though it cannot directly _prove_ such equality; and the same
philosophical principle, of the homogeneity of space, by which
congruence was proved, comes to our rescue here. But how about
right-handed and left-handed screws? Here we can no longer apply the
Identity of Indiscernibles, for the two are very well discernible.
But as with solids, so here, Free Mobility can help us much. It
can enable us, by ordinary measurement, to show that the internal
relations of both screws are the same, and that the difference lies
only in their relation to other things in space. Knowing these
internal relations, we can calculate, by the Geometry which Free
Mobility has rendered possible, all the geometrical properties of
these screws--radius, pitch, etc.--and can show them to be severally
equal in both. But this is all we require. Mediate comparison is
possible, though immediate comparison is not. Both can, for instance,
be compared with the cylinder on which both would fit, and thus their
equality can be proved. A precisely similar proof holds, of course,
for the other cases, right and left hands, spherical triangles, etc.
On the whole, these cases confirm my argument; for they show, as Kant
intended them to show[158], the essential relativity of space.
=151.= (2) As regards time, no congruence is here conceivable,
for to effect congruence requires always--as we saw in the case
of solids--one more dimension than belongs to the magnitudes
compared. No day can be brought into temporal coincidence with any
other day, to show that the two exactly cover each other; we are
therefore reduced to the arbitrary assumption that some motion or
set of motions, given us in experience, is uniform. Fortunately,
we have a large set of motions which all roughly agree; the swing
of the pendulum, the rotation and revolution of the earth and the
planets, etc. These do not exactly agree, but they lead us to the
laws of motion, by which we are able, on our arbitrary hypothesis,
to estimate their small departures from uniformity; just as the
assumption of Free Mobility enabled us to measure the departures
of actual bodies from rigidity. But here, as there, another
possibility is mathematically open to us, and can only be excluded
by its philosophic absurdity; we might have assumed that the above
set of approximately agreeing motions all had velocities which
varied approximately as some arbitrarily assumed function of the
time, _f(t)_ say, measured from some arbitrary origin. Such an
assumption would still keep them as nearly synchronous as before,
and would give an equally possible, though more complex, system
of Mechanics; instead of the first law of motion, we should have
the following: A particle perseveres in its state of rest, or of
rectilinear motion with velocity varying as _f(t)_, except in so far
as it is compelled to alter that state by the action of external
forces. Such a hypothesis _is_ mathematically possible, but, like the
similar one for space, it is excluded logically by the comparative
nature of the judgment of quantity, and philosophically by the fact
that it involves absolute time, as a determining agent in change,
whereas time can never, philosophically, be anything but a passive
form, abstracted from change. I have introduced this parallel from
time, not as directly bearing on the argument, but as a simpler case
which may serve to illustrate my reasoning in the more complex case
of space. For since time, in mathematics, is one-dimensional, the
mathematical difficulties are simpler than in Geometry; and although
nothing accurately corresponds to congruence, there is a very similar
mixture of mathematical and philosophical necessity, giving, finally,
a thoroughly definite axiom as the basis of time-measurement,
corresponding to congruence as the basis of space-measurement[159].
=152.= (3) The case of time-measurement suggests the third of the
above objections to the absolute necessity of the axiom of Free
Mobility. Psycho-physics has shown that we have an approximate
power, by means of what may be called the sense of duration, of
immediately estimating equal short times. This establishes a rough
measure independent of any assumed uniform motion, and in space also,
it may be said, we have a similar power of immediate comparison.
We can see, by immediate inspection, that the sub-divisions on
a foot rule are not grossly inaccurate; and so, it may be said,
we both have a measure independent of congruence, and also could
discover, by experience, any gross departure from Free Mobility.
Against this view, however, there is at the outset a very fundamental
psychological objection. It has been urged that all our comparison
of spatial magnitudes proceeds by ideal superposition. Thus James
says (Psychology, Vol. II. p. 152): "Even where we only feel one
sub-division to be vaguely larger or less, the mind must pass rapidly
between it and the other sub-division, and receive the immediate
sensible shock of the more," and "so far as the sub-divisions of
a sense-space are to be _measured_ exactly against each other,
objective forms occupying one sub-division must be directly or
indirectly superposed upon the other[160]."
Even if we waive this fundamental objection, however, others remain.
To begin with, such judgments of equality are only very rough
approximations, and cannot be applied to lines of more than a certain
length, if only for the reason that such lines cannot well be seen
together. Thus this method can only give us any security in our own
immediate neighbourhood, and could in no wise warrant such operations
as would be required for the construction of maps &c., much less the
measurement of astronomical distances. They might just enable us to
say that some lines were longer than others, but they would leave
Geometry in a position no better than that of the Hedonical Calculus,
in which we depend on a purely subjective measure. So inaccurate, in
fact, is such a method acknowledged to be, that the foot-rule is as
much a need of daily life as of science. Besides, no one would trust
such immediate judgments, but for the fact that the stricter test of
congruence to some extent confirms them; if we could not apply this
test, we should have no ground for trusting them even as much as
we do. Thus we should have, here, no real escape from our absolute
dependence upon the axiom of Free Mobility.
=153.= (4) One last elucidatory remark is necessary before our proof
of this axiom can be considered complete. We spoke above of the
Geometry on an egg, where Free Mobility does not hold. What, I may
be asked, is there about a thoroughly non-congruent Geometry, more
impossible than this Geometry on the egg? The answer is obvious. The
Geometry of non-congruent surfaces is _only_ possible by the use of
infinitesimals, and in the infinitesimal all surfaces become plane.
The fundamental formula, that for the length of an infinitesimal
arc, is only obtained on the assumption that such an arc may be
treated as a straight line, and that Euclidean Plane Geometry may be
applied in the immediate neighbourhood of any point. If we had not
our Euclidean measure, which could be moved without distortion, we
should have no method of comparing small arcs in different places,
and the Geometry of non-congruent surfaces would break down. Thus
the axiom of Free Mobility, as regards three-dimensional space, is
necessarily implied and presupposed in the Geometry of non-congruent
surfaces; the possibility of the latter, therefore, is a dependent
and derivative possibility, and can form no argument against the _à
priori_ necessity of congruence as the test of equality.
=154.= It is to be observed that the axiom of Free Mobility, as I
have enunciated it, includes also the axiom to which Helmholtz gives
the name of Monodromy. This asserts that a body does not alter its
dimensions in consequence of a complete revolution through four
right angles, but occupies at the end the same position as at the
beginning. The supposed mathematical necessity of making a separate
axiom of this property of space has been disproved by Sophus Lie (v.
Chap. I. § 45); philosophically, it is plainly a particular case of
Free Mobility[161], and indeed a particularly obvious case, for a
translation really does make some change in a body, namely, a change
in position, but a rotation through four right angles may be supposed
to have been performed any number of times without appearing in the
result, and the absurdity of ascribing to space the power of making
bodies grow in the process is palpable; everything that was said
above on congruence in general applies with even greater evidence to
this special case.
=155.= The axiom of Free Mobility involves, if it is to be true, the
homogeneity of space, or the complete relativity of position. For if
any shape, which is possible in one part of space, be always possible
in another, it follows that all parts of space are qualitatively
similar, and cannot, therefore, be distinguished by any intrinsic
property. Hence positions in space, if our axiom be true, must be
wholly defined by external relations, _i.e._ _Position is not an
intrinsic, but a purely relative, property of things in space_. If
there could be such a thing as absolute position, in short, metrical
Geometry would be impossible. This relativity of position is the
fundamental postulate of all Geometry, to which each of the necessary
metrical axioms leads, and from which, conversely, each of these
axioms can be deduced.
=156.= This converse deduction, as regards Free Mobility, is not very
difficult, and follows from the argument of Section A[162], which
I will briefly recapitulate. In the first place, externality is an
essentially relative conception--nothing can be external to itself.
To be external to something is to be an other with some relation to
that thing. Hence, when we abstract a form of externality from all
material content, and study it in isolation, position will appear
of necessity as purely relative--it can have no intrinsic quality,
for our form consists of pure externality, and externality contains
no shadow or trace of an intrinsic quality. Hence we derive our
fundamental postulate, the relativity of position. From this follows
the homogeneity of our form, for any quality in one position, which
marked out that position from another, would be necessarily more or
less intrinsic, and would contradict the pure relativity. Finally
Free Mobility follows from homogeneity, for our form would not be
homogeneous unless it allowed, in every part, shapes or systems
of relations, which it allowed in any other part. Free Mobility,
therefore, is a necessary property of every possible form of
externality.
=157.= In summing up the argument we have just concluded, we may
exhibit it, in consequence of the two preceding paragraphs, in the
form of a completed circle. Starting from the conditions of spatial
measurement, we found that the comparison, required for measurement,
could only be effected by superposition. But we found, further, that
the result of such comparison will only be unambiguous, if spatial
magnitudes and shapes are unaltered by motion in space, if, in other
words, shapes do not depend upon absolute position in space. But this
axiom can only be true if space is homogeneous and position merely
relative. Conversely, if position is assumed to be merely relative,
a change of magnitude in motion--involving as it does, the assertion
of absolute position--is impossible, and our test of spatial equality
is therefore adequate. But position in any form of externality must
be purely relative, since externality cannot be an intrinsic property
of anything. Our axiom, therefore, is _à priori_ in a double sense.
It is presupposed in all spatial measurement, and it is a necessary
property of any form of externality. A similar double apriority, we
shall see, appears in our other necessary axioms.
II. _The Axiom of Dimensions[163]._
=158.= We have seen, in discussing the axiom of Free Mobility, that
all position is relative, that is, a position exists only by virtue
of relations[164]. It follows that, if positions can be defined at
all, they must be uniquely and exhaustively defined by some finite
number of such relations. If Geometry is to be possible, it must
happen that, after enough relations have been given to determine a
point uniquely, its relations to any fresh known point are deducible
from the relations already given. Hence we obtain, as an _à priori_
condition of Geometry, logically indispensable to its existence, the
axiom that _Space must have a finite integral number of Dimensions_.
For every relation required in the definition of a point constitutes
a dimension, and a fraction of a relation is meaningless. The number
of relations required must be finite, since an infinite number of
dimensions would be practically impossible to determine. If we
remember our axiom of Free Mobility, and remember also that space
is a continuum, we may state our axiom, for metrical Geometry, in
the form given by Helmholtz (v. Chap. I. § 25): "In a space of n
dimensions, the position of every point is uniquely determined by the
measurement of n continuous independent variables (coordinates).[165]"
=159.= So much, then, is _à priori_ necessary to metrical Geometry.
The restriction of the dimensions to three seems, on the contrary,
to be wholly the work of experience[166]. This restriction cannot
be logically necessary, for as soon as we have formulated any
analytical system, it appears wholly arbitrary. Why, we are driven
to ask, cannot we add a fourth coordinate to our _x_, _y_, _z_, or
give a geometrical meaning to _x^{4}_? In this more special form, we
are tempted to regard the axiom of dimensions, like the number of
inhabitants of a town, as a purely statistical fact, with no greater
necessity than such facts have.
Geometry affords intrinsic evidence of the truth of my division of
the axiom of dimensions into an _à priori_ and empirical portion.
For while the extension of the number of dimensions to four, or to
_n_, alters nothing in plane and solid Geometry, but only adds a
new branch which interferes in no way with the old, _some_ definite
number of dimensions is assumed in all Geometries, nor is it
possible to conceive of a Geometry which should be free from this
assumption[167].
