192 THE CONCEPT Ol^ r^ A lUKu i^n.
the same time-system intersecting M. Similarly the
order of parallel planes and of event-particles on straight
lines arises from the time-order of these intersecting
moments. The explanation is not given here^. It is
sufficient now merely to mention the sources from which
the whole of geometry receives its physical explanation.
The correlation of the various momentary spaces of
one time-system is achieved by the relation of cogre-
dience. Evidently motion in an instantaneous space is
unmeaning. Motion expresses a comparison between
position in one instantaneous space with positions in
other instantaneous spaces of the same time-system.
Cogredience yields the simplest outcome of such com-
parison, namely, rest.
Motion and rest are immediately observed facts.
They are relative in the sense that they depend on the
time-system which is fundamental for the observation.
A string of event-particles whose successive occupation
means rest in the given time-system forms a timeless
point in the timeless space of that time-system. In this
way each time-system possesses its own permanent
timeless space peculiar to it alone, and each such space
is composed of timeless points which belong to that
time-system and to no other. The paradoxes of rela-
tivity arise from neglecting the fact that different as-
sumptions as to rest involve the expression of the facts
of physical science in terms of radically different spaces
and times, in which points and moments have different
meanings.
The source of order has already been indicated and that of congruence is now found. It depends on motion.
^ Cf. Principles of Natural Knowledge, and previous chapters of the present work.
Ix] the Ultimate Physical Concepts 193
From cogredience, perpendicularity arises; and from
perpendicularity in conjunction with the reciprocal
symmetry between the relations of any two time-systems
congruence both in time and space is completely defined
(tf. loc. cit.).
The resulting formulae are those for the electro- magnetic theory of relativity, or, as it is now termed, the restricted theory. But there is this vital difference: the critical velocity c which occurs in these formulae has now no connexion whatever with light or with any other fact of the physical field (in distinction from the extensional structure of events). It simply marks the fact that our congruence determination embraces both times and spaces in one universal system, and therefore if two arbitrary units are chosen, one for all spaces and one for all times, their ratio will be a velocity which is a fundamental property of nature expressing the fact that times and spaces are really comparable.
The physical properties of nature are expressed in terms of material objects (electrons, etc.). The physical character of an event arises from the fact that it belongs to the field of the whole complex of such objects. From another point of view we can say that these objects are nothing else than our way of expressing the mutual correlation of the physical characters of events.
The spatio-temporal measurableness of nature arises
from (i) the relation of extension between events, and
(ii) the stratified character of nature arising from each of
the alternative time-systems, and (iii) rest and motion,
as exhibited in the relations of finite events to time-
systems. None of these sources of measurement depend
on the physical characters of finite events as exhibited
by the situated objects. They are completely signified
W.N. ^3
194 THE CONCEPT OF NATURE L^h.
for events whose physical characters are unknown. Thus
the spatio-temporal measurements are independent of
the objectival physical characters. Furthermore the
character of our knowledge of a whole duration, which
is essentially derived from the significance of the part
within the immediate field of discrimination, constructs
it for us as a uniform whole independent, so far as its
extension is concerned, of the unobserved characters
of remote events. Namely, there is a definite whole of
nature, simultaneously now present, whatever may be
the character of its remote events. This consideration
reinforces the previous conclusion. This conclusion
leads to the assertion of the essential uniformity of the
momentary spaces of the various time-systems, and
thence to the uniformity of the timeless spaces of which
there is one to each time-system.
The analysis of the general character of observed nature set forth above affords explanations of various fundamental observational facts: (a) It explains the differentiation of the one quality of extension into time and space. (j8) It gives a meaning to the observed facts of geometrical and temporal position, of geometrical and temporal order, and of geometrical straightness and planeness. (y) It selects one definite system of congruence embracing both space and time, and thus explains the concordance as to measurement which is in practice attained. (S) It explains (consistently with the theory of relativity) the observed phenomena of rotation, e.g. Foucault's pendulum, the equatorial bulge of the earth, the fixed senses of rotation of cyclones and anticyclones, and the gyro-compass. It does this by its admission of definite stratifications of nature which are disclosed by the very character of our knowledge of it. (e) Its ex-
Ix] the Ultimate Physical Concepts 195
planations of motion are more fundamental than those expressed in (S) ; for it explains what is meant by motion itself. The observed motion of an extended object is the relation of its various situations to the stratification of nature expressed by the time-system fundamental to the observation. This motion expresses a real relation of the object to the rest of nature. The quantitative expression of this relation will vary according to the time-system selected for its^xpression.
This theory accords no peculiar character to light
beyond that accorded to other physical phenomena such
as sound. There is no ground for such a differentiation.
Some objects we know by sight only, and other objects
we know by sound only, and other objects we observe
neither by light nor by sound but by touch or smell or
otherwise. The velocity of light varies according to its
medium and so does that of sound. Light moves in
curved paths under certain conditions and so does
sound. Both light and sound are waves of disturbance
in the physical characters of events ; and (as has been
stated above, p. 188) the actual course of the light
is of no more importance for perception than is the
actual course of the sound. To base the whole philo-
sophy of nature upon light is a baseless assumption.
The Michelson-Morley and analogous experiments
show that within the limits of our inexactitude of
observation the velocity of light is an approximation to
the critical velocity 'c' which expresses the relation
between our space and time units. It is provable that
the assumption as to light by which these experiments
and the influence of the gravitational field on the light-
rays are explained is deducible as an approximation
from the equations of the electromagnetic field. This
196 THE CONCEFl Ut JNAlUKii l^«.
completely disposes of any necessity for differentiating
light from other physical phenomena as possessing
any peculiar fundamental character.
