The Analysis of Matter

from perception. We no longer have to contend with what used to seem mysterious in the causal theory of perception: a series of light-waves or sound-waves or what not suddenly producing a mental event apparently totally different from themselves in character. As to intrinsic character, we do not know enough about it in the physical world to have a right to say that it is very different from that of percepts; while as to structure we have reason to hold that it is similar in the stimulus and the percept. This has become possible owing to the facts that "matter" can be[Pg 401] regarded as a system of events, not as part of the stuff of the world, and that space-time, as it occurs in physics, has been found to be much more different from perceptual space than was formerly imagined. This brings us to Part III., in which we endeavour to discover a possible structure of the physical world which shall at once justify physics and take account of the connection with perception demanded by the necessity for an empirical basis for physics. Here we are concerned first with the construction of points as systems of events which overlap, or are "co-punctual," in space-time, and then with the purely ordinal properties of space-time. The method employed is very general, and can be adapted to a discrete or to a continuous order; it is proved that , events are sufficient to generate a continuum of points, given certain laws as to the manner of their overlapping. The whole of this theory, however, aims only at constructing such properties of space-time as belong to analysis situs; everything appertaining to intervals and metrics is omitted at this stage, since causal considerations are required for the theory of intervals. The conception of one unit of matter—say one electron—as a "substance," i.e. a single simple entity persisting through time, is not one which we are justified in adopting, since we have no evidence whatever as to whether it is false or true. We define a single material unit as a "causal line," i.e. as a series of events connected with each other by an intrinsic differential causal law which determines first-order changes, leaving second-order changes to be determined by extrinsic causal laws. (In this we are for the moment ignoring quantum phenomena.) If there are light-quanta, these will more or less fulfil this definition of matter, and we shall have returned to a corpuscular theory of light; but this is at present an open question. The whole conception of matter is less fundamental to physics than it used to be, since energy has more and more[Pg 402] taken its place. We find that under terrestrial conditions electrons and protons persist, but there is nothing in theoretical physics to lead us to expect this, and physicists are quite prepared to find that matter can be annihilated. This view is even, put forward to account for the energy of the stars. The question of interval presents great difficulties, when we attempt to construct a picture of the world which shall make its importance seem not surprising. The same may be said of the quantum. I have endeavoured, not, I fear, with much success, to suggest hypotheses which would link these two curious facts into one whole. I suggest that the world consists of steady events accompanied by rhythms, like a long note on the violin while arpeggios are played on the piano, or of rhythms alone. Steady events are of various sorts, and many sorts have their appropriate rhythmic