=160.= Let us, since the point seems of some interest, repeat our
proof of the apriority of this axiom from a slightly different point
of view. We will begin, this time, from the most abstract conception
of space, such as we find in Riemann's dissertation, or in Erdmann's
extents. We have here, an ordered manifold, infinitely divisible and
allowing of Free Mobility[168]. Free Mobility involves, as we saw,
the power of passing continuously from any one point to any other, by
any course which may seem pleasant to us; it involves, also, that,
in such a course, no changes occur except changes of mere position,
_i.e._, positions do not differ from one another in any qualitative
way. (This absence of qualitative difference is the distinguishing
mark of space as opposed to other manifolds, such as the colour- and
tone-systems: in these, every element has a definite qualitative
sensational value, whereas in space, the sensational value of a
position depends wholly on its spatial relation to our own body, and
is thus not intrinsic, but relative.) From the absence of qualitative
differences among positions, it follows logically that positions
exist only by virtue of other positions; one position differs from
another just because they are two, not because of anything intrinsic
in either. Position is thus defined simply and solely by relation to
other positions. Any position, therefore, is completely defined when,
and only when, enough such relations have been given to enable us to
determine its relation to any new position, this new position being
defined by the same number of relations. Now, in order that such
definition may be at all possible, a finite number of relations must
suffice. But every such relation constitutes a dimension. Therefore,
if Geometry is to be possible, it is _à priori_ necessary that space
should have a finite integral number of dimensions.
=161.= The limitation of the dimensions to three is, as we have
seen, empirical; nevertheless, it is not liable to the inaccuracy
and uncertainty which usually belong to empirical knowledge. For
the alternatives which logic leaves to sense are discrete--if the
dimensions are not three, they must be two or four or some other
number--so that _small_ errors are out of the question[169]. Hence
the final certainty of the axiom of three dimensions, though in part
due to experience, is of quite a different order from that of (say)
the law of Gravitation. In the latter, a small inaccuracy might exist
and remain undetected; in the former, an error would have to be so
considerable as to be utterly impossible to overlook. It follows that
the certainty of our whole axiom, that the number of dimensions is
three, is almost as great as that of the _à priori_ element, since
this element leaves to sense a definite disjunction of discrete
possibilities.
III. _The Axiom of Distance._
=162.= We have already seen, in discussing projective Geometry, that
two points must determine a unique curve, the straight line. In
metrical Geometry, the corresponding axiom is, that two points must
determine a unique spatial quantity, distance. I propose to prove,
in what follows, (1) that if distance, as a quantity completely
determined by two points, did not exist, spatial magnitude would
not be measurable; (2) that distance can only be determined by two
points, if there is an actual curve in space determined by those two
points; (3) that the existence of such a curve can be deduced from
the conception of a form of externality, and (4) that the application
of quantity to such a curve necessarily leads to a certain magnitude,
namely distance, uniquely determined by any two points which
determine the curve. The conclusion will be, if these propositions
can be successfully maintained, that the axiom of distance is _à
priori_ in the same double sense as the axiom of Free Mobility,
_i.e._ it is presupposed in the possibility of measurement, and it is
necessarily true of any possible form of externality.
=163.= (1) The possibility of spatial measurement allows us to
infer the existence of a magnitude uniquely determined by any two
points. The proof of this depends on the axiom of Free Mobility, or
its equivalent, the homogeneity of space. We have seen that these
are involved in the possibility of spatial measurement; we may
employ them, therefore, in any argument as to the conditions of this
possibility.
Now to begin with, two points must, if Geometry is to be possible,
have _some_ relation to each other, for we have seen that such
relations alone constitute position or localization. But if two
points have a relation to each other, this must be an intrinsic
relation. For it follows, from the axiom of Free Mobility, that
two points, forming a figure congruent with the given pair, can be
constructed in any part of space. If this were not possible, we have
seen that metrical Geometry could not exist. But both the figures may
be regarded as composed of two points and their relation; if the
two figures are congruent, therefore, it follows that the relation
is quantitatively the same for both figures, since congruence is the
test of spatial equality. Hence the two points have a quantitative
relation, which is such that they can traverse all space in a
combined motion without in any way altering that relation. But in
such a general motion, any external relation of the two points,
any relation involving other points or figures in space, must be
altered[170]. Hence the relation between the two points, being
unaltered, must be an intrinsic relation, a relation involving no
other point or figure in space; and this intrinsic relation we call
distance[171].
=164.= It might be objected, to the above argument, that it involves
a _petitio principii_. For it has been assumed that the two points
and their relation form a figure, to which other figures can be
congruent. Now if two points have no intrinsic relation, it would
seem that they cannot form such a figure. The argument, therefore,
apparently assumes what it had to prove. Why, it may be asked, should
not three points be required, before we obtain any relation, which
Free Mobility allows us to construct afresh in other parts of space?
The answer to this, as to the corresponding question in the first
section of this chapter, lies, I think, in the passivity of space,
or the mutual independence of its parts. For it follows, from this
independence, that any figure, or any assemblage of points, may
be discussed without reference to other figures or points. This
principle is the basis of infinite divisibility, of the use of
quantity in Geometry, and of all possibility of isolating particular
figures for discussion. It follows that two points cannot be
dependent, as to their relation, on any other points or figures, for
if they were so dependent, we should have to suppose some action
of such points or figures on the two points considered, which
would contradict the mutual independence of different positions.
To illustrate by an example: the relation of two given points does
not depend on the other points of the straight line on which the
given points lie. For only through their relation, _i.e._ through
the straight line which they determine, can the other points of the
straight line be known to have any peculiar connection with the given
pair.
=165.= But why, it may be asked, should there be only one such
relation between two points? Why not several? The answer to this lies
in the fact that points are wholly constituted by relations, and have
no intrinsic nature of their own[172]. A point is defined by its
relations to other points, and when once the relations necessary for
definition have been given, no fresh relations to the points used in
definition are possible, since the point defined has no qualities
from which such relations could flow. Now one relation to any one
other point is as good for definition as more would be, since however
many we had, they would all remain unaltered in a combined motion of
both points. Hence there can only be one relation determined by any
two points.
=166.= (2) We have thus established our first proposition--two points
have one and only one relation uniquely determined by those two
points. This relation we call their distance apart. It remains to
consider the conditions of the measurement of distance, _i.e._, how
far a unique value for distance involves a curve uniquely determined
by the two points.
In the first place, some curve joining the two points is involved
in the above notion of a combined motion of the two points, or of
two other points forming a figure congruent with the first two.
For without some such curve, the two point-pairs cannot be known
as congruent, nor can we have any test by which to discover when
a point-pair is moving as a single figure[173]. Distance must be
measured, therefore, by some line which joins the two points. But
need this be a line which the two points completely determine?
=167.= We are accustomed to the definition of the straight line
as the _shortest_ distance between two points, which implies that
distance might equally well be measured by curved lines. This
implication I believe to be false, for the following reasons. When we
speak of the length of a curve, we can give a meaning to our words
only by supposing the curve divided into infinitesimal rectilinear
arcs, whose sum gives the length of an equivalent straight line;
thus unless we presuppose the straight line, we have no means of
comparing the lengths of different curves, and can therefore never
discover the applicability of our definition. It might be thought,
perhaps, that some other line, say a circle, might be used as the
basis of measurement. But in order to estimate in this way the length
of any curve other than a circle, we should have to divide the curve
into infinitesimal circular arcs. Now two successive points do not
determine a circle, so that an arc of two points would have an
indeterminate length. It is true that, if we exclude infinitesimal
radii for the measuring circles, the lengths of the infinitesimal
arcs would be determinate, even if the circles varied, but that is
only because all the small circular arcs through two consecutive
points coincide with the straight line through those two points.
Thus, even with the help of the arbitrary restriction to a finite
radius, all that happens is that we are brought back to the straight
line. If, to mend matters, we take three consecutive points of our
curve, and reckon distance by the arc of the circle of curvature,
the notion of distance loses its fundamental property of being a
relation between _two_ points. For two consecutive points of the
arc could not then be said to have any corresponding distance
apart--three points would be necessary before the notion of distance
became applicable. Thus the circle is not a possible basis for
measurement, and similar objections apply, of course, with increased
force, to any other curve. All this argument is designed to show,
in detail, the logical impossibility of measuring distance by any
curve not completely defined by the two points whose distance apart
is required. If in the above we had taken distance as measured
by circles of _given radius_, we should have introduced into its
definition a relation to other points besides the two whose distance
was to be measured, which we saw to be a logical fallacy. Moreover,
how are we to know that all the circles have equal radii, until we
have an independent measure of distance?
=168.= A straight line, then, is not the _shortest_ distance, but is
simply _the_ distance between two points--so far, this conclusion
has stood firm. But suppose we had two or more curves through
two points, and that all these curves were congruent _inter se_.
We should then say, in accordance with the definition of spatial
equality, that the lengths of all these curves were equal. Now
it might happen that, although no one of the curves was uniquely
determined by the two end-points, yet the common length of all the
curves was so determined. In this case, what would hinder us from
calling this common length the distance apart, although no unique
figure in space corresponded to it? This is the case contemplated by
spherical Geometry, where, as on a sphere, antipodes can be joined by
an infinite number of geodesics, all of which are of equal length.
The difficulty supposed is, therefore, not a purely imaginary one,
but one which modern Geometry forces us to face. I shall consequently
discuss it at some length.
=169.= To begin with, I must point out that my axiom is not quite
equivalent to Euclid's. Euclid's axiom states that two straight lines
cannot enclose a space, _i.e._, cannot have more than one common
point. Now if every two points, without exception, determine a unique
straight line, it follows, of course, that two different straight
lines can have only one point in common--so far, the two axioms are
equivalent. But it may happen, as in spherical space, that two points
_in general_ determine a unique straight line, but fail to do so
when they have to each other the special relation of being antipodes.
In such a system every pair of straight lines in the same plane meet
in two points, which are each other's antipodes; but two points, _in
general_, still determine a unique straight line. We are still able,
therefore, to obtain distances from unique straight lines, except in
limiting cases; and in such cases, we can take any point intermediate
between the two antipodes, join it by the _same_ straight line to
both antipodes, and measure its distance from those antipodes in the
usual way. The sum of these distances then gives a unique value for
the distance between the antipodes.
Thus even in spherical space, we are greatly assisted by the axiom
of the straight line; all linear measurement is effected by it, and
exceptional cases can be treated, through its help, by the usual
methods for limits. Spherical space, therefore, is not so adverse
as it at first appeared to be to the _à priori_ necessity of the
axiom. Nevertheless we have, so far, not attacked the kernel of the
objection which spherical space suggested. To this attack it is now
our duty to proceed.
=170.= It will be remembered that, in our _à priori_ proof that
two points must have one definite relation, we held it impossible
for those two points to have, to the rest of space, any relation
which would be unaltered by motion. Now in spherical space, in the
particular case where the two points are antipodes, they _have_ a
relation, unaltered by motion, to the rest of space--the relation,
namely, that their distance is half the circumference of the
universe. In our former discussion, we assumed that any relation
to outside space must be a relation of position--and a relation of
position must be altered by motion. But with a finite space, in
which we have absolute magnitude, another relation becomes possible,
namely, a relation of magnitude. Antipodal points, accordingly,
like coincident points, no longer determine a unique straight line.
