to the question of a fourth dimension, and thence to spherical and
pseudo-spherical space. As before, he appears to know only the
more careless and popular utterances of Helmholtz and Riemann, and
to have taken no trouble to understand even the foundations of
mathematical Metageometry. By this neglect, much of what he says
is rendered wholly worthless. To begin with, he regards, as the
purpose of Helmholtz's fairy tale, the suggestion of a possible
fourth dimension, whereas the real purpose was quite the opposite--to
make intelligible a purely three-dimensional non-Euclidean space.
Helmholtz introduced Flatland only because its relation to Sphereland
is analogous to the relation of ours to spherical space[106]. But
Lotze says: The Flatlanders would find no difficulty in a third
dimension, since it would in no way contradict their own Geometry,
while the people in Sphereland, from the contradictions in their
two-dimensional system, would already have been led to it. The
latter contention I have already tried to answer; the former has an
odd sound, in view of the attempt, a few pages later, to prove _à
priori_ that all forms of intuition, in any way analogous to space,
_must_ have three dimensions. One cannot help suspecting that the
Flatlanders, with two instead of three dimensions, would make a
similar attempt. But to return to Lotze's argument: Neither analogy
can be used, he says, to prove that we ought perhaps to set up a
fourth dimension, since, for us, no contradictions or otherwise
inexplicable phenomena exist. The only people, so far as I know, who
have used this analogy, are Dr Abbot and a few Spiritualists--the
former in joke, the latter to explain certain phenomena more simply
explained, perhaps, by Maskelyne and Cooke. But although Lotze's
conclusion in this matter is sound, and one with which Helmholtz
might have agreed, his arguments, to my mind, are irrelevant and
unconvincing. There is this difference, he says, between us and the
Spherelanders: the latter were logically forced to a new dimension,
and found it possible; we are not forced to it, and find it, in our
space, impossible. I have contended that, on the contrary, nothing
would force the Spherelanders to assume a third dimension, while they
would find it impossible exactly as we find a fourth impossible--not
logically, that is to say, but only as a presentable construction in
given space.
After a somewhat elephantine piece of humour, about socialistic
whales in a four-dimensional sea of Fourrier's _eau sucrée_, Lotze
proceeds to a proof, by logic, that every form of intuition, which
embraces the whole system of ordered relations of a coexisting
manifold, _must_ have three dimensions. One might object, on _à
priori_ grounds, to any such attempt: what belongs to pure intuition
could hardly, one would have thought, be determined by _à priori_
reasoning[107]. I will not, however, develop this argument here,
but endeavour to point out, as far as its obscurity will allow, the
particular fallacy of the proof in question.
[Illustration]
Lotze's argument is as follows. In this discussion, though our
terminology is necessarily taken from space, we are really concerned
with a much more general conception. We assume, in order to preserve
the homogeneity of dimensions, that the difference (distance) between
any two elements (points) of our manifold--to borrow Riemann's
word--is of the same kind as, and commensurable with, the difference
between any other two elements. Let us take a series of elements at
successive distances _x_ such that the distance between any two is
the sum of the distances between intermediate elements. Such a series
corresponds to a straight line, which is taken as the _x_-axis. Then
a series _OY_ is called perpendicular to the _x_-axis _OX_, when
the distances of any element _y_, on _OY_, from +_mx_ and -_mx_ are
equal. By our hypothesis, these distances are comparable with, and
qualitatively similar to, _x_ and _y_. So long as _OY_ is defined
only by relation to _OX_, it is conceptually unique. But now let
us suppose the same relation as that between _OX_ and _OY_, to be
possible between _OY_ and a new series _OZ_; we then get a third
series _OZ_ perpendicular to _OY_, and again conceptually unique, so
long as it is defined by relation to _OY_ alone. We might proceed,
in the same way, to a fourth line _OU_ perpendicular to _OZ_. But it
is necessary, for our purposes, that _OZ_ should be perpendicular to
_OX_ as well as _OY_. Without this condition, _OZ_ might extend into
another world, and have no corresponding relation to _OX_--this is a
possibility only excluded by our unavoidable spatial images. At this
point comes the crux of the argument. _That OZ_, says Lotze, which,
besides being perpendicular to _OY_, is also perpendicular to _OX_,
must be among the series of _OY_'s, for these were defined only by
perpendicularity to _OX_. _Hence_, he concludes, there can only be
even a third dimension if _OZ_ coincides with one, and--as soon as
_OX_ is considered fixed--with _only_ one, of the many members of the
_OY_ series.
In this argument it is difficult--to me at any rate--to see any
force at all. The only way I can account for it is, to suppose that
Lotze has neglected the possibility of any but single infinities.
On this interpretation, the argument might be stated thus: There is
an infinite series of continuously varying _OY_'s; to the common
property of these, we add another property, which will divide their
total number by infinity. The remaining _OZ_, therefore, must be
uniquely determined. The same form of argument, however, would prove
that two surfaces can only cut one another in a single point, and
numberless other absurdities. The fact is, that infinities may be of
different orders. For example, the number of points in a line may be
taken as a single infinity, and so may the number of lines in a plane
through any point; hence, by multiplication, the number of points in
a plane is a double infinity, ∞^{2}, and if we divide this number
by a single infinity, we get still an infinite number left. Thus
Lotze's argument assumes what he has to prove, that the number of
lines perpendicular to a given line, through any point, is a single
infinity, which is equivalent to the axiom of three dimensions. The
whole passage is so obscure, that its meaning may have escaped me. It
is obvious _à priori_, however, as I pointed out in the beginning,
that any proof of the axiom must be fallacious somewhere, and the
above interpretation of the argument is the only one I have been able
to find.
=95.= The rest of the Chapter is devoted to an attack on spherical
and pseudo-spherical space, on the ground that they interfere with
the homogeneity of the three dimensions, and with the similarity
of all parts of space. This is simply false. Such spaces, like the
surface of a sphere, _are_ exactly alike throughout. Lotze shows,
here and elsewhere, that he has not taken the pains to find out
what Metageometry really is. I hold myself, and have tried to prove
in this Essay, that Congruence is an _à priori_ axiom, without
which Geometry would be impossible; but the wish to uphold this
axiom is, as Lotze ought to have known, the precise motive which
led Metageometry to limit itself to spaces of constant measure of
curvature. We see here the importance of distinguishing between
Helmholtz the philosopher and Helmholtz the mathematician. Though the
philosopher wished to dispense with Congruence, the mathematician, as
we saw in Chapter I., retained and strongly emphasized it. A little
later Lotze shows, again, how he has been misled by the unfortunate
analogy of Sphereland. A spherical _surface_, he says, he can
understand; but how are we to pass from this to a spherical space?
Either this surface is the whole of our space, as in Sphereland, or
it generates space by a gradually growing radius. Such concentric
spheres, as Lotze triumphantly points out, of course generate
Euclidean space. His disjunction, however, is utterly and entirely
false, and could never have been suggested by any one with even a
superficial knowledge of Metageometry. This point is less laboured
than the former, which, in all its nakedness, is thus re-stated in
the last sentence of the Chapter: "I cannot persuade myself that one
could, without the elements of homogeneous space, even form or define
the presentation of heterogeneous spaces, or of such as had variable
measures of curvature." As though such spaces were ever set up by
non-Euclidean mathematics!
In conclusion, Lotze expresses a hope that Philosophy, on this
point, will not allow itself to be imposed upon by Mathematics. I
must, instead, rejoice that Mathematics has not been imposed upon by
Philosophy, but has developed freely an important and self-consistent
system, which deserves, for its subtle analysis into logical and
factual elements, the gratitude of all who seek for a philosophy of
space.
=96.= The objections to non-Euclidean Geometry which have just been
discussed fall under four heads:
I. Non-Euclidean spaces are not homogeneous; Metageometry therefore
unduly reifies space.
II. They involve a reference to a fourth dimension.
III. They cannot be set up without an implicit reference to Euclidean
space, or to the Euclidean straight line, on which they are therefore
dependent.
IV. They are self-contradictory in one or more ways.
The reader who has followed me in regarding these four objections as
fallacious, will have no difficulty in disposing of any other critic
of Metageometry, as these are the only mathematical arguments, so
far as I know, ever urged against non-Euclideans[108]. The logical
validity of Metageometry, and the mathematical possibility of
three-dimensional non-Euclidean spaces, will therefore be regarded,
throughout the remainder of the work, as sufficiently established.
=97.= Two other objections may, indeed, be urged against
Metageometry, but these are rather of a philosophical than of a
strictly mathematical import. The first of these, which has been
made the base of operations by Delbœuf, applies equally to all
non-Euclidean spaces. The second, which has not, so far as I know,
been much employed, but yet seems to me deserving of notice, bears
directly against spaces of positive curvature alone; but if it could
discredit these, it might throw doubt on the method by which all
alike are obtained. The two objections are:
I. Space must be such as to allow of similarity, _i.e._ of the
increase or diminution, in a constant ratio, of all the lines in a
figure, without change of angles; whereas in non-Euclid, lines, like
angles, have absolute magnitude.
II. Space must be infinite, whereas spherical and elliptic spaces are
finite.
I will discuss the first objection in connection with Delbœuf's
articles referred to above. The second, which has not, to my
knowledge, been widely used in criticism, will be better deferred to
Chapter III.
Delbœuf.
=98.= M. Delbœuf's four articles in the Revue Philosophique contain
much matter that has already been dealt with in the criticism of
Lotze, and much that is irrelevant for our present purpose. The only
point, which I wish to discuss here, is the question of absolute
magnitude, as it is called--the question, that is, whether the
possibility of similar but unequal geometrical figures can be known
_à priori_[109].
In discussing this question, it is important, to begin with, to
distinguish clearly the sense in which absolute magnitude _is_
required in non-Euclidean Geometry, from another sense, in which
it would be absurd to regard any magnitude as absolute. Judgments
of magnitude can only result from comparison, and if Metageometry
required magnitudes which could be determined without comparison, it
would certainly deserve condemnation. But this is not required. All
we require is, that it shall be impossible, while the rest of space
is unaffected, to alter the magnitude of any figure, as compared with
other figures, while leaving the relative internal magnitudes of its
parts unchanged. This construction, which is possible in Euclid,
is impossible in Metageometry. We have to discuss whether such an
impossibility renders non-Euclidean spaces logically faulty.
M. Delbœuf's position on this axiom--which he calls the postulate
of homogeneity[110]--is, that all Geometry must presuppose it, and
that Metageometry, consequently, though logically sound, is logically
subsequent to Euclid, and can only make its constructions within a
Euclidean "homogeneous" space (Rev. Phil. Vol. XXXVII., pp. 380-1).
He would appear to think, nevertheless, that homogeneity (in his
sense) is learnt from experience, though on this point he is not
very explicit. (See Vol. XXXVIII., p. 129.) No _à priori_ proof, at
any rate, is offered in his articles. As a result of experience,
every one would admit, similarity is known to be possible within the
limits of observation; but the fact that this possibility extends to
Ordnance maps, which deal with a spherical surface, should make us
chary of inferring, from such a datum, the certainty of Euclid for
large spaces. Moreover if homogeneity be empirical, Metageometry,
which dispenses with it, is not necessarily in _logical_ dependence
upon Euclid, since homogeneity and isogeneity are _logically_
separable. I shall assume, therefore, as the only contention which
can be interesting to our argument, that homogeneity is regarded as
_à priori_, and as logically essential to Geometry.
