, no experience
has ever interrupted it. Under the actual conditions of existence, this
association is indissoluble.... Yet we can conceive that, given other
conditions of existence, it might be possible to transport ourselves
to the limits of space, and that after there receiving impressions of
a kind totally unknown to our present state, we might instantly become
capable of conceiving the fact and of stating the truth of it. After some
experience of the new conception, the fact would seem as natural to us as
the revelations of sight to the blind man whose cure is of long standing.’
This argument is founded upon an equivocation. Mill appears to admit
as the basis of his discussion the semi-concrete, semi-abstract notion
of space, described above; namely, that of common sense. Consequently
he confounds and mixes up two perfectly distinct questions; that of
Extension, the concrete fact perceived or imagined, and Space, abstract
and conceived. In the case of the former, the problem is cosmological and
objective, and we are not concerned with it; it is, under another form,
the repetition of the discussion on infinite number,—are we, or are we
not, to admit a continuous, real magnitude? In the latter, the problem
is psychological, subjective, relative to abstraction alone, and will be
answered later on.
Up to this point, in fact, the concept of Space corresponds to the state
of evolution that we have so frequently denoted as concrete-abstract.
The true concept of space—of purely abstract space—was only constituted
when the geometricians (Greek and otherwise) disengaged from different
extensions those essential features which they termed dimensions, showing
by their science that elements thus abstracted and considered separately
can be substituted for all the rest. Stallo justly observes that the
geometrical elements are neither real, nor ideal, nor hypothetical;
they are _conceptual_, the result of abstraction. “In the processes of
discursive thought the intellect never has before it either sensible
objects or the whole complement of relations which make up their mental
images or representations, but only some single relation or class of
relations. It operates along _lines_ of abstraction, the final synthesis
of whose result never yields anything more than outlines of the objects
represented. During all its operations the intellect is fully aware
that neither any one link in the chain of abstraction nor the group
of abstract results which we call a concept (in the narrower sense
of a collection of attributes representing an object of intuition or
sensation) is a copy or an exact counterpart of the object represented.
It is always conscious that to bring about true conformity between
concepts or any of their constituent attributes with forms of objective
reality, the group of relations embodied in these concepts would have
to be supplemented with an indeterminate number of other relations
which have not been apprehended and possibly are insusceptible of
apprehension....”[103]
No one imagines that there are in Nature points, surfaces, lines,
volumes, such as geometry proposes them, nor that its concepts are
copies of these, but it is not therefore necessary to take refuge in the
_à priori_: abstraction suffices, the act, i. e., by which fundamental
qualities are abstracted, to be subsequently fixed by definition. It is
strange that Stuart Mill in his long and untoward discussion of this
subject should content himself with saying that “we have a power, when a
perception is present to our senses, or a conception to our intellect, of
attending to a part only of that perception or conception.”[104] In this
remark upon attention he is very near recognising the rôle of abstraction
(which for the rest he fails to name), but instead of insisting upon it
he returns to his thesis, that “the foundation of all sciences, even
deductive and demonstrative sciences, is induction....”
The concept of space such as the geometers have made it, namely at
its highest degree of abstraction, is thus the result of association.
It is extension emptied of all its constitutive qualities, save the
necessary conventions which determine it. This schema (apart from
all transcendental considerations) appears to us as the total of the
conditions of bodily existence in so far as they are endowed with
extension. Thus constituted, with the marks which are proper to it, and
distinguish it from all others, this concept, like that of number, is
susceptible of multiple application, while moreover it has no assignable
limits in any direction (i. e., according to the time-honored expression,
it is infinite).
Just as concrete number represents real unities or collections, while
abstract number detached from discontinuous realities admits of infinite
numeration; so concrete space (extension) corresponds to the intuition
of certain bodies, while abstract space, by an unrepresentable concept,
if not by words, implies an unlimited extension.
If, hypothetically, it were possible to count all the leaves of all the
trees in the world, this prodigious number corresponding to concrete
unities would be as nothing to the mind that can count for ever beyond
that. So for the extension determined by the movement of our arms or
legs, by days of railway travelling or sailing, by balloon ascensions,
and finally by the most powerful telescopes that can scrutinise the
infinity of the heavens,—in all these concrete, fixed, _measured_
extensions we can always imagine a beyond, because the end of one
extension is the beginning of the next. All that, however, is but the
work of the imagination manipulating abstraction. The law of construction
for infinite space is the same as for infinite number: this infinity is
only in the operation of our mind, it is a pure psychological process;
we believe we are dealing with real magnitudes, and we are only acting
upon our own judgment: we are but adding states of consciousness one
upon another. Space is only potentially infinite, and this potentiality
is in us, and in nothing but ourselves; it is a virtuality which can
neither be exhausted nor achieved. To erect it into an entity is to reify
an abstraction, to attribute an undue objective value to an entirely
subjective concept. The journey to the end of space as suggested by
Stuart Mill in the passage above cited (if by space he means the simple
possibility of containing extended bodies) would really be a journey to
the end of our minds: if he means a journey to the end of the real world,
i. e., determinable and measurable extension—which has no limits beyond
the imperfection of our instruments—then he admits implicitly that the
universe has bounds, he takes sides in a debate in which experimental
psychology (we repeat) sees nothing, and which it is even totally
incompetent to decide.