And it is instructive to observe that there is, in consequence, an
ambiguity in the expression for distance, like the ordinary ambiguity
in angular measurement. If 1/k^{2} be the space constant, and _d_
be one value for the distance between two points, 2πkn ± d, where
_n_ is any integer, is an equally good value. Distance is, in short,
a periodic function like angle. Thus such a state of things rather
confirms than destroys my contention, that distance depends on a
curve uniquely determined by two points. For as soon as we drop this
unique determination, we see ambiguities creeping into our expression
for distance. Distance still has a set of _discrete_ values,
corresponding to the fact that, given one point, the straight line
is uniquely determined for all other points but one, the antipodal
point. It is tempting to go on, and say: If through _every_ pair of
points there were an infinite number of the curves used in measuring
distance, distance would be able, for the same pair of points, to
take, not only a discrete series, but an infinite _continuous_ series
of values.
=171.= This, however, is mere speculation. I come now to the _pièce
de résistance_ of my argument. The ambiguity in spherical space
arose, as we saw, from a relation of _magnitude_ to the rest of
space--such a relation being unaltered by a motion of the two points,
and therefore falling outside our introductory reasoning. But what
is this relation of magnitude? Simply a relation of the _distance_
between the two points to a _distance_ given in the nature of the
space in question. It follows that such a relation _presupposes_ a
measure of distance, and need not, therefore, be contemplated in
any argument which deals with the _à priori_ requisites for the
possibility of definite distances[174].
=172.= I have now shown, I hope conclusively, that spherical space
affords no objection to the apriority of my axiom. Any two points
have one relation, their distance, which is independent of the
rest of space, and this relation requires, as its measure, a curve
uniquely determined by those two points. I might have taken the bull
by the horns, and said: Two points _can_ have no relation but what
is given by lines which join them, and therefore, if they have a
relation independent of the rest of space, there must be one line
joining them which they completely determine. Thus James says[175]:
"Just as, in the field of quantity, the relation between two numbers
is another number, so in the field of space the relations are facts
of the same order with the facts they relate.... When we speak of
the relation of direction of two points towards each other, we mean
simply the sensation of the line that joins the two points together.
_The line is the relation...._ The relation of position between the
top and bottom points of a vertical line is that line, and nothing
else."
If I had been willing to use this doctrine at the beginning, I
might have avoided all discussion. A unique relation between two
points _must_ in this case, involve a unique line between them. But
it seemed better to avoid a doctrine not universally accepted, the
more so as I was approaching the question from the logical, not the
psychological, side. After disposing of the objections, however, it
is interesting to find this confirmation of the above theory from so
different a standpoint. Indeed, I believe James's doctrine could be
proved to be a logical necessity, as well as a psychological fact.
For what sort of thing can a spatial relation between two distinct
points be? It must be something spatial, and it must, since points
are wholly constituted by their relations, be something at least
as real and tangible as the points it relates. There seems nothing
which can satisfy these requirements, except a line joining them.
Hence, once more, a unique relation must involve a unique line. That
is, linear magnitude is logically impossible, unless space allows of
curves uniquely determined by any two of their points.
=173.= (3) But farther, the existence of curves uniquely determined
by two points can be deduced from the nature of any form of
externality[176]. For we saw, in discussing Free Mobility, that this
axiom, together with homogeneity and the relativity of position,
can be so deduced, and we saw in the beginning of our discussion on
distance, that the existence of a unique relation between two points
could be deduced from the homogeneity of space. Since position is
relative, we may say, any two points must have _some_ relation to
each other: since our form of externality is homogeneous, this
relation can be kept unchanged while the two points move in the
form, _i.e._, change their relations to other points; hence their
relation to each other is an intrinsic relation, independent of their
relations to other points. But since our form _is_ merely a complex
of relations, a relation of externality must appear in the form, with
the same evidence as anything else in the form; thus if the form be
intuitive or sensational, the relation must be immediately presented,
and not a mere inference. Hence the intrinsic relation between two
points must be a unique figure in our form, _i.e._ in spatial terms,
the straight line joining the two points.
=174.= (4) Finally, we have to prove that the existence of such a
curve necessarily leads, when quantity is applied to the relation
between two points, to a unique magnitude, which those two points
completely determine. With this, we shall be brought back to
distance, from which we started, and shall complete the circle of our
argument.
We saw, in section A § 119, that the figure formed by two points is
projectively indistinguishable from that formed by any two other
points in the same straight line; the figure, in both cases, is,
from the projective standpoint, simply the straight line on which
the two points lie. The difference of relation, in the two cases,
is not qualitative, since projective Geometry cannot deal with it;
nevertheless, there is some difference of relation. For instance, if
one point be kept fixed, while the other moves, there is obviously
some change of relation. This change, since all parts of the straight
line are qualitatively alike, must be a change of quantity. If two
points, therefore, determine a unique figure, there must exist, for
the distinction between the various other points of this figure, a
unique quantitative relation between the two determining points, and
therefore, since these points are arbitrary, between only two points.
This relation is _distance_, with which our argument began, and to
which it at least returns.
=175.= To sum up: If points are defined simply by relations to other
points, _i.e._, if all position is relative, _every point must have
to every other point one, and only one, relation independent of
the rest of space. This relation is the distance between the two
points._ Now a relation between two points can only be defined by a
line joining them--nay further, it may be contended that a relation
can only _be_ a line joining them. Hence a unique relation involves
a unique line, _i.e._, a line determined by any two of its points.
Only in a space which admits of such a line is linear magnitude a
logically possible conception. But when once we have established the
possibility, _in general_, of drawing such lines, and therefore of
measuring linear magnitudes, we may find that a certain magnitude has
a peculiar relation to the constitution of space. The straight line
may turn out to be of finite length, and in this case its length will
give a certain peculiar magnitude, the space-constant. Two antipodal
points, that is, points which bisect the entire straight line, will
then have a relation of magnitude which, though unaltered by motion,
is rendered peculiar by a certain constant relation to the rest of
space. This peculiarity presupposes a measure of linear magnitude in
general, and cannot, therefore, upset the apriority of the axiom of
the straight line. But it destroys, for points having the peculiar
antipodal relation to each other, the argument which proved that the
relation between two points could not, since it was unchanged by
motion, have reference to the rest of space. Thus it is intelligible
that, for such special points, the axiom breaks down, and an infinite
number of straight lines are possible between them; but unless we had
started with assuming the general validity of the axiom, we could
never have reached a position in which antipodal points could have
been known to be peculiar, or, indeed, a position which would have
enabled us to give any quantitative definition whatever of particular
points.
Distance and the straight line, as relations uniquely determined
by two points, are thus _à priori_ necessary to metrical Geometry.
But further, they are properties which must belong to any form of
externality. Since their necessity for Geometry was deduced from
homogeneity and the relativity of position, and since these are
necessary properties of any form of externality, the same argument
proves both conclusions. We thus obtain, as in the case of Free
Mobility, a double apriority: The axiom of Distance, and its
implication, the axiom of the Straight Line, are, on the one hand,
presupposed in the possibility of spatial magnitude, and cannot,
therefore, be contradicted by any experience resulting from the
measurement of space; while they are consequences, on the other hand,
of the necessary properties of any form of externality which is to
render possible experience of an external world.
=176.= In connection with the straight line, it will be convenient
to discuss the conditions of a metrical coordinate system. The
projective coordinate system, as we have seen, aims only at a
convenient nomenclature for different points, and can be set up
without introducing the notion of spatial quantity. But a metrical
coordinate system does much more than this. It defines every point
quantitatively, by its quantitative spatial relations to a certain
coordinate figure. Only when the system of coordinates is thus
metrical, _i.e._, when every coordinate represents some spatial
magnitude, which is itself a relation of the point defined to some
other point or figure--can operations with coordinates lead to a
metrical result. When, as in projective Geometry, the coordinates
are not spatial magnitudes, no amount of transformation can give a
metrical result. I wish to prove, here, that a metrical coordinate
system necessarily involves the straight line, and cannot, without a
logical fallacy, be set up on any other basis. The projective system
of coordinates, as we saw, is entirely based on the straight line;
but the metrical system is more important, since its quantities
embody actual information as to spatial magnitudes, which, in
projective Geometry, is not the case.
In the first place, a point's metrical coordinates constitute a
complete quantitative definition of it; now a point can only be
defined, as we have seen, by its relations to other points, and
these relations can only be defined by means of the straight line.
Consequently, any metrical system of coordinates must involve the
straight line, as the basis of its definitions of points.
This _à priori_ argument, however, though I believe it to be quite
sound, is not likely to carry conviction to any one persuaded of the
opposite. Let us, therefore, examine metrical coordinate systems in
detail, and show, in each case, their dependence on the straight line.
We have already seen that the notion of distance is impossible
without the straight line. We cannot, therefore, define our
coordinates in any of the ordinary ways, as the distances from three
planes, lines, points, spheres, or what not. Polar coordinates
are impossible, since,--waiving the straightness of the radius
vector--the length of the radius vector becomes unmeaning. Triangular
coordinates involve not only angles, which must in the limit be
rectilinear, but straight lines, or at any rate some well-defined
curves. Now curves can only be metrically defined in two ways:
_Either_ by relation to the straight line, as, _e.g._, by the
curvature at any point, _or_ by purely analytical equations, which
presuppose an intelligible system of metrical coordinates. What
methods remain for assigning these arbitrary values to different
points? Nay, how are we to get any estimate of the difference--to
avoid the more special notion of distance--between two points?
The very notion of a point has become illusory. When we have a
coordinate system, we may define a point by its three coordinates;
in the absence of such a system, we may define the notion of point
_in general_ as the intersection of three surfaces or of two curves.
Here we take surfaces and curves as notions which intuition makes
plain, but if we wish them to give us a precise numerical definition
of _particular_ points, we must specify the kind of surface or
curve to be used. Now this, as we have seen, is only possible when
we presuppose either the straight line, or a coordinate system. It
follows that every coordinate system presupposes the straight line,
and is logically impossible without it.
=177.= The above three axioms, we have seen, are _à priori_ necessary
to metrical Geometry. No others can be necessary, since metrical
systems, logically as unassailable as Euclid's, and dealing with
spaces equally homogeneous and equally relational, have been
constructed by the metageometers, without the help of any other
axioms. The remaining axioms of Euclidean Geometry--the axiom of
parallels, the axiom that the number of dimensions is three, and
Euclid's form of the axiom of the straight line (two straight lines
cannot enclose a space)--are not essential to the possibility of
metrical Geometry, _i.e._, are not deducible from the fact that
a science of spatial magnitudes is possible. They are rather
to be regarded as empirical laws, obtained, like the empirical
laws of other sciences, by actual investigation of the given
subject-matter--in this instance, experienced space.
=178.= In summing up the distinctive argument of this Section,
we may give it a more general form, and discuss the conditions
of measurement in any continuous manifold, _i.e._, the qualities
necessary to the manifold, in order that quantities in it may be
determinable, not only as to the more or less, but as to the precise
_how much_.
Measurement, we may say, is the application of number to continua,
or, if we prefer it, the transformation of mere quantity into number
of units. Using _quantity_ to denote the vague more or less, and
_magnitude_ to denote the precise number of units, the problem of
measurement may be defined as the transformation of quantity into
magnitude.
