<h2><span class="pagenum" title="Page 120"> </span><SPAN name="Page_120" id="Page_120"></SPAN><SPAN name="CHAPTER_VI" id="CHAPTER_VI"></SPAN>CHAPTER VI <br/> CONGRUENCE</h2>
<p>The aim of this lecture is to establish a theory of congruence. You must
understand at once that congruence is a controversial question. It is
the theory of measurement 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.</p>
<p>First let us fix attention on the purely mathematical question. When the
segment between two points <i>A</i> and <i>B</i> is congruent to that between the
two points <i>C</i> and <i>D</i>, the quantitative measurements of the two
segments 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<span class="pagenum" title="Page 121"> </span><SPAN name="Page_121" id="Page_121"></SPAN>
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
congruence 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.</p>
<p>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 conditions 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.</p>
<p>Poincaré, 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 Poincaré to
mean that there is nothing inherent in nature itself giving any peculiar
<i>rôle</i> to one of these<span class="pagenum" title="Page 122"> </span><SPAN name="Page_122" id="Page_122"></SPAN> congruence relations, and that the choice of one
particular 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.</p>
<p>This position has been misunderstood by many of Poincaré’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 statement
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.</p>
<p>This is an entirely different question and it presupposes a rejection of
Poincaré’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.</p>
<p>Poincaré’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 undeniably
the position is very paradoxical. Bertrand Russell had a controversy
with him on this question, and pointed out that on Poincaré’s principles
there was nothing in nature to determine whether the earth is larger or
smaller than some assigned billiard ball. Poincaré 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<span class="pagenum" title="Page 123"> </span><SPAN name="Page_123" id="Page_123"></SPAN> ship in the ocean by counting the crew and observing the colour of
the captain’s eyes.</p>
<p>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. Poincaré asks
for information as to the factor in nature which might lead any
particular congruence relation to play a preeminent <i>rôle</i> 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 theory nature at an instant in space is an independent
fact. Thus we have to look for our preeminent congruence relation amid
nature in instantaneous space; and Poincaré is undoubtedly right in
saying that nature on this hypothesis gives us no help in finding it.</p>
<p>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 mankind
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 discovered till the close of the
nineteenth century by a few mathematical philosophers and philosophic
mathematicians. The case is not like that of our agreement on some
fundamental fact of nature such as the three<span class="pagenum" title="Page 124"> </span><SPAN name="Page_124" id="Page_124"></SPAN> 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.</p>
<p>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.</p>
<p>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.</p>
<p>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
samenesses in nature. Recognition is not primarily an intellectual act
of comparison; it is in its essence merely<span class="pagenum" title="Page 125"> </span><SPAN name="Page_125" id="Page_125"></SPAN> 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.</p>
<p>Factors in nature which are without passage will be called objects.
There are radically different kinds of objects which will be considered
in the succeeding lecture.</p>
<p>Recognition is reflected into the intellect as comparison. The
recognised objects of one event are compared with the recognised objects
of another event. The comparison 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 considered 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.</p>
<p>Objects which are not posited by sense-awareness may be known to the
intellect. For example, relations<span class="pagenum" title="Page 126"> </span><SPAN name="Page_126" id="Page_126"></SPAN> 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.</p>
<p>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.</p>
<p>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 expressed
by the symmetrical geometrical relations of the line to the family of
planes normal to it.</p>
<p>Also another symmetry in the theory of motion arises from the fact that
rest in the points of β corresponds to uniform motion along a definite
family of parallel straight lines in the space of α. We must note the
three<span class="pagenum" title="Page 127"> </span><SPAN name="Page_127" id="Page_127"></SPAN> characteristics, (i) of the uniformity of the motion
corresponding to any point of β along its correlated straight line in α,
and (ii) of the equality in magnitude of the velocities along the
various lines of α correlated to rest in the various points of β, and
