curvature are liable to important differences. For example, the
cylinder is a surface of zero curvature, but since its lines of
curvature in one direction are finite, its Geometry coincides with
that of the plane only for lengths smaller than the circumference of
its generating circle (see Veronese, op. cit. p. 644). Two geodesics
on a cylinder may meet in many points. For surfaces of zero curvature
on which this is not possible, the identity with the plane may
be allowed to stand. Otherwise, the identity extends only to the
properties of figures not exceeding a certain size.
[25] For we may consider two different parts of the same surface as
corresponding parts of different surfaces; the above proposition then
shows that a figure can be reproduced in one part when it has been
drawn in another, if the measures of curvature correspond in the two
parts.
[26] Crelle, Vols, XIX., XX., 1839-40.
[27] In this formula, _u_, _v_ may be the lengths of lines, or the
angles between lines, drawn on the surface, and having thus no
necessary reference to a third dimension.
[28] In what follows, I have given rather Klein's exposition of
Riemann, than Riemann's own account. The former is much clearer
and fuller, and not substantially different in any way. V. Klein,
Nicht-Euklid, I. pp. 206 ff.
[29] See §§ 69-73.
[30] Grundlagen der Geometrie, I. and II., Leipziger Berichte, 1890;
v. end of present chapter, § 45.
[31] Nicht-Euklid, I. pp. 258-9.
[32] Giornale di Matematiche, Vol. VI., 1868. Translated into
French by J. Hoüel in the "Annales Scientifiques de l'École Normale
Supérieure," Vol. VI. 1869.
[33] Crelle's Journal, Vols. XIX. XX., 1839-40.
[34] Nicht-Euklid, I. p. 190.
[35] This article is more trigonometrical and analytical than the
German book, and therefore makes the above interpretation peculiarly
evident.
[36] Such surfaces are by no means particularly remote. One of them,
for example, is formed by the revolution of the common Tractrix
x = asin φ, y = a(log tan φ/2 + cos φ).
[37] "Teoria fondamentale degli spazii di curvatura costanta," Annali
di Matematica, II. Vol. 2, 1868-9. Also translated by J. Hoüel, _loc.
cit._
[38] See Klein, Nicht-Euklid, I. p. 47 ff., and the references there
given.
[39] See quotation below, from his British Association Address.
[40] Compare the opening sentence, due to Cayley, of Salmon's Higher
Plane Curves.
[41] V. Nicht-Euklid, I. Chaps. I. and II.
[42] See p. 9 of Cayley's address to the Brit. Ass. 1883. Also a
quotation from Klein in Erdmann's Axiome der Geometrie, p. 124 note.
[43] Nature, Vol. XLV. p. 407.
[44] Nicht-Euklid, I. p. 200.
[45] I.e. the equation _AB_ + _BC_ = _AC_, for three points in one
straight line.
[46] The formula substituted by Klein for Cayley's inverse sine or
cosine. The two are equivalent, but Klein's is mathematically much
the more convenient.
[47] Elements of Projective Geometry, Second Edition, Oxford, 1893,
Chap. IX.
[48] Chap. III. Section B.
[49] See Nicht-Euklid, I. p. 338 ff.
[50] See his Geometrie der Lage, § 8, Harmonische Gebilde.
[51] The anharmonic ratio of four numbers, _p_, _q_, _r_, _s_, is
defined as
(p - q).(r - s) / (p - r).(q - s).
[52] _I.e._ as transformable into each other by a collineation. See
Chap. III. Sec. A, § 110.
[53] See Chap. III. Sec. A.
[54] It follows from this, that the reduction of metrical to
projective properties, even when, as in hyperbolic Geometry, the
Absolute is real, is only apparent, and has a merely technical
validity.
[55] Sir R. Ball does not regard his non-Euclidean content as a
possible space (_v. op. cit._ p. 151). In this important point I
disagree with his interpretation, holding such a content to be a
space as possible, _à priori_, as Euclid's, and perhaps actually true
within the margin due to errors of observation.
[56] See Nicht-Euklid, I. p. 97 ff. and p. 292 ff.
[57] Newcomb says (_loc. cit._ p. 293): "The system here set forth is
founded on the following three postulates.
"1. I assume that space is triply extended, unbounded, without
properties dependent either on position or direction, and possessing
such planeness in its smallest parts that both the postulates of the
Euclidean Geometry, and our common conceptions of the relations of
the parts of space are true for every indefinitely small region in
space.
"2. I assume that this space is affected with such curvature that a
right line shall always return into itself at the end of a finite and
real distance 2_D_ without losing, in any part of its course, that
symmetry with respect to space on all sides of it which constitutes
the fundamental property of our conception of it.
"3. I assume that if two right lines emanate from the same point,
making the indefinitely small angle _a_ with each other, their
distance apart at the distance _r_ from the point of intersection
will be given by the equation
s = 2aD/π sin rπ/2D.
The right line thus has this property in common with the Euclidean
right line that two such lines intersect only in a single point.
It may be that the number of points in which two such lines can
intersect admit of being determined from the laws of curvature,
but not being able so to determine it, I assume as a postulate the
fundamental property of the Euclidean right line."
It is plain that in the absence of the determination spoken of, the
possibility of elliptic space is not established. It may be possible,
for example, to prove that, in a space where there is a maximum to
distance, there must be an infinite number of straight lines joining
two points of maximum distance. In this event, elliptic space would
become impossible.
[58] For an elucidation of this term, see Klein, Nicht-Euklid, I. p.
99 ff.
[59] Cf. p. 9 of Report: "My own view is that Euclid's twelfth axiom,
in Playfair's form of it, does not need demonstration, but is part
of our notion of space, of the physical space of our experience,
but which is the representation lying at the bottom of all external
experience."
[60] The exception to this axiom, in spherical space, presupposes
metrical Geometry, and does not destroy the validity of the axiom for
projective Geometry. See Chap. III. Sec. B, § 171.
[61] Mathematicians of Lie's school have a habit, at first somewhat
confusing, of speaking of motions of space instead of motions of
bodies, as though space as a whole could move. All that is meant is,
of course, the equivalent motion of the coordinate axes, _i.e._ a
change of axes in the usual elementary sense.
[62] "Ueber die Grundlagen der Geometrie," Leipziger Berichte, 1890.
The problem of these two papers is really metrical, since it is
concerned, not with collineations in general, but with motions. The
problem, however, is dealt with by the projective method, motions
being regarded as collineations which leave the Absolute unchanged.
It seemed impossible, therefore, to discuss Lie's work, until some
account had been given of the projective method.
[63] Lie's premisses, to be accurate, are the following:
Let
x{1} = f(x, y, z, a{1}, a{2}...)
x{2} = φ(x, y, z, a{1}, a{2}...)
x{3} = ψ(x, y, z, a{1}, a{2}...)
give an infinite family of real transformations of space, as to which
we make the following hypotheses:
A. The functions f, φ, ψ, are _analytical_ functions of
x, y, z, a{1}, a{2}....
B. Two points x{1}y{1}z{1}, x{2}y{2}z{2} possess an
invariant, _i.e._
Ω(x{1}, y{1}, z{1}, x{2}, y{2}, z{2}) =
Ω(x{1′}, y{1′}, z{1′}, x{2′}, y{2′}, z{2′})
where x{1′}..., x{2′}..., are the transformed coordinates of
the two points.
C. Free Mobility: _i.e._, any point can be moved into any other
position; when one point is fixed, any other point of general
position can take up ∞^{2} positions; when two points are fixed, any
other of general position can take up ∞^{1} positions; when three, no
motion is possible--these limitations being results of the equations
given by the invariant Ω.
[64] On this point, cf. Klein, Höhere Geometrie, Göttingen, 1893, II.
pp. 225-244, especially pp. 230-1.
[65] Axiom II. of the metrical triad corresponds to Axiom III. of the
projective, and _vice versâ_.
[66] Cf. Helmholtz, Wiss. Abh. Vol. II. p. 640, note: "Die Bearbeiter
der Nicht-Euklidischen Geometrie (haben) deren objective Wahrheit nie
behauptet."
CHAPTER II.
CRITICAL ACCOUNT OF SOME PREVIOUS PHILOSOPHICAL THEORIES OF GEOMETRY.
=51.= We have now traced the mathematical development of the theory
of geometrical axioms, from the first revolt against Euclid to the
present day. We may hope, therefore, to have at our command the
technical knowledge required for the philosophy of the subject. The
importance of Geometry, in the theories of knowledge which have
arisen in the past, can scarcely be exaggerated. In Descartes, we
find the whole theory of method dominated by analytical Geometry, of
whose fruitfulness he was justly proud. In Spinoza, the paramount
influence of Geometry is too obvious to require comment. Among
mathematicians, Newton's belief in absolute space was long supreme,
and is still responsible for the current formulation of the laws of
motion. Against this belief on the one hand, and against Leibnitz's
theory of space on the other, and not, as Caird has pointed out[67],
against Hume's empiricism, was directed that keystone of the Critical
Philosophy, the Kantian doctrine of space. Thus Geometry has been,
throughout, of supreme importance in the theory of knowledge.
But in a criticism of representative modern theories of Geometry,
which is designed to be, not a history of the subject, but an
introduction to, and defence of, the views of the author, it will
not be necessary to discuss any more ancient theory than that of
Kant. Kant's views on this subject, true or false, have so dominated
subsequent thought, that whether they were accepted or rejected,
they seemed equally potent in forming the opinions, and the manner of
exposition, of almost all later writers.
Kant.
=52.= It is not my purpose, in this chapter, to add to the voluminous
literature of Kantian criticism, but only to discuss the bearing of
Metageometry on the argument of the Transcendental Aesthetic, and
the aspect under which this argument must be viewed in a discussion
of Geometry[68]. On this point several misunderstandings seem to me
to have had wide prevalence, both among friends and foes, and these
misunderstandings I shall endeavour, if I can, to remove.
In the first place, what does Kant's doctrine mean for Geometry?
Obviously not the aspect of the doctrine which has been attacked by
psychologists, the "Kantian machine-shop" as James calls it--at any
rate, if this can be clearly separated from the logical aspect. The
question whether space is given in sensation, or whether, as Kant
maintained, it is given by an intuition to which no external matter
corresponds, may for the present be disregarded. If, indeed, we
held the view which seems crudely to sum up the standpoint of the
Critique, the view that all certain knowledge is self-knowledge,
then we should be committed, if we had decided that Geometry was
apodeictic, to the view that space is subjective. But even then, the
psychological question could only arise when the epistemological
question had been solved, and could not, therefore, be taken into
account in our first investigation. The question before us is
precisely the question whether, or how far, Geometry is apodeictic,
and for the moment we have only to investigate this question, without
fear of psychological consequences.
=53.= Now on this question, as on almost all questions in the
Aesthetic or the Analytic, Kant's argument is twofold. On the one
hand, he says, Geometry is known to have apodeictic certainty:
therefore space must be _à priori_ and subjective. On the other hand,
it follows, from grounds independent of Geometry, that space is
subjective and _à priori_; therefore Geometry must have apodeictic
certainty. These two arguments are not clearly distinguished in the
Aesthetic, but a little analysis, I think, will disentangle them.
Thus in the first edition, the first two arguments deduce, from
non-geometrical grounds, the apriority of space; the third deduces
the apodeictic certainty of Geometry, and maintains, conversely,
that no other view can account for this certainty[69]; the last two
arguments only maintain that space is an intuition, not a concept.
In the second edition, the double argument is clearer, the apriority
of space being proved independently of Geometry in the metaphysical
deduction, and deduced from the certainty of Geometry, as the only
possible explanation of this, in the transcendental deduction. In the
Prolegomena, the latter argument alone is used, but in the Critique
both are employed.