Now a number, to begin with, is a whole consisting of smaller units,
all of these units being qualitatively alike. In order, therefore,
that a continuous quantity may be expressible as a number, it must,
on the one hand, be itself a whole, and must, on the other hand,
be divisible into qualitatively similar parts. In the aspect of a
whole, the quantity is _intensive_; in the aspect of an aggregate of
parts, it is _extensive_. A purely intensive quantity, therefore,
is not numerable--a purely extensive quantity, if any such could
be imagined, would not be a single quantity at all, since it would
have to consist of wholly unsynthesized particulars. A measurable
quantity, therefore, is a whole divisible into similar parts. But
a continuous quantity, if divisible at all, must be _infinitely_
divisible. For otherwise the points at which it could be divided
would form natural barriers, and so destroy its continuity. But
further, it is not sufficient that there should be a possibility
of division into mutually external parts; while the parts, to be
perceptible as parts, must be mutually external, they must also, to
be knowable as _equal_ parts, be capable of overcoming their mutual
externality. For this, as we have seen, we require superposition,
which involves Free Mobility and homogeneity--the absence of Free
Mobility in time, where all other requisites of measurement are
fulfilled, renders direct measurement of time impossible. Hence
infinite divisibility, free mobility, and homogeneity are necessary
for the possibility of measurement in _any_ continuous manifold, and
these, as we have seen, are equivalent to our three axioms. These
axioms are necessary, therefore, not only for spatial measurement,
but for all measurement. The only manifold given in experience, in
which these conditions are satisfied, is space. All other exact
measurement--as could be proved, I believe, for every separate
case--is effected, as we saw in the case of time, by reduction to
a spatial correlative. This explains the paramount importance, to
exact science, of the mechanical view of nature, which reduces
all phenomena to motions in time and space. For number is, of
all conceptions, the easiest to operate with, and science seeks
everywhere for an opportunity to apply it, but finds this opportunity
only by means of spatial equivalents to phenomena[177].
=179.= We have now seen in what the _à priori_ element of Geometry
consists. This _à priori_ element may be defined as the axioms common
to Euclidean and non-Euclidean spaces, as the axioms deducible from
the conception of a form of externality, or--in metrical Geometry--as
the axioms required for the possibility of measurement. It remains
to discuss, in a final chapter, some questions of a more general
philosophic nature, in which we shall have to desert the firm ground
of mathematics and enter on speculations which I put forward very
tentatively, and with little faith in their ultimate validity. The
chief questions for this final chapter will be two: (1) How is such
_à priori_ and purely logical necessity possible, as applied to an
actually given subject-matter like space? (2) How can we remove the
contradictions which have haunted us in this chapter, arising out of
the relativity, infinite divisibility, and unbounded extension of
space? These two questions are forced upon us by the present chapter,
but as they open some of the fundamental problems of philosophy, it
would be rash to expect a conclusive or wholly satisfactory answer. A
few hints and suggestions may be hoped for, but a complete solution
could only be obtained from a complete philosophy, of which the
prospects are far too slender to encourage a confident frame of mind.
FOOTNOTES:
[116] See infra, Axiom of Distance, in Sec. B. of this Chapter.
[117] Thus on a cylinder, two geodesics, _e.g._ a generator and
a helix, may have any number of intersections--a very important
difference from the plane.
[118] Cf. Cremona, Projective Geometry (Clarendon Press, 2nd ed.
1893) p. 50: "Most of the propositions in Euclid's Elements are
metrical, and it is not easy to find among them an example of a
purely descriptive theorem."
[119] Op. cit. p. 226.
[120] Some ground for this choice will appear when we come to
metrical Geometry.
[121] The straight line _σa_ denotes the straight line common to the
planes _σ_ and _a_, the point _σa_ denotes the point common to the
plane _σ_ and the straight line _a_, and similarly for the rest of the
notation.
[122] Cremona (op. cit. Chap. IX. p. 50) defines anharmonic ratio as
a metrical property which is unaltered by projection. This, however,
destroys the logical independence of projective Geometry, which can
only be maintained by a purely descriptive definition.
[123] There is no corresponding property of _three_ points on a line,
because they can be projectively transformed into any other three
points on the same line. See § 120.
[124] Due to v. Staudt's "Geometrie der Lage."
[125] See Cremona, op. cit. Chapter VIII.
[126] The corresponding definitions, for the two-dimensional manifold
of lines through a point, follow by the principle of duality.
[127] It is important to observe that this definition of the Point
introduces metrical ideas. Without metrical ideas, we saw, nothing
appears to give the Point precedence of the straight line, or indeed
to distinguish it conceptually from the straight line. A reference
to quantity is therefore inevitable in defining the Point, if the
definition is to be geometrical. A non-metrical definition would have
to be also non-geometrical. See Chap. IV. §§ 196-199.
[128] §§ 163-175.
[129] On this axiom, however, compare § 131.
[130] For the proof of this proposition, see Chap. III. Sec. B, Axiom
of Dimensions.
[131] The straight line and plane, in all discussions of general
Geometry, are not necessarily Euclidean. They are simply figures
determined, in general, by two and by three points respectively;
whether they conform to the axiom of parallels and to Euclid's form
of the axiom of the straight line, is not to be considered in the
general definition.
[132] That projective Geometry must have existential import, I shall
attempt to prove in Chapter IV.
[133] Logic, Book I. Chapter II.
[134] Cf. Bradley's Logic, p. 63. It will be seen that the sense in
which I have spoken of space as a principle of differentiation is not
the sense of a "principle of individuation" which Bradley objects to.
[135] Chap. IV. §§ 186-191.
[136] Chap. IV. § 201 ff.
[137] It is important to observe, however, that this way of regarding
spatial relations is metrical; from the projective standpoint, the
relation between two points is the whole unbounded straight line on
which they lie, and need not be regarded as divisible into parts or
as built up of points.
[138] §§ 207, 208. Cf. Hegel, Naturphilosophie, § 254.
[139] See Chap. IV. §§ 196-199.
[140] See a forthcoming article on "The relations of number and
quantity" by the present writer in _Mind_, July, 1897.
[141] Logic, Vol. II. Chap. VII. p. 211.
[142] Real, as opposed to logical, diversity is throughout intended.
Diverse aspects may coexist in a thing at one time and place, but two
diverse real things cannot so coexist.
[143] On the insufficiency of time alone, see Chapter IV. § 191.
[144] Geometrically, the axiom of the plane is, not that three
points determine a figure at all, which follows from the axiom of
the straight line, but that the straight line joining two casual
points of the plane lies wholly in the plane. This axiom requires a
projective method of constructing the plane, _i.e._ of finding all
the triads of points which determine the same projective figure as
the given triad. The required construction will be obtained if we
can find any projective figure determined by three points, and any
projective method of reaching other points which determine the same
figure.
[Illustration]
Let _O_, _P_, _Q_ be the three points whose projective relation
is required. Then we have given us the three straight lines _PQ_,
_QO_, _OP_. Metrically, the relation between these points is made
up of the area, and the magnitude of the sides and angles, of the
triangle _OPQ_, just as the relation between two points is distance.
But projectively, the figure is unchanged when _P_ and _Q_ travel
along _OP_ and _OQ_, or when _OP_ and _OQ_ turn about _O_ in such a
way as still to meet _PQ_. This is a result of the general principle
of projective equivalence enunciated above (§§ 108, 109). Hence the
projective relation between _O_, _P_, _Q_ is the same as that between
_O_, _p_, _q_ or _O_, _P′_, _Q′_; that is, _p_, _q_ and _P′_, _Q′_
lie in the plane _OPQ_. In this way, any number of points on the
plane may be obtained, and by repeating the construction with fresh
triads, every point of the plane can be reached. We have to prove
that, when the plane is so constructed, the straight line joining any
two points of the plane lies wholly in the plane.
It is evident, from the manner of construction, that any point of
_PQ_, _OP_, _OQ_, _OP′_ or _OQ′_ lies in the plane. If we can prove
that any point of _pq_ lies in the plane, we shall have proved all
that is required, since _pq_ may be transformed, by successive
repetitions of the same construction, into any straight line joining
two points of the plane. But we have seen that the same plane is
determined by _O_, _p_, _q_ and by _O_, _P_, _Q_. The straight lines
_PQ_, _pq_ have, therefore, the same relation to the plane. But _PQ_
lies wholly in the plane; therefore _pq_ also lies wholly in the
plane. Hence our axiom is proved.
[145] A detailed proof has been given above, Chap. I. 3rd period. It
is to be observed that any reference to infinitely distant elements
involves metrical ideas.
[146] Cf. Section A, §§ 115-117.
[147] Contrast Erdmann, op. cit. p. 138.
[148] Cf. Erdmann, op. cit. p. 164.
[149] Strictly speaking, this method is only applicable where the two
magnitudes are commensurable. But if we take infinite divisibility
rigidly, the units can theoretically be taken so small as to obtain
any required degree of approximation. The difficulty is the universal
one of applying to continua the essentially discrete conception of
number.
[150] Cf. Erdmann, op. cit. p. 50.
[151] Also called the axiom of congruence. I have taken congruence to
be the _definition_ of spatial equality by superposition, and shall
therefore generally speak of the _axiom_ as Free Mobility.
[152] For the sense in which these figures are to be regarded as
material, see criticism of Helmholtz, Chapter II. §§ 69 ff.
[153] Op. cit. p. 60.
[154] The view of Helmholtz and Erdmann, that mechanical experience
suffices here, though geometrical experience fails us, has been
discussed above, Chapter II. §§ 73, 82.
[155] Chapter II. § 81.
[156] Chapter II. § 72.
[157] Contrast Delbœuf, L'ancienne et les nouvelles géométries, II.
Rev. Phil. 1894, Vol. xxxvii. p. 354.
[158] Prolegomena, § 13. See Vaihinger's Commentar, II. pp. 518-532
esp. pp. 521-2. The above was Kant's whole purpose in 1768, but only
part of his purpose in the Prolegomena, where the intuitive nature of
space was also to be proved.
[159] On the subject of time measurement, cf. Bosanquet's Logic, Vol.
i. pp. 178-183. Since time, in the above account, is measured by
motion, its measurement presupposes that of spatial magnitudes.
[160] Cf. Stumpf. Ursprung der Raumvorstellung, p. 68.
[161] As is Helmholtz's other axiom, that the possibility of
superposition is independent of the course pursued in bringing it
about.
[162] Cf. §§ 129, 130.
[163] This deduction is practically the same as that in Sec. A, but
I have stated it here with more special reference to space and to
metrical Geometry.
[164] The question: "Relations to what?" is a question involving
many difficulties. It will be touched on later in this chapter, and
answered, as far as possible, in the fourth chapter. For the present,
in spite of the glaring circle involved, I shall take the relations
as relations to other positions.
[165] Wiss. Abh. Vol. II. p. 614.
[166] Cp. Grassmann, Ausdehnungslehre von 1844, 2nd ed. p. XXIII.
[167] Delbœuf, it is true, speaks of Geometries with _m_/_n_
dimensions, but gives no reference (Rev. Phil. T. xxxvi. p. 450).
[168] In criticizing Erdmann, it will be remembered, we saw that Free
Mobility is a necessary property of his extents, though he does not
regard it as such.
[169] Cf. Riemann, Hypothesen welche der Geometrie zu Grunde liegen,
Gesammelte Werke, p. 266; also Erdmann, op. cit. p. 154.
[170] This is subject, in spherical space, to the modification
pointed out below, in dealing with the exception to the axiom of the
straight line. See §§ 168-171.