(iii) of the parallelism of the lines of this family.</p>
<p>We are now in possession of a theory of parallels and a theory of
perpendiculars and a theory of motion, and from these theories the
theory of congruence can be constructed. It will be remembered that a
family of parallel levels in any moment is the family of levels in which
that moment is intersected by the family of moments of some other
time-system. Also a family of parallel moments is the family of moments
of some one time-system. Thus we can enlarge our concept of a family of
parallel levels so as to include levels in different moments of one
time-system. With this enlarged concept we say that a complete family of
parallel levels in a time-system α is the complete family of levels in
which the moments of α intersect the moments of β. This complete family
of parallel levels is also evidently a family lying in the moments of
the time-system β. By introducing a third time-system γ, parallel rects
are obtained. Also all the points of any one time-system form a family
of parallel point-tracks. Thus there are three types of parallelograms
in the four-dimensional manifold of event-particles.</p>
<p>In parallelograms of the first type the two pairs of parallel sides are
both of them pairs of rects. In parallelograms of the second type one
pair of parallel sides is a pair of rects and the other pair is a pair
of point-tracks. In parallelograms of the third type the two pairs of
parallel sides are both of them pairs of point-tracks.</p>
<p><span class="pagenum" title="Page 128"> </span><SPAN name="Page_128" id="Page_128"></SPAN>The first axiom of congruence is that the opposite sides of any
parallelogram are congruent. This axiom enables us to compare the
lengths of any two segments either respectively on parallel rects or on
the same rect. Also it enables us to compare the lengths of any two
segments either respectively on parallel point-tracks or on the same
point-track. It follows from this axiom that two objects at rest in any
two points of a time-system β are moving with equal velocities in any
other time-system α along parallel lines. Thus we can speak of the
velocity in α due to the time-system β without specifying any particular
point in β. The axiom also enables us to measure time in any
time-system; but does not enable us to compare times in different
time-systems.</p>
<p>The second axiom of congruence concerns parallelograms on congruent
bases and between the same parallels, which have also their other pairs
of sides parallel. The axiom asserts that the rect joining the two
event-particles of intersection of the diagonals is parallel to the rect
on which the bases lie. By the aid of this axiom it easily follows that
the diagonals of a parallelogram bisect each other.</p>
<p>Congruence is extended in any space beyond parallel rects to all rects
by two axioms depending on perpendicularity. The first of these axioms,
which is the third axiom of congruence, is that if <i>ABC</i> is a triangle
of rects in any moment and <i>D</i> is the middle event-particle of the base
<i>BC</i>, then the level through <i>D</i> perpendicular to <i>BC</i> contains <i>A</i> when
and only when <i>AB</i> is congruent to <i>AC</i>. This axiom evidently expresses
the symmetry of perpendicularity, and is the essence of the famous pons
asinorum expressed as an axiom.</p>
<p>The second axiom depending on perpendicularity,<span class="pagenum" title="Page 129"> </span><SPAN name="Page_129" id="Page_129"></SPAN> and the fourth axiom of
congruence, is that if <i>r</i> and <i>A</i> be a rect and an event-particle in
the same moment and <i>AB</i> and <i>AC</i> be a pair of rectangular rects
intersecting <i>r</i> in <i>B</i> and <i>C</i>, and <i>AD</i> and <i>AE</i> be another pair of
rectangular rects intersecting <i>r</i> in <i>D</i> and <i>E</i>, then either <i>D</i> or
<i>E</i> lies in the segment <i>BC</i> and the other one of the two does not lie
in this segment. Also as a particular case of this axiom, if <i>AB</i> be
perpendicular to <i>r</i> and in consequence <i>AC</i> be parallel to <i>r</i>, then
<i>D</i> and <i>E</i> lie on opposite sides of <i>B</i> respectively. By the aid of
these two axioms the theory of congruence can be extended so as to
compare lengths of segments on any two rects. Accordingly Euclidean
metrical geometry in space is completely established and lengths in the
spaces of different time-systems are comparable as the result of
definite properties of nature which indicate just that particular method
of comparison.</p>
<p>The comparison of time-measurements in diverse time-systems requires two
other axioms. The first of these axioms, forming the fifth axiom of
congruence, will be called the axiom of ‘kinetic symmetry.’ It expresses
the symmetry of the quantitative relations between two time-systems when
the times and lengths in the two systems are measured in congruent
units.</p>
<p>The axiom can be explained as follows: Let α and β be the names of two
time-systems. The directions of motion in the space of α due to rest in
a point of β is called the ‘β-direction in α’ and the direction of
motion in the space of β due to rest in a point of α is called the
‘α-direction in β.’ Consider a motion in the space of α consisting of a
certain velocity in the β-direction of α and a certain velocity at