=54.= Now it must be admitted, I think, that Metageometry has
destroyed the legitimacy of the argument from Geometry to space; we
can no longer affirm, on purely geometrical grounds, the apodeictic
certainty of Euclid. But unless Metageometry has done more than
this--unless it has proved, what I believe it alone cannot prove,
that Euclid has _not_ apodeictic certainty--then Kant's other line
of argument retains what force it may ever have had. The actual
space we know, it may say, is admittedly Euclidean, and is proved,
without any reference to Geometry, to be _à priori_; _hence_ Euclid
has apodeictic certainty, and non-Euclid stands condemned. To this
it is no answer to urge, with the Metageometers, that non-Euclidean
systems are _logically_ self-consistent; for Kant is careful to argue
that geometrical reasoning, by virtue of our intuition of space, is
synthetic, and cannot, though _à priori_, be upheld by the principle
of contradiction alone[70]. Unless non-Euclideans can prove, what
they have certainly failed to prove up to the present, that we can
frame an _intuition_ of non-Euclidean spaces, Kant's position cannot
be upset by Metageometry alone, but must also be attacked, if it is
to be successfully attacked, on its purely philosophical side.
=55.= For such an attack, two roads lie open: either we may disprove
the first two arguments of the Aesthetic, or we may criticize, from
the standpoint of general logic, the Kantian doctrine of synthetic
_à priori_ judgments and their connection with subjectivity. Both
these attacks, I believe, could be conducted with some success;
but if we are to disprove the apodeictic certainty of Geometry,
one or other is essential, and both, I believe, will be found only
partially successful. It will be my aim to prove, in discussing
these two lines of attack, (1) that the distinction of synthetic and
analytic judgments is untenable, and further, that the principle of
contradiction can only give fruitful results on the assumption that
experience in general, or, in a particular science, some special
branch of experience, is to be formally possible; (2) that the first
two arguments of the Transcendental Aesthetic suffice to prove,
not Euclidean space, but _some_ form of externality--which may be
sensational or intuitional, but not merely conceptual--a necessary
prerequisite of experience of an external world. In the third and
fourth chapters, I shall contend, as a result of these conclusions,
that those axioms, which Euclid and Metageometry have in common,
coincide with those properties of any form of externality which are
deducible, by the principle of contradiction, from the possibility of
experience of an external world. These properties, then, may be said,
though not quite in the Kantian sense, to be _à priori_ properties of
space, and as to these, I think, a modified Kantian position may be
maintained. But the question of the subjective or objective nature
of space may be left wholly out of account during the course of this
discussion, which will gain by dealing exclusively with logical, as
opposed to psychological points of view.
=56.= (1) _Kant's logical position._ The doctrine of synthetic and
analytic judgments--at any rate if this is taken as the corner-stone
of Epistemology--has been so completely rejected by most modern
logicians[71], that it would demand little attention here, but
for the fact that an enthusiastic French Kantian, M. Renouvier,
has recently appealed to it, with perfect confidence, on the very
question of Geometry[72]. And it must be owned, with M. Renouvier,
that if such judgments existed, in the Kantian sense, non-Euclidean
Geometry, which makes no appeal to intuition, could have nothing to
say against them. M. Renouvier's contention, therefore, forces us
briefly to review the arguments against Kant's doctrine, and briefly
to discuss what logical canon is to replace it.
Every judgment--so modern logic contends--is both synthetic and
analytic; it combines parts into a whole, and analyses a whole into
parts[73]. If this be so, the distinction of analysis and synthesis,
whatever may be its importance in pure Logic, can have no value in
Epistemology. But such a doctrine, it must be observed, allows full
scope to the principle of contradiction: this criterion, since all
judgments, in one aspect at least, are analytic, is applicable to
all judgments alike. On the other hand, the whole which is analysed
must be supposed already given, before the parts can be mutually
contradictory: for only by connection in a given whole can two parts
or adjectives be incompatible. Thus the principle of contradiction
remains barren until we already have some judgments, and even some
inference: for the parts may be regarded, to some extent, as an
inference from the whole, or _vice versâ_. When once the arch of
knowledge is constructed, the parts support one another, and the
principle of contradiction is the keystone: but until the arch is
built, the keystone remains suspended, unsupported and unsupporting,
in the empty air. In other words, knowledge once existent can be
analysed, but knowledge which should have to win every inch of the
way against a critical scepticism, could never begin, and could never
attain that circular condition in which alone it can stand.
But Kant's doctrine, if true, is designed to restrain a critical
scepticism even where it might be effective. Certain fundamental
propositions, he says, are not deducible from logic, _i.e._
their contradictories are not self-contradictory; they combine a
subject and predicate which cannot, in any purely logical way,
be shewn to have any connection, and yet these judgments have
apodeictic certainty. But concerning such judgments, Kant is
generally careful not to rely upon the mere subjective conviction
that they are undeniable: he proves, with every precaution, that
without them experience would be impossible. Experience consists
in the combination of terms which formal logic leaves apart, and
presupposes, therefore, certain judgments by which a framework is
made for bringing such terms together. Without these judgments--so
Kant contends--all synthesis and all experience would be impossible.
If, therefore, the detail of the Kantian reasoning be sound, his
results may be obtained by the principle of contradiction _plus_ the
possibility of experience, as well as by his distinction of synthetic
and analytic judgments.
Logic, at the present day, arrogates to itself at once a wider and a
narrower sphere than Kant allowed to it. Wider, because it believes
itself capable of condemning any false principle or postulate;
narrower, because it believes that its law of contradiction,
without a given whole or a given hypothesis, is powerless, and
that two terms, _per se_, though they may be different, cannot be
contradictories, but acquire this relation only by combination
in a whole about which something is known, or by connection with
a postulate which, for some reason, must be preserved. Thus no
judgment, _per se_, is either analytic or synthetic, for the
severance of a judgment from its context robs it of its vitality, and
makes it not truly a judgment at all. But in its proper context it
is neither purely synthetic nor purely analytic; for while it is the
further determination of a given whole, and thus in so far analytic,
it also involves the emergence of _new_ relations within this whole,
and is so far synthetic.
=57.= We may retain, however, a distinction roughly corresponding to
the Kantian _à priori_ and _à posteriori_, though less rigid, and
more liable to change with the degree of organisation of knowledge.
Kant usually endeavoured to prove, as observed above, that his
synthetic _à priori_ propositions were necessary prerequisites of
experience; now although we cannot retain the term synthetic, we
can retain the term _à priori_, for those assumptions, or those
postulates, from which alone the possibility of experience follows.
Whatever can be deduced from these postulates, without the aid of the
matter of experience, will also, of course, be _à priori_. From the
standpoint of general logic, the laws of thought and the categories,
with the indispensable conditions of their applicability, will be
alone _à priori_; but from the standpoint of any special science,
we may call _à priori_ whatever renders possible the experience
which forms the subject-matter of our science. In Geometry, to
particularize, we may call _à priori_ whatever renders possible
experience of externality as such.
It is to be observed that this use of the term is at once more
rationalistic and less precise than that of Kant. Kant would seem
to have supposed himself immediately aware, by inspection, that
some knowledge was apodeictic, and its subject-matter, therefore,
_à priori_: but he did not always deduce its apriority from any
further principle. Here, however, it is to be shown, before admitting
apriority, that the falsehood of the judgment in question would not
be effected by a mere change in the _matter_ of experience, but only
by a change which should render some branch of experience formally
impossible, _i.e._ inaccessible to our methods of cognition. The
above use is also less precise, for it varies according to the
specialization of the experience we are assuming possible, and with
every progress of knowledge some new connection is perceived, two
previously isolated judgments are brought into logical relation, and
the _à priori_ may thus, at any moment, enlarge its sphere, as more
is found deducible from fundamental postulates.
=58.= (2) _Kant's arguments for the apriority of space._ Having now
discussed the logical canon to be used as regards the _à priori_, we
may proceed to test Kant's arguments as regards space. The argument
from Geometry, as remarked above, is upset by Metageometry, at least
so far as those properties are concerned, which belong to Euclid
but not to non-Euclidean spaces; as regards the common properties
of both kinds of space, we cannot decide on their apriority till we
have discussed the consequences of denying them, which will be done
in Chapter III. As regards the two arguments which prove that space
is an intuition, not a concept, they would call for much discussion
in a special criticism of Kant, but here they may be passed by
with the obvious comment that infinite homogeneous Euclidean space
is a concept, not an intuition--a concept invented to explain an
intuition, it is true, but still a pure concept[74]. And it is this
pure concept which, in all discussions of Geometry, is primarily to
be dealt with; the intuition need only be referred to where it throws
light on the functions or the nature of the concept. The second
of Kant's arguments, that we can imagine empty space, though not
the absence of space, is false if it means a space without matter
anywhere, and irrelevant if it merely means a space between matters
and regarded as empty[75]. The only argument of importance, then,
is the first argument. But I must insist, at the outset, that our
problem is purely logical, and that all psychological implications
must be excluded to the utmost possible extent. Moreover, as will be
proved in Chapter IV., the proper function of space is to distinguish
between different presented things, not between the Self and the
object of sensation or perception. The argument then becomes the
following: consciousness of a world of mutually external things
demands, in presentations, a cognitive but non-inferential element
leading to the discrimination of the objects presented. This element
must be non-inferential, for from whatever number or combination of
presentations, which did not of themselves demand diversity in their
objects, I could never be led to infer the mutual externality of
their objects. Kant says: "In order that sensations may be ascribed
to something external to me ... and similarly in order that I may
be able to present them as outside and beside one another, ... the
presentation of space must be already present." But this goes rather
too far: in the first place, the question should be only as to the
mutual externality of presented things, not as to their externality
to the Self[76]; and in the second place, things will appear mutually
external if I have the presentation of _any_ form of externality,
whether Euclidean or non-Euclidean. Whatever may be true of the
_psychological_ scope of this argument--whose validity is here
irrelevant--the _logical_ scope extends, not to Euclidean space, but
only to any form of externality which could exist intuitively, and
permit knowledge, in beings with our laws of thought, of a world of
diverse but interrelated things.
Moreover externality, to render the scope of the argument wholly
logical, must not be left with a sensational or intuitional meaning,
though it must be supposed given in sensation or intuition. It must
mean, in this argument, the fact of Otherness[77], the fact of being
different from some other thing: it must involve the distinction
between different things, and must be that element, in a cognitive
state, which leads us to discriminate constituent parts in its
object. So much, then, would appear to result from Kant's argument,
that experience of diverse but interrelated things demands, as a
necessary prerequisite, some sensational or intuitional element, in
perception, by which we are led to attribute complexity to objects
of perception[78]; that this element, in its isolation may be called
a form of externality; and that those properties of this form, if
any such be found, which can be deduced from its mere function of
rendering experience of interrelated diversity possible, are to
be regarded as _à priori_. What these properties are, and how the
various lines of argument here suggested converge to a single result,
we shall see in Chapters III. and IV.
=59.= In the philosophers who followed Kant, Metaphysics, for the
most part, so predominated over Epistemology, that little was added
to the theory of Geometry. What was added, came indirectly from
the one philosopher who stood out against the purely ontological
speculations of his time, namely _Herbart_. Herbart's actual views on
Geometry, which are to be found chiefly in the first section of his
_Synechologie_, are not of any great value, and have borne no great
fruit in the development of the subject. But his psychological theory
of space, his construction of extension out of series of points, his
comparison of space with the tone and colour-series, his general
preference for the discrete above the continuous, and finally his
belief in the great importance of classifying space with other forms
of series (_Reihenformen_[79]), gave rise to many of Riemann's
epoch-making speculations, and encouraged the attempt to explain the
nature of space by its analytical and quantitative aspect alone[80].
Through his influence on Riemann, he acquired, indirectly, a great
importance in geometrical philosophy. To Riemann's dissertation,
which we have already discussed in its mathematical aspect, we must
now return, considering, this time, only its philosophical views.
Riemann.
=60.= The aim of Riemann's dissertation, as we saw in Chapter I.,
was to define space as a species of manifold, _i.e._ as a particular
kind of collection of magnitudes. It was thus assumed, to begin with,
that spatial figures could be regarded as magnitudes, and the axioms
which emerged, accordingly, determined only the particular place of
these among the many algebraically possible varieties of magnitudes.
The resulting formulation of the axioms--while, from the mathematical
standpoint of metrical Geometry, it was almost wholly laudable--must,
from the standpoint of philosophy, be regarded, in my opinion, as a
_petitio principii_. For when we have arrived at regarding spatial
figures as magnitudes, we have already traversed the most difficult
part of the ground. The axioms of metrical Geometry--and it is
metrical Geometry, exclusively, which is considered in Riemann's
Essay--will appear, in Chapter III., to be divisible into two
classes. Of these, the first class--which contains the axioms common
to Euclid and Metageometry, the only axioms seriously discussed by
Riemann--are not the results of measurement, nor of any conception
of magnitude, but are conditions to be fulfilled before measurement
becomes possible. The second class only--those which express the
difference between Euclidean and non-Euclidean spaces--can be
deduced as results of measurement or of conceptions of magnitude.