[171] In speaking of distance at once as a quantity and as an
intrinsic relation, I am anxious to guard against an apparent
inconsistency. I have spoken of the judgment of quantity, throughout,
as one of comparison; how, then, can a quantity be intrinsic? The
reply is that, although measurement and the judgment of quantity
express the result of comparison, yet the terms compared must exist
before the comparison; in this case, the terms compared in measuring
distances, _i.e._ in comparing them _inter se_, are intrinsic
relations between points. Thus, although the _measurement_ of
distance involves a reference to other distances, and its expression
as a magnitude requires such a reference, yet its existence does not
depend on any external reference, but exclusively on the two points
whose distance it is.
[172] See the end of the argument on Free Mobility, § 155 ff.
[173] In Frischauf's "Absolute Geometrie nach Johann Bolyai," Anhang,
there is a series of definitions, starting from the sphere, as the
locus of congruent point-pairs when one point of the pair is fixed,
and hence obtaining the circle and the straight line. From the above
it follows, that the sphere so defined already involves a curve
between the points of the point-pair, by which various point-pairs
can be known as congruent; and it will appear, as we proceed, that
this curve must be a straight line. Frischauf's definition by means
of the sphere involves, therefore, a vicious circle, since the sphere
presupposes the straight line, as the test of congruent point-pairs.
[174] Nor in any argument which, like those of projective Geometry,
avoids the notion of magnitude or distance altogether. It follows
that the propositions of projective Geometry apply, without reserve,
to spherical space, since the exception to the axiom of the straight
line arises only on metrical ground.
[175] Psychology, Vol. II. pp. 149-150.
[176] This step in the argument has been put very briefly, since it
is a mere repetition of the corresponding argument in Section A, and
is inserted here only for the sake of logical completeness. See § 137
ff.
[177] Cf. Hannequin, Essai critique sur l'hypothèse des atomes,
Paris, 1895, passim.
CHAPTER IV.
PHILOSOPHICAL CONSEQUENCES.
=180.= In the present chapter, we have to discuss two questions
which, though scarcely geometrical, are of fundamental importance
to the theory of Geometry propounded above. The first of these
questions is this: What relation can a purely logical and deductive
proof, like that from the nature of a form of externality, bear to an
experienced subject-matter such as space? You have merely framed a
general conception, I may be told, containing space as a particular
species, and you have then shown, what should have been obvious from
the beginning, that this general conception contained some of the
attributes of space. But what ground does this give for regarding
these attributes as _à priori_? The conception Mammal has some of the
attributes of a horse; but are these attributes therefore _à priori_
adjectives of the horse? The answer to this obvious objection is so
difficult, and involves so much general philosophy, that I have kept
it for a final chapter, in order not to interrupt the argument on
specially geometrical topics.
=181.= I have already indicated, in general terms, the ground for
regarding as _à priori_ the properties of any form of externality.
This ground is transcendental, _i.e._ it is to be found in the
conditions required for the possibility of experience. The form of
externality, like Riemann's manifolds, is a general class-conception,
including time as well as Euclidean and non-Euclidean spaces. It
is not motived, however, like the manifolds, by a _quantitative_
resemblance to space, but by the fact that it fulfils, if it has
more than one dimension, all those functions which, in our actual
world, are fulfilled by space. But a form of externality, in order
to accomplish this, must be, not a mere conception, but an actually
experienced intuition. Hence the conception of such a form is the
general _conception_, containing under it every logically possible
_intuition_ which can fulfil the function actually fulfilled by
space. And this function is, to render possible experience of
diverse but interrelated things. Some form in sense-perception,
then, whose conception is included under our form of externality,
is _à priori_ necessary to experience of diversity in relation, and
without experience of this, we should, as modern logic shows, have no
experience at all. This still leaves untouched the relation of the
_à priori_ to the subjective: the form of externality is necessary
to experience, but is not, _on that account_, to be declared purely
subjective. Of course, necessity for experience can only arise from
the nature of the mind which experiences; but it does not follow that
the necessary conditions could be fulfilled, unless the objective
world had certain properties. The _ground_ of necessity, we may
safely say, arises from the mind; but it by no means follows that
the _truth_ of what is necessary depends only on the constitution of
the mind. Where this is not the case, our conclusion, when a piece
of knowledge has been declared _à priori_, can only be: Owing to the
constitution of the _mind_, experience will be impossible unless the
_world_ accepts certain adjectives.
Such, in outline, will be the argument of the first half of this
chapter, and such will be the justification for regarding as _à
priori_ those axioms of Geometry, which were deduced above from the
conception of a form of externality. For these axioms, and these
only, are necessarily true of any world in which experience is
possible.
=182[178].= The view suggested has, obviously, much in common with
that of the Transcendental Aesthetic. Indeed the whole of it, I
believe, can be obtained by a certain limitation and interpretation
of Kant's classic arguments. But as it differs, in many important
points, from the conclusions aimed at by Kant, and as the agreement
may easily seem greater than it is, I will begin by a brief
comparison, and endeavour, by reference to authoritative criticisms,
to establish the legitimacy of my divergence from him.
=183.= In the first place, the psychological element is much larger
in Kant's thesis than in mine. I shall contend, it is true, that a
form of externality, if it is to do its work, must not be a mere
conception or a mere inference, but must be a given element in
sense-perception--not, of course, originally given in isolation,
but discoverable, through analysis, by attention to the object of
sense-perception[179]. But Kant contended, not only that this element
is given, but also that it is subjective. Space, for him, is, on the
one hand, not conceptual, but on the other hand, not sensational.
It forms, for him, no part of the data of sense, but is added by a
subjective intuition, which he regards as not only logically, but
psychologically, prior to objects in space[180].
This part of Kant's argument is wholly irrelevant for us. Whether a
form of externality be given in sense, or in a pure intuition, is for
us unimportant, since we neglect the question as to the connection
of the _à priori_ and the subjective; while the temporal priority of
space to objects in it has been generally recognized as irrelevant
to Epistemology, and has often been regarded as forming no part of
Kant's thesis[181]. If we call intuitional whatever is given in
sense-perception, then we may contend that a form of externality must
be intuitional; but whether it is a pure intuition, in Kant's sense,
or not, is irrelevant to us, as is its priority to the objects in it.
That the non-sensational nature of space is no essential part of
Kant's _logical_ teaching, appears from an examination of his
argument. He has made, in the introduction, the purely logical
distinction of matter and form, but has given to this distinction,
in the very moment of suggesting it, a psychological implication.
This he does by the assertion that the form, in which the matter
of sensations is ordered, cannot itself be sensational. From this
assumption it follows, of course, that space cannot be sensational.
But the assumption is totally unsupported by argument, being set
forth, apparently, as a self-evident axiom; it has been severely
criticized by Stumpf[182] and others[183], and has been described by
Vaihinger as a fatal _petitio principii_[184]; it is irrelevant to
the logical argument, when this argument is separated, as we have
separated it, from all connection with psychological subjectivity;
and finally, it leaves us a prey to psychological theories of space,
which have seemed, of late, but little favourable to the pure Kantian
doctrine.
=184.= We have a right, therefore, in an epistemological inquiry, to
neglect Kant's psychological teaching--in so far, at any rate, as it
distinguishes spatial intuition from sensation--and attend rather to
the logical aspect alone. That part of his psychological teaching,
which maintains that space is not a mere conception, is, with certain
limitations, sufficiently evident as applied to actual space; but for
us, it must be transformed into a much more difficult thesis, namely,
that _no_ form of externality, which renders experience of diversity
in relation possible, can be merely conceptual. This question, to
which we must return later, is no longer psychological, but belongs
wholly to Epistemology.
=185.= What, then, remains the kernel, for our purposes, of Kant's
first argument for the apriority of space? His argument, in the form
in which he gave it, is concerned with the eccentric projection
of sensations. In order that I may refer sensations, he says,
to something outside myself, I must already have the subjective
space-form in the mind. In this shape, as Vaihinger points out
(Commentar, II. pp. 69, 165), the argument rests on a _petitio
principii_, for only if sensations are necessarily non-spatial does
their projection demand a subjective space-form. But, further, is the
logical apriority of space concerned with the externality of things
to ourselves?
Space _seems_ to perform two functions: on the one hand, it reveals
things, by the eccentric projection of sensations, as external
to the self, while, on the other hand, it reveals simultaneously
presented things as mutually external. These two functions, though
often treated as coordinate and almost equivalent[185], seem to me
widely different. Before we discuss the apriority of space, we must
carefully distinguish, I think, between these two functions, and
decide which of them we are to argue about.
Now externality to the Self, it would seem, must necessarily raise
the whole question of the nature and limits of the Ego, and what is
more, it cannot be derived from spatial presentation, unless we give
the Self a definite position in space. But things acquire a position
in space only when they can appear in sense-perception; we are
forced, therefore, if we adopt this view of the function of space, to
regard the Self as a phenomenon presented to sense-perception. But
this reduces externality to the Self to externality to the body. The
body, however, is a presented object like any other, and externality
of objects to it is, therefore, a special case of the mutual
externality of presented things. Hence we cannot regard space as
giving, primarily at any rate, externality to the Self, but only the
mutual externality of the things presented to sense-perception[186].
=186.= This, then, is the kind of externality we are to expect
from space, and our question must be: Would the existence of
diverse but interrelated things be unknowable, if there were not,
in sense-perception, some form of externality? This is the crucial
question, on which turns the apriority of our form, and hence of the
necessary axioms of Geometry.
=187.= The converse argument to mine, the argument from the
spatio-temporal element in perception to a world of interrelated
but diverse things, is developed at length in Bradley's Logic. It
is put briefly in the following sentence (p. 44, note): "If space
and time are continuous, and if all appearance must occupy some time
or space--and it is not hard to support both these _theses_--we can
at once proceed to the conclusion, no mere particular exists. Every
phenomenon will exist in more times or spaces than one; and against
that diversity will be itself an universal[187]." The importance of
this fact appears, when we consider that, if any _mere_ particular
existed, all judgment and inference as to that particular would be
impossible, since all judgment and inference necessarily operate
by means of universal. But all reality is constructed from the
_This_ of immediate presentation, from which judgment and inference
necessarily spring. Owing, however, to the continuity and relativity
of space and time, no _This_ can be regarded either as simple or
as self-subsistent. Every _This_, on the one hand, can be analyzed
into _Thises_, and on the other hand, is found to be necessarily
related to other things, outside the limits of the given object of
sense-perception. This function of space and time is presupposed in
the following statement from Bosanquet's Logic (Vol. I. pp. 77-78):
"Reality is given for me _in_ present sensuous perception, and _in_
the immediate feeling of my own sentient existence that goes with
it. The real world, as a definite organized system, is _for me_ an
extension of this present sensation and self feeling by means of
judgment, and it is the essence of judgment to effect and sustain
such an extension.... The subject in every judgment of Perception
is some given spot or point in sensuous contact with the percipient
self. But, as all reality is continuous, the subject is not _merely_
this given spot or point."