=99.= Now we saw, in discussing Erdmann's views of the judgment of
quantity, that in non-Euclidean space, as in Euclidean, a change
of all spatial magnitudes, in the same ratio, would be no change
at all; the ratios of all magnitudes to the space-constant would
be unchanged, and the space-constant, as the ultimate standard of
comparison, cannot, in any intelligible sense, be said to have any
particular magnitude. The absolute magnitudes of Metageometry,
therefore, are absolute only as against any other _particular_
magnitude, not as against other magnitudes in general. If this were
not the case, the comparative nature of the judgment of magnitude
would be contradicted, and metrical Metageometry would become absurd.
But as it is, the difference from Euclid consists only in this: that
in Metageometry we have, while in Euclid we have not, a standard of
comparison involved in the nature of our space as a whole, which we
call the space-constant. We have to discuss whether the assertion of
such a standard involves an undue reification of space.
I do not believe that this is the case. For an undue reification of
space would only arise, if we were no longer able to regard position
as wholly relative, and as geometrically definable only by departure
from other positions. But the relativity of position, as we have
abundantly seen, is preserved by all spaces of constant curvature--in
all of these, positions can only be defined, geometrically, by
relations to fresh positions[111]. This series of definitions may
lead to an infinite regress, but it may also, as in spherical space,
form a vicious circle, and return again to the position from which
it started. No reification of space, no independent existence of
mere relations, seems involved in such a procedure. The whole of
Metageometry, in short, is a proof that the relativity of position
is compatible with absolute magnitude, in the only sense required
by non-Euclidean spaces. We must conclude, therefore, that there
is nothing incompatible, in a denial of homogeneity (in Delbœuf's
sense), either with the relational nature of space, or with the
comparative nature of magnitude. This last _à priori_ objection to
Metageometry, therefore, cannot be maintained, and the issue must be
decided on empirical grounds alone.
=100.= The foundations of Geometry have been the subject of much
recent speculation in France, and this seems to demand some notice.
But in spite of the splendid work which the French have done on the
allied question of number and continuous quantity, I cannot persuade
myself that they have succeeded in greatly advancing the subject
of geometrical philosophy. The chief writers have been, from the
mathematical side, _Calinon_ and _Poincaré_, from the philosophical,
_Renouvier_ and _Delbœuf_; as a mediator between mathematics and
philosophy, _Lechalas_.
_Calinon_, in an interesting article on the geometrical
indeterminateness of the universe, maintains that any Geometry may be
applied to the actual world by a suitable hypothesis as to the course
of light-rays. For the earth only is known to us otherwise than by
Optics, and the earth is an infinitesimal part of the universe.
This line of argument has been already discussed in connection with
Lotze, but Calinon adds a new suggestion, that the space-constant may
perhaps vary with the time. This would involve a causal connection
between space and other things, which seems hardly conceivable, and
which, if regarded as possible, must surely destroy Geometry, since
Geometry depends throughout on the irrelevance of Causation[112].
Moreover, in all operations of measurement, some time is spent;
unless we knew that space was unchanging throughout the operation,
it is hard to see how our results could be trustworthy, and how,
consequently, a change in the parameter could be discovered. The
same difficulties would arise, in fact, as those which result from
supposing space not homogeneous.
_Poincaré_ maintains that the question, whether Euclid or
Metageometry should be accepted, is one of convenience and
convention, not of truth; axioms are definitions in disguise, and the
choice between definitions is arbitrary. This view has been discussed
in Chapter I., in connection with Cayley's theory of distance, on
which it depends.
_Lechalas_ is a philosophical disciple of Calinon. He is a
rationalist of the pre-Kantian type, but a believer in the validity
of Metageometry. He holds that Geometry can dispense with all purely
spatial postulates, and work with axioms of magnitude alone[113],
which, in his opinion, are purely analytic. The principle of
contradiction, to him, is the sole and only test of truth; we make
long chains of reasoning from our premisses to see if contradictions
will emerge. It might be objected that this view, though it saves
general Geometry from being logically empirical, leaves it only
empirically logical; this must, in fact, be the fate of every piece
of _à priori_ knowledge, if M. Lechalas's were the only test of
truth. However, he concludes that general Geometry is apodeictic,
while the space of our actual world, like all other phenomena, is
contingent.
_Delbœuf_ criticizes non-Euclidean space from an ultra-realist
standpoint: he holds that _real_ space is neither homogeneous nor
isogeneous, but that _conceived_ space, as abstracted from real
space, has both these properties. He offers no justification for
his real space, which seems to be maintained in the spirit of naïve
realism, nor does he show how he has acquired his intimate knowledge
of its constitution[114]. His arguments against Metageometry, in so
far as they are not repetitions of Lotze, have been discussed above.
_Renouvier_, finally, is a pure Kantian, of the most orthodox type.
His views as to the importance, for Geometry, of the distinction
between synthetic and analytic judgments, have been discussed, in
connection with Kant, at the beginning of the present Chapter[115].
=101.= Before beginning the constructive argument of the next
Chapter, let us endeavour briefly to sum up the theories which
have been polemically advocated throughout the criticisms we have
just concluded. We agreed to accept, with Kant, necessity for any
possible experience as the test of the _à priori_, but we refused,
for the present, to discuss the connection of the _à priori_ with
the subjective, regarding the purely logical test as sufficient for
our immediate purpose. We also refused to attach importance to the
distinction of analytic and synthetic, since it seemed to apply, not
to different judgments, but only to different aspects of any judgment.
We then discussed Riemann's attempt to identify the empirical element
in Geometry with the element not deducible from ideas of magnitude,
and we decided that this identification was due to a confusion as
to the nature of magnitude. For judgments of magnitude, we said,
require always some qualitative basis, which is not quantitatively
expressible.
In criticizing Helmholtz, we decided that Mechanics logically
presupposes Geometry, though space presupposes matter; but that the
matter which space presupposes, and to which Geometry indirectly
refers, is a more abstract matter than that of Mechanics, a matter
destitute of force and of causal attributes, and possessed only
of the purely spatial attributes required for the possibility
of spatial figures. But we conceded that Geometry, when applied
to mixed mathematics or to daily life, demands more than this,
demands, in fact, some means of discovering, in the more concrete
matter of Mechanics, either a rigid body, or a body whose departure
from rigidity follows some empirically discoverable law. _Actual_
measurement, therefore, we agreed to regard as empirical.
Our conclusions, as regards the empiricism of Riemann and Helmholtz,
were reinforced by a criticism of Erdmann. We then had an opposite
task to perform, in defending Metageometry against Lotze. Here we
saw that there are two senses in which Metageometry is possible. The
first concerns our actual space, and asserts that it may have a very
small space-constant; the second concerns philosophical theories
of space, and asserts a purely logical possibility, which leaves
the decision to experience. We saw also that Lotze's mathematical
strictures arose from insufficient knowledge of the subject, and
could all be refuted by a better acquaintance with Metageometry.
Finally, we discussed the question of absolute magnitude, and found
in it no logical obstacle to non-Euclidean spaces. Our conclusion,
then, in so far as we are as yet entitled to a conclusion, is
that all spaces with a space-constant are _à priori_ justifiable,
and that the decision between them must be the work of experience.
Spaces without a space-constant, on the other hand, spaces, that is,
which are not homogeneous throughout, we found logically unsound
and impossible to know, and therefore to be condemned _à priori_.
The constructive proof of this thesis will form the argument of the
following chapter.
FOOTNOTES:
[67] The Critical Philosophy of Kant, Vol. I. p. 287.
[68] For a discussion of Kant from a less purely mathematical
standpoint, see Chap. IV.
[69] Cf. Vaihinger's Commentar, II. pp. 202, 265. Also p. 336 ff.
[70] E.g. second edition, p. 39: "So werden auch alle geometrischen
Grundsätze, z. B. dass in einem Triangel zwei Seiten zusammen grösser
sind als die dritte, niemals aus allgemeinen Begriffen von Linie
und Triangel, sondern aus der Anschauung, und zwar _à priori_ mit
apodiktischer Gewissheit abgeleitet."
[71] Cf. Bradley's Logic, Bk. III. Pt. I. Chap. VI.; Bosanquet's
Logic, Bk. I. Chap. I. pp. 97-103.
[72] Philosophie de la Règle et du Compas, Année Philosophique, II.
pp. 1-66.
[73] I have stated this doctrine dogmatically, as a proof would
require a whole treatise on Logic. I accept the proofs offered by
Bradley and Bosanquet, to which the reader is referred.
[74] For a further discussion of this point, see Chaps. III. and IV.
[75] See Chap. IV. for a discussion of this argument.
[76] See Chap. IV. § 185.
[77] An Otherness of substance, rather than of attribute, is here
intended; an Otherness which may perhaps be called real as opposed to
logical diversity.
[78] This proposition will be argued at length in Chap. IV.
[79] See Psychologie als Wissenschaft, I. Section III. Chap. VII.;
II. Section I. Chap. III. and Section II. Chap. III. Compare also
Synechologie, Section I. Chaps. II. and III.
[80] On the influence of Herbart on Riemann, compare Erdmann, Die
Axiome der Geometrie, p. 30.
[81] I do not mean that measurement of colours is effected without
reference to their relations, since all measurement is essentially
comparison. But in colours, it is the elements which are compared,
while in space, it is the relations between elements.
[82] For a discussion of this point, see Chap. III. Sec. B, § 176.
[83] The works of Helmholtz on geometrical philosophy comprise, in
addition to the articles quoted in Chap. I., the following articles:
"Ursprung und Sinn der geometrischen Axiome, gegen Land," Wiss. Abh.
Vol. II. p. 640, 1878. (Also Mind, Vol. III.: an answer to Land in
Mind, Vol. II.) "Ursprung und Bedeutung der geometrischen Axiome,"
1870, Vorträge und Reden, Vol. II. p. 1. (Also Mind, Vol. I.) Two
Appendices to "Die Thatsachen in der Wahrnehmung," entitled: II. "Der
Raum kann transcendental sein, ohne dass es die Axiome sind"; and
III. "Die Anwendbarkeit der Axiome auf die physische Welt," 1878,
Vorträge und Reden, Vol. II. p. 256 ff.
The two Appendices last mentioned are popularizings and expansions of
the article in Mind, Vol. III. The most widely read, though also, to
my mind, the least valuable, of all Helmholtz's writings on Geometry,
is the article in Mind, Vol. I. This contains the famous and much
misunderstood analogies of Flatland and Sphereland, which will be
discussed, and as far as possible defended, in answering Lotze's
attack on Metageometry--an attack based, apparently, almost entirely
on this one popular article. The present discussion, therefore, may
be confined almost entirely to Mind, Vol. III., and the philosophical
portions of the two papers quoted in Chap. I., _i.e._ to the articles
in Wiss. Abh. Vol. II. pp. 610-660. His other works are popular, and
important only because of the large public to which they appeal.
[84] In the answer to Land, Mind, Vol. III. and Wiss. Abh. II. p. 640.
[85] See also Die Thatsachen in der Wahrnehmung, Zusatz II., Der Raum
kann transcendental sein, ohne dass es die Axiome sind. Vorträge und
Reden, Vol. II.
[86] See below, criticism of Erdmann, § 84.
[87] See Prof. Land, in Mind, Vol. II.
[88] See concluding paragraph of Helmholtz's article in Mind, Vol.
III.
[89] Cf. Veronese, Grundzüge der Geometrie (German translation), p.
ix. Also pp. xxxiv, 304, and Note II. pp. 692-4.
[90] See Chap. IV. § 197 ff.
[91] Cf. the opinion of Bolyai, quoted by Erdmann, Axiome, p. 26; cf.
also ib. p. 60.