As regards the first class, on the contrary, we shall see that
the relativity of position--by which space is distinguished from
all other known manifolds, except time--leads logically to the
necessity of three of the most distinctive axioms of Geometry, and
yet this relativity cannot be called a deduction from conceptions of
magnitude. In analytical Geometry, owing to the fact that coordinate
systems start from points, and hence build up lines and surfaces, it
is easy to suppose that points can be given independently of lines
and of each other, and thus the relativity of position is lost sight
of. The error thus suggested by mathematics was probably reinforced
by Herbart's theory of space, which, by its serial character, as
we have seen, appeared to him to facilitate a construction out of
successive points, and to which Riemann acknowledges his indebtedness
both in his Dissertation and elsewhere. The same error reappears
in Helmholtz, in whom it is probably due wholly to the methods of
analytical Geometry. It is a striking fact that, throughout the
writings of these two men, there is not, so far as I know, one
allusion to the relativity of position, that property of space
from which, as our next chapter will shew, the richest quarry of
consequences can be extracted. This is not a result of any conception
of magnitude, but follows from the nature of our space-intuition; yet
no one, surely, could call it empirical, since it is bound up in the
very possibility of locating things _there_ as opposed to _here_.
=61.= Indeed we can see, from a purely logical consideration of
the judgment of quantity, that Riemann's manner of approaching the
problem can never, by legitimate methods, attain to a philosophically
sound formulation of the axioms. For quantity is a result of
comparison of two qualitatively similar objects, and the judgment of
quantity neglects altogether the qualitative aspect of the objects
compared. Hence a knowledge of the essential properties of space can
never be obtained from judgments of quantity, which neglect these
properties, while they yet presuppose them. As well might one hope
to learn the nature of man from a census. Moreover, the judgment of
quantity is the result of comparison, and therefore presupposes
the possibility of comparison. To know whether, or by what means,
comparison is possible, we must know the qualities of the things
compared and of the medium in which comparison is effected; while to
know that _quantitative_ comparison is possible, we must know that
there is a qualitative identity between the things compared, which
again involves a previous qualitative knowledge. When spatial figures
have once been reduced to quantity, their quality has already been
neglected, as known and similar to the quality of other figures. To
hope, therefore, for the qualities of space, from a comparison of its
expression as pure quantity with other pure quantities, is an error
natural to an analytical geometer, but an error, none the less, from
which there is no return to the qualitative basis of spatial quantity.
=62.= We must entirely dissent, therefore, from the disjunction which
underlies Riemann's philosophy of space. Either the axioms must be
consequences of general conceptions of magnitude, he thinks, or
else they can only be proved by experience (p. 255). Whatever _can_
be derived from general conceptions of magnitude, we may retort,
cannot be an _à priori_ adjective of space: for all the necessary
adjectives of space are presupposed in any judgment of spatial
quantity, and cannot, therefore, be consequences of such a judgment.
Riemann's disjunction, accordingly, since one of its alternatives is
obviously impossible, really begs the question. In formulating the
axioms of metrical Geometry, our question should be: What axioms,
_i.e._ what adjectives of space, must be presupposed, in order that
quantitative comparison of the parts of space may be possible at
all? And only when we have determined these conditions, which are
_à priori_ necessary to any quantitative science of space, does the
second question arise: what inferences can we draw, as to space,
from the observed results of this quantitative science, _i.e._ of
this measurement of spatial figures? The conditions of measurement
themselves, though not results of any conception of magnitude, will
be _à priori_, if it can be shown that, without them, experience of
externality would be impossible.
After this initial protest against Riemann's general philosophical
position, let us proceed to examine, in detail, his use of the notion
of a manifold.
=63.= In the first place there is, if I am not mistaken, considerable
obscurity in the definition of a manifold, of which an almost verbal
rendering was given in Chapter I. What is meant, to begin with, by
a general conception capable of various determinations? Does not
this property belong to all conceptions? It affords, certainly,
a basis for counting, but if continuous quantity is to arise, we
must, surely, have some less discrete formulation. It might afford
a basis, for example, for the distinction of points in projective
Geometry, but projective Geometry has nothing to do with quantity.
Something more fluid and flexible than a conception, one would think,
is necessary as the basis of continua. Then, again, what is meant by
a quantum of a manifold? In space, the answer is obvious: what is
meant is a piece of volume. But how about Riemann's other continuous
manifold, colour? Does a quantum of colour mean a single line in the
spectrum, or a band of finite thickness? In either case, what are
the magnitudes to be compared? And how is superposition necessary,
or even possible? A colour is fixed by its position in the spectrum:
two lines in the same spectrum cannot be superposed, and two lines in
different spectra need not be--their positions in their respective
spectra suffice, or even, roughly, their immediate sense-quality. The
fact is, Riemann had space in his mind from the start, and many of
the properties, which he enunciates as belonging to all manifolds,
belong, as a matter of fact, only to space. It is far from clear what
the magnitudes are which the various determinations make possible.
Do these magnitudes measure the elements of the manifold, or the
relations between elements? This is surely a very fundamental point,
but it is one which Riemann never touches on. In the former case,
the superposition which he speaks of becomes unnecessary, since the
magnitude is inherent in the element considered. We do not require
superposition to measure quantities corresponding to different tones
or colours; these can be discovered by analysis of single tones or
colours. With space, on the other hand, if we seek for elements,
we can find none except points, and no analysis of a point will
find magnitudes inherent in it--such magnitudes are a fiction of
coordinate Geometry. The magnitudes which space deals with, as we
shall see in Chapter III., are relations between points, and it is
for this reason that superposition is essential to space-measurement.
There is no inherent quality in a single point, as there is in a
single colour, by which it can be quantitatively distinguished from
another. Thus the conception of a manifold, as defined by Riemann,
either does not include colours, or does not involve superposition as
the only means of measurement. From this dilemma there is no escape.
=64.= But if "measurement _consists_ in a superposition of the
magnitudes compared" (p. 256), does it not follow immediately that
measurement is logically possible _only_ where such superposition
leaves the magnitudes unchanged? And therefore that measurement, as
above defined, involves, as an _à priori_ condition, that magnitudes
are unchanged by motion? This consequence is not drawn by Riemann;
indeed he proceeds immediately (pp. 256-7) to consider what he calls
a general portion of the doctrine of magnitude (_Grössenlehre_),
independent of measurement. But how is any doctrine of magnitude
possible, in which the magnitudes cannot be measured? The reason
of the confusion is, that Riemann's definition of measurement is
applicable to no single manifold except space, since it depends
on the noteworthy property that what we measure in Geometry is
not points, but relations between points, and the latter, though
not the former, may of course be unaltered by motion. Let us try,
in illustration, to apply Riemann's definition of measurement to
colours. We must remember that motion, in dealing with the colour
manifold, means--not motion in space but--motion in the colour
manifold itself. Now since every point of the colour manifold is
completely determined by three magnitudes, which are given in fact,
and cannot be arbitrarily chosen, it is plain that measurement
by superposition--involving, as it does, motion, and therefore
change in these determining magnitudes--is totally out of the
question. The superposition of one colour on another, as a means
of measurement, is sheer nonsense. And yet measurement is possible
in the colour-manifold, by means of Helmholtz's law of mixture
(_Mischungsgesetz_); but the measurement is of every separate
element, not of the relations between elements, and is thus radically
different from space-measurement[81]. The elements are not, like
points in space, qualitatively alike, and distinguished by the
mere fact of their mutual externality. What we have, in colours, is
three fundamental qualitatively distinct elements, out of certain
proportions of which we can build up all the other elements of the
manifold--each of the resulting elements having the same combination
of qualitative diversity and similarity as the three original
elements. But in space, what could we make of such a procedure? Given
three points, how are we to combine them in certain proportions? The
phrase is meaningless. If some one makes the obvious retort, that we
have to combine lines, not points, my rejoinder is equally obvious.
To begin with, lines are not elements. Metaphysically, space has _no_
elements, being, as the sequel will show, mere relations between
non-spatial elements. Mathematically, this fact exhibits itself in
the self-contradictory notion of the point, or zero magnitude in
space, as the limit in our vain search for spatial elements. But
even if we allow the line to pass as the spatial element, what does
the combination of three lines in definite proportions give us? It
gives us, simply, the coordinates of a _point_. Here again we see a
great difference between the colour and space-manifolds. In colours,
the combination of magnitudes gives a new magnitude of the same
kind; in space, it defines, not a magnitude at all, but a would-be
element of a different kind from the defining magnitudes. In the
tone-manifold, we should find still different conditions. Here, no
one of the measuring magnitudes can vanish without the tone vanishing
too, and all three are so bound up together, in the single resulting
sensation, that none can exist without a finite quantity of the
others. They are all qualitatively different, both from each other,
and from any possible tone, being constituents of it, as mass and
velocity are constituents of momentum. All these different conditions
require to be examined, before a manifold can be completely defined;
and until we have conducted such an examination in detail, we cannot
pronounce as to the _à priori_ or empirical nature of the laws of the
manifold. As regards space, I have attempted such an examination in
the third and fourth chapters of this Essay.
=65.= I do not wish to deny, however, the great value of the
conception of space as a manifold. On the contrary, this conception
seems to have become essential to any treatment of the question.
I only wish to urge that the purely algebraical treatment of any
manifold, important as it may be in deducing fresh consequences from
known premisses, tends rather to conceal than to make clear the
basis of the premisses themselves, and is therefore misleading in a
philosophical investigation. For mathematics, where quantity reigns
supreme, Riemann's conception has proved itself abundantly fruitful;
for philosophy, on the contrary, where quantity appears rather as
a cloak to conceal the qualities it abstracts from, the conception
seems to me more productive of error and confusion than of sound
doctrine.
We are thus brought back to the point from which we started, namely,
the falsity of Riemann's initial disjunction, and the consequent
fallacy in his proof of the empirical nature of the axioms. His
philosophy is chiefly vitiated, to my mind, by this fallacy, and
by the uncritical assumption that a metrical coordinate system can
be set up independently of any axioms as to space-measurement[82].
Riemann has failed to observe, what I have endeavoured to prove in
the next chapter, that, unless space had a strictly constant measure
of curvature, Geometry would become impossible; also that the absence
of constant measure of curvature involves absolute position, which is
an absurdity. Hence he is led to the conclusion that all geometrical
axioms are empirical, and may not hold in the infinitesimal, where
observation is impossible. Thus he says (p. 267): "Now the empirical
conceptions, on which spatial measurements are based, the conceptions
of the rigid body and the light-ray, appear to lose their validity
in the infinitesimal: it is therefore quite conceivable that
the relations of spatial magnitudes in the infinitesimal do not
correspond to the presuppositions of Geometry, and this would, in
fact, have to be assumed, as soon as it would enable us to explain
the phenomena more simply." From this conclusion I must entirely
dissent. In very large spaces, there might be a departure from
Euclid; for they depend upon the axiom of parallels, which is not
contained in the axiom of Free Mobility; but in the infinitesimal,
departures from Euclid could only be due to the absence of Free
Mobility, which, as I hope my third chapter will show, is once for
all impossible.
Helmholtz.
=66.= Helmholtz, like Riemann, was important both in the mathematics
and in the philosophy of Geometry. From the mathematical point
of view, his work has been already considered in Chapter I.; the
consideration of his philosophy, which must occupy us here, will
be a more serious task. Like Riemann, he endeavoured to prove that
all the axioms are empirical, and like Riemann, he based his proof
chiefly on Metageometry. He had an additional resource, however,
in the physiology of the senses, which first led him to reject the
Transcendental Aesthetic, and enabled him to attack Kant from the
psychological as well as the mathematical side[83].
The principal topics, for a criticism of Helmholtz, are three: First,
his criterion of the _à priori_; second, his discussion with Land as
to the "imaginability" of non-Euclidean spaces; third--and this is by
far the most important of the three--his theory of the dependence of
Geometry on Mechanics. Let us discuss these three points successively.