=188.= This doctrine of Bradley and Bosanquet is the converse of
the epistemological doctrine I have to advocate. Owing to the
continuity and relativity of space and time, they say, we are able
to construct a systematic world, by judgment and inference, out of
that fragmentary and yet necessarily complex existence which is
given in sense-perception. My contention is, conversely, that since
all knowledge is necessarily derived by an extension of the _This_
of sense-perception, and since such extension is only possible if
the _This_ has that fragmentary and yet complex character conferred
by a form of externality, therefore some form of externality,
given with the _This_, is essential to all knowledge, and is thus
logically _à priori_. Bradley's argument, if sound, already proves
this contention; for while, on the one hand, he uses no properties of
space and time but those which belong to every form of externality,
he proves, on the other hand, that judgment and inference require
the _This_ to be neither single nor self-subsistent. But I will
endeavour, since the point is of fundamental importance, to
reproduce the proof, in a form more suited than Bradley's to the
epistemological question.
=189.= The essence of my contention is that, if experience is to be
possible, every sensational _This_ must, when attended to, be found,
on the one hand, resolvable into _Thises_, and on the other hand
dependent, for some of its adjectives, on external reference. The
second of these theses follows from the first, for if we take one
of the _Thises_ contained in the first _This_, we get a new _This_
necessarily related to the other _Thises_ which make up the original
_This_. I may, therefore, confine myself to the first proposition,
which affirms that the object of perception must contain a diversity,
not only of conceptual content, but of existence, and that this can
only be known if sense-perception contains, as an element, some form
of externality.
My premiss, in this argument, is that all knowledge involves a
recognition of diversity in relation, or, if we prefer it, of
identity in difference. This premiss I accept from Logic, as
resulting from the analysis of judgment and inference. To prove
such a premiss, would require a treatise on Logic; I must refer the
reader, therefore, to the works of Bradley and Bosanquet on the
subject. It follows at once, from my premiss, that knowledge would be
impossible, unless the object of attention could be complex, _i.e._
not a _mere_ particular. Now could the mental object--_i.e._, in this
connection, the object of a cognition--be complex, if the object of
immediate perception were always simple?
=190.= We might be inclined, at first sight, to answer this question
affirmatively. But several difficulties, I think, would prevent
such an answer. In the first place, knowledge must start from
perception. Hence, either we could have no knowledge except of our
present perception, or else we must be able to contrast and compare
it with some other perception. Now in the first case, since the
present perception, by hypothesis, is a mere particular, knowledge
of it is impossible, according to our premiss. But in the second
case, the other perception, with which we compare our first, must
have occurred at some other time, and with time, we have at once a
form of externality. But what is more, our present perception is
no longer a mere particular. For the power of comparing it with
another perception involves a point of identity between the two, and
thus renders both complex. Moreover, time must be continuous, and
the present, as Bradley points out, is no mere point of time[188].
Thus our present perception contains the complexity involved in
duration throughout the specious present: its mere particularity
and its simplicity are lost. Its self-subsistence is also lost, for
beyond the specious present, lie the past and the future, to which
our present perception thus unavoidably refers us. Time at least,
therefore, is essential to that identity in difference, which all
knowledge postulates.
=191.= But we have derived, from all this, no ground for affirming a
multiplicity of real things, or a form of externality of more than
one dimension, which, we saw, was necessary for the truth of two out
of our three axioms. This brings us to the question: Have we enough,
with time alone as a form of externality, for the possibility of
knowledge?
This question we must, I think, answer in the negative. With time
alone, we have seen, our presented object must be complex, but its
complexity must, if I may use such a phrase, be merely adjectival.
Without a second form of externality, only one thing can be given at
one moment[189], and this one thing, therefore, must constitute the
whole of our world. The object of past perception must--since our one
thing has nothing external to it, by which it could be created or
destroyed--be regarded as the same thing in a different state. The
complexity, therefore, will lie only in the changing states of our
one thing--it will be adjectival, not substantival. Moreover we have
the following dilemma: Either the one thing must be ourselves, or
else self-consciousness could never arise. But the chief difficulty
of such a world would lie in the changes of the thing. What could
cause these changes, since we should know of nothing external to our
thing? It would be like a Leibnitzian monad, without any God outside
it to prearrange its changes. Causality, in such a world, could not
be applied, and change would be wholly inexplicable.
Hence we require also the possibility of a diversity of
simultaneously existing things, not merely of successive adjectives;
and this, we have seen, cannot be given by time alone, but only by
a form of externality for simultaneous parts of one presentation.
We could never, in other words, infer the existence of diverse but
interrelated things, unless the object of sense-perception could have
substantival complexity, and for such complexity we require a form
of externality other than time. Such a form, moreover, as was shown
in Chapter III., Section A (§ 135), can only fulfil its functions
if it has more than one dimension. In our actual world, this form
is given by space; in any world, knowable to beings with our laws
of thought, some such form, as we have now seen, must be given in
sense-perception.
This argument may be briefly summed up, by assuming the doctrine
of Bradley, that all knowledge is obtained by inference from the
_This_ of sense-perception. For, if this be so, the _This_--in
order that inference, which depends on identity in difference, may
be possible at all--must itself be complex, and must, on analysis,
reveal adjectives having a reference beyond itself. But this, as was
shown above, can only happen by means of a form of externality. This
establishes the à priori axioms of Geometry, as necessarily having
existential import and validity in any intelligible world.
=192.= The above argument, I hope, has explained why I hold it
possible to deduce, from a mere conception like that of a form
of externality, the logical apriority of certain axioms as to
experienced space. The Kantian argument--which was correct, if our
reasoning has been sound, in asserting that real diversity, in our
actual world, could only be known by the help of space--was only
mistaken, so far as its purely logical scope extends, in overlooking
the possibility of other forms of externality, which could, if they
existed, perform the same task with equal efficiency. In so far as
space differs, therefore, from these other conceptions of possible
intuitional forms, it is a mere experienced fact, while in so far as
its properties are those which all such forms must have, it is _à
priori_ necessary to the possibility of experience.
I cannot hope, however, that no difficulty will remain, for the
reader, in such a deduction, from abstract conceptions, of the
properties of an actual _datum_ in sense-perception. Let us consider,
for example, such a property as impenetrability. To suppose two
things simultaneously in the same position in a form of externality,
is a logical contradiction; but can we say as much of actual space
and time? Is not the impossibility, here, a matter of experience
rather than of logic? Not if the above argument has been sound,
I reply. For in that case, we infer real diversity, _i.e._ the
existence of different things, only from difference of position in
space or time. It follows, that to suppose two things in the same
point of space and time, is still a logical contradiction: not
because we have constructed the data of sense out of logic, but
because logic is dependent, as regards its application, on the nature
of these data. This instance illustrates, what I am anxious to make
plain, that my argument has not attempted to construct the living
wealth of sense-perception out of "bloodless categories," but only to
point out that, unless sense-perception contained a certain element,
these categories would be powerless to grapple with it.
=193.= How we are to account for the fortunate realization of these
requirements--whether by a pre-established harmony, by Darwinian
adaptation to our environment, by the subjectivity of the necessary
element in sense-perception, or by a fundamental identity and unity
between ourselves and the rest of reality--is a further question,
belonging rather to metaphysics than to our present line of argument.
The _à priori_, we have said throughout, is that which is necessary
for the possibility of experience, and in this we have a purely
logical criterion, giving results which only Logic and Epistemology
can prove or disprove. What is subjective in experience, on the
contrary, is primarily a question for psychology, and should be
decided on psychological grounds alone. When these two questions have
been separately answered, but not till then, we may frame theories as
to the connection of the _à priori_ and the subjective; to allow such
theories to influence our decision, on either of the two previous
questions, is liable, surely, to confuse the issue, and prevent a
clear discrimination between fundamentally different points of view.
=194.= I come now to the second question with which this chapter
has to deal, the question, namely: What are we to do with the
contradictions which obtruded themselves in Chapter III., whenever we
came to a point which seemed fundamental? I shall treat this question
briefly, as I have little to add to answers with which we are all
familiar. I have only to prove, first, that the contradictions are
inevitable, and therefore form no objection to my argument; secondly,
that the first step in removing them is to restore the notion of
matter, as that which, in the data of sense-perception, is localized
and interrelated in space.
=195.= The contradictions in space are an ancient theme--as ancient,
in fact, as Zeno's refutation of motion. They are, roughly, of two
kinds, though the two kinds cannot be sharply divided. There are
the contradictions inherent in the notion of the continuum, and the
contradictions which spring from the fact that space, while it must,
to be knowable, be pure relativity, must also, it would seem, since
it is immediately experienced, be something more than mere relations.
The first class of contradictions has been encountered more
frequently in this essay, and is also, I think, the more definite,
and the more important for our present purpose. I doubt, however,
whether the two classes are really distinct; for any continuum,
I believe, in which the elements are not data, but intellectual
constructions resulting from analysis, can be shown to have the same
relational and yet not wholly relational character as belongs to
space.
The three following contradictions, which I shall discuss
successively, seem to me the most prominent in a theory of Geometry.
(1) Though the parts of space are intuitively distinguished, no
conception is adequate to differentiate them. Hence arises a
vain search for elements, by which the differentiation could be
accomplished, and for a whole, of which the parts of space are to be
components. Thus we get the point, or zero extension, as the spatial
element, and an infinite regress or a vicious circle in the search
for a whole.
(2) All positions being relative, positions can only be defined by
their relations, _i.e._ by the straight lines or planes through
them; but straight lines and planes, being all qualitatively similar,
can only be defined by the positions they relate. Hence, again, we
get a vicious circle.
(3) Spatial figures must be regarded as relations. But a relation
is necessarily indivisible, while spatial figures are necessarily
divisible _ad infinitum_.
=196.= (1) _Points._ The antinomy of the point--which arises wherever
a continuum is given, and elements have to be sought in it--is
fundamental to Geometry. It has been given, perhaps unintentionally,
by Veronese as the first axiom, in the form: "There are different
points. All points are identical" (_op. cit._ p. 226). We saw, in
discussing projective Geometry, that straight lines and planes must
be regarded, on the one hand as relations between points, and on the
other hand as made up of points[190]. We saw again, in dealing with
measurement, how space must be regarded as infinitely divisible,
and yet as mere relativity. But what is divisible and consists of
parts, as space does, must lead at last, by continued analysis, to
a simple and unanalyzable part, as the unit of differentiation. For
whatever can be divided, and has parts, possesses some thinghood,
and must, therefore, contain two ultimate units, the whole namely,
and the smallest element possessing thinghood. But in space this is
notoriously not the case. After hypostatizing space, as Geometry
is compelled to do, the mind imperatively demands elements, and
insists on having them, whether possible or not. Of this demand,
all the geometrical applications of the infinitesimal calculus are
evidence[191]. But what sort of elements do we thus obtain? Analysis,
being unable to find any earlier halting-place, finds its elements
in points, that is, in zero quanta of space. Such a conception is
a palpable contradiction, only rendered tolerable by its necessity
and familiarity. A point must be spatial, otherwise it would not
fulfil the function of a spatial element; but again it must contain
no space, for any finite extension is capable of further analysis.
Points can never be given in intuition, which has no concern with the
infinitesimal: they are a purely conceptual construction, arising
out of the need of terms between which spatial relations can hold.
If space be more than relativity, spatial relations must involve
spatial relata; but no relata appear, until we have analyzed our
spatial data down to nothing. The contradictory notion of the point,
as a thing in space without spatial magnitude, is the only outcome
of our search for spatial relata. This _reductio ad absurdum_ surely
suffices, by itself, to prove the essential relativity of space.