accompaniments. Quantum changes consist of "transactions," i.e. of the substitution, suddenly, of one rhythm for another. When two events have a time-like interval, if space-time is discrete, this interval is the greatest number of transitions on any causal route leading from the one event to the other. The definition of space-like intervals is derived from that of time-like intervals. The whole process of nature may, so far as present evidence goes, be conceived as discontinuous; even the periodic rhythms may consist of a finite number of events per period. The periodic rhythms are required in order to give an account of the uses of the quantum principle. A percept, at any rate when it is visual, will be a steady event, or system of steady events, following upon a transaction. Percepts are the only part of the physical world that we know otherwise than abstractly. As regards the world in general, both physical and mental, everything that we know of its intrinsic character is derived from the mental side, and almost everything that we know of its causal laws is derived from the physical side. But from the standpoint of philosophy the distinction between physical and mental is superficial and unreal. [Pg 403] INDEX A priori and empirical knowledge: 173 Abstract geometry: 76 Acceleration: 231, 368 „ absolute: 18, 67 Accuracy: 99 Action: 36, 341, 379, 394 „ at a distance: 18, 108, 162, 327, 356, 359, 374, 380 Action, principle of least: 19 Adjectives: 242 Æther: 19, 121, 276, 355 Analogy: 399 Analysis: 4 „ logical: 2 Analysis situs: 55, 113, 114, 295, 298, 303, 311, 401 Analytic and synthetic propositions: 169 Ancestor: 373, 378 Apes: 148 Aristotle: 161 Arithmetic: 4, 276 Aspects: 347 Astronomy: 163, 234, 317, 397 Atom: 24, 234 „ Heisenberg's: 42 „ nucleus of: 25 „ Rutherford-Bohr: 24, 27, 42 Atomic number: 234 „ weight: 234 Atomism, spatial: 22 Axioms: 2 Bach: 392 Beauty: 392 Behaviourism: 142 Beliefs: 142, 174, 178 Bergson: 101, 279, 387 Berkeley: 6, 7, 136, 156, 159, 192, 213 Between: 305 Bifurcation of nature: 257 Biography: 212 Black-body radiation: 30 Bohr: 32, 33, 35, 133, 134, 195, 286, 360 Bolyai: 20, 103 Born: 42, 343 Boscovitch: 13 Bradley: 387 Brain: 382 Braithwaite: 167 Broad: 197, 259, 260, 286 Camera: 209, 265 Cantor, Georg: 14 Cantorian continuum: 58, 279 Cartesianism: 160 Category: 238, 243 Causal ancestor: 314 „ chain: 285, 314, 320, 371 „ laws: 107, 121, 168, 191, 200, 214, 245, 285, 340, 355, 366, 381 „ „ extrinsic: 324 „ lines: 313, 401 „ relation: 367 „ route: 379 „ unit: 315 Causality: 313, 367, 381, 398, 399 Causation: 108, 383 „ at a distance: 214 Cause: 150, 159 Cause-and-effect: 346 Change: 281, 362 Chuang-Tze: 180 Clarke: 14 Classes: 116, 387 Collinear: 306 Collinearity: 308 Colour: 288 Colour-similarity: 288 Colours: 253 Common sense: 141, 156 Comparison: 278 Compresent: 294, 336, 349, 356, 372, 374, 376, 378, 383, 385 Conjunctions: 242 Constants: 172, 251 Continuity: 24, 71, 168, 214, 217, 231, 279, 322, 362 „ perceptual: 280 „ qualitative: 81 Continuous, space-time is: 42, 311 Continuum: 401 „ mathematical: 280 Contravariant: 65 Co-ordinates: 56, 57, 60, 63, 71, [Pg 404]110, 334, 337, 397 Co-punctual: 307, 314, 321, 373, 401 „ events: 315 Co-punctuality: 299, 376, 386 Co-regional: 311 Correlation: 188, 204, 205 Correspondence: 239 Co-superficial: 311 Co-variant: 65 „ derivative: 96 Curve of pursuit: 101 Data: 141, 187, 189 De Broglie: 46 Debye: 32 Descartes: 6, 9, 19, 71, 156, 160, 205 Determinism: 390 „ physical: 