right-angles to it. This motion represents rest in the space of another
time-system<span class="pagenum" title="Page 130"> </span><SPAN name="Page_130" id="Page_130"></SPAN>—call it π. Rest in π will also be represented in the space
of β by a certain velocity in the α-direction in β and a certain
velocity at right-angles to this α-direction. Thus a certain motion in
the space of α is correlated to a certain motion in the space of β, as
both representing the same fact which can also be represented by rest in
π. Now another time-system, which I will name σ, can be found which is
such that rest in its space is represented by the same magnitudes of
velocities along and perpendicular to the α-direction in β as those
velocities in α, along and perpendicular to the β-direction, which
represent rest in π. The required axiom of kinetic symmetry is that rest
in σ will be represented in α by the same velocities along and
perpendicular to the β-direction in α as those velocities in β along and
perpendicular to the α-direction which represent rest in π.</p>
<p>A particular case of this axiom is that relative velocities are equal
and opposite. Namely rest in α is represented in β by a velocity along
the α-direction which is equal to the velocity along the β-direction in
α which represents rest in β.</p>
<p>Finally the sixth axiom of congruence is that the relation of congruence
is transitive. So far as this axiom applies to space, it is superfluous.
For the property follows from our previous axioms. It is however
necessary for time as a supplement to the axiom of kinetic symmetry. The
meaning of the axiom is that if the time-unit of system α is congruent
to the time-unit of system β, and the time-unit of system β is congruent
to the time-unit of system γ, then the time-units of α and γ are also
congruent.</p>
<p>By means of these axioms formulae for the trans<span class="pagenum" title="Page 131"> </span><SPAN name="Page_131" id="Page_131"></SPAN>formation of
measurements made in one time-system to measurements of the same facts
of nature made in another time-system can be deduced. These formulae
will be found to involve one arbitrary constant which I will call <i>k</i>.</p>
<p>It is of the dimensions of the square of a velocity. Accordingly four
cases arise. In the first case <i>k</i> is zero. This case produces
nonsensical results in opposition to the elementary deliverances of
experience. We put this case aside.</p>
<p>In the second case <i>k</i> is infinite. This case yields the ordinary
formulae for transformation in relative motion, namely those formulae
which are to be found in every elementary book on dynamics.</p>
<p>In the third case, <i>k</i> is negative. Let us call it −<i>c</i><sup>2</sup>, where <i>c</i>
will be of the dimensions of a velocity. This case yields the formulae
of transformation which Larmor discovered for the transformation of
Maxwell’s equations of the electromagnetic field. These formulae were
extended by H. A. Lorentz, and used by Einstein and Minkowski as the
basis of their novel theory of relativity. I am not now speaking of
Einstein’s more recent theory of general relativity by which he deduces
his modification of the law of gravitation. If this be the case which
applies to nature, then <i>c</i> must be a close approximation to the
velocity of light <i>in vacuo</i>. Perhaps it is this actual velocity. In
this connexion ‘<i>in vacuo</i>’ must not mean an absence of events, namely
the absence of the all-pervading ether of events. It must mean the
absence of certain types of objects.</p>
<p>In the fourth case, <i>k</i> is positive. Let us call it <i>h</i><sup>2</sup>, where <i>h</i>
will be of the dimensions of a velocity. This gives a perfectly possible
type of transformation formulae,<span class="pagenum" title="Page 132"> </span><SPAN name="Page_132" id="Page_132"></SPAN> but not one which explains any facts
of experience. It has also another disadvantage. With the assumption of
this fourth case the distinction between space and time becomes unduly
blurred. The whole object of these lectures has been to enforce the
doctrine that space and time spring from a common root, and that the
ultimate fact of experience is a space-time fact. But after all mankind
does distinguish very sharply between space and time, and it is owing to
this sharpness of distinction that the doctrine of these lectures is
somewhat of a paradox. Now in the third assumption this sharpness of
distinction is adequately preserved. There is a fundamental distinction
between the metrical properties of point-tracks and rects. But in the
fourth assumption this fundamental distinction vanishes.</p>
<p>Neither the third nor the fourth assumption can agree with experience
unless we assume that the velocity <i>c</i> of the third assumption, and the
velocity <i>h</i> of the fourth assumption, are extremely large compared to
the velocities of ordinary experience. If this be the case the formulae
of both assumptions will obviously reduce to a close approximation to
the formulae of the second assumption which are the ordinary formulae of
dynamical textbooks. For the sake of a name, I will call these textbook
formulae the ‘orthodox’ formulae.</p>
<p>There can be no question as to the general approximate correctness of