=197.= Thus Geometry is forced, since it wishes to regard space
as independent, to hypostatize its abstractions, and therefore to
invent a self-contradictory notion as the spatial element. A similar
absurdity appears, even more obviously, in the notion of a whole of
space. The antinomy may, therefore, be stated thus: Space, as we have
seen throughout, must, if knowledge of it is to be possible, be mere
relativity; but it must also, if _independent_ knowledge of it, such
as Geometry seeks, is to be possible, be something more than mere
relativity, since it is divisible and has parts. But we saw, in Chap.
III., Section A (§ 133) that knowledge of a form of externality must
be logically independent of the particular matter filling the form.
How then are we to extricate ourselves from this dilemma?
The only way, I think, is, not to make Geometry dependent on
Physics, which we have seen to be erroneous[192], but to give every
geometrical proposition a certain reference to matter in general. And
at this point an important distinction must be made. We have hitherto
spoken of space as relational, and of spatial figures as relations.
But space, it would seem, is rather relativity than relations--itself
not a relation, it gives the bare possibility of relations between
diverse things[193]. As applied to a spatial figure, which can only
arise by a differentiation of space, and hence by the introduction
of some differentiating matter, the word relation is, perhaps, less
misleading than any other; as applied to empty undifferentiated
space, it seems by no means an accurate description.
But a bare possibility cannot exist, or be given in sense-perception!
What becomes, then, of the arguments of the first part of this
chapter? I reply, it is not empty space, but spatial figures, which
sense-perception reveals, and spatial figures, as we have just seen,
involve a differentiation of space, and therefore a reference to the
matter which is in space. It is spatial figures, also, and not empty
space, with which Geometry has to deal. The antinomy discussed above
arises then--so it would seem--from the attempt to deal with empty
space, rather than with spatial figures and the matter to which they
necessarily refer.
=198.= Let us see whether, by this change, we can overcome the
antinomy of the point. Spatial figures, we shall now say, are
relations between the matter which differentiates empty space. Their
divisibility, which seemed to contradict their relational character,
may be explained in two ways: first, as holding of the figures
considered as parts of empty space, which is itself not a relation;
second, as denoting the possibility of continuous change in the
relation expressed by the spatial figure. These two ways are, at
bottom, the same; for empty space is a possibility of relations, and
the figure, when viewed in connection with empty space, thus becomes
a _possible_ relation, with which other possible relations may be
contrasted or compared. But the second way of regarding divisibility
is the better way, since it introduces a reference to the matter
which differentiates empty space, without which, spatial figures,
and therefore Geometry, could not exist. It is empty space, then--so
we must conclude--which gives rise to the antinomy in question; for
empty space is a bare possibility of relations, undifferentiated and
homogeneous, and thus wholly destitute of parts or of thinghood.
To speak of parts of a possibility is nonsense; the parts and
differentiations arise only through a reference to the matter which
is differentiated in space.
=199.= But what nature must we ascribe to this matter, which is
to be involved in all geometrical propositions? In criticizing
Helmholtz (Chap. II. § 73), it may be remembered, we decided that
Geometry refers to a peculiar and abstract kind of matter, which is
not regarded as possessing any causal qualities, as exerting or as
subject to the action of forces. And this is the matter, I think,
which we require for the needs of the moment. Not that we affirm,
of course, that actual matter can be destitute of the properties
with which Physics is cognizant, but that we abstract from these
properties, as being irrelevant to Geometry. All that we require, for
our immediate purpose, is a subject of that diversity which space
renders possible, or terms for those relations by which empty space,
if space is to be studied at all, must be differentiated. But how
must a matter, which is to fulfil this function, be regarded?
Empty space, we have said, is a possibility of diversity in
relation, but spatial figures, with which Geometry necessarily
deals, are the actual relations rendered possible by empty space.
Our matter, therefore, must supply the terms for these relations.
It must be differentiated, since such differentiation, as we have
seen, is the special work of space. We must find, therefore, in
our matter, that unit of differentiation, or atom[194], which in
space we could not find. This atom must be simple, _i.e._ it must
contain no real diversity; it must be a _This_ not resolvable into
_Thises_. Being simple, it can contain no relations within itself,
and consequently, since spatial figures are mere relations, it cannot
appear as a spatial figure; for every spatial figure involves some
diversity of matter. But our atom must have spatial relations with
other atoms, since to supply terms for these relations is its only
function. It is also capable of having these relations, since it is
differentiated from other atoms. Hence we obtain an unextended term
for spatial relations, precisely of the kind we require. So long as
we sought this term without reference to anything more than space,
the self-contradictory notion of the point was the only outcome
of our search; but now that we allow a reference to the matter
differentiated by space, we find at once the term which was needed,
namely, a non-spatial simple element, with spatial relations to other
elements. To Geometry such a term will appear, owing to its spatial
relations, as a point; but the contradiction of the point, as we now
see, is a result only of the undue abstraction with which Geometry
deals.
=200.= (2) _The circle in the definition of straight lines and
planes._ This difficulty need not long detain us, since we have
already, with the material atom, broken through the relativity
which caused our circle. Straight lines, in the purely geometrical
procedure, are defined only by points, and points only by straight
lines. But points, now, are replaced by material atoms: the duality
of points and lines, therefore, has disappeared, and the straight
line may be defined as the spatial relation between two unextended
atoms. These atoms have spatial adjectives, derived from their
relations to other atoms; but they have no _intrinsic_ spatial
adjectives, such as could belong to them if they had extension or
figure. Thus straight lines and planes are the true spatial units,
and points result only from the attempt to find, within space, those
terms for spatial relations which exist only in a more than spatial
matter. Straight lines, planes and volumes are the spatial relations
between two, three or four unextended atoms, and points are a merely
convenient geometrical fiction, by which possible atoms are replaced.
For, since space, as we saw, is a possibility, Geometry deals not
with actually realized spatial relations, but with the whole scheme
of possible relations.
=201.= (3) _Space is at once relational and more than relational._
We have already touched on the question how far space is other than
relations, but as this question is quite fundamental, as _relation_
is an ambiguous and dangerous word, as I have made constant use of
the relativity of space without attempting to define a relation, it
will be necessary to discuss this antinomy at length.
=202.= Now for this discussion it is essential to distinguish clearly
between empty space and spatial figures. Empty space, as a form
of externality, is not actual relations, but the possibility of
relations: if we ascribe existential import to it, as the ground,
in reality, of all diversity in relation, we at once have space as
something not itself relations, though giving the possibility of all
relations. In this sense, space is to be distinguished from spatial
order. Spatial order, it may be said, presupposes space, as that in
which this order is possible. Thus Stumpf says[195]: "There is no
order or relation without a positive absolute content, underlying
it, and making it possible to order anything in this manner. Why and
how should we otherwise distinguish one order from another?... To
distinguish different orders from one another, we must everywhere
recognize a particular absolute content, in relation to which the
order takes place. And so space, too, is not a mere order, but just
that by which the spatial order, side-by-sideness (_Nebeneinander_)
distinguishes itself from the rest."
May we not, then, resolve the antinomy very simply, by a reference to
this ambiguity of space? Bradley contends (Appearance and Reality,
pp. 36-7) that, on the one hand, space has parts, and is therefore
not mere relations, while on the other hand, when we try to say what
these parts are, we find them after all to be mere relations. But
cannot the space which has parts be regarded as empty space, Stumpf's
absolute underlying content, which is not mere relations, while the
parts, in so far as they turn out to be mere relations, are those
relations which constitute spatial order, not empty space? If this
can be maintained, the antinomy no longer exists.
But such an explanation, though I believe it to be a first step
towards a solution, will, I fear, itself demand almost as much
explanation as the original difficulty. For the connection of empty
space with spatial order is itself a question full of difficulty, to
be answered only after much labour.
=203.= Let us consider what this empty space is. (I speak of "empty"
space without necessarily implying the absence of matter, but only
to denote a space which is not a mere order of material things.)
Stumpf regards it as given in sense; Kant, in the last two arguments
of his metaphysical deduction, argues that it is an intuition, not
a concept, and must be known before spatial order becomes possible.
I wish to maintain, on the contrary, that it is wholly conceptual;
that space is given only as spatial order; that spatial relations,
being given, appear as more than mere relations, and so become
hypostatized; that when hypostatized, the whole collection of
them is regarded as contained in empty space; but that this empty
space itself, if it means more than the logical possibility of
space-relations, is an unnecessary and self-contradictory assumption.
Let us begin by considering Kant's arguments on this point.
Leibnitz had affirmed that space was only relations, while Newton
had maintained the objective reality of absolute space. Kant adopted
a middle course: he asserted absolute space, but regarded it as
purely subjective. The assertion of absolute space is the object of
his second argument; for if space were mere relations between things,
it would necessarily disappear with the disappearance of the things
in it; but this the second argument denies[196]. Now spatial order
obviously does disappear with matter, but absolute or empty space may
be supposed to remain. It is this, then, which Kant is arguing about,
and it is this which he affirms to be a pure intuition, necessarily
presupposed by spatial order[197].
=204.= But can we agree in regarding empty space, the "infinite
given whole," as really given? Must we not, in spite of Kant's
argument, regard it as wholly conceptual? It is not required, in
the first place, by the argument of the first half of this chapter,
which required only that every _This_ of sense-perception should
be resolvable into _Thises_, and thus involved only an order among
_Thises_, not anything given originally without reference to them
at all. In the second place, Kant's two arguments[198] designed to
prove that empty space is not conceptual, are inadequate to their
purpose. The argument that the parts of space are not contained
_under_ it, but _in_ it, proves certainly that space is not a general
conception, of which spatial figures are the instances; but it by
no means follows that empty space is not a conception. Empty space
is undifferentiated and homogeneous; parts of space, or spatial
figures, arise only by reference to some differentiating matter, and
thus belong rather to spatial order than to empty space. If empty
space be the pre-condition of spatial order, we cannot expect it
to be connected with spatial relations as genus with species. But
empty space may nevertheless be a universal conception; it may be
related to spatial order as the state to the citizens. These are not
instances of the state, but are contained in it; they also, in a
sense, presuppose it, for a man can only become a citizen by being
related to other citizens in a state[199].
The uniqueness of space, again, seems hardly a valid argument for
its intuitional nature; to regard it as an argument implies, indeed,
that all conceptions are abstracted from a series of instances--a
view which has been criticized in Chapter II. (§ 77), and need not be
further discussed here[200]. There is no ground, therefore, in Kant's
two arguments for the intuitional nature of empty space, which can be
maintained against criticism.
=205.= Another ground for condemning empty space is to be found in
the mathematical antinomies. For it is no solution, as Lotze points
out (Metaphysik, Bk. II. Chap. I., § 106), to regard empty space as
purely subjective: contradictions in a necessary subjective intuition
form as great a difficulty as in anything else. But these antinomies
arise only in connection with empty space, not with spatial order
as an aggregate of relations. For only when space is regarded as
possessed of some thinghood, can a whole or a true element be
demanded. This we have seen already in connection with the Point.
When space is regarded, so far as it is valid, as only spatial order,
unbounded extension and infinite divisibility both disappear. What
is divided is not spatial relations, but matter; and if matter, as
we have seen that Geometry requires, consists of unextended atoms
with spatial relations, there is no reason to regard matter either as
infinitely divisible, or as consisting of atoms of finite extension.