[92] Die Axiome der Geometrie: Eine philosophische Untersuchung der
Riemann-Helmholtz'schen Raumtheorie, Leipzig, 1877.
[93] On the influence of Mill, cf. Stallo, Concepts of Modern
Physics, p. 216.
[94] This view seems to be derived, through Riemann, from Herbart.
See Psych. als Wiss. ed. Hart. Vol. V. p. 262.
[95] The same irreducibility of space to mere magnitude is proved by
Kant's hands and spherical triangles, in which a difference persists
in spite of complete quantitative equality.
[96] See §§ 146-7.
[97] "Jeder Versuch, Kant's Lehre von der Apriorität als
des subjectiven, von aller Erfahrung absolut unabhängigen
Erkenntnissfactors, trotzdem zu halten, ist deshalb von voruherein
aussichtslos."
[98] Ausdehnungslehre von 1844, 2nd edition, pp. xxii. xxiii.
[99] See § 129 ff.
[100] Metaphysik, Book II. Chap. II. My references are to the
original.
[101] See Lectures and Essays, Vol. I. p. 261.
[102] On the meaning of geometrical possibility, cf. Veronese,
Grundzüge der Geometrie (German translation), pp. xi.-xiii.
[103] Compare Calinon, "Sur l'Indétermination géométrique de
l'Univers," Revue Philosophique, 1893, Vol. XXXVI. pp. 595-607.
[104] Vorträge und Reden, Vol. II. p. 9: "Parallele Linien würden
die Bewohner der Kugel gar nicht kennen. Sie würden behaupten, dass
jede beliebige zwei _geradeste_ Linien, gehörig verlangert, sich
schliesslich nicht nur in einem, sondern in zwei Punkten schneiden
müssten." (The italics are mine.) The omission of _straight_ in such
phrases is a frequent laxity of mathematicians.
[105] It has been suggested to me that Lotze regards the meridians
as projected on to a plane, as in a map. If this be so, there is an
obviously illegitimate introduction of the third dimension.
[106] This is proved by Helmholtz's remark at the end of a detailed
attempt to make spherical and pseudo-spherical spaces imaginable
(l.c. p. 28): "Anders ist es mit den drei Dimensionen des Raumes. Da
alle unsere Mittel sinnlicher Anschauung sich nur auf einen Raum von
drei Dimensionen erstrecken, und die vierte Dimension nicht bloss
eine Abänderung von Vorhandenem, sondern etwas vollkommen Neues wäre,
so befinden wir uns schon wegen unserer körperlichen Organisation in
der absoluten Unmöglichkeit, uns eine Anschauungsweise einer vierten
Dimension vorzustellen."
[107] Cf. Grassmann, Ausdehnungslehre von 1844, 2nd Edition, p. xxiii.
[108] See especially Stallo, Concepts of Modern Physics,
International Science Series, Vol. XLII. Chaps. XIII. and
XIV.; Renouvier, "Philosophie de la règle et du compas," Année
Philosophique, II.; Delbœuf, "L'ancienne et les nouvelles
géométries," Revue Philosophique, Vols. XXXVI.-XXXIX.
[109] M. Delbœuf deserves credit for having based Euclid, already in
1860, in his "Prolégomènes Philosophiques de la Géométrie," on this
axiom--certainly a better basis, at first sight, than the axiom of
parallels.
[110] This meaning of homogeneity must not be confounded with the
sense in which I have used the word. In Delbœuf's sense, it means
that figures may be similar though of different sizes; in my sense it
means that figures may be similar though in different places. This
property of space is called by Delbœuf isogeneity.
[111] For a full proof of this proposition, see Chap. III.
[112] See Chap. III., especially § 133.
[113] For a criticism of this view, see the above discussions on
Riemann and Erdmann.
[114] Cf. Couturat, "De l'Infini Mathématique," Paris, Félix Alcan,
1896, p. 544.
[115] The following is a list of the most important recent French
philosophical writings on Geometry, so far as I am acquainted with
them.
Andrade: "Les bases expérimentales de la géométrie euclidienne";
Rev. Phil. 1890, II., and 1891, I.
Bonnel: "Les hypothèses dans la géométrie"; Gauthier-Villars,
1897.
L'Abbé de Broglie: "La géométrie non-euclidienne," two articles;
Annales de Phil. Chrét. 1890.
Calinon: "Les espaces géométriques"; Rev. Phil. 1889, I., and
1891, II. "Sur l'indétermination géométrique de l'univers"; ib.
1893, II.
Couturat: "L'Année Philosophique de F. Pillon," Rev. de Mét. et
de Morale, Jan. 1893.
"Note sur la géométrie non-euclidienne et la relativité de
l'espace"; ib., May, 1893.
"Études sur l'espace et le temps," ib. Sep. 1896.
Delbœuf: "L'ancienne et les nouvelles géométries," four articles;
Rev. Phil. 1893-5.
Lechalas: "La géométrie générale"; Crit. Phil. 1889.
"La géométrie générale et les jugements synthétiques à priori"
and "Les bases expérimentales de la géométrie"; Rev. Phil. 1890,
II.
"M. Delbœuf et Le problème des mondes semblables"; ib. 1894, I.
"Note sur la géométrie non-euclidienne et le principe de
similitude"; Rev. de Mét. et de Morale, March, 1893.
"La courbure et la distance en géométrie générale"; ib., March,
1896.
"La géométrie générale et l'intuition"; Annales de Phil. Chrét.,
1890.
"Etude sur l'espace et le temps"; Paris, Alcan, 1896.
Liard: "Des définitions géométrie et des définitions empiriques,"
2nd ed.; Paris, Alcan, 1888.
Mansion: "Premiers principes de la métagéométrie"; two
articles in Rev. Néo-Scholastique, 1896. Separately published,
Gauthier-Villars, 1896.
Milhaud: "La géométrie non-euclidienne et la théorie de la
connaissance"; Rev. Phil. 1888, I.
Poincaré: "Non-Euclidian Geometry"; Nature, Vol. XLV., 1891-2.
"L'espace et la géométrie"; Rev. de Mét. et de Morale, Nov. 1895.
"Résponse à quelques critiques," ib. Jan. 1897.
Renouvier: "Philosophie de la règle et du compas"; Crit. Phil.,
1889, and L'Année Phil., II^{me} année, 1891.
Sorel: "Sur la géométrie non-euclidienne"; Rev. Phil., 1891, I.
Tannery: "Théorie de la connaissance mathématique"; Rev. Phil.,
1894, II.
CHAPTER III.
Section A.
THE AXIOMS OF PROJECTIVE GEOMETRY.
=102.= Projective Geometry proper, as we saw in Chapter I., does not
employ the conception of magnitude, and does not, therefore, require
those axioms which, in the systems of the second or metrical period,
were required solely to render possible the application of magnitude
to space. But we saw, also, that Cayley's reduction of metrical
to projective properties was purely technical and philosophically
irrelevant. Now it is in metrical properties alone--apart from
the exception to the axiom of the straight line, which itself,
however, presupposes metrical properties[116]--that non-Euclidean
and Euclidean spaces differ. The properties dealt with by projective
Geometry, therefore, in so far as these are obtained without the use
of imaginaries, are properties common to all spaces. Finally, the
differences which appear between the Geometries of different spaces
of the same curvature--_e.g._ between the Geometries of the plane
and the cylinder--are differences in projective properties[117].
Thus the necessity which arises, in metrical Geometry, for further
qualifications besides those of constant curvature, disappears when
our general space is defined by purely projective properties.
=103.= We have good ground for expecting, therefore, that the axioms
of projective Geometry will be the simplest and most complete
expression of the indispensable requisites of any geometrical
reasoning: and this expectation, I hope, will not be disappointed.
Projective Geometry, in so far as it deals only with the properties
common to all spaces, will be found, if I am not mistaken, to be
wholly _à priori_, to take nothing from experience, and to have, like
Arithmetic, a creature of the pure intellect for its object. If this
be so, it is that branch of pure mathematics which Grassmann, in his
_Ausdehnungslehre_ of 1844, felt to be possible, and endeavoured, in
a brilliant failure, to construct without any appeal to the space of
intuition.
=104.= But unfortunately, the task of discovering the axioms of
projective Geometry is far from easy. They have, as yet, found
no Riemann or Helmholtz to formulate them philosophically. Many
geometers have constructed systems, which they intended to be, and
which, with sufficient care in interpretation, really are, free from
metrical presuppositions. But these presuppositions are so rooted
in all the very elements of Geometry, that the task of eliminating
them demands a reconstruction of the whole geometrical edifice. Thus
Euclid, for example, deals, from the start, with spatial equality--he
employs the circle, which is necessarily defined by means of
equality, and he bases all his later propositions on the congruence
of triangles as discussed in Book I.[118] Before we can use any
elementary proposition of Euclid, therefore, even if this expresses a
projective property, we have to prove that the property in question
can be deduced by projective methods. This has not, in general,
been done by projective geometers, who have too often assumed, for
example, that the quadrilateral construction--by which, as we saw in
Chap. I., they introduce projective coordinates--or anharmonic ratio,
which is _primâ facie_ metrical, could be satisfactorily established
on their principles. Both these assumptions, however, can be
justified, and we may admit, therefore, that the claims of projective
Geometry to logical independence of measurement or congruence are
valid. Let us see, then, how it proceeds.
=105.= In the first place, it is important to realize that
when coordinates are used, in projective Geometry, they are not
coordinates in the ordinary metrical sense, _i.e._ the numerical
measures of certain spatial magnitudes. On the contrary, they are a
set of numbers, arbitrarily but systematically assigned to different
points, like the numbers of houses in a street, and serving only,
from a philosophical standpoint, as convenient designations for
points which the investigation wishes to distinguish. But for the
brevity of the alphabet, in fact, they might, as in Euclid, be
replaced by letters. How they are introduced, and what they mean, has
been discussed in Chapter I. Here we have only to repeat a caution,
whose neglect has led to much misunderstanding.
=106.= The distinction between various points, then, is not a result,
but a condition, of the projective coordinate system. The coordinate
system is a wholly extraneous, and merely convenient, set of marks,
which in no way touches the essence of projective Geometry. What we
must begin with, in this domain, is the possibility of distinguishing
various points from one another. This may be designated, with
Veronese, as the first axiom of Geometry[119]. How we are to define
a point, and how we distinguish it from other points, is for the
moment irrelevant; for here we only wish to discover the nature
of projective Geometry, and the kind of properties which it uses
and demonstrates. How, and with what justification, it uses and
demonstrates them, we will discuss later.
=107.= Now it is obvious that a mere collection of points,
distinguished one from another, cannot found a Geometry: we must
have some idea of the manner in which the points are interrelated,
in order to have an adequate subject-matter for discussion. But
since all ideas of quantity are excluded, the relations of points
cannot be relations of distance in the ordinary sense, nor even, in
the sense of ordinary Geometry, anharmonic ratios, for anharmonic
ratios are usually defined as the ratios of four distances, or of
four sines, and are thus quantitative. But since all quantitative
comparison presupposes an identity of quality, we may expect to find,
in projective Geometry, the qualitative substrata of the metrical
superstructure.
And this, we shall see, is actually the case. We have not distance,
but we _have_ the straight line; we have not quantitative anharmonic
ratio, but we _have_ the property, in any four points on a line,
of being the intersections with the rays of a given pencil. And
from this basis, we can build up a qualitative science of abstract
externality, which is projective Geometry. How this happens, I shall
now proceed to show.