214 Dictaphone: 209, 265, 400 Differential equations: 231 „ law: 245 „ „ intrinsic: 323 Diffraction: 395 Dimensions: 312, 341 Dirac: 42, 43 Discontinuity: 24, 395 Distance: 166, 303, 350 Diversity: 386 Doppler effect: 165 Dreams: 205, 222 Eddington: 57, 64, 75, 78, 82, 84, 91, 96, 97, 103, 106, 110, 117, 122, 129, 131, 136, 168, 341, 377, 395 Einstein: 13, 32, 42, 52, 75, 77, 129, 161, 187, 233, 323, 395 Electromagnetism: 328 Electrons: 9, 24, 80, 90, 135, 153, 168, 234, 244, 276, 287, 319, 320, 325, 347, 386, 394 Ellis: 123, 125, 126 Emergent: 286 Empirical: 141, 176 Empiricists: 169 Enclosure-series: 291 Energy: 84, 122, 234, 318, 345, 360, 362, 394, 401 „ kinetic: 123 „ radiant: 365 Entities, unperceived: 191, 206 „ unperceivable: 216 Epistemology: 179 Error: 183, 184, 186 Euclid: 61, 70, 71, 103 Euclid's first axiom: 94, 99 Euler's diagrams: 387 Events: 9, 243, 247, 275, 284, 286, 345, 356, 377, 384 Events: group of: 244 „ luminous: 352, 333, 354, 369, 371, 373 „ maximum: 293 „ minimum: 292 „ persistent: 347 „ steady: 355, 356, 363, 368, 402 Evidence, empirical: 6 Existents, unperceived: 213, 226 Expectations: 181, 182, 183, 189 Experience: 173 Facts: 239 Fallacy of simple location: 340 Faraday: 20 Fijians: 102 FitzGerald contraction: 50, 100 Force: 18, 19, 76, 161 Form: 172 Foucault's pendulum: 358 Fowler: 33, 42 Franck: 38 Freewill: 38 Frege: 4, 242 Frequency: 357, 364 Galileo: 6, 13, 101, 161, 395 Gauge-system: 97, 98 Gauss: 58, 59, 113 Geodesics: 55, 62, 69, 72, 313, 327, 371, 372, 379, 396 Geometry: 2, 3, 21, 102, 117, 174, 227 „ abstract: 76 „ affine: 98 „ descriptive: 210 „ Euclidean, 77 „ natural: 76 „ non-Euclidean: 20, 59, 396 „ projective: 103, 109 Gestalt-psychologie: 345 Gramophone: 205, 249 Gravitation: 17, 52, 60, 74, 77, 85, 89, 163, 176, 187, 324, 326, 371 Group, ideal: 212 : 30, 33, 45, 317, 330, 341, 343, 360, 364, 365, 368, 394 Habit: 184, 191 Haldane, J. B.: 233 Hallucinations: 222 Hand-eye co-ordination: 143, 147 185 Hausdorff: 296, 297, 303, 312 Hearing: 316 Hegel: 55, 199 Heisenberg: 24, 27, 42, 43, 45, 246, 283, 321, 322, 326, 332, 343, 338, [Pg 405]360, 361 Helium: 24, 26 Helmholtz: 138 Heraclitus: 62, 176 Hertz: 19, 20, 22 Holt: 10 Hume: 159, 190, 398 Hydrogen: 25 „ spectrum: 32 Hypothesis: 194 Idealism: 7, 28, 215, 382, 387, 388 Idealists: 215 Impenetrable: 385 Implication, strict: 199 Indestructible: 385 Indistinguishability: 282 Individuals: 387 Induction: 153, 154, 167, 175, 194, 230, 233, 398 „ mathematical: 71 Inference: 150, 187, 190 „ physiological: 150, 190 Infinite divisibility: 279 Infinitesimals: 104, 107 Instants: 294 Integers, finite: 3 Interference: 352, 395 Interpenetration, logical: 386 „ spatio-temporal: 386 Interpretation: 4, 88, 137, 141, 188, 225, 288 Interval: 51, 58, 69, 70, 88, 110, 117, 129, 314, 327, 330, 354, 367, 377, 396, 397, 402 „ discontinuous: 367 „ time-like; space-like: 51 Introspection: 173 Invariants: 82, 84 Jeans: 31, 32, 123, 129, 341, 381, 386 Jeremiah: 240 Johnson, Dr.: 136 Jordan, P.: 38, 42, 343 Kant: 14, 20, 55, 78, 157, 159, 160, 169, 171, 174, 175, 279 Keynes: 167, 233, 399 Kinematics: 361 King, L. V.: 47, 129 Kirchoff: 19 Knowledge: 174, 178 „ subjective factor in: 334 Köhler: 148 Kropotkin: 74 Language: 152, 239 Laws, causal: 101 Laws: differential: 101 „ general: 229 „ of contradiction: 171 „ of motion, first: 323 „ simple: 232, 236 „ scientific: 388 „ statistical: 231, 235, 236 Learned reactions: 154 Leibniz: 8, 14, 17, 18, 156, 157, 160, 200, 238, 328 Leonardo: 