the orthodox formulae. It would be merely silly to raise doubts on this
point. But the determination of the status of these formulae is by no
means settled by this admission. The independence of time and space is
an unquestioned presupposition of the orthodox thought which has
produced the orthodox formulae. With this presupposition and given the<span class="pagenum" title="Page 133"> </span><SPAN name="Page_133" id="Page_133"></SPAN>
absolute points of one absolute space, the orthodox formulae are
immediate deductions. Accordingly, these formulae are presented to our
imaginations as facts which cannot be otherwise, time and space being
what they are. The orthodox formulae have therefore attained to the
status of necessities which cannot be questioned in science. Any attempt
to replace these formulae by others was to abandon the <i>rôle</i> of
physical explanation and to have recourse to mere mathematical formulae.</p>
<p>But even in physical science difficulties have accumulated round the
orthodox formulae. In the first place Maxwell’s equations of the
electromagnetic field are not invariant for the transformations of the
orthodox formulae; whereas they are invariant for the transformations of
the formulae arising from the third of the four cases mentioned above,
provided that the velocity <i>c</i> is identified with a famous
electromagnetic constant quantity.</p>
<p>Again the null results of the delicate experiments to detect the earth’s
variations of motion through the ether in its orbital path are explained
immediately by the formulae of the third case. But if we assume the
orthodox formulae we have to make a special and arbitrary assumption as
to the contraction of matter during motion. I mean the
Fitzgerald-Lorentz assumption.</p>
<p>Lastly Fresnel’s coefficient of drag which represents the variation of
the velocity of light in a moving medium is explained by the formulae of
the third case, and requires another arbitrary assumption if we use the
orthodox formulae.</p>
<p>It appears therefore that on the mere basis of physical explanation
there are advantages in the formulae<span class="pagenum" title="Page 134"> </span><SPAN name="Page_134" id="Page_134"></SPAN> of the third case as compared with
the orthodox formulae. But the way is blocked by the ingrained belief
that these latter formulae possess a character of necessity. It is
therefore an urgent requisite for physical science and for philosophy to
examine critically the grounds for this supposed necessity. The only
satisfactory method of scrutiny is to recur to the first principles of
our knowledge of nature. This is exactly what I am endeavouring to do in
these lectures. I ask what it is that we are aware of in our
sense-perception of nature. I then proceed to examine those factors in
nature which lead us to conceive nature as occupying space and
persisting through time. This procedure has led us to an investigation
of the characters of space and time. It results from these
investigations that the formulae of the third case and the orthodox
formulae are on a level as possible formulae resulting from the basic
character of our knowledge of nature. The orthodox formulae have thus
lost any advantage as to necessity which they enjoyed over the serial
group. The way is thus open to adopt whichever of the two groups best
accords with observation.</p>
<p>I take this opportunity of pausing for a moment from the course of my
argument, and of reflecting on the general character which my doctrine
ascribes to some familiar concepts of science. I have no doubt that some
of you have felt that in certain aspects this character is very
paradoxical.</p>
<p>This vein of paradox is partly due to the fact that educated language
has been made to conform to the prevalent orthodox theory. We are thus,
in expounding an alternative doctrine, driven to the use of either
strange terms or of familiar words with unusual meanings. This<span class="pagenum" title="Page 135"> </span><SPAN name="Page_135" id="Page_135"></SPAN> victory
of the orthodox theory over language is very natural. Events are named
after the prominent objects situated in them, and thus both in language
and in thought the event sinks behind the object, and becomes the mere
play of its relations. The theory of space is then converted into a
theory of the relations of objects instead of a theory of the relations
of events. But objects have not the passage of events. Accordingly space
as a relation between objects is devoid of any connexion with time. It
is space at an instant without any determinate relations between the
spaces at successive instants. It cannot be one timeless space because
the relations between objects change.</p>
<p>A few minutes ago in speaking of the deduction of the orthodox formulae
for relative motion I said that they followed as an immediate deduction
from the assumption of absolute points in absolute space. This reference
to absolute space was not an oversight. I know that the doctrine of the
relativity of space at present holds the field both in science and
philosophy. But I do not think that its inevitable consequences are
understood. When we really face them the paradox of the presentation of
the character of space which I have elaborated is greatly mitigated. If
there is no absolute position, a point must cease to be a simple entity.