=108.= All geometrical reasoning is, in the last resort, circular:
if we start by assuming points, they can only be defined by the
lines or planes which relate them; and if we start by assuming lines
or planes, they can only be defined by the points through which
they pass. This is an inevitable circle, whose ground of necessity
will appear as we proceed. It is, therefore, somewhat arbitrary to
start either with points or with lines, as the eminently projective
principle of duality mathematically illustrates; nevertheless we will
elect, with most geometers, to start with points[120]. We suppose,
therefore, as our datum, a set of discrete points, for the moment
without regard to their interconnections. But since connections are
essential to any reasoning about them as a system, we introduce, to
begin with, the axiom of the straight line. Any two of our points,
we say, lie on a line which those two points completely define.
This line, being determined by the two points, may be regarded as
a relation of the two points, or an adjective of the system formed
by both together. This is the only purely qualitative adjective--as
will be proved later--of a system of two points. Now projective
Geometry can only take account of qualitative adjectives, and can
distinguish between different points only by their relations to
other points, since all points, _per se_, are qualitatively similar.
Hence it comes that, for projective Geometry, when two points only
are given, they are qualitatively indistinguishable from any two
other points on the same straight line, since any two such other
points have the same qualitative relation. Reciprocally, since one
straight line is a figure determined by any two of its points, and
all points are qualitatively similar, it follows that all straight
lines are qualitatively similar. We may regard a point, therefore, as
determined by two straight lines which meet in it, and the point,
on this view, becomes the only qualitative relation between the two
straight lines. Hence, if the point only be regarded as given, the
two straight lines are qualitatively indistinguishable from any other
pair through the point.
=109.= The extension of these two reciprocal principles is the
essence of all projective transformations, and indeed of all
projective Geometry. The fundamental operations, by which figures
are projectively transformed, are called projection and section. The
various forms of projection and section are defined in Cremona's
"Projective Geometry," Chapter I., from which I quote the following
account.
"_To project from a fixed point S_ (the _centre of projection_) a
figure (_ABCD_ ... _abcd_ ...) composed of points and straight lines,
is to construct the straight lines or _projecting rays SA_, _SB_,
_SC_, _SD_, ... and the planes (_projecting planes_) _Sa_, _Sb_,
_Sc_, _Sd_, ... We thus obtain a new figure composed of straight
lines and planes which all pass through the centre _S_.
"_To cut by a fixed plane σ (transversal plane_) a figure (_αβγδ_ ...
_abcd_ ...) made up of planes and straight lines, is to construct the
straight lines or _traces σα, σβ, σγ_ ... and the points or _traces
σa, σb, σc_....[121] By this means we obtain a new figure
composed of straight lines and points lying in the plane _σ_.
"_To project from a fixed straight line s_ (the _axis_) a figure
_ABCD_ composed of points, is to construct the planes _sA_, _sB_,
_sC_.... The figure thus obtained is composed of planes which all
pass through the axis _s_.
"_To cut by a fixed straight line s_ (a _transversal_) a figure
_αβγδ_ ... composed of planes, is to construct the points _sα_, _sβ_,
_sγ_.... In this way a new figure is obtained, composed of points all
lying on the fixed transversal _s_.
"If a figure is composed of straight lines _a_, _b_, _c_ ... which
all pass through a fixed point or _centre S_, it can be _projected_
from a straight line or _axis s_ passing through _S_; the result is a
figure composed of planes _sa_, _sb_, _sc_....
"If a figure is composed of straight lines _a_, _b_, _c_ ...
all lying in a fixed plane, it may be cut by a straight line
(transversal) _s_ lying in the same plane; the figure which results
is formed by the points _sa_, _sb_, _sc_...."
=110.= The successive application, to any figure, of two reciprocal
operations of projection and section, is regarded as producing a
figure protectively indistinguishable from the first, provided only
that the dimensions of the original figure were the same as those
of the resulting figure, that, for example, if the second operation
be section by a plane, the original figure shall have been a plane
figure. The figures obtained from a given figure, by projection
or section alone, are related to that figure by the principle of
duality, of which we shall have to speak later on.
I shall endeavour to show, in what follows, first, in what sense
figures obtained from each other by projective transformation are
qualitatively alike; secondly, what axioms, or adjectives of space,
are involved in the principle of projective transformation; and
thirdly, that these adjectives must belong to any form of externality
with more than one dimension, and are, therefore, _à priori_
properties of any possible space.
For the sake of simplicity, I shall in general confine myself to two
dimensions. In so doing, I shall introduce no important difference of
principle, and shall greatly simplify the mathematics involved.
=111.= The two mathematically fundamental things in projective
Geometry are anharmonic ratio, and the quadrilateral construction.
Everything else follows mathematically from these two. Now what is
meant, in projective Geometry, by anharmonic ratio?
[Illustration]
If we start from anharmonic ratio as ordinarily defined, we are
met by the difficulty of its quantitative nature[122]. But among
the properties deduced from this definition, many, if not most,
are purely qualitative. The most fundamental of these is that, if
through any four points in a straight line we draw four straight
lines which meet in a point, and if we then draw a new straight line
meeting these four, the four new points of intersection have the
same anharmonic ratio as the four points we started with. Thus, in
the figure, _abcd_, _a′b′c′d′_, _a″b″c″d″_, all have the same
anharmonic ratio. The reciprocal relation holds for the anharmonic
ratio of four straight lines. Here we have, plainly, the required
basis for a qualitative definition. The definition must be as follows:
Two sets of four points each are defined as having the same
anharmonic ratio, when (1) each set of four lies in one straight
line, and (2) corresponding points of different sets lie two by two
on four straight lines through a single point, or when both sets have
this relation to any third set[123]. And reciprocally: Two sets of
four straight lines are defined as having the same anharmonic ratio
when (1) each set of four passes through a single point, and (2)
corresponding lines of different sets pass, two by two, through four
points in one straight line, or when both sets have this relation to
any third set.
Two sets of points or of lines, which have the same anharmonic ratio,
are treated by projective Geometry as equivalent: this qualitative
equivalence replaces the quantitative equality of metrical Geometry,
and is obviously included, by its definition, in the above account of
projective transformations in general.
=112.= We have next to consider the quadrilateral construction[124].
This has a double purpose: first, to define the important special
case known as a harmonic range; and secondly, to afford an
unambiguous and exhaustive method of assigning different numbers to
different points. This last method has, again, a double purpose:
first, the purpose of giving a convenient symbolism for describing
and distinguishing different points, and of thus affording a means
for the introduction of analysis; and secondly, of so assigning
these numbers that, if they had the ordinary metrical significance,
as distances from some point on the numbered straight line, they
would yield -1 as the anharmonic ratio of a harmonic range, and
that, if four points have the same anharmonic ratio as four
others, so have the corresponding numbers. This last purpose is
due to purely technical motives: it avoids the confusion with our
preconceptions which would result from any other value for a harmonic
range; it allows us, when metrical interpretations of projective
results are desired, to make these interpretations without tedious
numerical transformations, and it enables us to perform projective
transformations by algebraical methods. At the same time, from the
strictly projective point of view, as observed above, the numbers
introduced have a purely conventional meaning; and until we pass to
metrical Geometry, no reason can be shown for assigning the value -1
to a harmonic range. With this preliminary, let us see in what the
quadrilateral construction consists.
[Illustration]
=113.= A harmonic range, in elementary Geometry, is one whose
anharmonic ratio is -1, or one in which the three segments formed by
the four points are in harmonic progression, or again, one in which
the ratio of the two internal segments is equal to the ratio of the
two external segments. If _a_, _b_, _c_, _d_ be the four points, it
is easily seen that these definitions are equivalent to one another:
they give respectively:
(ab/bc)/(ad/dc) = -1, (1/ab) - (1/ac) = (1/ac) - (1/ad),
and (ab/bc) = (ad/cd).
But as they are all quantitative, they cannot be used for our present
purpose. Nor are any definitions which involve bisection of lines or
angles available. We must have a definition which proceeds entirely
by the help of straight lines and points, without measurement of
distances or angles. Now from the above definitions of a harmonic
range, we see that _a_, _b_, _c_, _d_ have the same anharmonic ratio
as _c_, _b_, _a_, _d_. This gives us the property we require for our
definition. For it shows that, in a harmonic range, we can find a
projective transformation which will interchange _a_ and _c_. This is
a necessary and sufficient condition for a harmonic range, and the
quadrilateral construction is the general method for giving effect to
it.
[Illustration]
Given any three points _A_, _B_, _D_ in one straight line, the
quadrilateral construction finds the point _C_ harmonic to _A_ with
respect to _B_, _D_ by the following method: Take any point _O_
outside the straight line _ABD_, and join it to _B_ and _D_. Through
_A_ draw any straight line cutting _OD_, _OB_ in _P_ and _Q_. Join
_DQ_, _BP_, and let them intersect in _R_. Join _OR_, and let _OR_
meet _ABD_ in _C_. Then _C_ is the point required.
To prove this, let _DRQ_ meet _OA_ in _T_, and draw _AR_, meeting
_OD_ in _S_. Then a projective transformation of _A_, _B_, _C_, _D_
from _R_ on to _OD_ gives the points _S_, _P_, _O_, _D_, which,
projected from _A_ on to _DQ_, give _R_, _Q_, _T_, _D_. But these
again, projected from _O_ on to _ABD_, give _C_, _B_, _A_, _D_.
Hence _A_, _B_, _C_, _D_ can be projectively transformed into _C_,
_B_, _A_, _D_, and therefore form a harmonic range. From this point,
the proof that the construction is unique and general follows
simply[125].
The introduction of numbers, by this construction, offers no
difficulties of principle--except, indeed, those which always
attend the application of number to continua--and may be studied
satisfactorily in Klein's Nicht-Euklid (I. p. 337 ff.). The principle
of it is, to assign the numbers 0, 1, ∞ to _A_, _B_, _D_ and
therefore the number 2 to _C_, in order that the differences _AB_,
_AC_, _AD_ may be in harmonic progression. By taking _B_, _C_, _D_ as
a new triad corresponding to _A_, _B_, _D_, we find a point harmonic
to _B_ with respect to _C_, _D_ and assign to it the number 3, and so
on. In this way, we can obtain any number of points, and we are sure
of having no number and no point twice over, so that our coordinates
have the essential property of a unique correspondence with the
points they denote, and _vice versa_.
=114.= The point of importance in the above construction,
however, and the reason why I have reproduced it in detail, is
that it proceeds entirely by means of the general principles of
transformation enunciated above. From this stage onwards, everything
is effected by means of the two fundamental ideas we have just
discussed, and everything, therefore, depends on our general
principle of projective equivalence. This principle, as regards two
dimensions, may be stated more simply than in the passage quoted from
Cremona. It starts, in two dimensions, from the following definitions:
To project the points _A_, _B_, _C_, _D_ ... from a centre _O_, is to
construct the straight lines _OA_, _OB_, _OC_, _OD_....
To cut a number of straight lines _a_, _b_, _c_, _d_ ... by a
transversal _s_, is to construct the points _sa_, _sb_, _sc_,
_sd_....[126]
The successive application of these two operations, provided the
original figure consisted of points on one straight line or of
straight lines through one point, gives a figure projectively
indistinguishable from the former figure; and hence, by extension,
if any points in one straight line in the original figure lie in one
straight line in the derived figure, and reciprocally for straight
lines through points, the two operations have given projectively
similar figures. This general principle may be regarded as consisting
of two parts, according to the order of the operations: if we begin
with projection and end with section, we transform a figure of
points into another figure of points; by the converse order, we
transform a figure of lines into another figure of lines.