161 Lewis, G. I.: 199 „ G. N.: 125, 126, 134, 164 Light: 123, 131, 132, 155, 163, 216, 235, 244, 267, 276, 314, 317, 328, 339, 345, 351, 364, 369, 373, 395 „ frequency of: 353 Light-quantum: 124, 268, 316, 318, 395, 401 Light-ray: 70 Limit: 309 Limitation of variety: 233 Limits: 71, 117 Line: 306 „ spectra: 32 „ "straight": 61 Lobatchevsky: 20, 103 Location: 258, 320 Locke: 132, 159, 257, 339 Logic: 2, 9, 171, 239, 247, 250, 387 „ mathematical: 2, 9, 71, 107, 138, 350 „ traditional: 387 Logical property: 251 „ symbols: 289 Lorentz: 124 „ transformation: 49, 54 Mach: 15, 19 Mass: 53, 86, 122, 163, 284, 318, 386 „ gravitational: 341 „ invariant: 122 „ relative: 122 Material energy-tensor: 86 Materialism: 7, 162, 383, 387, 388 Materialists: 215 Mathematics: 174, 176 „ applied: 5 Matrices: 44, 46 Matter: 131, 134, 156, 160, 182, 192, 215, 243, 321, 324, 355, 372, 379, 384, 392, 401 „ atomicity of: 30 „ indestructibility of: 168 „ unperceived: 232 Matters of fact: 176, 178[Pg 406] Maxwell, Clerk: 15, 20, 22 Maxwell's equations: 49, 54, 97, 121, 258, 359 Meaning: 240, 243 Measurement: 90, 97, 99, 107, 109, 337, 377, 397 Memory: 186, 266, 267, 389 Mendelians: 233 Menger, Karl: 312 Mental states: 320, 322 Mercury: 40 Metre: 91 Mill, J. S.: 21, 101, 171 Mind: 156, 160, 192, 336, 382, 387 Minkowski: 15, 135, 395 Mnemic phenomena: 265, 315 Modality: 169 Momentum: 36, 84, 86, 156, 162 Monad: 157 Moore, G. E.: 210 Motion: 244, 246, 278, 317, 326, 355 „ absolute: 356, 358 „ continuous: 380 „ percept of: 279 „ periodic: 344, 348 „ relative: 359, 378 Motor habits: 148 Muscular physics: 160 Naive realism: 155, 218, 262, 398 Names: 152 Napoleon: 186 Necessary and contingent propositions: 169 Neighbourhoods: 295 ff., 303, 312 Neutral monism: 382 „ stuff: 10 Newman, M. H. A.: 290 Newton: 13, 18, 161, 356, 358, 395 Nicod: 167 Nisbet: 167 Nucleus: 347 Numbers: 4 „ ordinal: 250 „ real: 290 Object: 142, 181, 182, 186, 204, 207, 218 Objectivity: 218, 222 Objects, external: 198 Occam: 15, 290 Odour: 133 Ontology: 9 Orbits, elliptical: 38 Order: 109, 231, 242 „ macroscopic: 304 „ microscopic: 305 Overlapping: 294, 356 Paneth: 24 Parallel displacement: 96, 106 Parallelism, psycho-cerebral: 391 Parallelogram: 99, 104, 117, 377 Particulars: 275, 319, 386 Pasch: 210 Peano: 3, 240 Perception: 137, 159, 164, 173, 175, 177, 178, 186, 187, 188, 189, 215, 218, 219, 247, 398 „ causal theory of: 181, 193, 197, 320 „ causes of: 8 „ cognitive efficacy of: 262 Percepts: 218 „ "ideal": 211 „ physical status of: 257 „ structure of: 281 Percipients, "ideal": 210 Periodicity: 343, 363, 365 Periodic process: 267, 343, 348, 208, 216 Perspective: 198, 208, 216, 221, 323, 333, 334, 348 Peters: 324 Phenomenalism: 209, 324, 333, 388, 398, 399 Philosophy: 194 Photo-electric effect: 31, 329, 395 Photographs: 334 Photon: 126 Physical time: 254 Physics, abstractness of: 130 „ and perception: 6 „ and pure mathematics: 1 Physiology: 137 Place: 217 Planck: 30 Planck's constant. See "" Pluralism: 242 Poincaré: 233, 312 Points: 9, 22, 276, 286, 290, 299, 321, 348, 376, 401 „ "connected": 304 „ material: 321 Possibility: 170 Postulates: 167, 249 Predicate: 239, 242 Prepositions: 242 Primitive religion: 149 Principle of identification: 86 Probability: 16, 141, 167, 170, 398 Progression: 4 Proper names: 242 Propositional functions: 170 Propositions, general: 185 Protons: 9, 24, 80, 153, 234, 244, 394[Pg 407] Psychologist's fallacy: 179 Psychology: 137 Pythagoras: 22, 30, 58, 59, 61, 71, 235 Qualitative