The theory of perpendicularity in the timeless space of any time-system a follows immediately from this theory of perpendicularity in each of its instantaneous spaces. Let p be any rect in the moment M of « and let A be a level in M which is perpendicular to p . The locus of those points of the space of a which intersect M in event-particles on p is the straight line r of space a, and the locus of those points of the space of a which intersect M in event-particles on A is the plane / of space a. Then the plane / is perpendicular to the line r.
In this way we have pointed out unique and definite
properties in nature which correspond to perpen-
dicularity. We shall find that this discovery of definite
unique properties defining perpendicularity is of
critical importance in the theory of congruence which
is the topic for the next lecture.
V] Space and Motion 119
I regret that it has been necessary for me in this lecture to administer such a large dose of four-dimen- sional geometry. I do not apologise, because I am really not responsible for the fact that nature in its most fundamental aspect is four-dimensional. Things are what they are ; and it is useless to disguise the fact that ' what things are ' is often very difficult for our intellects to follow. It is a mere evasion of the ultimate problems to shirk such obstacles.
CHAPTER VI CONGRUENCE
The aim of this lecture is to establish a theory of con-
gruence. You must understand at once that congruence
is a controversial question. It is the theory of measure-
ment in space and in time. The question seems simple.
In fact it is simple enough for a standard procedure to
have been settled by act of parliament ; and devotion to
metaphysical subtleties is almost the only crime which
has never been imputed to any English parliament.
But the procedure is one thing and its meaning is
another.
First let us fix attention on the purely mathematical
question. When the segment between two points A
and B is congruent to that between the two points C
and D, the quantitative measurements of the two seg-
ments are equal . The equality of the numerical measures
and the congruence of the two segments are not always
clearly discriminated, and are lumped together under
the term equality. But the procedure of measurement
presupposes congruence. For example, a yard measure
is applied successively to measure two distances between
two pairs of points on the floor of a room. It is of the
essence of the procedure of measurement that the
yard measure remains unaltered as it is transferred from
one position to another. Some objects can palpably
alter as they move — for example, an elastic thread;
but a yard measure does not alter if made of the proper
material. What is this but a judgment of congruence
applied to the train of successive positions of the yard
CH.vi] CONGRUENCE 121
measure? We know that it does not alter because we
judge it to be congruent to itself in various positions.
In the case of the thread we can observe the loss of
self-congruence. Thus immediate judgments of con-
gruence are presupposed in measurement, and the
process of measurement is merely a procedure to extend
the recognition of congruence to cases where these
immediate judgments are not available. Thus we cannot
define congruence by measurement.
In modern expositions of the axioms of geometry certain conditions are laid down which the relation of congruence between segments is to satisfy. It is supposed that we have a complete theory of points, straight lines, planes, and the order of points on planes — in fact, a complete theory of non-metrical geometry. We then enquire about congruence and lay down the set of conditions — or axioms as they are called — which this relation satisfies. It has then been proved that there are alternative relations which satisfy these con- ditions equally well and that there is nothing intrinsic in the theory of space to lead us to adopt any one of these relations in preference to any other as the relation of congruence which we adopt. In other words there are alternative metrical geometries which all exist by an equal right so far as the intrinsic theory of space is concerned.
Poincare, the great French mathematician, held that
our actual choice among these geometries is guided
purely by convention, and that the effect of a change of
choice would be simply to alter our expression of the
physical laws of nature. By 'convention' I understand
Poincare to mean that there is nothing inherent in
nature itself giving any peculiar rdle to one of these
122 THE CONCEPT OF NATUKli l^«.
congruence relations, and that the choice of one par-
ticular relation is guided by the volitions of the mind at
the other end of the sense-awareness. The principle of
guidance is intellectual convenience and not natural
fact.
This position has been misunderstood by many of
Poincare's expositors. They have muddled it up with
another question, namely that owing to the inexactitude
of observation it is impossible to make an exact state-
ment in the comparison of measures. It follows that a
certain subset of closely allied congruence relations can
be assigned of which each member equally well agrees
with that statement of observed congruence when the
statement is properly qualified with its limits of
error.
This is an entirely different question and it pre- supposes a rejection of Poincare's position. The absolute indetermination of nature in respect of all the relations of congruence is replaced by the indetermination of observation with respect to a small subgroup of these relations.
Poincare's position is a strong one. He in effect
challenges anyone to point out any factor in nature
which gives a preeminent status to the congruence
relation which mankind has actually adopted. But un-
deniably the position is very paradoxical. Bertrand
Russell had a controversy with him on this question,
and pointed out that on Poincare's principles there was
nothing in nature to determine whether the earth is
larger or smaller than some assigned billiard ball.
Poincare replied that the attempt to find reasons in
nature for the selection of a definite congruence relation
in space is like trying to determine the position of a
Vi] Congruence 123
ship in the ocean by counting the crew and observing the colour of the captain's eyes.
In my opinion both disputants were right, assuming
the grounds on which the discussion was based.