=115.= Before we can be clear as to the meaning of our principle, we
must have some notion as to our definition of points and straight
lines. But this definition, in projective Geometry, cannot be given
without some discussion of the principle of duality, the mathematical
form of the philosophical circle involved in geometrical definitions.
Confining ourselves for the moment to two dimensions, the principle
asserts, roughly speaking, that any theorem, dealing with lines
through a point and points on a line, remains true if these two
terms, wherever they occur, are interchanged. Thus: two points
lie on one straight line which they completely determine; and two
straight lines meet in one point, which they completely determine.
The four points of intersection of a transversal with four lines
through a point have an anharmonic ratio independent of the
particular transversal; and the four lines joining four points on one
straight line to a fifth point have an anharmonic ratio independent
of that fifth point. So also our general principle of projective
transformation has two sides: one in which points move along fixed
lines, and one in which lines turn about fixed points.
This duality suggests that any definition of points must be effected
by means of the straight line, and any definition of the straight
line must be effected by means of points. When we take the third
dimension into account, it is true, the duality is no longer so
simple; we have now to take account also of the plane, but this only
introduces a circle of three terms, which is scarcely preferable to
a circle of two terms. We now say: Three points, or a line and a
point, determine a plane: but conversely, three planes, or a line
and plane, determine a point. We may regard the straight line as a
relation between two of its points, but we may also regard the point
as a relation between two straight lines through it. We may regard
the plane as a relation between three points, or between a point and
a line, but we may also regard the point as a relation between three
planes, or between a line and a plane, which meet in it.
=116.= How are we to get outside this circle? The fact is that, in
pure Geometry, we cannot get outside it. For space, as we shall see
more fully hereafter, is nothing but relations; if, therefore, we
take any spatial figure, and seek for the terms between which it is a
relation, we are compelled, in Geometry, to seek these terms within
space, since we have nowhere else to seek them, but we are doomed,
since anything purely spatial is a mere relation, to find our terms
melting away as we grasp them.
Thus the relativity of space, while it is the essence of the
principle of duality, at the same time renders impossible the
expression of that principle, or of any other principle of pure
Geometry, in a manner which shall be free from contradictions.
Nevertheless, if we are to advance at all with our analysis of
geometrical reasoning and with our definitions of lines and points,
we must, for a while, ignore this contradiction; we must argue
as though it did not exist, so as to free our science from any
contradictions which are not inevitable.
=117.= In accordance with this procedure, then, let us define our
points as the terms of spatial relations, regarding whatever is not
a point as a relation between points. What, on this view, must our
points be taken to be? Obviously, if extension is mere relativity,
they must be taken to contain no extension; but if they are to
supply the terms for spatial relations, _e.g._ for straight lines,
these relations must exhibit them as the terms of the figures they
relate. In other words, since what can really be taken, without
contradiction, as the term of a spatial relation, is unextended, we
must take, as the term to be used in Geometry, where we cannot go
outside space, the least spatial thing which Geometry can deal with,
the thing which, though _in_ space, _contains_ no space; and this
thing we define as the point[127].
Neglecting, then, the fundamental contradiction in this definition,
the rest of our definitions follow without difficulty. The straight
line is the relation between two points, and the plane is the
relation between three. These definitions will be argued and defended
at length in section B of this Chapter[128], where we can discuss at
the same time the alternative metrical definitions; for our present
purpose, it is sufficient to observe that projective Geometry, from
the first, regards the straight line as determined by two points, and
the plane as determined by three, from which it follows, if we take
points as possible terms for spatial relations, that the straight
line and the plane may be regarded as relations between two and three
points respectively. If we agree on these definitions, we can proceed
to discuss the fundamental principle of projective Geometry, and to
analyse the axioms implicated in its truth.
=118.= Projective Geometry, we have seen, does not deal with
quantity, and therefore recognizes no difference where the
difference is purely quantitative. Now quantitative comparison
depends on a recognized identity of quality; the recognition of
qualitative identity, therefore, is logically prior to quantity, and
presupposed by every judgment of quantity. Hence all figures, whose
differences can be exhaustively described by quantity, _i.e._ by
pure measurement, must have an identity of quality, and this must be
recognizable without appeal to quantity. It follows that, by defining
the word quality in geometrical matters, we shall discover what
sets of figures are projectively indiscernible. If our definition
is correct, it ought to yield the general projective principle with
which we set out.
=119.= We agreed to regard points as the terms of spatial relations,
and we agreed that different points could be distinguished. But
we postponed the discussion of the conditions under which this
distinction could be effected. This discussion will yield us the
definition of quality and the proof of our general projective
principle.
Points, to begin with, have been defined as nothing but the terms for
spatial relations. They have, therefore, no intrinsic properties;
but are distinguished solely by means of their relations. Now the
relation between two points, we said, is the straight line on which
they lie. This gives that identity of quality for all pairs of points
on the same straight line, which is required both by our projective
principle and by metrical Geometry. (For only where there is identity
of quality can quantity be properly applied.) If only two points are
given, they cannot, without the use of quantity, be distinguished
from any two other points on the same straight line; for the
qualitative relation between any two such points is the same as for
the original pair, and only by a difference of relation can points be
distinguished from one another.
But conversely, one straight line is nothing but the relation
between two of its points, and all points are qualitatively alike.
Hence there can be nothing to distinguish one straight line from
another except the points through which it passes, and these are
distinguished from other points only by the fact that it passes
through them. Thus we get the reciprocal transformation: if we are
given only one point, any pair of straight lines through that point
is qualitatively indistinguishable from any other. This again is, on
the one hand, the basis of the second part of our general projective
principle, and on the other hand the condition of applying quantity,
in the measurement of angles, to the departure of two intersecting
straight lines.
=120.= We can now see the reason for what may have hitherto seemed a
somewhat arbitrary fact, namely, the necessity of _four_ collinear
points for anharmonic ratio. Recurring to the quadrilateral
construction and the consequent introduction of number, we see
that anharmonic ratio is an intrinsic projective relation of four
collinear points or concurrent straight lines, such that given three
terms and the relation, the fourth term can be uniquely determined
by projective methods. Now consider first a pair of points. Since
all straight lines are projectively equivalent, the relation between
one pair of points is precisely equivalent to that between another
pair. Given one point only, therefore, no projective relation, to
any second point, can be assigned, which shall in any way limit our
choice of the second point. Given two points, however, there is such
a relation--the third point may be given collinear with the first
two. This limits its position to one straight line, but since two
points determine nothing but one straight line, the third point
cannot be further limited. Thus we see why no intrinsic projective
relation can be found between three points, which shall enable us,
from two, uniquely to determine the third. With three given collinear
points, however, we have more given than a mere straight line, and
the quadrilateral construction enables us uniquely to determine any
number of fresh collinear points. This shows why anharmonic ratio
must be a relation between four points, rather than between three.
=121.= We can now prove, I think, that two figures, which are
projectively related, are qualitatively similar. Let us begin with
a collection of points on a straight line. So long as these are
considered without reference to other points or figures, they are
all qualitatively similar. They can be distinguished by immediate
intuition, but when we endeavour, without quantity, to distinguish
them conceptually, we find the task impossible, since the only
qualitative relation of any two of them, the straight line, is
the same for any other two. But now let us choose, at hap-hazard,
some point outside the straight line. The points of our line now
acquire new adjectives, namely their relations to the new point,
_i.e._ the straight lines joining them to this new point. But these
straight lines, reciprocally, alone define our external point,
and all straight lines are qualitatively similar. If we take some
other external point, therefore, and join it to the same points
of our original straight line, we obtain a figure in which, so
long as quantity is excluded, there is no conceptual difference
from the former figure. Immediate intuition can distinguish the
two figures, but qualitative discrimination cannot do so. Thus we
obtain a projective transformation of four lines into four other
lines, as giving a figure qualitatively indistinguishable from the
original figure. A similar argument applies to the other projective
transformations. Thus the only reason, within projective Geometry,
for not regarding projective figures as actually identical, is the
intuitive perception of difference of position. This is fundamental,
and must be accepted as a _datum_. It is presupposed in the
distinction of various points, and forms the very life of Geometry.
It is, in fact, the essence of the notion of a form of externality,
which notion forms the subject-matter of projective Geometry.
=122.= We may now sum up the results of our analysis of projective
Geometry, and state the axioms on which its reasoning is based. We
shall then have to prove that these axioms are necessary to any form
of externality, with which we shall pass, from mere analysis, to a
transcendental argument.
The axioms which have been assumed in the above analysis, and which,
it would seem, suffice to found projective Geometry, may be roughly
stated as follows:
I. We can distinguish different parts of space, but all parts are
qualitatively similar, and are distinguished only by the immediate
fact that they lie outside one another.
II. Space is continuous and infinitely divisible; the result of
infinite division, the zero of extension, is called a _point_[129].
III. Any two points determine a unique figure, called a straight
line, any three in general determine a unique figure, the plane.
Any four determine a corresponding figure of three dimensions, and
for aught that appears to the contrary, the same may be true of any
number of points. But this process comes to an end, sooner or later,
with some number of points which determine the whole of space. For
if this were not the case, no number of relations of a point to a
collection of given points could ever determine its relation to fresh
points, and Geometry would become impossible[130].
This statement of the axioms is not intended to have any exclusive
precision: other statements equally valid could easily be made. For
all these axioms, as we shall see hereafter, are philosophically
interdependent, and may, therefore, be enunciated in many ways. The
above statement, however, includes, if I am not mistaken, everything
essential to projective Geometry, and everything required to prove
the principle of projective transformation. Before discussing the
apriority of these axioms, let us once more briefly recapitulate the
ends which they are intended to attain.
=123.= From the exclusively mathematical standpoint, as we have
seen, projective Geometry discusses only what figures can be obtained
from each other by projective transformations, _i.e._ by the
operations of projection and section. These operations, in all their
forms, presuppose the point, straight line, and plane[131], whose
necessity for projective Geometry, from the purely mathematical point
of view, is thus self-evident from the start. But philosophically,
projective Geometry has, as we saw, a wider aim. This wider aim,
which gives, to the investigation of projectively equivalent figures,
its chief importance, consists in the determination of qualitative
spatial similarity, in the determination, that is, of all the figures
which, when any one figure is given, can be distinguished from the
given figure, so long as quantity is excluded, only by the mere fact
that they are external to it.
=124.= Now when we consider what is involved in such absolute
qualitative equivalence, we find at once, as its most obvious
prerequisite, the perfect homogeneity of space. For it is assumed
that a figure can be completely defined by its internal relations,
and that the external relations, which constitute its position,
though they suffice to distinguish it from other figures, in
no way affect its internal properties, which are regarded as
qualitatively identical with those of figures with quite different
external relations. If this were not the case, anything analogous
to projective transformation would be impossible. For such
transformation always alters the position, _i.e._ the external
relations, of a figure, and could not, therefore, if figures were
dependent on their relations to other figures or to empty space,
be studied without reference to other figures, or to the absolute
position of the original figure. We require for our principle,
in short, what may be called the mutual passivity and reciprocal
independence of two parts or figures of space.
This passivity and this independence involve the homogeneity of
space, or its equivalent, the relativity of position. For if
the internal properties of a figure are the same, whatever its
external relations may be, it follows that all parts of space are
qualitatively similar, since a change of external relation is a
change in the part of space occupied. It follows, also, that all
position is relative and extrinsic, _i.e._, that the position of a
point, or the part of space occupied by a figure, is not, and has
no effect upon, any intrinsic property of the point or figure, but
is exclusively a relation to other points or figures in space, and
remains without effect except where such relations are considered.