=67.= Helmholtz's criterion of apriority is difficult to discover,
as he never, to my knowledge, gives a precise statement of it.
From his discussion of physical and transcendental Geometry[84],
however, it would appear that he regards as empirical whatever
applies to empirical matter. For he there maintains, that even if
space were an _à priori_ form, yet any Geometry, which aimed at an
application to Physics, would, since the actual places of bodies are
not known _à priori_, be necessarily empirical[85]. It seems the
more probable that he regards this as a possible criterion, as it
is adopted, in several passages, by his disciple Erdmann[86], and
so strange a test could hardly be accepted by a philosopher, unless
he had found it in his master. I have called this a strange test,
because it seems to me completely to ignore the work of the Critical
Philosophy. For if there is one thing which, one might have hoped,
had been made sufficiently clear by Kant's Critique, it is this,
that knowledge which is _à priori_, being itself the condition of
possible experience, applies--and in Kant's view, applies only--to
empirical matter. Helmholtz and Erdmann, therefore, in setting up
this test without discussion, simply ignore the existence of Kant
and the possibility of a transcendental argument. Helmholtz assumes
always that empirical knowledge must be wholly empirical, that there
can be no _à priori_ conditions of the experience in question,
that experience will always be possible, and may give any kind of
result. Thus in discussing "physical" Geometry, he assumes that the
possibility of empirical measurement involves no _à priori_ axioms,
and that no _à priori_ element can be contained in the process. This
assumption, as we shall see in Chapter III., is quite unwarrantable:
certain properties of _space_, in fact, are involved in the
possibility of measuring _matter_. In spite of the fact, therefore,
that we apply measurement to empirical matter, and that our results
are therefore empirical, there may well be an _à priori_ element
in measurement, which is presupposed in its possibility. Such a
criterion, therefore, must pronounce everything empirical, but must
itself be pronounced worthless.
Another and a better criterion, it is true, is also to be found in
Helmholtz, and has also been adopted by Erdmann. Whatever might,
by a different experience, have been rendered different--so this
criterion contends--must itself be dependent on experience, and so
empirical. This criterion seems perfectly sound, but Helmholtz's use
of it is usually vitiated by his neglecting to prove the possibility
of the different experience in question. He says, for example, that
if our experience showed us only bodies which changed their shapes
in motion, we should not arrive at the axiom of Congruence, which
he pronounces accordingly to be empirical. But I shall endeavour
to prove, in Chapter III., that without the axiom of Congruence,
experience of spatial magnitude would be impossible. If my proof
be correct, it follows that no experience can ever reveal spatial
magnitudes which contradict this axiom--a possibility which Helmholtz
nowhere discusses, in setting up his hypothetical experience. Thus
this second criterion, though perfectly sound, requires always
an accompanying transcendental argument, as to the conditions of
possible experience. But this accompaniment is seldom to be found in
Helmholtz.
=68.= One of the few cases, in which Helmholtz has attempted such
an accompaniment, occurs in connection with our second point, the
imaginability of non-Euclidean spaces. The argument on this point
was elicited by Helmholtz's Kantian opponents, who maintained that
the merely logical possibility of these spaces was irrelevant, since
the basis of Geometry was not logic, but intuition. The axioms,
they said, are synthetic propositions, and their contraries are,
therefore, not self-contradictory; they are nevertheless apodeictic
propositions, since no other _intuition_ than the Euclidean is
possible to us[87]. I have already criticized this line of argument
in the beginning of the present chapter. Helmholtz's criticism,
however, was different: admitting the internal consistency of
the argument, he denied one of its premisses. We _can_ imagine
non-Euclidean spaces, he said, though their unfamiliarity makes
this difficult. From this view it followed, of course, that Kant's
argument, even if it were formally valid, could not prove the
apriority of Euclidean space in particular, but only of that general
space which included Euclid and non-Euclid alike[88].
Although I agree with Helmholtz in thinking the distinction between
Euclidean and non-Euclidean spaces empirical, I cannot think his
argument on the "imaginability" of the latter a very happy one. The
validity of any proof must turn, obviously, on the definition of
imaginability. The definition which Helmholtz gives in his answer
to Land is as follows: Imaginability requires "die vollständige
Vorstellbarkeit derjenigen Sinneseindrücke, welche das betreffende
Object in uns nach den bekannten Gesetzen unserer Sinnesorgane
unter allen denkbaren Bedingungen der Beobachtung erregen, und
wodurch es sich von anderen ähnlichen Objecten unterscheiden würde"
(Wiss. Abh. II. p. 644). This definition is not very clear, owing
to the ambiguity of the word "_Vorstellbarkeit_." The following
definition seems less ambiguous: "Wenn die Reihe der Sinneseindrücke
vollständig und eindeutig angegeben werden kann, muss man m. E.
die Sache für _anschaulich vorstellbar_ erklären" (Vorträge und
Reden, II. p. 234). This makes clear, what also appears from his
manner of proof, that he regards things as imaginable which can be
_described_ in conceptual terms. Such, as Land remarks (Mind, Vol.
II. p. 45), "is not the sense required for argumentation in this
case." That Land's criticism is just, is shown by Helmholtz's proof
for non-Euclidean spaces, for it consists only in an analogy to
the volume inside a sphere, which is mathematically obtained thus:
We take the symbols representing magnitudes in "pseudo-spherical"
(hyperbolic) space, and give them a new Euclidean meaning; thus all
our symbolic propositions become capable of two interpretations,
one for pseudo-spherical space, and one for the volume inside a
sphere. It is, however, sufficiently obvious that this procedure,
though it enables us to _describe_ our new space, does not enable
us to _imagine_ it, in the sense of calling up images of the way
things would look in it. We really derive, from this analogy, no
more knowledge than a man born blind may derive, as to light,
from an analogy with heat. The dictum "Nihil est in intellectu
quod non fuerit ante in sensu," would unquestionably be true, if
for _intellect_ we were to substitute _imagination_; it is vain,
therefore, _if_ our actual space be Euclidean, to hope for a power of
_imagining_ a non-Euclidean space. What Helmholtz might, I believe
with perfect truth, have urged against Land, is that the image we
actually have of space is not sufficiently accurate to exclude, in
the actual space we know, all possibility of a slight departure from
the Euclidean type. But in maintaining that we cannot imagine, though
we can conceive and describe, a space different from that we actually
have, Land is, in my opinion, unquestionably in the right. For a
pure Kantian, who maintains, with Land, that none of the axioms can
be proved, this question is of great importance. But if, as I have
maintained, some of the axioms are susceptible of a transcendental
proof, while the others can be verified empirically, the question
is freed from psychological implications, and the imaginability or
non-imaginability of metageometrical spaces becomes unimportant.
=69.= We come now to the third and most important question, the
relation of Geometry to Mechanics. There are three senses in which
Helmholtz's appeal to rigid bodies may be taken: the first, I think,
is the sense in which he originally intended it; the second seems
to be the sense which he adopted in his defence against Land; while
the third is admitted by Land, and will be admitted in the following
argument. These three senses are as follows:
(1) It may be asserted that the actual meaning of the axiom of Free
Mobility lies in the assertion of empirical rigid bodies, and that
the two propositions are equivalent to one another. This is certainly
false.
(2) The axiom of Free Mobility, it may be said, is logically
distinguishable from the assertion of rigid bodies, and may even be
not empirical; but it is barren, even for pure Geometry, without the
aid of measures, which must themselves be empirical rigid bodies.
This sense is more plausible than the first, but I believe we can
show that, in this sense also, the proposition is false.
(3) For pure Geometry and the abstract study of space, it may be
said, Free Mobility, as applied to an abstract geometrical matter,
gives a sufficient possibility of quantitative comparison; but
the moment we extend our results to mixed mathematics, and apply
them to empirically given matter, we require also, as measures,
empirically given rigid bodies, or bodies, at least, whose departures
from rigidity are empirically known. In this sense, I admit, the
proposition is correct[89].
In discussing these three meanings, I shall not confine myself
strictly to the text of Helmholtz or Land: if I endeavoured to do so,
I should be met by the difficulty that neither of them defines the
_à priori_, and that each is too much inclined, in my opinion, to
test it by psychological criteria. I shall, therefore, take the three
meanings in turn, without laying stress on their historical adequacy
to the views of Land or Helmholtz.
=70.= (1) Congruence may be taken to mean--as Helmholtz would
certainly seem to desire--that we find actual bodies, in our
mechanical experience, to preserve their shapes with approximate
constancy, and that we infer, from this experience, the homogeneity
of space. This view, in my opinion, radically misconceives the
nature of measurement, and of the axioms involved in it. For what
is meant by the non-rigidity of a body? We mean, simply, that
it has changed its shape. But this involves the possibility of
comparison with its former shape, in other words, of measurement.
In order, therefore, that there may be any question of rigidity
or non-rigidity, the measurement of spatial magnitudes must be
already possible. It follows that measurement cannot, without a
vicious circle, be itself derived from experience of rigid bodies.
Geometrical measurement, in fact, is the comparison of spatial
magnitudes, and such comparison involves, as will be proved at length
in Chapter III., the homogeneity of space. This is, therefore, the
logical prerequisite of all experience of rigid bodies, and cannot
be the result of such experience. Without the homogeneity of space,
the very notion of rigidity or non-rigidity could not exist, since
these mean, respectively, the constancy or inconstancy of spatial
magnitude in pieces of matter, and both alike, therefore, presuppose
the possibility of spatial measurement. From the homogeneity of
space, we learn that a body, when it moves, will not change its shape
without some physical cause; that it actually does not change its
shape, is never asserted, and is indeed known to be false. As soon as
measurement is possible, actual changes of shape can be estimated,
and their empirical causes can be sought. But if space were not
homogeneous, measurement would be impossible, constant shape would
be a meaningless phrase, and rigidity could never be experienced.
Congruence asserts, in short, that a body can, so far as mere space
is concerned, move without change of shape; rigidity asserts that
it actually does so move--a very different proposition, involving
obviously, as its logical prius, the former geometrical proposition.
This argument may be summed up by the following disjunction: If
bodies change their shapes in motion--and to some extent, since no
body is perfectly rigid, they must all do so--then one of two cases
must occur. _Either_ the changes of shape, as bodies move from
place to place, follow no geometrical law, are not, for instance,
functions of the amount or direction of motion; in which case the
law of causation requires that they should not be effects of the
change of place, but of some simultaneous non-geometrical change,
such as temperature. _Or_ the changes are regular, and the shape _S_
becomes, in a new position _p_, _Sf_(_p_). In this case, the law of
concomitant variations leads us to attribute the change of shape
to the mere motion, and shape thus becomes a function of absolute
position. But this is absurd, for position _means_ merely a relation
or set of relations; it is impossible, therefore, that mere position
should be able to effect changes in a body. Position is one term in a
relation, not a thing _per se_; it cannot, therefore, act on a thing,
nor exist by itself, apart from the other terms of the relation. Thus
Helmholtz's view, that Congruence depends on the existence of rigid
bodies, must, since it involves absolute position, be condemned as
a logical fallacy. Congruence, in fact, as I shall prove more fully
in Chapter III., is an _à priori_ deduction from the relativity of
position.
=71.= (2) The above argument seems to me to answer satisfactorily
Helmholtz's contention in the precise form which he first gave it.
The axiom of Congruence, we must agree, is logically distinguishable
from the existence of rigid bodies. Nevertheless some reference
to matter is logically involved in Geometry[90], but whether this
reference makes Geometry empirical, or does not, rather, show an _à
priori_ element in dynamics, is a further question.
The reference to matter is necessitated by the homogeneity of empty
space. For so long as we leave matter out of account, one position
is perfectly indistinguishable from another, and a science of
the relations of positions is impossible. Indeed, before spatial
relations can arise at all, the homogeneity of empty space must
be destroyed, and this destruction must be effected by matter.
The blank page is useless to the geometer until he defaces its
homogeneity by lines in ink or pencil. No spatial figures, in short,
are conceivable, without a reference to a not purely spatial matter.