Russell in effect pointed out that apart from minor
inexactitudes a determinate congruence relation is
among the factors in nature which our sense-awareness
posits for us. Poincare asks for information as to the
factor in nature which might lead any particular con-
gruence relation to play a preeminent role among the
factors posited in sense-awareness. I cannot see the
answer to either of these contentions provided that you
admit the materialistic theory of nature. With this
heory nature at an instant in space is an independent
fact. Thus we have to look for our preeminent con-
gruence relation amid nature in instantaneous space;
and Poincare is undoubtedly right in saying that nature
on this hypothesis gives us no help in finding it.
On the other hand Russell is in an equally strong
position when he asserts that, as a fact of observation,
we do find it, and what is more agree in finding the same
congruence relation. On this basis it is one of the most
extraordinary facts of human experience that all man-
kind without any assignable reason should agree in
fixing attention on just one congruence relation amid
the indefinite number of indistinguishable competitors
for notice. One would have expected disagreement on
this fundamental choice to have divided nations and to
have rent families. But the difficulty was not even dis-
covered till the close of the nineteenth century by a
few mathematical philosophers and philosophic mathe-
maticians. The case is not like that of our agreement
on some fundamental fact of nature such as the three
124 THE CONCEPT OF NATURE [cH.
dimensions of space. If space has only three dimensions
we should expect all mankind to be aware of the fact,
as they are aware of it. But in the case of congruence,
mankind agree in an arbitrary interpretation of sense-
awareness when there is nothing in nature to guide it.
I look on it as no slight recommendation of the theory
of nature which I am expounding to you that it gives
a solution of this difficulty by pointing out the factor
in nature which issues in the preeminence of one
congruence relation over the indefinite herd of other
such relations.
The reason for this result is that nature is no longer
confined within space at an instant. Space and time
are now interconnected ; and this peculiar factor of time
which is so immediately distinguished among the
deliverances of our sense-awareness, relates itself to
one particular congruence relation in space.
Congruence is a particular example of the fundamental
fact of recognition. In perception we recognise. This
recognition does not merely concern the comparison of
a factor of nature posited by memory with a factor
posited by immediate sense-awareness. Recognition
takes place within the present without any intervention
of pure memory. For the present fact is a duration with
its antecedent and consequent durations which are
parts of itself. The discrimination in sense-awareness
of a finite event with its quality of passage is also
accompanied by the discrimination of other factors of
nature which do not share in the passage of events.
Whatever passes is an event. But we find entities in
nature which do not pass ; namely we recognise same-
nesses in nature. Recognition is not primarily an
intellectual act of comparison ; it is in its essence merely
Vi] Congruence 125
sense-awareness in its capacity of positing before us
factors in nature which do not pass. For example,
green is perceived as situated in a certain finite event
within the present duration. This green preserves its
self-identity throughout, whereas the event passes and
thereby obtains the property of breaking into parts.
The green patch has parts. But in talking of the green
patch we are speaking of the event in its sole capacity of
being for us the situation of green. The green itself is
numerically one self-identical entity, without parts
because it is without passage.
Factors in nature which are without passage will be called objects. There are radically different kinds of ob- jects which will be considered in the succeeding lecture. Recognition is reflected into the intellect as comparison . The recognised objects of one event are compared with the recognised objects of another event. The com- parison may be between two events in the present, or it may be between two events of which one is posited by memory-awareness and the other by immediate sense-awareness. But it is not the events which are compared. For each event is essentially unique and incomparable. What are compared are the objects and relations of objects situated in events. The event con- sidered as a relation between objects has lost its passage and in this aspect is itself an object. This object is not the event but only an intellectual abstraction. The same object can be situated in many events ; and in this sense even the whole event, viewed as an object, can recur, though not the very event itself with its passage and its relations to other events.
Objects which are not posited by sense-awareness may be known to the intellect. For example, relations
126 THE CONCEPT OF NATURE [CH.
between objects and relations between relations may
be factors in nature not disclosed in sense-awareness
but known by logical inference as necessarily in being.
Thus objects for our knowledge may be merely logical
abstractions. For example, a complete event is never
disclosed in sense-awareness, and thus the object which
is the sum total of objects situated in an event as thus
inter-related is a mere abstract concept. Again a right-
angle is a perceived object which can be situated in
many events; but, though rectangularity is posited by
sense-awareness, the majority of geometrical relations
are not so posited. Also rectangularity is in fact often
not perceived when it can be proved to have been there
for perception. Thus an object is often known merely
as an abstract relation not directly posited in sense-
awareness although it is there in nature.
The identity of quality between congruent segments
is generally of this character. In certain special cases
this identity of quality can be directly perceived. But
in general it is inferred by a process of measurement
depending on our direct sense-awareness of selected
cases and a logical inference from the transitive character
of congruence.
Congruence depends on motion, and thereby is
generated the connexion between spatial congruence
and temporal congruence. Motion along a straight line
has a symmetry round that line. This symmetry is ex-
pressed by the symmetrical geometrical relations of the
line to the family of planes normal to it.