=125.= The homogeneity of space and the relativity of position,
therefore, are presupposed in the qualitative spatial comparison
with which projective Geometry deals. The latter, as we saw, is
also the basis of the principle of duality. But these properties,
as I shall now endeavour to prove, belong of necessity to any form
of externality, and are thus _à priori_ properties of all possible
spaces. To prove this, however, we must first define the notion of a
form of externality in general.
Let us observe, to begin with, that the distinction between
Euclidean and non-Euclidean Geometries, so important in metrical
investigations, disappears in projective Geometry proper. This
suggests that projective Geometry, though originally invented as
the science of Euclidean space, and subsequently of non-Euclidean
spaces also, deals really with a wider conception, a conception which
includes both, and neglects the attributes in which they differ. This
conception I shall speak of as a form of externality.
=126.= In Grassmann's profound philosophical introduction to his
_Ausdehnungslehre_ of 1844, he suggested that Geometry, though
improperly regarded as pure, was really a branch of applied
mathematics, since it dealt with a subject-matter not created,
like number, by the intellect, but given to it, and therefore not
wholly subject to its laws alone. But it must be possible--so he
contended--to construct a branch of pure mathematics, a science, that
is, in which our object should be wholly a creature of the intellect,
which should yet deal, as Geometry does, with extension--extension
as conceived, however, not as empirically perceived in sensation or
intuition.
From this point of view, the controversy between Kantians and
anti-Kantians becomes wholly irrelevant, since the distinction
between pure and mixed mathematics does not lie in the distinction
between the subjective and the objective, but between the purely
intellectual on the one hand, and everything else on the other.
Now Kant had contended, with great emphasis, that space was not an
intellectual construction, but a subjective intuition. Geometry,
therefore, with Grassmann's distinction, belongs to mixed mathematics
as much on Kant's view as on that of his opponents. And Grassmann's
distinction, I contend, is the more important for Epistemology,
and the one to be adopted in distinguishing the _à priori_ from
the empirical. For what is merely intuitional can change, without
upsetting the laws of thought, without making knowledge formally
impossible: but what is purely intellectual cannot change, unless the
laws of thought should change, and all our knowledge simultaneously
collapse. I shall therefore follow Grassmann's distinction in
constructing an _à priori_ and purely conceptual form of externality.
=127.= The pure doctrine of extension, as constructed by Grassmann,
need not be discussed--it included much empirical material, and was
philosophically a failure. But his principles, I think, will enable
us to prove that projective Geometry, abstractly interpreted, is
the science which he foresaw, and deals with a matter which can be
constructed by the pure intellect alone. If this be so, however,
it must be observed that projective Geometry, for the moment, is
rendered purely hypothetical[132]. All necessary truth, as Bradley
has shown, is hypothetical[133], and asserts, _primâ facie_, only
the ground on which rests the necessary connection of premisses
and conclusion. If we construct a mere conception of externality,
and thus abandon our actually given space, the result of our
construction, until we return to something actually given, remains
without existential import--if there _be_ experienced externality, it
asserts, then there must be a form of externality with such and such
properties. That there must be experienced externality, Kant's first
argument about space proves, I think, to those who admit experience
of a world of diverse but interrelated things. But this is a question
which belongs to the next Chapter.
What we have to do here is, not to discuss whether there is a form of
externality, but whether, if there be such a form, it must possess
the properties embodied in the axioms of projective Geometry. Now
first of all, what do we mean by such a form?
=128.= In any world in which perception presents us with various
things, with discriminated and differentiated contents, there must
be, in perception, at least one "principle of differentiation[134],"
an element, that is, by which the things presented are distinguished
as various. This element, taken in isolation, and abstracted from the
content which it differentiates, we may call a form of externality.
That it must, when taken in isolation, appear as a form, and not as
a mere diversity of material content, is, I think, fairly obvious.
For a diversity of material content cannot be studied apart from
that material content; what we wish to study here, on the contrary,
is the bare possibility of such diversity, which forms the residuum,
as I shall try to prove hereafter[135], when we abstract from any
sense-perception all that is distinctive of its particular matter.
This possibility, then, this principle of bare diversity, is our form
of externality. How far it is necessary to assume such a form, as
distinct from interrelated things, I shall consider later on[136].
For the present, since space, as dealt with by Geometry, is certainly
a form of this kind, we have only to ask: What properties must such a
form, when studied in abstraction, necessarily possess?
=129.= In the first place, externality is an essentially relative
conception--nothing can be external to itself. To be external to
something is to be another with some relation to that thing. Hence,
when we abstract a form of externality from all material content,
and study it in isolation, position will appear, of necessity,
as purely relative--a position can have no intrinsic quality, for
our form consists of pure externality, and externality contains
no shadow or trace of an intrinsic quality. Thus we obtain our
fundamental postulate, the relativity of position, or, as we may put
it, the complete absence, on the part of our form, of any vestige of
thinghood.
The same argument may also be stated as follows: If we abstract the
conception of externality, and endeavour to deal with it _per se_, it
is evident that we must obtain an object alike destitute of elements
and of totality. For we have abstracted from the diverse matter
which filled our form, while any element, or any whole, would retain
some of the qualities of a matter. Either an element or a whole, in
fact, would have to be a thing not external to itself, and would
thus contain something not pure externality. Hence arise infinite
divisibility, with the self-contradictory notion of the point, in the
search for elements, and unbounded extension, with the contradiction
of an infinite regress or a vicious circle, in the search for a
completed whole. Thus again, our form contains neither elements nor
totality, but only endless relations--the terms of these relations
being excluded by our abstraction from the matter which fills our
form.
=130.= In like manner we can deduce the homogeneity of our form. The
diversity of content, which was possible only within the form of
externality, has been abstracted from, leaving nothing but the bare
possibility of diversity, the bare principle of differentiation,
itself uniform and undifferentiated. For if diversity presupposes
such a form, the form cannot, unless it were contained in a fresh
form, be itself diverse or differentiated.
Or we may deduce the same property from the relativity of position.
For any quality in one position, by which it was marked out from
another, would be necessarily more or less intrinsic, and would
contradict the pure relativity. Hence all positions are qualitatively
alike, _i.e._ the form is homogeneous throughout.
=131.= From what has been said of homogeneity and relativity,
follows one of the strangest properties of a form of externality.
This property is, that the relation of externality between any two
things is infinitely divisible, and may be regarded, consequently,
as made up of an infinite number of the would-be elements of our
form, or again as the sum of two relations of externality[137]. To
speak of dividing or adding relations may well sound absurd--indeed
it reveals the impropriety of the word relation in this connexion.
It is difficult, however, to find an expression which shall be less
improper. The fact seems to be, that externality is not so much a
relation as bare relativity, or the bare possibility of a relation.
On this subject, I shall enlarge in Chapter IV.[138] At this point
it is only important to realize, what the subsequent argument will
assume, that the relation--if we may so call it--of externality
between two or more things must, since our form is homogeneous, be
capable of continuous alteration, and must, since our infinitely
divisible form is constituted by such relations, be capable of
infinite division. But the result of infinite division is defined
as the element of our form. (Our form has no elements, but we have
to imagine elements in order to reason about it, as will be shown
more fully in Chapter IV.) Hence it follows, that every relation
of externality may be regarded, for scientific purposes, as an
infinite congeries of elements, though philosophically, the relations
alone are valid, and the elements are a self-contradictory result
of hypostatizing the form of externality. This way of regarding
relations of externality is important in understanding the meaning of
such ideas as three or four collinear points.
As this point is difficult and important, I will repeat, in somewhat
greater detail, the explanation of the manner in which straight
lines and planes come to be regarded as congeries of points. From
the strictly projective standpoint, though all other figures _are_
merely a collection of any required number of points, lines or
planes, given by some projective construction, straight lines and
planes themselves are given integrally, and are not to be considered
as divisible or composed of parts. To say that a point lies on
a straight line means, for projective Geometry proper, that the
straight line is a relation between this and some other point.
Here the points concerned, if our statement is to be freed from
contradictions, must be regarded, if I may use such an expression, as
_real_ points--_i.e._ as unextended material centres[139]. Straight
lines and planes are then relations between these material atoms.
They are relations, however, which may undergo a metrical alteration
while remaining projectively unchanged. When the projective relation
between the two points _A_, _B_ is the same as that between the two
points _A_, _C_, while the metrical relation (distance) is different,
the three points _A_, _B_, _C_ are said to be collinear. Now the
metrical manner of regarding spatial figures demands that they should
be hypostatized, and no longer regarded as mere relations. For when
we regard a quantity as extensive, _i.e._ as divisible into parts,
we necessarily regard it as more than a mere relation or adjective,
since no mere relation or adjective can be divided. For quantitative
treatment, therefore, spatial relations must be hypostatized[140].
When this is done, we obtain, as we saw above, a homogeneous
and infinitely divisible form of externality. We find now that
distance, for example, may be continuously altered without changing
the straight line on which it is measured. We thus obtain, on the
straight line in question, a continuous series of points, which,
since it is continuous, we regard as constituting our straight line.
It is thus solely from the hypostatizing of relations, which metrical
Geometry requires, that the view of straight lines and planes as
_composed_ of points arises, and it is from this hypostatizing that
the difficulties of metrical Geometry spring.
=132.= The next step, in defining a form of externality, is obtained
from the idea of _dimensions_. Positions, we have seen, are defined
solely by their relations to other positions. But in order that
such definition may be possible, a finite number of relations must
suffice, since infinite numbers are philosophically inadmissible. A
position must be definable, therefore, if knowledge of our form is
to be possible at all, by some finite integral number of relations
to other positions. Every relation thus necessary for definition
we call a dimension. Hence we obtain the proposition: _Any form of
externality must have a finite integral number of dimensions_.
=133.= The above argument, it may be urged, has overlooked a
possibility. It has used a transcendental argument, so an opponent
may contend, without sufficiently proving that knowledge about
externality must be possible without reference to the matters
external to each other. The definition of a position may be
impossible, so long as we neglect the matter which fills the form,
but may become possible when this matter is taken into account.
Such an objection can, I think, be successfully met, by a reference
to the passivity and homogeneity of our form. For any dependence
of the definition of a position on the particular matter filling
that position, would involve some kind of interaction between the
matter and its position, some effect of the diverse content on
the homogeneous form. But since the form is totally destitute of
thinghood, perfectly impassive, and perfectly void of differences
between its parts, any such effect is inconceivable. An effect on
a position would have to alter it in some way, but how could it be
altered? It has no qualities except those which make it the position
it is, as opposed to other positions; it cannot change, therefore,
without becoming a different position. But such a change contradicts
the law of identity. Hence it is not the position which has changed,
but the content which has moved in the form. Thus it must be
possible, if knowledge of our form can be obtained at all, to obtain
this knowledge in logical independence of the particular matter which
fills it. The above argument, therefore, granted the possibility of
knowledge in the department in question, shows the necessity of a
finite integral number of dimensions.
=134.= Let us repeat our original argument in the light of this
elucidation. A position is completely defined when, and only when,
enough relations are known to enable us to determine its relation
to any fresh known position. Only by relations within the form of
externality, as we have just seen, and never by relations which
involve a reference to the particular matter filling the form,
can such a definition be effected. But the possibility of such a
definition follows from the Law of Excluded Middle, when this law is
interpreted to mean, as Bosanquet makes it mean, that "Reality ... is
a system of reciprocally determinate parts[141]." For this implies
that, given the relations of a part _A_ to other parts _B_, _C_ ...,
a sufficient wealth of such relations throws light on the relations
of _B_ to _C_, etc. If this were not the case, the parts _A_, _B_,
_C_ ... could not be said to form such a system; for in such a
system, to define _A_ is to define, at the same time, all the other
members, and to give an adjective to _A_, is to give an adjective to
_B_ and _C_. But the relations between positions are, when we restore
the matter from which the positions were abstracted, relations
between the things occupying those positions, and these relations, we
have seen, can be studied without reference to the particular nature,
in other respects, of the related things. It follows that, when we
apply the general principle of systematic unity to these relations in
particular, we find these relations to be dependent on each other,
since they are not dependent, for their definition, on anything else.