Again, if Congruence is ever to be used, there must be motion: but
a purely geometrical point, being defined solely by its spatial
attributes, cannot be supposed to move without a contradiction in
terms. What moves, therefore, must be matter. Hence, in order that
motion may afford a test of equality, we must have some _matter_
which is known to be unaffected throughout the motion, that is, we
must have some rigid bodies. And the difficulty is, that these bodies
must not only undergo no change due solely to the nature of space,
but must, further, be unchanged by their changing relation to other
bodies. And here we have a requisite which can no longer be fulfilled
_à priori_: which, indeed, we know to be, in strictness, untrue.
For the forces acting on a body depend upon its spatial relations
to other bodies, and changing forces are liable to produce changing
configuration. Hence, it would seem, actual measurement must be
purely empirical, and must depend on the degree of rigidity to be
obtained, during the process of measurement, in the bodies with which
we are conversant.
This conclusion, I believe, is valid of all actual measurement.
But the possibility of such empirical and approximate rigidity,
I must insist, depends on the _à priori_ law that _mere_ motion,
apart from the action of other matter, cannot effect a change of
shape. For without this law, the effect of other matter would not be
discoverable; the laws of motion would be absurd, and Physics would
be impossible. Consider the second law, for example: How could we
measure the change of motion, if motion itself produced a change
in our measures? Or consider the law of gravitation: How could we
establish the inverse square, unless we were able, independently
of Dynamics, to measure distances? The whole science of Dynamics,
in short, is fundamentally dependent on Geometry, and but for the
independent possibility of measuring spatial magnitudes, none of the
magnitudes of Dynamics could be measured. Time, force, and mass are
alike measured by spatial correlates: these correlates are given, for
time, by the first law, for force and mass, by the second and third.
It is true, then, that an empirical element appears unavoidably in
all actual measurement, inasmuch as we can only know empirically
that a given piece of matter preserves its shape throughout the
necessary change of dynamical relations to other matter involved in
motion; but it is further true that, for Geometry--which regards
matter simply as supplying the necessary breach in the homogeneity
of space, and the necessary term for spatial relations, not as the
bearer of forces which change the configuration of other material
systems--for Geometry, which deals with this abstract and merely
kinematical matter, rigidity is _à priori_, in so far as the only
changes with which it is cognizant--changes of mere position,
namely--are incapable of affecting the shapes of the imaginary and
abstract bodies with which it deals. To use a scholastic distinction,
we may say that matter is the _causa essendi_ of space, but Geometry
is the _causa cognoscendi_ of Physics. Without a Geometry independent
of Physics, Physics itself, which necessarily assumes the results of
Geometry, could never arise; but when Geometry is used in Physics, it
loses some of its _à priori_ certainty, and acquires the empirical
and approximate character which belongs to all accounts of actual
phenomena.
=72.= (3) This argument leads us to Land's distinction of physical
and geometrical rigidity. The distinction may be expressed--and
I think it is better expressed--by distinguishing between
the conceptions of matter proper to Dynamics and to Geometry
respectively. In Dynamics, we are concerned with matter as subject
to and as causing motion, as affected by and as exerting _force_. We
are therefore concerned with the changes of spatial configuration to
which material systems are liable: the description and explanation
of these changes is the proper subject-matter of all Dynamics. But
in order that such a science may exist, it is obviously necessary
that spatial configuration should be already measurable. If this
were not the case, motion, acceleration and force would remain
perfectly indeterminate. Geometry, therefore, must already exist
before Dynamics becomes possible: to make Geometry dependent for its
possibility on the laws of motion or any of their consequences, is
a gross ὕστερον πρότερον. Nevertheless, as we have seen, some sort
of matter is essential to Geometry. But this geometrical matter is a
more abstract and wholly different matter from that of Dynamics. In
order to study space by itself, we reduce the properties of matter
to a bare minimum: we avoid entirely the category of causation,
so essential to Dynamics, and retain nothing, in our matter, but
its spatial adjectives[91]. The kind of rigidity affirmed of this
abstract matter--a kind which suffices for the theory of our
science, though not for its application to the objects of daily
life--is purely geometrical, and asserts no more than this: That
since our matter is devoid, _ex hypothesi_, of causal properties,
there remains nothing, in mere empty space, which is capable of
changing the configuration of any geometrical system. A change of
absolute position, it asserts, is nothing; therefore the only real
change involved in motion is a change of relation to other matter;
but such other matter, for the purposes of our science, is regarded
as destitute of causal powers; hence no change can occur, in the
configuration of our system, by the mere effect of motion through
empty space. The necessity of such a principle may be shown by a
simple _reductio ad absurdum_, as follows. A motion of translation
of the universe as a whole, with constant direction and velocity, is
dynamically negligeable; indeed it is, philosophically, no motion at
all, for it involves no change in the condition or mutual relations
of the things in the universe. But if our geometrical rigidity
were denied, the change in the parameter of space might cause all
bodies to change their shapes owing to the mere change of absolute
position, which is obviously absurd.
To make quite plain the function of rigid bodies in Geometry, let us
suppose a liquid geometer in a liquid world. We cannot suppose the
liquid perfectly homogeneous and undifferentiated, in the first place
because such a liquid would be indistinguishable from empty space,
in the second place because our geometer's body--unless he be a
disembodied spirit--will itself constitute a differentiation for him.
We may therefore assume
"dim beams,
Which amid the streams
Weave a network of coloured light,"
and we may suppose this network to form the occasion for our
geometer's reflections. Then he will be able to imagine a network
in which the lines are straight, or circular, or parabolic, or any
other shape, and he will be able to infer that such a network, if it
can be woven in one part of the fluid, can be woven in another. This
will form sufficient basis for his deductions. The superposition he
is concerned with--since not actual equality, but only the formal
conditions of equality, are the subject-matter of Geometry--is purely
ideal, and is unaffected by the impossibility of congealing any
actual network. But in order to apply his Geometry to the exigencies
of life, he would need some standard of comparison between actual
networks, and here, it is true, he would need either a rigid body,
or a knowledge of the conditions under which similar networks arose.
Moreover these conditions, being necessarily empirical, could hardly
be known apart from previous measurement. Hence for applied, though
not for pure Geometry, one rigid body at least seems essential.
=73.= The utility, for Dynamics, of our abstract geometrical matter,
is sufficiently evident. For having, by its means, a power of
determining the configurations of material systems in whatever part
of space, and knowing that changes of configuration are not due to
mere change of place, we are able to attribute these changes to the
action of other matter, and thus to establish the notion of force,
which would be impossible if change of shape might be due to empty
space.
Thus, to conclude: Geometry requires, if it is to be _practically_
possible, some body or bodies which are either rigid (in the
dynamical sense), or known to undergo some definite changes of shape
according to some definite law. (These changes, we may suppose,
are known by the laws of Physics, which have been experimentally
established, and which throughout assume the truth of Geometry.)
One or more such bodies are necessary to applied Geometry--but only
in the sense in which rulers and compasses are necessary. They are
necessary as, in making the Ordnance Survey, an elaborate apparatus
was necessary for measuring the base line on Salisbury Plain. But
for the _theory_ of Geometry, geometrical rigidity suffices, and
geometrical rigidity means only that a shape, which is possible in
one part of space, is possible in any other. The empirical element
in practice, arising from the purely empirical nature of physical
rigidity, is comparable to the empirical inaccuracies arising from
the failure to find straight lines or circles in the world--which no
one but Mill has regarded as rendering Geometry itself empirical or
inaccurate. But to make Geometry await the perfection of Physics,
is to make Physics, which depends throughout on Geometry, forever
impossible. As well might we leave the formation of numbers until we
had counted the houses in Piccadilly.
Erdmann.
=74.= In connection with Riemann and Helmholtz, it is natural to
consider Erdmann's philosophical work on their theories[92]. This is
certainly the most important book on the subject which has appeared
from the philosophical side, and in spite of the fact that, like
the whole theory of Riemann and Helmholtz, it is inapplicable to
projective Geometry, it still deserves a very full discussion.
Erdmann agrees throughout with the conclusions of Riemann and
Helmholtz, except on a few points of minor importance; and his views,
as this agreement would lead one to expect, are ultra-empirical.
Indeed his logic seems--though I say this with hesitation--to be
incompatible with any system but that of Mill: there is apparently
no distinction, to him, between the general and the universal, and
consequently no concept not embodied in a series of instances. Such
a theory of logic, to my mind, vitiates most of his work, as it
vitiated Riemann's philosophy[93]. This general criticism will find
abundant illustration in the course of our account of Erdmann's views.
=75.= After a general introduction, and a short history of the
development of Metageometry, Erdmann proceeds, in his second
chapter, to discuss what are the axioms of Euclidean Geometry. The
arithmetical axioms, as they are called, he leaves aside, as applying
to magnitude in general; what we want here, he says, is a definition
of space, for which the geometrical axioms are alone relevant.
But a definition of space, he says--following Riemann--demands a
genus of which space shall be a species, and this, since our space
is psychologically unique, can only be furnished by analytical
mathematics (p. 36). Now the space-forms dealt with by Geometry are
magnitudes, and conceptions of magnitude are everywhere applied in
Geometry. But before Riemann, only particular determinations of space
could be exhibited as magnitudes, and thus the desired definition
was impossible to obtain. Now, however, we can subsume space as a
whole under a general conception of magnitude, and thus obtain,
besides the space-intuition and the space-conception, a third form,
namely, the conception of space as a magnitude (_Grössenbegriff
vom Raum_, pp. 38-39). The definition of this will give us the
complete, but not redundant, system of axioms, which could not be
obtained by transforming the general intuition of space into the
space-conception, for want of a plurality of instances (p. 40).
=76.= Before considering the subsequent method of definition, let
us reflect on the theories involved in the above account of the
conception of space as a magnitude. In the first place, it is assumed
that conceptions cannot be formed unless we have a series of separate
objects from which to abstract a common property--in other words,
that the universal is always the general. In the second place, it
is assumed that all definition is classification under a genus. In
the third place, the conception of magnitude, if I am not mistaken,
is fundamentally misunderstood when it is supposed applicable to
space as a whole. But in the fourth place, even if such a conception
existed, it could give none of the essential properties of space. Let
us consider these four points successively.
=77.= As regards the first point, it is to be observed that people
certainly had some conception of space before Riemann invented
the notion of a manifold, and that this conception was certainly
something other than the common qualities of all the points, lines
or figures in space. In the second place, Erdmann's view would
make it impossible to conceive God, unless one were a polytheist,
or the universe--unless, like Leibnitz, one imagined a series of
possible worlds, set over against God, and none of them, therefore,
a true Universe--or, to take an instance more likely to appeal to an
empiricist, the necessarily unique centre of mass of the material
universe. Any universal, in short, which is a bond or unity between
things, and not merely a common property among independent objects,
becomes impossible on Erdmann's view. We cannot, therefore, unless
we adopt Mill's philosophy intact, regard the conception of space
as demanding a series of instances from which to abstract. But even
if we did so regard it, Riemann's manifolds would leave us without
resources. For Euclidean space still appears as unique, at the end of
his series of determinations. We have instances of manifolds, but not
instances of Euclidean space. Thus if Erdmann's theory of conceptions
were correct, he would still be left searching in vain for the
conception of Euclidean space.
=78.= The second point, the view that all definition is
classification, is closely allied to the first, and the two together
plunge us into the depths of scholastic formal logic. The same
instances of things which could not, on Erdmann's view, be conceived,
may now be adduced as things which cannot be defined. Whatever was
said above applies here also, and the point need not, therefore, be
further discussed[94].
=79.= As regards the third point, the impossibility of applying
conceptions of magnitude to space as a whole, a longer argument will
be necessary, for we are concerned, here, with the whole question
of the logical nature of judgments of magnitude. As we had before
too much comparison for our needs, so we have now too little. I will
endeavour to explain this point, which is of great importance, and
underlies, I think, most of the philosophical fallacies of Riemann's
school.