* * * * *
During this century certain illustrious mathematicians,—Gauss, 1792, in
an unpublished work, Lobachévski in 1829, Riemann, Beltrami, Helmholtz
and many others after them, constructed a new geometry known under
various names: astral, imaginary, pangeometric, metageometric, and lastly
non-Euclidean geometry. Its fundamental principle is that our Euclidean
space is only one particular case among several possible cases, and our
Euclidean geometry one species of which pangeometry is the genus—that the
sole determining reason in its favor is that Euclidean geometry alone is
practically applicable to, and verified by, experience. These essays,
beyond their direct interest to mathematicians, have already given rise
to a considerable number of philosophical considerations. While they have
only very distant relations with psychology, they deserve notice, because
they enable us the better to understand the genesis of the concepts of
space, and are moreover a striking proof of the constructive power of the
mind, emancipated from experimental data, and subject only to the rules
of logic.
Our space being of three dimensions, the neo-geometers speculated in the
first place as to the hypothesis of a space of 4, 5, or _n_-dimensions;
later on they chose as their base of operations a space of three
dimensions, considered no longer as plane (Euclidean space) but as
spherical or pseudo-spherical, having, i. e., instead of a zero
curvature, either a positive (spherical space) or a negative curvature
(pseudo-spherical space). Their point of departure is the rejection
of Euclid’s postulate—they do not admit that it is impossible to draw
through a point more than one parallel to a given straight line. In
spherical space there is nothing analogous to the Euclidean axiom of
parallels; in pseudo-spherical space two parallels to a line can be drawn
through any point. In the first hypothesis, the sum of the three angles
of a triangle is greater than two right angles; in the second it is
smaller. Thus by deduction after deduction, the neo-geometers constructed
an edifice very different from ordinary geometry, subject to no other
conditions than that of being free from internal contradiction.
In our connexion, the sole utility of the invention of imaginary
geometries is to have reinforced, as if by a magnifying process, the
distinction between space _perceived_ and _conceived_; this assumes
various forms according to the process of abstraction employed and fixed
in definitions. “Euclidean” space has only one advantage, that it is the
simplest, the most practical, the best adapted to facts: in short, that
which involves the least disparity between the ideal and our experience,
and consequently the most useful. “Certain neo-geometers have in fact
maintained that it is uncertain whether space can, or cannot, have the
same properties throughout the whole universe ... and that it is possible
that in the rapid march of the solar system across space we might
gradually pass into regions in which space has not the same properties as
those we know”; yet this thesis, which, fundamentally, reifies an entity,
does not seem to have gained many partisans. Stallo criticises it at
length (_op. cit._, Chap. XIII).
There is no agreement as to the measure in which the new concepts
agree or disagree with the theory of space, “the _à priori_ form of
sensibility.” Some hold them to be indifferent, others to be unfavorable
to Kantism: this discussion which, for the rest, does not concern us is
still in progress.
* * * * *
In conclusion, extension is a primary datum of perception and cannot be
further reduced: it is multiple, full, heterogeneous, continuous (at
least in appearance), variable, perhaps finite; while space (concept) is
void, unified, homogeneous, continuous, and without limits.
Many men and races never get beyond this stage of concrete
representation, which corresponds with the first moment of evolution in
the individual and in the species. The first step towards the concept of
space (concrete-abstract period) consists in representing it to oneself
as the place, the receptacle of all bodies. This is the direct result
of primitive reflexion: image rather than concept, to which the mind
attributes an illusive reality.
The true concept, resultant of abstraction, has been the elaboration of
geometricians. It is actually constituted by a synthesis of abstracts or
extracts which are, according to Riemann, size, continuity, dimension,
simplicity, distance, measure. This synthesis or association of abstracts
has nothing necessary about it; its elements may be combined in several
ways; hence the possibility of different concepts of space (Euclidean,
non-Euclidean). Space conceived as infinity reduces itself to the power
that the human mind has of forming sequences, and it forms them thanks
to abstraction, which admits of its seizing the law of their formation.
Intuition is the common basis of all concepts of space. Euclidean space
rests directly upon this, and upon definitions. Non-Euclidean space rests
directly upon it, but more particularly upon definitions.
Although inapplicable to the real world, these last—which are
constructions in which the mind is submitted to no other laws than
agreement with itself—are brilliant examples of the power of abstraction,
when it attains its highest degree of development.