This gives the axiom of dimensions, in the above general form, as
the result, on our abstract geometrical level, of the relativity of
position and the law of excluded middle.
=135.= Before proceeding further, it is necessary to discuss the
important special case where a form of externality has only one
dimension. Of the two such forms, given in experience, one, namely
time, presents an instance of this special case. But it may be shown,
I think, that the function, in constituting the possibility of
experience, which we demand of such forms, could not be accomplished
by a one-dimensional form alone. For in a one-dimensional form, the
various contents may be arranged in a series, and cannot, without
interpenetration, change the order of contents in the series.
But interpenetration is impossible, since a form of externality
is the mere expression of diversity among things, from which it
follows that things cannot occupy the same position in a form,
unless there is another form by which to differentiate them. For
without externality, there is no diversity[142]. Thus two bodies
may occupy the same space, but only at different times: two things
may exist simultaneously, but only at different places. A form of
one dimension, therefore, could not, by itself, allow that change
of the relations of externality, by which alone a varied world
of interrelated things can be brought into consciousness. In a
one-dimensional space, for example, only a single object, which must
appear as a point, or two objects at most, one in front and one
behind, could ever be perceived. Thus two or more dimensions seem
an essential condition of anything worth calling an experience of
interrelated things.
=136.= It may be objected, to this argument, that its validity
depends upon the assumption that the change of a relation of
externality must be continuous. Both to make and to meet this
objection, in a manner which shall not imply time, seems almost
impossible. For we cannot speak of change, whether continuous or
discrete, without imagining time. Let us, therefore, allow time to
be known, and discuss whether the temporal change, in any other form
of externality, is necessarily continuous[143]. We must reply, I
think, that continuity is necessary. The change of relation, in our
non-temporal form, may be safely described as motion, and the law
of Causality--since we have already assumed time--may be applied to
this motion. It then follows that discrete motion would involve a
finite effect from an infinitesimal cause, for a cause acting only
for a moment of time would be infinitesimal. It involves, also, a
validity in the point of time, whereas what is valid in any form of
externality is not, as we have already seen, the infinitesimal and
self-contradictory element resulting from infinite division, but the
finite relation which mathematics analyzes into vanishing elements.
Hence change must be continuous, and the possibility of serial
arrangement holds good.
In a one-dimensional form other than time, the same argument must
hold. For something analogous to Causality would be necessary to
experience, and the relativity of the form would still necessarily
hold. Hence, since only these two properties of time have been
assumed, the above contention would remain valid of any second form
whose relations were correlated with those of the first, as the
analogue of Causality would require them to be.
=137.= The next step in the argument, which assumes two or more
dimensions, is concerned with the general analogues of straight
lines and planes, _i.e._ with figures--which may be regarded either
as relations between positions or as series of positions--uniquely
determined by two or by three positions. If this step can be
successfully taken, our deduction of the above projective axioms will
be complete, and descriptive Geometry will be established as the
abstract _à priori_ doctrine of forms of externality.
To prove this contention, consider of what nature the relations
can be by which positions are defined. We have seen already that
our form is purely relational and infinitely divisible, and that
positions (points) are the self-contradictory outcome of the search
for something other than relations. What we really mean, therefore,
by the relations defining a position, is, when we undo our previous
abstraction, the relations of externality by which some thing is
related to other things. But how, when we remain in the abstract
form, must such relations appear?
=138.= We have to prove that two positions must have a relation
independent of any reference to other positions. To prove this, let
us recur to what was said, in connection with dimensions, as to the
passivity and homogeneity of our form. Since positions are defined
only by relations, there must be relations, within the form, between
positions. But if there are such relations, there must be a relation
which is intrinsic to two positions. For to suppose the contrary,
is to attribute an interaction or causal connection, of some kind,
between those two positions and other positions--a supposition
which the perfect homogeneity of our form renders absurd, since all
positions are qualitatively similar, and cannot be changed without
losing their identity. We may put this argument thus: since positions
are only defined by their relations, such definition could never
begin, unless it began with a relation between only two positions.
For suppose three positions _A_, _B_, _C_ were necessary, and gave
rise to the relation _abc_ between the three. Then there would remain
no means of defining the different pairs _BC_, _CA_, _AB_, since the
only relation defining them would be one common to all three pairs.
Nothing would be gained, in this case, by reference to fresh points,
for it follows, from the homogeneity and passivity of the form, that
these fresh points could not affect the internal relations of our
triad, which relations, if they can give definiteness at all, must
give it without the aid of external reference. Two positions must,
therefore, if definition is to be possible, have some relation which
they by themselves suffice to define. Precisely the same argument
applies to three positions, or to four; the argument loses its scope
only when we have exhausted the dimensions of the form considered.
Thus, in three dimensions, five positions have no fresh relation,
not deducible from those already known, for by the definition of
dimensions, all the relations involved can be deduced from those of
the fourth point to the first three, together with those of the fifth
to the first three.
We may give the argument a more concrete, and perhaps a more
convincing shape, by considering the matter arranged in our form. If
two things are mutually external, they must since they belong to the
same world, have some relation of externality; there is, therefore,
a relation of externality between two things. But since our form is
homogeneous, the same relation of externality may subsist in other
parts of the form, _i.e._ while the two things considered alter their
relations of externality to other things. The relation of externality
between two things is, therefore, independent of other things. Hence,
when we return to the abstract language of the form, two positions
have a relation determined by those two positions alone, and
independent of other positions.
Precisely the same argument applies to the relations of three
positions, and in each case the relation must appear in the form as
not a mere inference from the positions it relates. For relations,
as we have seen, actually constitute a form of externality, and are
not mere inferences from terms, which are nowhere to be found in the
form[144].
To sum up: Since position is relative, two positions must have
_some_ relation to each other; and since our form of externality
is homogeneous, this relation can be kept unchanged while the two
positions change their relations to other positions. Hence their
relation is intrinsic, and independent of other positions. Since
the form is a mere complex of relations, the relation in question
must, if the form is sensuous or intuitive, be itself sensuous or
intuitive, and not a mere inference. In this case, a unique relation
must be a unique figure--in spatial terms, the straight line joining
the two points.
=139.= With this, our deduction of projective Geometry from the
_à priori_ conceptual properties of a form of externality is
completed. That such a form, when regarded as an independent thing,
is self-contradictory, has been abundantly evident throughout the
discussion. But the science of the form has been founded on the
opposite way of regarding it: we have held it throughout to be a mere
complex of relations, and have deduced its properties exclusively
from this view of it. The many difficulties, in applying such an _à
priori_ deduction to intuitive space, and in explaining, as logical
necessities, properties which appear as sensuous or intuitional data,
must be postponed to Chapter IV. For the present, I wish to point out
that projective Geometry is wholly _à priori_; that it deals with an
object whose properties are logically deduced from its definition,
not empirically discovered from data; that its definition, again, is
founded on the possibility of experiencing diversity in relation,
or multiplicity in unity; and that our whole science, therefore, is
logically implied in, and deducible from, the possibility of such
experience.
=140.= In metrical Geometry, on the contrary, we shall find a very
different result. Although the geometrical conditions which render
spatial measurement possible, will be found identical, except for
slight differences in the form of statement, with the _à priori_
axioms discussed above, yet the actual measurement--which deals
with actually given space, not the mere intellectual construction
we have been just discussing--gives results which can only be known
empirically and approximately, and can be deduced by no necessity
of thought. The Euclidean and non-Euclidean spaces give the various
results which are _à priori_ possible; the axioms peculiar to
Euclid--which are properly not axioms, but empirical results of
measurement--determine, within the errors of observation, which of
these _à priori_ possibilities is realized in our actual space. Thus
measurement deals throughout with an empirically given matter, not
with a creature of the intellect, and its _à priori_ elements are
only the conditions presupposed in the possibility of measurement.
What these conditions are, we shall see in the second section of this
chapter.
Section B.
THE AXIOMS OF METRICAL GEOMETRY.
=141.= We have now reviewed the axioms of projective Geometry, and
have seen that they are _à priori_ deductions from the fact that
we can experience externality, _i.e._ a coexistent multiplicity of
different but interrelated things. But projective Geometry, in spite
of its claims, is not the whole science of space, as is sufficiently
proved by the fact that it cannot discriminate between Euclidean and
non-Euclidean spaces[145]. For this purpose, spatial measurement
is required: metrical Geometry, with its quantitative tests, can
alone effect the discrimination. For all application of Geometry to
physics, also, measurement is required; the law of gravitation, for
example, requires the determination of actual distances. For many
purposes, in short, projective Geometry is wholly insufficient: thus
it is unable to distinguish between different kinds of conics, though
their distinction is of fundamental importance in many departments of
knowledge.
Metrical Geometry is, then, a necessary part of the science of
space, and a part not included in descriptive Geometry. Its _à
priori_ element, nevertheless, so far as this is spatial and not
arithmetical, is the same as the postulate of projective Geometry,
namely, the homogeneity of space, or its equivalent, the relativity
of position. We can see, in fact, that the _à priori_ element in
both is likely to be the same. For the _à priori_ in metrical
Geometry will be whatever is presupposed in the possibility of
spatial measurement, _i.e._ of quantitative spatial comparison. But
such comparison presupposes simply a known identity of quality,
the determination of which is precisely the problem of projective
Geometry. Hence the conditions for the possibility of measurement, in
so far as they are not arithmetical, will be precisely the same as
those for projective Geometry.
=142.= Metrical Geometry, therefore, though distinct from projective
Geometry, is not independent of it, but presupposes it, and arises
from its combination with the extraneous idea of _quantity_.
Nevertheless the mathematical form of the axioms, in metrical
Geometry, is slightly different from their form in projective
Geometry. The homogeneity of space is replaced by its equivalent, the
axiom of Free Mobility. The axiom of the straight line is replaced
by the axiom of distance: Two points determine a unique quantity,
distance, which is unaltered in any motion of the two points as a
single figure. This axiom, indeed, will be found to involve the axiom
of the straight line--such a quantity could not exist unless the
two points determined a unique curve--but its mathematical form is
changed. Another important change is the collapse of the principle
of duality: quantity can be applied to the straight line, because
it is divisible into similar parts, but cannot be applied to the
indivisible point. We thus obtain a reason, which was wanting in
descriptive Geometry, for preferring points, as spatial elements,
to straight lines or planes[146]. Finally, an entirely new idea is
introduced with quantity, namely, the idea of _Motion_. Not that we
study motion, or that any of our results have reference to motion,
but that they cannot, though in projective Geometry they could, be
obtained without at least an ideal motion of our figures through
space.
Let us now examine in detail the prerequisites of spatial
measurement. We shall find three axioms, without which such
measurement would be impossible, but with which it is adequate to
decide, empirically and approximately, the Euclidean or non-Euclidean
nature of our actual space. We shall find, further, that these three
axioms can be deduced from the conception of a form of externality,
and owe nothing to the evidence of intuition. They are, therefore,
like their equivalents the axioms of projective Geometry, _à priori_,
and deducible from the conditions of spatial experience. This
experience, accordingly, can never disprove them, since its very
existence presupposes them.