A judgment of magnitude is always a judgment of comparison, and what
is more, the comparison is never concerned with quality, but only
with quantity. Quality, in the judgment of magnitude, is supposed
identical, in the object whose magnitude is stated, and in the unit
with which it is compared. But quality, except in pure number, and in
pure quantity as dealt with by the Calculus, is always present, and
is partly absorbed into quantity, partly untouched by the judgment of
magnitude. As Bosanquet says (Logic, Vol. I. p. 124); "Quantitative
comparison is not _prima facie_ coordinate with qualitative,
but rather stands in its place as the _effect of comparison on
quality_, which so far as comparable _becomes quantity_, and so
far as incomparable furnishes the distinction of parts essential
to the quantitative whole" (italics in the original). Thus, if we
are to regard space as a magnitude, we must be able to adduce all
those series of instances of which Erdmann speaks, and which, for
the conception of space, seemed irrelevant. But it remains to be
proved that the comparison, which we _can_ institute between various
spaces, is capable of expression in a quantitative form. Rather it
would seem that the difference of quality is such as to preclude
quantitative comparison between different spaces, and therefore also
to preclude all judgments of magnitude about space as a whole. Here
an exception might seem to be demanded by non-Euclidean spaces,
whose space-constants give a definite magnitude, inherent in space
as a whole, and therefore, one might think, characterizing space as
a magnitude. But this is a mistake. For the space-constant, in such
spaces, is the ultimate unit, the fixed term in all quantitative
comparison; it is itself, therefore, destitute of quantity, since
there is no independently given magnitude with which to compare it.
A non-Euclidean world, in which the space-constant and all lines
and figures were suddenly multiplied in a constant ratio, would be
wholly unchanged; the lines, as measured against the space-constant,
would have the same magnitude as before, and the space-constant
itself, having no outside standard of comparison, would be destitute
of quantity, and therefore not subject to change of quantity. Such an
enlargement of a non-Euclidean world, in other words, is unmeaning;
and this proves how inapplicable is the notion of quantity to space
as a whole.
It might be objected that this only proves the absence of
quantitative difference between different spaces of positive
space-constant, or between those of negative space-constant: the
quantitative difference persists, it might be said, between those
of positive curvature in general and those of negative curvature
in general, or between both together and Euclidean space. This I
entirely deny. There is no qualitatively similar unit, in the three
kinds of space, by which quantitative comparison could be effected.
The straight lines of one space cannot be put into the other: the
two straight lines, in one space, whose product is the reciprocal
of the measure of curvature, have no corresponding curves in the
other space, and the measures of curvature cannot, therefore,
be quantitatively compared with each other. That the one may be
regarded as positive, the other negative, I admit, but their values
are indeterminate, and the units in the two cases are qualitatively
different. A debt of £300 may be represented as the asset of -£300,
and the height of the Eiffel Tower is +300 metres; but it does
not follow that the two are quantitatively comparable. So with
space-constants: the space-constant is itself the unit for magnitudes
in its own space, and differs qualitatively from the space-constant
of another kind of space.
Again, to proceed to a more philosophical argument, two different
spaces cannot co-exist in the same world: we may be unable to decide
between the alternatives of the disjunction, but they remain, none
the less, absolutely incompatible alternatives. Hence we cannot get
that coexistence of two spaces which is essential to comparison. The
fact seems to be that Erdmann, in his admiration for Riemann and
Helmholtz, has fallen in with their mathematical bias, and assumed,
as mathematicians naturally tend to assume, that quantity is
everywhere and always applicable and adequate, and can deal with more
than the mere comparison of things whose qualities are already known
as similar[95].
=80.= This suggests the fourth and last of the above points, that the
_qualities_ of space, even if space could be successfully regarded
as a magnitude, would have to be entirely omitted in such a manner
of regarding it, and that, therefore, none of its important or
essential properties would emerge from such treatment. For to regard
space as a magnitude involves, as we saw, a comparison with something
qualitatively similar, and an abstraction from the similar qualities.
To some extent and by the help of certain doubtful arguments, such a
comparison is instituted by Riemann and Erdmann; but when they have
instituted it, they forget all about the common qualities on which
its possibility depends. But these are precisely the fundamental
properties of space, and those from which, as I shall endeavour to
prove in Chapter III., the axioms common to Euclid and Metageometry
follow _à priori_. Such are the dangers of the quantitative bias.
=81.= After this protest against the initial assumptions in Erdmann's
deduction of space, let us return to consider the manner, in which
this deduction is carried out. Here there will be less ground
for criticism, as the deduction, given its presuppositions, is,
I think, as good as such a deduction can be. To define space as
a magnitude, he says, let us start with two of its most obvious
properties, continuity and the three dimensions. Tones and colours
afford other instances of a manifold with these two properties, but
differ from space in that their dimensions are not homogeneous and
interchangeable. To designate this difference, Erdmann introduces
a useful pair of terms: in the general case, he calls a manifold
_n_-determined (n-_bestimmt_); in the case where, as in space, the
dimensions are homogeneous, he calls the manifold _n_-extended
(n-_ausgedehnt_). Manifolds of the latter sort he calls extents
(_Ausgedehntheiten_). That the difference between the two kinds
is one of quality, not of quantity, he seems not to perceive; he
also overlooks the fact that, in the second kind, from its very
definition, the axiom of Congruence must hold, on account of the
qualitative similarity of different parts. In spite of this fact, he
defines space as an extent, and then regards Congruence as empirical,
and as possibly false in the infinitesimal. This is the more strange,
as he actually proves (p. 50) that measurement is impossible, in an
extent, unless the parts are independent of their place, and can be
carried about unaltered as measures. In spite of this, he proceeds
immediately to discuss whether the measure of curvature is constant
or variable, without investigating how, in the latter case, Geometry
could exist. We cannot know, he says, from geometrical superposition,
that geometrical bodies are independent of place, for if their
dimensions altered in motion according to any fixed law, two bodies
which could be superposed in one place could be superposed in any
other. That such a hypothesis involves absolute position, and denies
the qualitative similarity of the parts of space, which he declares
(p. 171) to be the principle of his theory of Geometry, is nowhere
perceived. But what is more, his notion that magnitude is something
absolute, independent of comparison, has prevented him from seeing
that such a hypothesis is unmeaning. He says himself that, even on
this hypothesis, a geometrical body can be defined as one whose
points retain constant distances from each other, for, since we have
no absolute measure, measurement could not reveal to us the change of
absolute magnitude (p. 60). But is not this a _reductio ad absurdum_?
For magnitude is nothing apart from comparison, and the comparison
here can only be effected by superposition; if, then, as on the above
hypothesis, superposition always gives the same result, by whatever
motion it is effected, there is no sense in speaking of magnitudes as
no longer equal when separated: absolute magnitude is an absurdity,
and the magnitude resulting from comparison does not differ from that
which would result if the dimensions of bodies were unchanged in
motion. Therefore, since magnitude is only intelligible as the result
of comparison, the dimensions of bodies _are_ unchanged in motion,
and the suggested hypothesis is unmeaning. On this subject I shall
have more to say in Chapter III.[96]
=82.= This hypothesis, however, is not introduced for its own sake,
but only to usher in the Helmholtzian _deus ex machina_, Mechanics.
For Mechanics proves--so Erdmann confidently continues--that
rigidity must hold, not merely as to ratios, in the above restricted
geometrical sense, but as to absolute magnitudes (p. 62). Hence we
get at last true Congruence, empirical as Mechanics is empirical, and
impossible to prove apart from Mechanics. I have already criticized
Helmholtz's view of the dependence of Geometry on Mechanics, and
need not here speak of it at length. It is a pity that Erdmann has
in no way specified the procedure by which Mechanics decides the
geometrical alternatives--indeed he seems to rely on the _ipse dixit_
of Helmholtz. How, if Geometry would be totally unable to discover a
change in dimensions of the kind suggested, the Laws of Motion, which
throughout depend on Geometry, should be able to discover it if it
existed, I am wholly at a loss to understand. Uniform motion in a
straight line, for example, presupposes geometrical measurement; if
this measurement is mistaken, what Mechanics imagines to be uniform
motion is not really such, but Mechanics can never discover the
discrepancy. If the Laws of Motion had been regarded as _à priori_,
Geometry might possibly have been reinforced by them; but so long
as they are empirical, they presuppose geometrical measurement, and
cannot therefore condition or affect it.
Erdmann's conclusion, in the second chapter, is that Congruence is
probable, but cannot be verified in the infinitesimal; that its truth
involves the actual existence of rigid bodies (though, by the way,
we know these to be, strictly speaking, non-existent), that rigid
bodies are freely moveable, and do not alter their size in rotation
(Helmholtz's Monodromy); that the axiom of three dimensions is
certain, since small errors are impossible; and that the remaining
axioms of Euclid--those of the straight line and of parallels--are
approximately, if not accurately, true of our actual space (pp.
78, 83). He does not discuss how Congruence, on the above view,
is compatible with the atomic theory, or even with the observed
deformations of approximately rigid bodies; nor how, if space, as he
assumes, is homogeneous, rigid bodies can fail to be freely moveable
through space. The axioms are all lumped together as empirical, and
it appears, in the following chapters, that Erdmann regards their
empirical nature as sufficiently proved by their applicability to
empirical material (cf. pp. 159, 165)--a strange criterion, which
would prove the same conclusion, with equal facility, of Arithmetic
and of the laws of thought.
=83.= The third chapter, on the philosophical consequences of
Metageometry, need not be discussed at length, since it deals rather
with space than with Geometry. At the same time, it will be worth
while to treat briefly of Erdmann's criterion of apriority. On this
subject it is very difficult to discover his meaning, since it
seems to vary with the topic he is discussing. Thus at one time (p.
147) he rejects most emphatically the Kantian connection of the _à
priori_ and the subjective[97], and yet at another time (p. 96) he
regards every presentation of external things as partly _à priori_,
partly empirical, merely because such a presentation is due to an
interaction between ourselves and things, and is therefore partly
due to subjective activity, partly due to outside objects. Hence,
he says, the distinction is not between different presentations,
but between different aspects of one and the same presentation.
This seems to return wholly to the Kantian psychological criterion
of subjectivity, with the added disadvantage that it makes the
distinction, like that of analytic and synthetic, epistemologically
worthless. And yet he never hesitates to pronounce every piece
of knowledge in turn empirical. The fact seems to be, that where
he wants a more logical criterion, he adopts a modification of
Helmholtz's criterion for sensations. If space be an _à priori_
form, he says, no experience could possibly change it (p. 108);
but this Metageometry has proved not to be the case, since we can
intuit the perceptions which non-Euclidean space would give us (p.
115). I have criticised this argument in discussing Helmholtz; at
present we are concerned with Erdmann's criterion of apriority. The
subjectivity-criterion--though he certainly uses it in discussing the
apriority of space, and solemnly decides, by its means, that space
is both _à priori_ and empirical since a change either in us or in
the outer world could change it (p. 97)--would seem, like several of
his other tests, to be a lapse on his part: the criterion which he
means to use is Helmholtz's. This criterion, I think, with a slight
change of wording, might be accepted; it seems to me a necessary,
but not a sufficient condition. The _à priori_, we may say, is not
only that which no experience can change, but that without which
experience would become impossible. It is the omission to discuss
the conditions which render geometrical (and mechanical) experience
possible, to my mind, which vitiates the empirical conclusions of
Helmholtz and Erdmann. Why certain conditions should be necessary
for experience--whether on account of the constitution of the mind,
or for some other reason--is a further question, which introduces
the relation of the _à priori_ to the subjective. But in discussing
the question as to what knowledge is _à priori_, as opposed to the
question concerning the further consequences of apriority, it is
well to keep to the purely logical criterion, and so preserve our
independence of psychological controversies. The fact, if it be a
fact, that the world might be such as to defy our attempts to know
it, will not, with the above criterion, invalidate the conclusion
that certain elements in knowledge are _à priori_; for whether
fulfilled or not, they remain necessary conditions for the existence
of any knowledge at all.
=84.= With this caution as to the meaning of apriority, we shall
find, I think, that the conclusions of Erdmann's final chapter, on
the principles of a theory of Geometry, are largely invalidated
by the diversity and inadequacy of his tests of the _à priori_.
He begins by asserting, in conformity with the quantitative bias
noticed above, that the question as to the nature of geometrical
axioms is completely analogous to the corresponding question of the
foundations of pure mathematics (p. 138). This is, I think, a radical
error: for the function of the axioms seems to be, to establish that
qualitative basis on which, as we saw, all qualitative comparison
must rest. But in pure mathematics, this qualitative basis is
irrelevant, for we deal there with pure quantity, _i.e._ with the
merely quantitative result of quantitative comparison, wherever it is
possible, independently of the qualities underlying the comparison.