It is to be observed that the measurement of extended nature by means of extended objects is meaningless apart from some observed fact of simultaneity inherent in nature and not merely a play of thought. Otherwise there is no meaning to the concept of one presentation of your extended measuring rod AB. Why not AB' where B' is the end B five minutes later ? Measurement presupposes for its possibility nature as a simultaneity, and an observed object present then and present now. In other words, measurement of extended nature re- quires some inherent character in nature affording a rule of presentation of events. Furthermore congruence cannot be defined by the permanence of the measuring rod. The permanence is itself meaningless apart from some immediate judgment of self-congruence. Other- wise how is an elastic string differentiated from a rigid measuring rod? Each remains the same self -identical object. Why is one a possible measuring rod and the other not so } The meaning of congruence lies beyond the self-identity of the object. In other words measure- ment presupposes the measurable, and the theory of the measurable is the theory of congruence.
Furthermore the admission of stratifications of nature
bears on the formulation of the laws of nature. It has
been laid down that these laws are to be expressed in
differential equations which, as expressed in any general
system of measurement, should bear no reference to
any other particular measure-system. This requirement
is purely arbitrary. For a measure-system measures
something inherent in nature; otherwise it has no
Ix] the Ultimate Physical Concepts 197
connexion with nature at all. And that something which
is measured by a particular measure-system may have a
special relation to the phenomenon whose law is being
formulated. For example the gravitational field due to
a material object at rest in a certain time-system may
be expected to exhibit in its formulation particular
reference to spatial and temporal quantities of that
time-system. The field can of course be expressed in
any measure-systems, but the particular reference will
remain as the simple physical explanation.
Note: on the Greek Concept of a Point
The preceding pages had been passed for press before I had the pleasure of seeing Sir T. L. Heath's Euclid in Greek^. In the original Euclid's first definition is
(TTjfjLelov icTTtv, ov fiepo<; ovOev. I have quoted it on p. 86 in the expanded form taught to me in childhood, ' without parts and without magnitude.' I should have consulted Heath's English edition — a classic from the moment of its issue — before committing myself to a statement about Euclid. This is however a trivial correction not affecting sense and not worth a note. I wish here to draw attention to Heath's own note to this definition in his Euclid in Greek. He summarises Greek thought on the nature of a point, from the Pythagoreans, through Plato and Aristotle, to Euclid. My analysis of the requisite character of a point on pp. 89 and 90 is in complete agreement with the outcome of the Greek discussion.
Note: on Significance and Infinite Events
The theory of significance has been expanded and made more definite in the present volume. It had already been introduced in the Principles of Natural Knowledge (cf. subarticles 3-3 to 3-8 and i6-i, 16-2, 19-4, and articles 20, 21). In reading over the proofs of the present volume, I come to the conclusion that in the 1 Camb. Univ. Press, 1920.
198 THE CONCEPT OF JNAlUKJi L^n. .^
light of this development my limitation of infinite events to dura- tions is untenable. This limitation is stated in article 33 of the Principles and at the beginning of Chapter IV (p. 74) of this book. There is not only a significance of the discerned events embracing the whole present duration, but there is a significance of a cogre- dient event involving its extension through a whole time-system backwards and forwards. In other words the essential 'beyond' in nature is a definite beyond in time as well as in space [cf. pp. 53, 194]. This follows from my whole thesis as to the assimila- tion of time and space and their origin in extension. It also has the same basis in the analysis of the character of our knowledge of nature. It follows from this admission that it is possible to define point-tracks \i.e. the points of timeless spaces] as abstrac- tive elements. This is a great improvement as restoring the balance between moments and points. I still hold however to the statement in subarticle 35-4 of the Principles that the intersection of a pair of non-parallel durations does not present itself to us as one event. This correction does not affect any of the subsequent reasoning in the two books.
I may take this opportunity of pointing out that the ' stationary events' of article 57 of the Principles are merely cogredient events got at from an abstract mathematical point of view.
Index
In the case of terms of frequent occurrence, only those occurrences are indexed which are of peculiar importance for the elucidation of meaning.
A \pr an], ii
Abraham, 105
Absolute position, 105, 106, 114,
Abstraction, 33, 37, 168, ryi, 173;
extensive, 65, 79, 85 Abstractive element, 84; set, 61,
Action at a distance, 159, 190
Action by transmission, 159, 190
Active conditions, 158
Activity, field of, 170, 181
Adjunction, loi
Aggregate, 23
Alexander, Prof., viii
Alexandria, 71
Alfred the Great, 137
Anticipation, 69
Anti-prime, 88
Apparent nature, 31, 39
Area, 99; momental, 103; vagrant,
Aristotelian logic, 150
Aristotle, 16, 17, 18, 24, 197
Associate-potential, 183
Atom, 17
Attribute, 21, 26, 150
Awareness, 3
Axiom, 36, 121
Axioms of congruence, 128 et seqq.
Bacon, Francis, 78
Behaviouristic, 185
Bergson, 54
Berkeley, 28
Between, 64
Beyond, 186, 198
Bifurcation, vi, 30, 185, 187
Boundary, 100; moment, 63; par-
ticle, 100
Broad, C. D., viii
Calculation, formula of, 45, 158
Cambridge, 97
Causal nature, 31, 39
Causation, 31, 146
Centrifugal force, 138
Change, uniformity of, 140
Character, extrinsic, 82, 8g, go, 113, 191; intrinsic, 80, 82,90, 113, 191