cycles: 359 „ differences: 346 „ series: 9, 115, 343 „ similarity: 120 Qualities: 345 „ primary, secondary: 132, 385 Quality: 288, 388, 391 „ intrinsic: 264 Quanta: 24, 26, 30, 329 Quantities, extensive: 116 „ intensive: 116 Quantum: 101, 168, 231, 234, 268, 326, 343, 345, 352, 355, 360, 366, 381, 393, 394, 402 „ laws: 331 „ number: 33, 36 „ phenomena: 322 Radiations: 283, 316, 326, 331 Radio-activity: 368, 393 Ramsey, F. P.: 299 Rationalists: 169 Realism, naive: 149 Recognition: 151 Reflexes: 143, 184 Region: 302, 311 Reichenbach: 381 Relation-number: 250 Relations: 238, 249 „ atomic; molecular: 116 Relativity: 337, 395 „ special theory of: 39, 48 „ theory of: 14, 17, 53, 100 Repetition: 346 Reversibility: 381 Rhythms: 347, 355, 356, 359, 363, 368, 374, 378, 402 Riemann: 21, 55, 58, 59 Robb, A. A.: 313 Rotation, absolute: 17, 358 Rutherford: 285, 325 Rydberg's constant: 32, 35, 195 Schrödinger: 46 Semi-similarity: 254, 265, 269 Sensation: 189 Sense organs: 259 Separation: 118, 377 Shakespeare: 392 Shapes: 202 Sheffer, H. M.: 10, 299 Sight: 316 „ physics: 160 Similar: 265 Similarity: 117, 120, 249 Simultaneity: 52, 63, 278 Skeleton, causal: 391 Smell: 165, 316 Solipsism: 8, 28, 158, 325, 398 Sommerfeld: 34, 35, 36, 39, 40, 41, 194, 343, 361 Sound: 133, 165, 209, 253, 261, 267 Space: 131, 143, 198, 286, 347 „ absolute: 321 „ empty: 324 „ Euclidean: 75, 298 „ metrical: 296 „ non-Euclidean: 42 „ perceptual: 252, 335, 338, 386 „ physical: 252, 336, 383, 385 „ tactual; visual: 144, 253 „ topological: 296, 312 Space-time: 49, 132, 244, 286, 338, 340, 341 „ genesis of: 376 „ order: 303 „ physical and perceptual: 333 Space and time, absolute: 14, 22 „ „ common: 207 Spatial relations:348, 359 Specific heat: 32, 195 Specious present: 278 Speech, parts of: 242 Spinoza: 157 Statistical averages: 191, 393 „ fact: 339 Stimulus: 227, 337, 338, 385, 390, 400 Straightness: 307 Structure: 227, 249, 276, 282, 285, 335, 338, 340, 386, 388, 390, 400 Stuff: 244, 388, 401 Subject: 239, 242 Subject-predicate logic: 287 Subjectivity: 223 „ cerebral (or psychological): 225 „ physical: 224 „ physiological (or sensory): 224 Substance: 121, 151, 192, 231, 238, 277, 283, 286, 318, 355, 401 „ indestructible: 238 „ permanent: 239 Suddenness: 347 Surface: 311 Surprise: 185 Symbols: 171[Pg 408] Syntax: 151 Tautologies: 171 Tensor equation: 233 Tensors: 18, 63, 96, 136, 396, 397 Testimony: 193, 203, 206, 399 Theory: 194, 195 Thermodynamics, second law of: 381 Thing, ideal: 213 Thomas, St.: 55 Time: 131, 132, 208, 253, 286, 381 „ perceptual: 338 „ proper: 350, 357 „ psychological: 254 Topography: 177 Touch: 260, 316 „ physics: 160 Transactions: 355, 356, 360, 362, 363, 368, 372, 378, 394, 402 Transitiveness: 251 Truth: 8 Truth-functions: 243 Unconscious: 385 Universals: 152 Urysohn: 297 Vagueness: 220, 223, 224, 280 Variables: 172 Variables, separation of: 36, 393 Velocities, composition of: 53 Velocity: 323, 374 Verbs: 242 Vietoris, L.: 296 Vis viva: 156 Watson, T. B.: 154, 192 Waves, electromagnetic: 20 Weierstrass: 3 Weight: 163 Wells: 226 Weyl: 87, 94, 95, 103, 106, 119, 328, 395, 396 Whitehead: 6, 22, 57, 75, 77, 78, 90, 130, 138, 144, 158, 199, 210, 230, 247, 237, 290, 291, 292, 294, 340, 397 Wilson: 35, 41, 343 Wittgenstein: 16, 171, 239 Words: 151 „ written: 241 World-lines: 135, 244, 253, 317 Writing: 241 Zeeman effect: 35 Zermelo's axiom: 299 Zones: 304 PRINTED IN GREAT BRITAIN BY BILLING AND SONS, LTD., GUILDFORD AND ESHER *** END OF THE PROJECT GUTENBERG EBOOK 77427 ***