Geometry, as Grassmann insists[98], ought not to be classed with
pure mathematics, for it deals with a matter which is given to the
intellect, not created by it. The axioms give the means by which
this matter is made amenable to quantity, and cannot, therefore, be
themselves deduced from purely quantitative considerations.
Leaving this point aside, however, let us return to Erdmann.
He distinguishes, within space, a form and a matter: the form
is to contain the properties common to all extents, the matter
the properties which distinguish space from other extents. This
distinction, he says, is purely logical, and does not correspond
with Kant's: matter and form, for Erdmann, are alike empirical. The
axioms and definitions of Geometry, he says, deal exclusively with
the matter of space. It seems a pity, having made this distinction,
to put it to so little use: after a few pages, it is dropped, and
no epistemological consequences are drawn from it. The reason is, I
think, that Erdmann has not perceived how much can be deduced from
his definition of an extent, as a manifold in which the dimensions
are homogeneous and interchangeable. For this property suffices to
prove the complete homogeneity of an extent, and hence--from the
absence of qualitative differences among elements--the relativity of
position and the axiom of Congruence. This deduction will be made at
length in the sequel[99]; at present, I have only to observe that
every extent, on this view, possesses all the properties (except
the three dimensions) common to Euclidean and non-Euclidean spaces.
The axioms which express these properties, therefore, apply to the
form of space, and follow from homogeneity alone, which Erdmann
allows (p. 171) as the principle of any theory of space. The above
distinction of form and matter, therefore, corresponds, when its
full consequences are deduced, to the distinction between the axioms
which follow from the homogeneity of space and those which do not.
Since, then, homogeneity is equivalent to the relativity of position,
and the relativity of position is of the very essence of a form of
externality, it would seem that his distinction of form and matter
can also be made coextensive with the distinction of the _à priori_
and empirical in Geometry. On this subject, I shall have more to say
in Chapter III.
In the remainder of the chapter, Erdmann insists that the straight
line, etc., though not abstracted from experience, which nowhere
presents straight lines, must yet, as applicable to admittedly
empirical sciences, be empirical (p. 159)--a criterion which he
appears to employ only when all other grounds for an empirical
opinion fail, and one which, obviously, can never refuse to do its
work, since all elements of knowledge are susceptible of employment
on some empirical material. He also defines the straight line (p.
155) as a line of constant curvature zero, as though curvature could
be measured independently of the straight line. Even the arithmetical
axioms are declared empirical (p. 165), since in a world where things
were all hopelessly different from one another, these axioms could
not be applied. After this reminder of Mill, we are not surprised, a
few pages later (p. 172), at a vague appeal to "English logicians"
as having proved Geometry to be an inductive science. Nevertheless,
Erdmann declares, almost on the last page of his book (p. 173), that
Geometry is distinguished from all other sciences by the homogeneity
of its material: a principle of which no single application occurs
throughout his book, and which, as we shall see in Chapter III.,
flatly contradicts the philosophical theories advocated throughout
his preceding pages.
On the whole, then, it cannot be said that Erdmann has done much to
strengthen the philosophical position of Riemann and Helmholtz. I
have criticized him at length, because his book has the appearance
of great thoroughness, and because it is undoubtedly the best
defence extant of the position which it takes up. We shall now have
the opposite task to perform, in defending Metageometry, on its
mathematical side, from the attacks of Lotze and others, and in
vindicating for it that measure of philosophical importance--far
inferior, indeed, to the hopes of Erdmann--which it seems really to
possess.
Lotze.
=85.= Lotze's argument as regards Geometry[100]--which follows a
metaphysical argument as to the ontological nature of space, and
assumes the results of this argument--consists of two parts: the
first discusses the various meanings logically assignable (pp.
233-247) to the proposition that other spaces than Euclid's are
possible, and the second criticizes, in detail, the procedure of
Metageometry. The first of these questions is very important, and
demands considerable care as to the logical import of a judgment
of possibility. Although Lotze's discussion is excellent in many
respects, I cannot persuade myself that he has hit on the only true
sense in which non-Euclidean spaces are possible. I shall endeavour
to make good this statement in the following pages.
=86.= Lotze opens with a somewhat startling statement, which,
though philosophically worthy to be true, does not appear to be
historically borne out. Euclidean Geometry has been chiefly shaken,
he says, by the Kantian notion of the exclusive subjectivity of
space--if space is only our private form of intuition, to which
there exists no analogue in the objective world, then other beings
may have other spaces, without supposing any difference in the world
which they arrange in these spaces (p. 233). This certainly seems
a legitimate deduction from the subjectivity of space, which, so
far from establishing the universal validity of Euclid, establishes
his validity only after an empirical investigation of the nature of
space as intuited by Tom, Dick or Harry. But as a matter of fact,
those who have done most to further non-Euclidean Geometry--with the
exception of Riemann, who was a disciple of Herbart--have usually
inherited from Newton a naïve realism as regards absolute space.
I might instance the passage quoted from Bolyai in Chapter I., or
Clifford, who seems to have thought that we actually see the images
of things on the retina[101], or again Helmholtz's belief in the
dependence of Geometry on the behaviour of rigid bodies. This belief
led to the view that Geometry, like Physics, is an experimental
science, in which objective truth can be attained, it is true, but
only by empirical methods. However, Lotze's ground for uncertainty
about Euclid is a philosophically tenable ground, and it will be
instructive to observe the various possibilities which arise from it.
If space is only a subjective form--so Lotze opens his
argument--other beings may have a different form. If this corresponds
to a different world, the difference, he says, is uninteresting:
for our world alone is relevant to any metaphysical discussion. But
if this different space corresponds to the same world which we know
under the Euclidean form, then, in his opinion, we get a question of
genuine philosophic interest. And here he distinguishes two cases:
_either_ the relations between things, which are presented to these
hypothetical beings under the form of some different space, are
relations which do not appear to us, or at any rate do not appear
spatial; _or_ they are the same relations which appear to us as
figures in Euclidean space (p. 235). The first possibility would be
illustrated, he says, by beings to whom the tone or colour-manifolds
appeared extended; but we cannot, in his opinion, imagine a manifold,
such as is required for this case, to have its dimensions homogeneous
and comparable _inter se_, and therefore the contents of the various
presentations constituting such a manifold could not be combined
into a single content containing them all. But the possibility of
such a combination is of the essence of anything worth calling a
space: therefore the first of the above possibilities is unmotived
and uninteresting. Lotze's conclusion on this point, I think, is
undeniable, but I doubt whether his argument is very cogent. However,
as this possibility has no connection with that contemplated by
non-Euclideans, it is not worth while to discuss it further.
The second possibility also, Lotze thinks, is not that of
Metageometry, but in truth it comes nearer to it than any of the
other possibilities discussed. If a non-Euclidean were at the same
time a believer in the subjectivity of space, he would have to be
an adherent of this view. Let us see more precisely what the view
is. In Book II., Chapter I., Lotze has accepted the argument of the
Transcendental Aesthetic, but rejected that of the mathematical
antinomies: he has decided that space is, as Kant believed,
subjective, but possesses nevertheless, what Kant denied it, an
objective counterpart. The relation of presented space to its
objective counterpart, as conceived by Lotze, is rather hard to
understand. It seems scarcely to resemble the relation of sensation
to its object--_e.g._ of light to ether-vibrations--for if it did,
space would not be in any peculiar sense subjective. It seems rather
to resemble the relation of a perceived bodily motion to the state
of mind of the person willing the motion. However this may be, the
objective counterpart of space is supposed to consist of certain
immediate interactions of monads, who experience the interactions
as modifications of their internal states. Such interactions, it
is plain, do not form the subject-matter of Geometry, which deals
only with our resulting perceptions of spatial figures. Now if
Lotze's construction of space be correct, there seems certainly
no reason why these resulting perceptions should not, for one and
the same interaction between monads, be very different in beings
differently constituted from ourselves. But if they were different,
says Lotze, they would have to be utterly different--as different,
for example, as the interval between two notes is from a straight
line. The possibility is, therefore, in his opinion, one about which
we can know nothing, and one which must remain always a mere empty
idea. This seems to me to go too far: for whatever the objective
counterpart may be, any argument which gives us information about
it must, when reversed, give us information about any possible form
of intuition in which this counterpart is presented. The argument
which Lotze has used in his former chapter, for example, deducing,
from the relativity of position, the merely relational nature of
the objective counterpart, allows us, conversely, to infer, from
this relational nature, the complete relativity of position in any
possible space-intuition--unless, indeed, it bore a wholly deceitful
relation to those interactions of monads which form its objective
counterpart. But the complete relativity of position, as I shall
endeavour to establish in Chapter III., suffices to prove that our
Geometry must be Euclidean, elliptic, spherical or pseudo-spherical.
We have, therefore, it would seem, very considerable knowledge,
on Lotze's theory of space, of the manner in which what appears
to us as space _must_ appear to any beings with our laws of
thought. We cannot know, it is true, what _psychological_ theory
of space-perception would apply to such beings: they might have a
sense different from any of ours, and they might have no sense in
any way resembling ours, but yet their Geometry would have points of
resemblance to ours, as that of the blind coincides with that of the
seeing. If space has any objective counterpart whatever, in short,
and if any inference is possible, as Lotze holds it to be, from space
to its counterpart, then a converse argument is also possible, though
it may give some only of the qualities of Euclidean space, since some
only of these qualities may be found to have a necessary analogue in
the counterpart.
=87.= Admitting, then, in Lotze's sense, the subjectivity of space,
the above possibility does not seem so empty as he imagines. He
discusses it briefly, however, in order to pass on to what he regards
as the real meaning of Metageometry. In this he is guilty of a
mathematical mistake, which causes much irrelevant reasoning. For
he believes that Metageometry constructs its spaces out of straight
lines and angles in all respects similar to Euclid's, whence he
derives an easy victory in proving that these elements can lead only
to the one space. In this he has been misled by the phraseology of
non-Euclideans, as well as by Euclid's separation of definitions
and axioms. For the fact is, of course, that straight lines are
only fully defined when we add to the formal definition the axioms
of the straight line and of parallels. Within Euclidean space,
Euclid's definition suffices to distinguish the straight line from
all other curves; the two axioms referred to are then absorbed into
the definition of space. But apart from the restriction to Euclidean
space, the definition has to be supplemented by the two axioms, in
order to define completely the Euclidean straight line. Thus Lotze
has misconceived the bearing of non-Euclidean constructions, and
has simply missed the point in arguing as he does. The possibility
contemplated by a non-Euclidean, if it fell under any of Lotze's
cases, would fall under the second case discussed above.
=88.= But the bearing of Metageometry is really, I think, different
from anything imagined by Lotze; and as few writers seem clear on
this point, I will enter somewhat fully into what I conceive to be
its purpose.
In the first place, there are some writers--notably Clifford--who,
being naïve realists as regards space, hold that our evidence is
wholly insufficient, as yet, to decide as to its nature in the
infinite or in the infinitesimal (cf. Essays, Vol. I. p. 320):
these writers are not concerned with any possibility of beings
different from ourselves, but simply with the everyday space we
know, which they investigate in the spirit of a chemist discussing
whether hydrogen is a metal, or an astronomer discussing the nebular
hypothesis.
But these are a minority: most, more cautious, admit that our space,
so far as observation extends, is Euclidean, and if not accurately
Euclidean, must be only slightly spherical or pseudo-spherical. Here
again, it is the space of daily life which is under discussion,
and here further the discussion is, I think, independent of any
philosophical assumption as to the nature of our space-intuition. For
even if this be purely subjective, the translation of an intuition
into a conception can only be accomplished approximately, within
the errors of observation incident to self-analysis; and until the
intuition of space has become a conception, we get no scientific
Geometry. The apodeictic certainty of the axiom of parallels shrinks
to an unmotived subjective conviction, and vanishes altogether in
those who entertain non-Euclidean doubts. To reinforce the Euclidean
faith, reason must now be brought to the aid of intuition; but
reason, unfortunately, abandons us, and we are left to the mercy of
approximate observations of stellar triangles--a meagre support,
indeed, for the cherished religion of our childhood.
