Bertrand Russell

“ It consists in this, that the slower will never be overtaken in its course by the quickest, for the pursuer must always come first to the point from which the pursued has just departed, so that the slower must necessarily be always still more or less in advance. ”

“ We have now seen that projective Geometry is logically prior to metrical Geometry, but cannot supersede it 46 45. Sophus Lie has applied projective methods to Helmholtz's formulation of the axioms, and has shown the axiom of Monodromy to be superfluous 46 46. Metageometry has gradually grown independent of philosophy, but has grown continually more interesting to philosophy 50 47. Metrical Geometry has three indispensable axioms, 50 48. Which we shall find to be not results, but conditions, of measurement, 51 49. And which are nearly equivalent to the three axioms of projective Geometry 52 50. ”

“ The quadrilateral construction is essential to the projective definition of points, 123 113. And can be projectively defined, 124 114. By the general principle of projective transformation 126 115. The principle of duality is the mathematical form of a philosophical circle, 127 116. Which is an inevitable consequence of the relativity of space, and makes any definition of the point contradictory 128 117. We define the point as that which is spatial, but contains no space, whence other definitions follow 128 118. What is meant by qualitative equivalence in Geometry? ”

“ Modern Logic regards every judgment as both synthetic and analytic, 57 57. But leaves the à priori, as that which is presupposed in the possibility of experience 59 58. Kant's first two arguments as to space suffice to prove some form of externality, but not necessarily Euclidean space, a necessary condition of experience 60 59. Among the successors of Kant, Herbart alone advanced the theory of Geometry, by influencing Riemann 62 60. Riemann regarded space as a particular kind of manifold, i.e. ”

“ Coordinates, in projective Geometry, are not spatial magnitudes, but convenient names for points 118 106. The possibility of distinguishing various points is an axiom 119 107. The qualitative relations between points, dealt with by projective Geometry, are presupposed by the quantitative treatment 119 108. The only qualitative relation between two points is the straight line, and all straight lines are qualitatively similar 120 109. ”

“ Geometry deals with an abstract matter, whose physical properties are disregarded; and Physics must presuppose Geometry 80 74. Erdmann accepted the conclusions of Riemann and Helmholtz, 81 [xi] 75. And regarded the axioms as necessarily successive steps in classifying space as a species of manifold 82 76. His deduction involves four fallacious assumptions, namely: 82 77. That conceptions must be abstracted from a series of instances; 83 78. That all definition is classification; 83 79. That conceptions of magnitude can be applied to space as a whole ”

“ Three axioms are used by projective Geometry, 132 [xiii] 123. And are required for qualitative spatial comparison, 132 124. Which involves the homogeneity, relativity and passivity of space 133 125. The conception of a form of externality, 134 126. Being a creature of the intellect, can be dealt with by pure mathematics 134 127. The resulting doctrine of extension will be, for the moment, hypothetical 135 128. But is rendered assertorical by the necessity, for experience, of some form of externality 136 129. ”

“ As to the distinction between t? p??????, continuous, and t? p?s??, discrete quantity, see Iambl., in Nicomachi Geraseni Arithmeticam introductionem, ed. Tennulius, p. 148.) ” Cf. p. 48. [25] Referred to by Burnet, op. cit., p. 120. [26] iv., 6. 213b, 22; H. Ritter and L. Preller, Historia Philosophiæ Græcæ, 8th ed., Gotha, 1898, p. 75 (this work will be referred to in future as “R. P.”) . [27] The Pythagorean proof is roughly as follows. If possible, let the ratio of the diagonal to the side of a square be m/n, where m and n are whole numbers having no common factor. ”

“ The comparison of volumes and of Kant's symmetrical objects 154 151. (2) The measurement of time, where congruence is impossible 156 152. (3) The immediate perception of spatial magnitude; and 157 153. (4) The Geometry of non-congruent surfaces 158 154. Free Mobility includes Helmholtz's Monodromy 159 155. Free Mobility involves the relativity of space 159 156. From which, reciprocally, it can be deduced 160 157. Our axiom is therefore à priori in a double sense 160 II. The Axiom of Dimensions. 158. Space must have a finite integral number of dimensions 161 159. ”

“ AS A FIELD FOR SCIENTIFIC METHOD IN PHILOSOPHY, By Bertrand Russell CONTENTS LECTURE PAGE I. Current Tendencies 3 II. Logic as the Essence of Philosophy 33 III. On our Knowledge of the External World 63 IV. The World of Physics and the World of Sense 101 V. The Theory of Continuity 129 VI. The Problem of Infinity considered Historically 155 VII. The Positive Theory of Infinity 185 VIII. On the Notion of Cause, with Applications to the Free-will Problem 211 Index 243 INDEX Absolute, 6, 39. Abstraction, principle of, 42, 124 ff. Achilles, Zeno's argument of, 173. ”