I. _The Axiom of Free Mobility._
=143.= Metrical Geometry, to begin with, may be defined as the
science which deals with the comparison and relations of spatial
magnitudes. The conception of magnitude, therefore, is necessary from
the start. Some of Euclid's axioms, accordingly, have been classed as
arithmetical, and have been supposed to have nothing particular to do
with space. Such are the axioms that equals added to or subtracted
from equals give equals, and that things which are equal to the same
thing are equal to one another. These axioms, it is said, are purely
arithmetical, and do not, like the others, ascribe an adjective to
space. As regards their use in arithmetic, this is of course true.
But if an arithmetical axiom is to be applied to spatial magnitudes,
it must have some spatial import[147], and thus even this class is
not, in Geometry, _merely_ arithmetical. Fortunately, the geometrical
element is the same in all the axioms of this class--we can see at
once, in fact, that it can amount to no more than a definition of
spatial magnitude[148]. Again, since the space with which Geometry
deals is infinitely divisible, a definition of spatial magnitude
reduces itself to a definition of spatial equality, for, as soon as
we have this last, we can compare two spatial magnitudes by dividing
each into a number of equal units, and counting the number of such
units in each[149]. The ratio of the number of units is, of course,
the ratio of the two magnitudes.
=144.= We require, then, at the very outset, some criterion of
spatial equality: without such a criterion metrical Geometry would
become wholly impossible. It might appear, at first sight, as though
this need not be an axiom, but might be a mere definition. In part
this is true, but not wholly. The part which is merely a definition
is given in Euclid's eighth axiom: "Magnitudes which exactly coincide
are equal." But this gives a sufficient criterion only when the
magnitudes to be compared already occupy the same position. When, as
will normally be the case, the two spatial magnitudes are external to
one another--as, indeed, must be the case, if they are distinct, and
not whole and part--the two magnitudes can only be made to coincide
by a motion of one or both of them. In order, therefore, that our
definition of spatial magnitude may give unambiguous results,
coincidence when superposed, if it can ever occur, must occur always,
whatever path be pursued in bringing it about. Hence, if mere motion
could alter shapes, our criterion of equality would break down.
It follows that the application of the conception of magnitude
to figures in space involves the following axiom[150]: _Spatial
magnitudes can be moved from place to place without distortion_; or,
as it may be put, _Shapes do not in any way depend upon absolute
position in space_.
The above axiom is the axiom of Free Mobility[151]. I propose to
prove (1) that the denial of this axiom would involve logical and
philosophical absurdities, so that it must be classed as wholly _à
priori_; (2) that metrical Geometry, if it refused this axiom, would
be unable, without a logical absurdity, to establish the notion of
spatial magnitude at all. The conclusion will be, that the axiom
cannot be proved or disproved by experience, but is an _à priori_
condition of metrical Geometry. As I shall thus be maintaining a
position which has been much controverted, especially by Helmholtz
and Erdmann, I shall have to enter into the arguments at some length.
=145.= A. _Philosophical Argument._ The denial of the axiom involves
absolute position, and an action of mere space, _per se_, on things.
For the axiom does not assert that real bodies, as a matter of
empirical fact, never change their shape in any way during their
passage from place to place: on the contrary, we know that such
changes do occur, sometimes in a very noticeable degree, and always
to some extent. But such changes are attributed, not to the change
of place as such, but to physical causes: changes of temperature,
pressure, etc. What our axiom has to deal with is not actual material
bodies, but geometrical figures[152], and it asserts that a figure
which is possible in any one position in space is possible in every
other. Its meaning will become clearer by reference to a case where
it does not hold, say the space formed by the surface of an egg.
Here, a triangle drawn near the equator cannot be moved without
distortion to the point, as it would no longer fit the greater
curvature of the new position: a triangle drawn near the point
cannot be fitted on to the flatter end, and so on. Thus the method
of superposition, such as Euclid employs in Book I. Prop. IV.,
becomes impossible; figures cannot be freely moved about, indeed,
given any figure, we can determine a certain series of possible
positions for it on the egg, outside which it becomes impossible.
What I assert is, then, that there is a philosophic absurdity in
supposing space in general to be of this nature. On the egg we have
marked points, such as the two ends; the space formed by its surface
is not homogeneous, and if things are moved about in it, it must of
itself exercise a distorting effect upon them, quite independently of
physical causes; if it did not exercise such an effect, the things
could not be moved. Thus such a space would not be homogeneous, but
would have marked points, by reference to which bodies would have
absolute position, quite independently of any other bodies. Space
would no longer be passive, but would exercise a definite effect upon
things, and we should have to accommodate ourselves to the notion
of marked points in empty space; these points being marked, not by
the bodies which occupied them, but by their effects on any bodies
which might from time to time occupy them. This want of homogeneity
and passivity is, however, absurd; space must, since it is a form of
externality, allow only of relative, not of absolute, position, and
must be completely homogeneous throughout. To suppose it otherwise,
is to give it a thinghood which no form of externality can possibly
possess. We must, then, on purely philosophical grounds, admit that a
geometrical figure which is possible anywhere is possible everywhere,
which is the axiom of Free Mobility.
=146.= B. _Geometrical Argument._ Let us see next what sort of
Geometry we could construct without this axiom. The ultimate standard
of comparison of spatial magnitudes must, as we saw in introducing
the axiom, be equality when superposed; but need we, from this
equality, infer equality when separated? It has been urged by Erdmann
that, for the more immediate purposes of Geometry, this would be
unnecessary[153]. We might construct a new Geometry, he thinks, in
which sizes varied with motion on any definite law. Such a view,
as I shall show below, involves a logical error as to the nature
of magnitude. But before pointing this out, let us discuss the
geometrical consequences of assuming its truth. Suppose the length
of an infinitesimal arc in some standard position were _ds_; then
in any other position _p_ its length would be _ds.f(p)_, where the
form of the function _f(p)_ must be supposed known. But how are we
to determine the position _p_? For this purpose, we require _p_'s
coordinates, _i.e._, some measurement of distance from the origin.
But the distance from the origin could only be measured if we assumed
our law _f(p)_ to measure it by. For suppose the origin to be _O_,
and _Op_ to be a straight line whose length is required. If we have
a measuring rod with which we travel along the line and measure
successive infinitesimal arcs, the measuring rod will change its
size as we move, so that an arc which appears by the measure to be
_ds_ will really be _f(s).ds_, where _s_ is the previously traversed
distance. If, on the other hand, we move our line _Op_ slowly through
the origin, and measure each piece as it passes through, our measure,
it is true, will not alter, but now we have no means of discovering
the law by which any element has changed its length in coming to
the origin. Hence, until we assume our function _f(p)_, we have no
means of determining _p_, for we have just seen that distances from
the origin can only be estimated by means of the law _f(p)_. It
follows that experience can neither prove nor disprove the constancy
of shapes throughout motion, since, if shapes were not constant, we
should have to _assume_ a law of their variation before measurement
became possible, and therefore measurement could not itself reveal
that variation to us[154].
Nevertheless, such an arbitrarily assumed law _does_, at first sight,
give a mathematically possible Geometry. The fundamental proposition,
that two magnitudes which can be superposed in any one position can
be superposed in any other, still holds. For two infinitesimal arcs,
whose lengths in the standard position are _ds{1}_ and _ds{2}_,
would, in any other position _p_, have lengths _f(p).ds{1}_ and
_f(p).ds{2}_, so that their ratio would be unaltered. From this
constancy of ratio, as we know through Riemann and Helmholtz, the
above proposition follows. Hence all that Geometry requires, it would
seem, as a basis for measurement, is an axiom that the alteration
of shapes during motion follows a definite known law, such as that
assumed above.
=147.= There is, however, in such a view, as I remarked above, a
logical error as to the nature of magnitude. This error has been
already pointed out in dealing with Erdmann[155], and need only be
briefly repeated here. A judgment of magnitude is essentially a
judgment of comparison: in unmeasured quantity, comparison as to
the mere more or less, but in measured magnitude, comparison as to
the precise how many times. To speak of differences of magnitude,
therefore, in a case where comparison cannot reveal them, is
logically absurd. Now in the case contemplated above, two magnitudes,
which appear equal in one position, appear equal also when compared
in another position. There is no sense, therefore, in supposing
the two magnitudes unequal when separated, nor in supposing,
consequently, that they have changed their magnitudes in motion.
This senselessness of our hypothesis is the logical ground of the
mathematical indeterminateness as to the law of variation. Since,
then, there is no means of comparing two spatial figures, as regards
magnitude, except superposition, the only logically possible axiom,
if spatial magnitude is to be self-consistent, is the axiom of Free
Mobility in the form first given above.
=148.= Although this axiom is _à priori_, its application to the
measurement of actual bodies, as we found in discussing Helmholtz's
views, always involves an empirical element[156]. Our axiom, then,
only supplies the _à priori_ condition for carrying out an operation
which, in the concrete, is empirical--just as arithmetic supplies the
_à priori_ condition for a census. As this topic has been discussed
at length in Chapter II., I shall say no more about it here.
=149.= There remain, however, a few objections and difficulties to be
discussed. First, how do we obtain equality in solids, and in Kant's
cases of right and left hands, or of right and left-handed screws,
where actual superposition is impossible? Secondly, how can we take
congruence as the only possible basis of spatial measurement, when we
have before us the case of time, where no such thing as congruence
is conceivable? Thirdly, it might be urged that we can immediately
estimate spatial equality by the eye, with more or less accuracy,
and thus have a measure independent of congruence. Fourthly, how is
metrical Geometry possible on non-congruent surfaces, if congruence
be the basis of spatial measurement? I will discuss these objections
successively.
=150.= (1) How do we measure the equality of solids? These could only
be brought into actual congruence if we had a fourth dimension to
operate in[157], and from what I have said before of the absolute
necessity of this test, it might seem as though we should be left
here in utter ignorance. Euclid is silent on the subject, and in
all works on Geometry it is assumed as self-evident that two cubes
of equal side are equal. This assumption suggests that we are not
so badly off as we should have been without congruence, as a test
of equality in one or two dimensions; for now we can at least be
sure that two cubes have all their sides and all their faces equal.
Two such cubes differ, then, in no sensible spatial quality save
position, for volume, in this case at any rate, is not a sensible
quality. They are, therefore, as far as such qualities are concerned,
indiscernible. If their places were interchanged, we might know the
change by their colour, or by some other non-geometrical property;
but so far as any property of which Geometry can take cognisance is
concerned, everything would seem as before. To suppose a difference
of volume, then, would be to ascribe an effect to mere position,
which we saw to be inadmissible while discussing Free Mobility.
Except as regards position, they are geometrically indiscernible, and
we may call to our aid the Identity of Indiscernibles to establish
their agreement in the one remaining geometrical property of volume.
This may seem rather a strange principle to use in Mathematics,
and for Geometry their equality is, perhaps, best regarded as
a definition; but if we demand a philosophical ground for this
definition, it is, I believe, only to be found in the Identity of
Indiscernibles. We can, without error, make our _definition_ of
three-dimensional equality rest on two-dimensional congruence. For
since direct comparison as to volume is impossible, we are at liberty
to _define_ two volumes as equal, when all their various lines,
surfaces, angles and solid angles are congruent, since there remains,
in such a case, no _measurable_ difference between the figures
composing the two volumes. Of course, as soon as we have established
this one case of equality of volumes, the rest of the theory follows;
as appears from the ordinary method of integrating volumes, by
dividing them into small cubes.