What is a point to one man in a balloon with his eyes fixed on an
instrument is a track of points to an observer on the earth who is
watching the balloon through a telescope, and is another track of points
to an observer in the sun who is watching the balloon through some
instrument suited to such a being. Accordingly if I am reproached with
the paradox of my theory of points as classes of event-particles, and of
my theory of event-particles as<span class="pagenum" title="Page 136"> </span><SPAN name="Page_136" id="Page_136"></SPAN> groups of abstractive sets, I ask my
critic to explain exactly what he means by a point. While you explain
your meaning about anything, however simple, it is always apt to look
subtle and fine spun. I have at least explained exactly what I do mean
by a point, what relations it involves and what entities are the relata.
If you admit the relativity of space, you also must admit that points
are complex entities, logical constructs involving other entities and
their relations. Produce your theory, not in a few vague phrases of
indefinite meaning, but explain it step by step in definite terms
referring to assigned relations and assigned relata. Also show that your
theory of points issues in a theory of space. Furthermore note that the
example of the man in the balloon, the observer on earth, and the
observer in the sun, shows that every assumption of relative rest
requires a timeless space with radically different points from those
which issue from every other such assumption. The theory of the
relativity of space is inconsistent with any doctrine of one unique set
of points of one timeless space.</p>
<p>The fact is that there is no paradox in my doctrine of the nature of
space which is not in essence inherent in the theory of the relativity
of space. But this doctrine has never really been accepted in science,
whatever people say. What appears in our dynamical treatises is Newton’s
doctrine of relative motion based on the doctrine of differential motion
in absolute space. When you once admit that the points are radically
different entities for differing assumptions of rest, then the orthodox
formulae lose all their obviousness. They were only obvious because you
were really thinking of something else. When discussing this topic you
can<span class="pagenum" title="Page 137"> </span><SPAN name="Page_137" id="Page_137"></SPAN> only avoid paradox by taking refuge from the flood of criticism in
the comfortable ark of no meaning.</p>
<p>The new theory provides a definition of the congruence of periods of
time. The prevalent view provides no such definition. Its position is
that if we take such time-measurements so that certain familiar
velocities which seem to us to be uniform are uniform, then the laws of
motion are true. Now in the first place no change could appear either as
uniform or non-uniform without involving a definite determination of the
congruence for time-periods. So in appealing to familiar phenomena it
allows that there is some factor in nature which we can intellectually
construct as a congruence theory. It does not however say anything about
it except that the laws of motion are then true. Suppose that with some
expositors we cut out the reference to familiar velocities such as the
rate of rotation of the earth. We are then driven to admit that there is
no meaning in temporal congruence except that certain assumptions make
the laws of motion true. Such a statement is historically false. King
Alfred the Great was ignorant of the laws of motion, but knew very well
what he meant by the measurement of time, and achieved his purpose by
means of burning candles. Also no one in past ages justified the use of
sand in hour-glasses by saying that some centuries later interesting
laws of motion would be discovered which would give a meaning to the
statement that the sand was emptied from the bulbs in equal times.
