di unità rettilinee . . . (Padua, 1891, German translation, Leipzig, 1894); G. Fontené, L’Hyperespace à (n − 1) dimensions (Paris, 1892); and A. N. Whitehead, loc. cit. Cf. also E. Study, “Über nicht-Euklidische und Liniengeometrie,” Jahr. d. Deutsch. Math. Ver. vol. xv. (1906); W. Burnside, “On the Kinematics of non-Euclidean Space,” Proc. Lond. Math. Soc. vol. xxvi. (1894). A bibliography on the subject up to 1878 has been published by G. B. Halsted, Amer. Journ. of Math. vols. i. and ii.; and one up to 1900 by R. Bonola, Index operum ad geometriam absolutam spectantium . . . (1902, and Leipzig, 1903). (B. A. W. R.; A. N. W.)
VII. Axioms of Geometry
Until the discovery of the non-Euclidean geometries (Lobatchewsky, 1826 and 1829; J. Bolyai, 1832; B. Riemann, 1854),
geometry was universally considered as being exclusively
the science of existent space. (See section VI. Non-Euclidean Geometry.) In respect to the Theories
of space.
science, as thus conceived, two controversies may be noticed.
First, there is the controversy respecting the absolute and
relational theories of space. According to the absolute theory,
which is the traditional view (held explicitly by Newton), space
has an existence, in some sense whatever it may be, independent
of the bodies which it contains. The bodies occupy space, and
it is not intrinsically unmeaning to say that any definite body
occupies this part of space, and not that part of space, without
reference to other bodies occupying space. According to the
relational theory of space, of which the chief exponent was
Leibnitz,[1] space is nothing but a certain assemblage of the relations
between the various particular bodies in space. The idea of
space with no bodies in it is absurd. Accordingly there can be
no meaning in saying that a body is here and not there, apart
from a reference to the other bodies in the universe. Thus, on
this theory, absolute motion is intrinsically unmeaning. It is
admitted on all hands that in practice only relative motion is
directly measurable. Newton, however, maintains in the
Principia (scholium to the 8th definition) that it is indirectly
measurable by means of the effects of “centrifugal force” as
it occurs in the phenomena of rotation. This irrelevance of
absolute motion (if there be such a thing) to science has led to
the general adoption of the relational theory by modern men
of science. But no decisive argument for either view has at
present been elaborated.[2] Kant’s view of space as being a form
of perception at first sight appears to cut across this controversy.
But he, saturated as he was with the spirit of the Newtonian
physics, must (at least in both editions of the Critique) be classed
with the upholders of the absolute theory. The form of perception
has a type of existence proper to itself independently
of the particular bodies which it contains. For example he
writes:[3] “Space does not represent any quality of objects by
themselves, or objects in their relation to one another, i.e. space
does not represent any determination which is inherent in the
objects themselves, and would remain, even if all subjective
conditions of intuition were removed.”
The second controversy is that between the view that the axioms applicable to space are known only from experience, and the view that in some sense these axioms are given a priori. Both these views, thus broadly stated, Axioms. are capable of various subtle modifications, and a discussion of them would merge into a general treatise on epistemology. The cruder forms of the a priori view have been made quite untenable by the modern mathematical discoveries. Geometers now profess ignorance in many respects of the exact axioms which apply to existent space, and it seems unlikely that a profound study of the question should thus obliterate a priori intuitions.
Another question irrelevant to this article, but with some relevance to the above controversy, is that of the derivation of our perception of existent space from our various types of sensation. This is a question for psychology.[4]
Definition of Abstract Geometry.—Existent space is the subject matter of only one of the applications of the modern science of abstract geometry, viewed as a branch of pure mathematics. Geometry has been defined[5] as “the study of series of two or more dimensions.” It has also been defined[6] as “the science of cross classification.” These definitions are founded upon the actual practice of mathematicians in respect to their use of the term “Geometry.” Either of them brings out the fact that geometry is not a science with a determinate subject matter. It is concerned with any subject matter to which the formal axioms may apply. Geometry is not peculiar in this respect. All branches of pure mathematics deal merely with types of relations. Thus the fundamental ideas of geometry (e.g. those of points and of straight lines) are not ideas of determinate entities, but of any entities for which the axioms are true. And a set of formal geometrical axioms cannot in themselves be true or false, since they are not determinate propositions, in that they do not refer to a determinate subject matter. The axioms are propositional functions.[7] When a set of axioms is given, we can ask (1) whether they are consistent, (2) whether their “existence theorem” is proved, (3) whether they are independent. Axioms are consistent when the contradictory of any axiom cannot be deduced from the remaining axioms. Their existence theorem is the proof that they are true when the fundamental ideas are considered as denoting some determinate subject matter, so that the axioms are developed into determinate propositions. It follows from the logical law of contradiction that the proof of the existence theorem proves also the consistency of the axioms. This is the only method of proof of consistency. The axioms of a set are independent of each other when no axiom can be deduced from the remaining axioms of the set. The independence of a given axiom is proved by establishing the consistency of the remaining axioms of the set, together with the contradictory of the given axiom. The enumeration of the axioms is simply the enumeration of the hypotheses[8] (with respect to the undetermined subject matter) of which some at least occur in each of the subsequent propositions.
Any science is called a “geometry” if it investigates the theory of the classification of a set of entities (the points) into classes (the straight lines), such that (1) there is one and only one class which contains any given pair of the entities, and (2) every such class contains more than two members. In the two geometries, important from their relevance to existent space, axioms which secure an order of the points on any line also occur. These geometries will be called “Projective Geometry” and “Descriptive Geometry.” In projective geometry any two straight lines in a plane intersect, and the straight lines are closed series which return into themselves, like the circumference of a circle. In descriptive geometry two straight lines in a plane do not necessarily intersect, and a straight line is an open series without beginning or end. Ordinary Euclidean geometry is a descriptive geometry; it becomes a projective geometry when the so-called “points at infinity” are added.
Projective Geometry.
Projective geometry may be developed from two undefined fundamental ideas, namely, that of a “point” and that of a “straight line.” These undetermined ideas take different specific meanings for the various specific subject matters to which projective geometry can be applied. The number of the axioms is always to some extent arbitrary, being dependent upon the verbal forms of statement which are adopted. They will
- ↑ For an analysis of Leibnitz’s ideas on space, cf. B. Russell, The Philosophy of Leibnitz, chs. viii.-x.
- ↑ Cf. Hon. Bertrand Russell, “Is Position in Time and Space Absolute or Relative?” Mind, n.s. vol. 10 (1901), and A. N. Whitehead, “Mathematical Concepts of the Material World,” Phil. Trans. (1906), p. 205.
- ↑ Cf. Critique of Pure Reason, 1st section: “Of Space,” conclusion A, Max Müller’s translation.
- ↑ Cf. Ernst Mach, Erkenntniss und Irrtum (Leipzig); the relevant chapters are translated by T. J. McCormack, Space and Geometry (London, 1906); also A. Meinong, Über die Stellung der Gegenstandstheorie im System der Wissenschaften (Leipzig, 1907).
- ↑ Cf. Russell, Principles of Mathematics, § 352 (Cambridge, 1903).
- ↑ Cf. A. N. Whitehead, The Axioms of Projective Geometry, § 3 (Cambridge, 1906).
- ↑ Cf. Russell, Princ. of Math., ch. i.
- ↑ Cf. Russell, loc. cit., and G. Frege, “Über die Grundlagen der Geometrie,” Jahresber. der Deutsch. Math. Ver. (1906).