=89.= But the possibility of an inaccuracy so slight, that our
finest instruments and our most distant parallaxes show no trace
of it, would trouble men's minds no more than the analogous chance
of inaccuracy in the law of gravitation, were it not for the
philosophical import of even the slenderest possibility in this
sphere. And it is the philosophical bearing of Metageometry alone,
I think, which constitutes its real importance. Even if, as we
will suppose for the moment, observation had established, beyond
the possibility of doubt, that our space might be safely regarded
as Euclidean, still Metageometry would have shown a philosophical
possibility, and on this ground alone it could claim, I think, very
nearly all the attention which it at present deserves.
But what is this possibility? A thing is possible, according to
Bradley (Logic, p. 187), when it would follow from a certain number
of conditions, some of which are known to be realized. Now the
conditions to which a form of externality must conform, in order to
be affirmed, are: first, of course, that it should be experienced,
or legitimately inferred from something experienced; but secondly,
that it should conform to certain logical conditions, detailed in
Chapter III., which may be summed up in the relativity of position.
Now what Metageometry has done, in any case, is to suggest the proof
that the second of these conditions is fulfilled by non-Euclidean
spaces. Euclid is affirmed, therefore, on the ground of immediate
experience alone, and his truth, as unmediated by logical necessity,
is merely assertorical, or, if we prefer it, empirical. This is
the most important sense, it seems to me, in which non-Euclidean
spaces are possible. They are, in short, a step in a philosophical
argument, rather than in the investigation of fact: they throw light
on the nature of the grounds for Euclid, rather than on the actual
conformation of space[102]. This import of Metageometry is denied by
Lotze, on the ground that non-Euclidean logic is faulty, a ground
which he endeavours, by much detail and through many pages, to make
good--with what success, we will now proceed to examine.
=90.= Lotze's attack on Metageometry--although it remains, so
far as I know, the best hostile criticism extant, and although
its arguments have become part of the regular stock-in-trade of
Euclidean philosophers--contains, if I am not mistaken, several
misunderstandings due to insufficient mathematical knowledge of the
subject. As these misunderstandings have been widely spread among
philosophers, and cannot be easily removed except by a critic who has
gone into non-Euclidean Geometry with some care, it seems desirable
to discuss Lotze's strictures point by point.
[Illustration]
=91.= The mathematical criticism begins (§ 131) with a somewhat
question-begging definition of parallel straight lines. Two straight
lines _aα_, _bβ_, according to this definition, are parallel
when--_a_ and _b_ being arbitrary points on the two lines--if _aα_
= _bβ_, then _ab_ = _αβ_, where _α_, _β_ are two other points on the
two straight lines respectively. This definition--which contains
Euclid's axiom and definition combined in a very convenient and
enticing form--is of course thoroughly suitable to Euclidean
Geometry, and leads immediately to all the Euclidean propositions
about parallels. But it is perhaps more honest to follow Euclid's
course; when an axiom is thus buried in a definition, it is apt to
seem, since definitions are supposed to be arbitrary, as though the
difficulty had been overcome, while in reality, the possibility of
parallels, as above defined, involves the very point in question,
namely, the disputed axiom of parallels. For what this axiom asserts
is simply the existence of lines conforming to Lotze's definition.
The deduction of the principal propositions on parallels, with
which Lotze follows up his definition, is of course a very simple
proceeding--a proceeding, however, in which the first step begs the
question.
=92.= The next argument for the apriority of Euclidean Geometry has,
oddly enough, an exactly opposite bearing, although it is a great
favourite with opponents of Metageometry. Measurements of stellar
triangles, and all similar attempts at an empirical determination
of the space-constant are, according to Lotze, beside the mark; for
any observed departure from two right angles, or any finite annual
parallax for distant stars, would be attributed to some new kind of
refraction, or, as in the case of aberration, to some other physical
cause, and never to the geometrical nature of space. This is a
strong argument for the empirical validity of Euclid, but as an
argument for the apodeictic certainty of the orthodox system, it has
an opposite tendency. For observations of the kind contemplated would
have to be due to departures from Euclidean straightness, hitherto
unknown, on the part of stellar light-rays. Such departure could,
in certain cases, be accounted for by a finite space-constant, but
it could also, probably, be accounted for by a change in Optics,
for example, by attributing refractive properties to the ether.
Such properties could only exist if ether were of varying density,
if (say) it were denser in the neighbourhood of any of the heavenly
bodies. But such an assumption would, I believe, destroy the utility
of ether for Physics; a slight alteration in our Geometry, so slight
as not appreciably to affect distances within the Solar System,
would probably be in the end, therefore, should such errors ever
be discovered, a simpler explanation than any that Physics could
offer. But this is not the point of my contention. The point is
that, if the physical explanation, as Lotze holds, be possible in
the above case, the converse must also hold: it must be possible to
explain the present phenomena by supposing ether refractive and space
non-Euclidean. From this conclusion there is no escape. If every
conceivable behaviour of light-rays can be explained, within Euclid,
by physical causes, it must also be possible, by a suitable choice
of hypothetical physical causes, to explain the actual phenomena
as belonging to a non-Euclidean space. Such a hypothesis would be
rightly rejected by Science, for the present, on account of its
unnecessary complexity. Nevertheless it would remain, for philosophy,
a possibility to be reckoned with, and the choice could only be
decided upon empirical grounds of simplicity. It may well be doubted
whether, in the world we know, the phenomena could be attributed
to a distinctly non-Euclidean space, but this conclusion follows
inevitably from the contention that no phenomena could force us to
assume such a space. Lotze's argument, therefore, if pushed home,
disproves his own view, and puts Euclidean space, as an empirical
explanation of phenomena, on a level with luminiferous ether[103].
=93.= Lotze now proceeds (§ 132) to a detailed criticism of
Helmholtz, whom he regards as a typical exponent of Metageometry.
It is possible that, at the time when he wrote, Helmholtz really
did occupy this position; but it is unfortunate that, in the minds
of philosophers, he should still continue to do so, after the very
material advances brought about by the projective treatment of the
subject. It is also unfortunate that his somewhat careless attempts
to popularise mathematical results have so often been disposed of,
without due attention to his more technical and solid contributions.
Thus his romances about Flatland and Sphereland--at best only
fairy-tale analogies of doubtful value--have been attacked as if they
formed an essential feature of Metageometry.
But to proceed to particulars: Lotze readily allows that the
Flatlanders would set up Plane Geometry, as we know it, but refuses
to admit that the Spherelanders could, without inferring the third
dimension, set up a two-dimensional spherical Geometry which should
be free from contradictions. I will endeavour to give a free
rendering of Lotze's argument on this point.
[Illustration]
Suppose, he says, a north and south pole, _N_ and _S_, arbitrarily
fixed, and an equator _EW_. Suppose a being, _B_, capable of
impressions only from things on the surface of the sphere, to move
in a meridian _NBS_. Let _B_ start from some point _a_, and finally,
after describing a great circle, return to the same point _a_. If
_a_ is known only by the quality of the impression it makes on _B_,
_B_ may imagine he has not reached the same point _a_, but another
similar point _a′_, bearing a relation to _a_ similar to that of
the octave in singing: he might even not arrange his impressions
spatially at all. In order that this may occur, we require the
further assumption, that every difference in the above-mentioned
feelings (as he describes the meridian) may be presented as a spatial
distance between two places. Even now, _B_ may think he is describing
a Euclidean straight line, containing similar points at certain
intervals. Allowing, however, that he realizes the identity of _a_
with his initial position, he will now seem, by motion in a straight
line, to have returned to the point from which he started, for his
motion cannot, without the third dimension, seem to him other than
rectilinear.
Up to this point, there seems little ground for objection, except,
perhaps, to the idea of a straight line with periodical similar
points--if _B_ were as philosophical as, in these discussions,
we usually suppose him to be, he would probably object to this
interpretation of his experiences, on the ground that it regards
empty space as something independent of the objects in it. It is
worth pointing out, also, that _B_ would not need to describe the
whole circle, in order suddenly to find himself home again with his
old friends. Accurate measurements of small triangles would suffice
to determine his space-constant, and show him the length of a great
circle (or straight line, as he would call it). We must admit, also,
that so hypothetical a being as _B_ might form no space-intuition at
all, but as he is introduced solely for the purposes of the analogy,
it is convenient to allow him all possible qualifications for his
post. But these points do not touch the kernel of the argument, which
lies in the statement that such a straight line, returning into
itself after a finite time, would appear to _B_ as an "unendurable
contradiction," and thus force him, for logical though not for
sensational purposes, into the assumption of a third dimension. This
assertion seems to me quite unwarranted: the whole of Metageometry
is a solid array in disproof of it. Helmholtz's argument is, it must
be remembered, only an analogy, and the contradiction would exist
_only_ for a Euclidean. A complete _three_-dimensional Geometry has,
we have seen in Chapter I., been developed on the assumption that
straight lines are of finite length. A _constant_ value for the
measure of curvature, as our discussion of Riemann showed, involves
neither reference to the fourth dimension, nor any kind of internal
contradiction. This fact disproves Lotze's contention, which arises
solely from inability to divest his imagination of Euclidean ideas.
Lotze next attacks Helmholtz for the assertion that _B_ would know
nothing of parallel lines--parallel _straight_ lines, as the context
shows, he meant to say[104]. Lotze, however, takes him as meaning,
apparently, mere curves of constant distance from a given straight
line, which are part of the regular stock-in-trade of Metageometry.
Parallels of latitude, in the geographical sense, would not--with
the exception of the equator--appear to _B_ as straight lines, but
as circles. _Great_ circles he _would_ call straight, and this fact
seems to have misled Lotze into thinking _all_ circles were to be
treated as straight lines. Parallels of latitude, therefore, though
_B_ might call them parallels, would not invalidate Helmholtz's
contention, which applies only to straight lines.
The argument that such small circles would be parallel, which we
have just disposed of, is only the preface to another proof that _B_
would need a third dimension. Let us call two of these parallels
of latitude _l{n}_ and _l{s}_, and let them be equidistant from
the equator, one in the northern, one in the southern hemisphere.
Consecutive tangent planes, along these parallels, converge, in
the one case northwards, in the other southwards. Either _B_ could
become aware of their difference, says Lotze, or he could not. In
the former case, which he regards as the more probable, he easily
proves that _B_ would infer a third dimension. But this alternative
is, I think, wholly inadmissible. Tangent planes, like Euclidean
planes in general, would have no meaning to _B_; unless, indeed,
he were a metageometrician, which, with all his metaphysical and
mathematical subtlety, the argument supposes him not to be--and to
such a supposition Lotze, surely, is the last person who has a right
to object. Lotze's attempted proof that this is the right alternative
rests, if I understand him aright, on a sheer error in ordinary
spherical Geometry. _B_ would observe, he says, that the meridians
made smaller angles with his path towards the nearer than towards the
further pole--as a matter of fact, they would be simply perpendicular
to his path in both directions. What Lotze means is, perhaps, that
all the meridians would meet sooner in one direction than in the
other, and this, of course, is true. But the poles, in which the
meridians meet, would appear to _B_ as the centres of the respective
parallels, while the parallels themselves would appear to be circles.
Now I am at a loss to see what difficulty would arise, to _B_, in
supposing two different circles to have different centres[105]. We
must, therefore, take the first alternative, that _B_ would have no
sort of knowledge as to the direction in which the tangent planes
converged. Here Lotze attempts, if I have not misunderstood him, to
prove a _reductio ad absurdum_: _B_ would think, he says, that he
was describing two paths wholly the same in direction, and then he
_might_ regard both paths as circles in a plane. It may be observed
that direction, when applied to a circle as a whole, is meaningless;
indeed direction, in all Metageometry, can only mean, even when
applied to straight lines, direction towards a point. To speak of
two lines, which do not meet, as having the same direction, is a
surreptitious introduction of the axiom of parallels. Apart from
this, I cannot conceive any objection, on _B_'s part, to such a
view--one should say _must_, not _might_. The whole argumentation,
therefore, unless its obscurity has led me astray, must be pronounced
fruitless and inconclusive.