=206.= But whence arises, on this view, the paradox that we cannot
but regard space as having more or less thinghood, and as divisible
_ad infinitum_? This must be explained, I think, as a psychological
illusion, unavoidably arising from the fact that spatial relations
are immediately presented. They thus have a peculiar psychical
quality, as immediate experiences, by which quality they can be
distinguished from time-relations or any other order in which things
may be arranged. To Stumpf, whose problem is psychological, such a
psychical quality would constitute an absolute underlying content,
and would fully justify his thesis; to us, however, whose problem is
epistemological, it would not do so, but would leave the _meaning_ of
the spatial element in sense-perception free from any implication
of an absolute or empty space[201]. May we not, then, abandon empty
space, and say: Spatial order consists of _felt_ relations, and _quâ_
felt has, for Psychology, an existence not wholly resolvable into
relations, and unavoidably _seeming_ to be more than mere relations.
But when we examine the information, as to space, which we derive
from sense-perception, we find ourselves plunged in contradictions,
as soon as we allow this information to consist of more than
relations. This leaves spatial order alone in the field, and reduces
empty space to a mere name for the logical possibility of spatial
relations.
=207.= The apparent divisibility of the relations which constitute
spatial order, then, may be explained in two ways, though these are
at bottom equivalent. We may take the relation as considered in
connection with empty space, in which case it becomes more than a
relation; but being falsely hypostatized, it appears as a complex
thing, necessarily composed of elements, which elements, however,
nowhere emerge until we analyze the pseudo-thing down to nothing,
and arrive at the point. In this sense, the divisibility of spatial
relations is an unavoidable illusion. Or again, we may take the
relation in connection with the material atoms it relates. In this
case, other atoms may be imagined, differently localized by different
spatial relations. If they are localized on the straight line joining
two of the original atoms, this straight line appears as divided by
them. But the original relation is not really divided: all that has
happened is, that two or more equivalent relations have replaced
it, as two compounded relations of father and son may replace the
equivalent relation of grandfather and grandson. These two ways of
viewing the apparent divisibility are equivalent: for empty space, in
so far as it is not illusion, is a name for the aggregate of possible
space-relations. To regard a figure in empty space as divided,
therefore, means, if it means anything, to regard two or more other
possible relations as substituted for it, which gives the second way
of viewing the question.
The same reference to matter, then, by which the antinomy of the
Point was solved, solves also the antinomy as to the relational
nature of space. Space, if it is to be freed from contradictions,
must be regarded exclusively as spatial order, as relations between
unextended material atoms. Empty space, which arises, by an
inevitable illusion, out of the spatial element in sense-perception,
may be regarded, if we wish to retain it, as the bare principle of
relativity, the bare logical possibility of relations between diverse
things. In this sense, empty space is wholly conceptual; spatial
order alone is immediately experienced.
=208.= But in what sense does spatial order consist of relations?
We have hitherto spoken of externality as a relation, and in a
sense such a manner of speaking is justified. Externality, when
predicated of anything, is an adjective of that thing, and implies
a reference to some other thing. To this extent, then, externality
is analogous to other relations; and only to this extent, in our
previous arguments, has it been regarded as a relation. But when we
take account of further qualities of relations, externality begins to
appear, not so much as a relation, but rather as a necessary aspect
or element in every relation. And this is borne out by the necessity,
for the existence of relations, of some given form of externality.
Every relation, we may say, involves a diversity between the
related terms, but also some unity. Mere diversity does not give
a ground for that interaction, and that interdependence, which a
relation requires. Mere unity leaves the terms identical, and thus
destroys the reference of one to another required for a relation.
Mere externality, taken in abstraction, gives only the element of
diversity required for a relation, and is thus more abstract than any
actual relation. But mere diversity does not give that indivisible
whole of which any actual relation must consist, and is thus, when
regarded abstractly, not subject to the restrictions of ordinary
relations.
But with mere diversity, we seem to have returned to empty space, and
abandoned spatial order. Mere diversity, surely, is either complete
or non-existent; degrees of diversity, or a quantitative measure of
it, are nonsense. We cannot, therefore, reduce spatial order to mere
diversity. Two things, if they occupy different positions in space,
are necessarily diverse, but are as necessarily something more;
otherwise spatial order becomes unmeaning.
Empty space, then, in the above sense of the possibility of spatial
relations, contains only one aspect of a relation, namely the aspect
of diversity; but spatial order, by its reference to matter, becomes
more concrete, and contains also the element of unity, arising out
of the connection of the different material atoms. Spatial order,
then, consists of relations in the ordinary sense; its merely
spatial element, however--if one may make such a distinction--the
element, that is, which can be abstracted from matter and regarded
as constituting empty space, is only one aspect of a relation, but
an aspect which, in the concrete, must be inseparably bound up
with the other aspect. Here, once more, we see the ground of the
contradictions in empty space, and the reason why spatial order is
free from these contradictions.
_Conclusion._
=209.= We have now completed our review of the foundations of
Geometry. It will be well, before we take leave of the subject,
briefly to review and recapitulate the results we have won.
In the first chapter, we watched the development of a branch of
Mathematics designed, at first, only to establish the logical
independence of Euclid's axiom of parallels, and the possibility
of a self-consistent Geometry which dispensed with it. We found
the further development of the subject entangled, for a while, in
philosophical controversy; having shown one axiom to be superfluous,
the geometers of the second period hoped to prove the same conclusion
of all the others, but failed to construct any system free from three
fundamental axioms. Being concerned with analytical and metrical
Geometry, they tended to regard Algebra as _à priori_, but held that
those properties of spatial magnitudes, which were not deducible from
the laws of Algebra, must be empirical. In all this, they aimed as
much at discrediting Kant as at advancing Mathematics. But with the
third period, the interest in Philosophy diminishes, the opposition
to Euclid becomes less marked, and most important of all, measurement
is no longer regarded as fundamental, and space is dealt with by
descriptive rather than quantitative methods. But nevertheless,
three axioms, substantially the same as those retained in the second
period, are still retained by all geometers.
In the second chapter, we endeavoured, by a criticism of some
geometrical philosophies, to prepare the ground for a constructive
theory of Geometry. We saw that Kant, in applying the argument of
the Transcendental Aesthetic to space, had gone too far, since
its logical scope extended only to some form of externality in
general. We saw that Riemann, Helmholtz and Erdmann, misled by the
quantitative bias, overlooked the qualitative substratum required
by all judgments of quantity, and thus mistook the direction in
which the necessary axioms of Geometry are to be found. We rejected,
also, Helmholtz's view that Geometry depends on Physics, because we
found that Physics must assume a knowledge of Geometry before it
can become possible. But we admitted, in Geometry, a reference to
matter--not, however, to matter as empirically known in Physics, but
to a more abstract matter, whose sole function is to appear in space,
and supply the terms for spatial relations. We admitted, however,
besides this, that all _actual_ measurement must be effected by means
of _actual_ matter, and is only empirically possible, through the
empirical knowledge of approximately rigid bodies. In criticizing
Lotze, we saw that the most important sense, in which non-Euclidean
spaces are possible, is a philosophical sense, namely, that they are
not condemned by any _à priori_ argument as to the necessity of space
for experience, and that consequently, if they are not affirmed,
this must be on empirical grounds alone. Lotze's strictures on the
mathematical procedure of Metageometry we found to be wholly due to
ignorance of the subject.
Proceeding, in the third chapter, to a constructive theory of
Geometry, we saw that projective Geometry, which has no reference to
quantity, is necessarily true of any form of externality. Its three
axioms--homogeneity, dimensions, and the straight line--were all
deduced from the conception of a form of externality, and, since some
such form is necessary to experience, were all declared _à priori_.
In metrical Geometry, on the contrary, we found an empirical element,
arising out of the alternatives of Euclidean and non-Euclidean space.
Three _à priori_ axioms, common to these spaces, and necessary
conditions of the possibility of measurement, still remained; these
were the axiom of Free Mobility, the axiom that space has a finite
integral number of dimensions, and the axiom of distance. Except for
the new idea of motion, these were found equivalent to the projective
triad, and thus necessarily true of any form of externality. But
the remaining axioms of Euclid--the axiom of three dimensions, the
axiom that two straight lines can never enclose a space, and the
axiom of parallels--were regarded as empirical laws, derived from the
investigation and measurement of our actual space, and true only, as
far as the last two are concerned, within the limits set by errors of
observation.
In the present chapter, we completed our proof of the apriority
of the projective and equivalent metrical axioms, by showing the
necessity, for experience, of some form of externality, given by
sensation or intuition, and not merely inferred from other data.
Without this, we said, a knowledge of diverse but interrelated
things, the corner-stone of all experience, would be impossible.
Finally, we discussed the contradictions arising out of the
relativity and continuity of space, and endeavoured to overcome
them by a reference to matter. This matter, we found, must consist
of unextended atoms, localized by their spatial relations, and
appearing, in Geometry, as points. But the non-spatial adjectives
of matter, we contended, are irrelevant to Geometry, and its
causal properties may be left out of account. To deal with the new
contradictions, involved in such a notion of matter, would demand
a fresh treatise, leading us, through Kinematics, into the domains
of Dynamics and Physics. But to discuss the special difficulties
of space is all that is possible in an essay on the Foundations of
Geometry.
FOOTNOTES:
[178] Compare, with the following paragraphs, the admirable
discussion in Mr Hobhouse's Theory of Knowledge (Methuen 1896), Part
I. Chapter II.
[179] I speak of sense-perception instead of sensation, so as not to
prejudge the issue as to the sensational nature of space.
[180] See Vaihinger's Commentar, II. pp. 86-7, 168-171.
[181] See Caird, Critical Philosophy of Kant, Vol. I. p. 287.
[182] Ursprung der Raumvorstellung, pp. 12-30.
[183] See the references in Vaihinger's Commentar, II. p. 76 ff.
[184] Commentar, II. p. 71 ff.
[185] _E.g._ by Caird, _op. cit._ Vol. I. p. 286.
[186] I have no wish to deny, however, that space is essential in the
subsequent distinction of Self and not-Self.
[187] See also Book I. Chap II. passim; especially p. 51 ff. and pp.
70-1.
[188] Logic, p. 51 ff.
[189] For the _This_, on such a hypothesis, has a purely temporal
complexity, and is not resolvable into coexisting _Thises_.
[190] Chapter III. Section A, (§ 131).
[191] Cf. Hannequin, Essai critique sur l'hypothèse des atomes, Paris
1895, Chap. I. Section III.; especially p. 43.
[192] See Chapter II. § 69 ff.
[193] See third antinomy below, § 201 ff.
[194] This atom, of course, must not be confounded with the atom of
Chemistry.
[195] Ursprung der Raumvorstellung, p. 15.
[196] See Vaihinger's Commentar, II. pp. 189-190.
[197] See ibid. p. 224 ff. for Kant's inconsistencies on this point.
[198] The fourth and fifth in the first edition, the third and fourth
in the second.
[199] Cf. Vaihinger's Commentar, II. p. 218.
[200] Cf. Vaihinger's Commentar, II. p. 207.
[201] Cf. James, Psychology, Vol. II., p. 148 ff.
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apodeictic; premisses.
§ 82, 'so Erdmannn confidently' replaced by 'so Erdmann confidently'.
§ 150 Footnote [157], 'Delboeuf' replaced by 'Delbœuf' for consistency.
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§ 159 Footnote [167], 'Delboeuf' replaced by 'Delbœuf' for consistency.
§ 204, 'and homogenous;' replaced by 'and homogeneous;'.