Uniformity in change is directly perceived, and it follows that mankind
perceives in nature factors from which a theory of temporal congruence
can be formed. The prevalent theory entirely fails to produce such
factors.</p>
<p><span class="pagenum" title="Page 138"> </span><SPAN name="Page_138" id="Page_138"></SPAN>The mention of the laws of motion raises another point where the
prevalent theory has nothing to say and the new theory gives a complete
explanation. It is well known that the laws of motion are not valid for
any axes of reference which you may choose to take fixed in any rigid
body. You must choose a body which is not rotating and has no
acceleration. For example they do not really apply to axes fixed in the
earth because of the diurnal rotation of that body. The law which fails
when you assume the wrong axes as at rest is the third law, that action
and reaction are equal and opposite. With the wrong axes uncompensated
centrifugal forces and uncompensated composite centrifugal forces
appear, due to rotation. The influence of these forces can be
demonstrated by many facts on the earth’s surface, Foucault’s pendulum,
the shape of the earth, the fixed directions of the rotations of
cyclones and anticyclones. It is difficult to take seriously the
suggestion that these domestic phenomena on the earth are due to the
influence of the fixed stars. I cannot persuade myself to believe that a
little star in its twinkling turned round Foucault’s pendulum in the
Paris Exhibition of 1861. Of course anything is believable when a
definite physical connexion has been demonstrated, for example the
influence of sunspots. Here all demonstration is lacking in the form of
any coherent theory. According to the theory of these lectures the axes
to which motion is to be referred are axes at rest in the space of some
time-system. For example, consider the space of a time-system α. There
are sets of axes at rest in the space of α. These are suitable dynamical
axes. Also a set of axes in this space which is moving with uniform
velocity without rotation is<span class="pagenum" title="Page 139"> </span><SPAN name="Page_139" id="Page_139"></SPAN> another suitable set. All the moving
points fixed in these moving axes are really tracing out parallel lines
with one uniform velocity. In other words they are the reflections in
the space of α of a set of fixed axes in the space of some other
time-system β. Accordingly the group of dynamical axes required for
Newton’s Laws of Motion is the outcome of the necessity of referring
motion to a body at rest in the space of some one time-system in order
to obtain a coherent account of physical properties. If we do not do so
the meaning of the motion of one portion of our physical configuration
is different from the meaning of the motion of another portion of the
same configuration. Thus the meaning of motion being what it is, in
order to describe the motion of any system of objects without changing
the meaning of your terms as you proceed with your description, you are
bound to take one of these sets of axes as axes of reference; though you
may choose their reflections into the space of any time-system which you
wish to adopt. A definite physical reason is thereby assigned for the
peculiar property of the dynamical group of axes.</p>
<p>On the orthodox theory the position of the equations of motion is most
ambiguous. The space to which they refer is completely undetermined and
so is the measurement of the lapse of time. Science is simply setting
out on a fishing expedition to see whether it cannot find some procedure
which it can call the measurement of space and some procedure which it
can call the measurement of time, and something which it can call a
system of forces, and something which it can call masses, so that these
formulae may be satisfied. The only reason—on this theory—why anyone
should want to satisfy these formulae is a sentimental regard for
Galileo,<span class="pagenum" title="Page 140"> </span><SPAN name="Page_140" id="Page_140"></SPAN> Newton, Euler and Lagrange. The theory, so far from founding
science on a sound observational basis, forces everything to conform to
a mere mathematical preference for certain simple formulae.</p>
<p>I do not for a moment believe that this is a true account of the real
status of the Laws of Motion. These equations want some slight
adjustment for the new formulae of relativity. But with these
adjustments, imperceptible in ordinary use, the laws deal with
fundamental physical quantities which we know very well and wish to
correlate.</p>
<p>The measurement of time was known to all civilised nations long before
the laws were thought of. It is this time as thus measured that the laws
are concerned with. Also they deal with the space of our daily life.
When we approach to an accuracy of measurement beyond that of
observation, adjustment is allowable. But within the limits of
observation we know what we mean when we speak of measurements of space
and measurements of time and uniformity of change. It is for science to
give an intellectual account of what is so evident in sense-awareness.
It is to me thoroughly incredible that the ultimate fact beyond which
there is no deeper explanation is that mankind has really been swayed by
an unconscious desire to satisfy the mathematical formulae which we call
the Laws of Motion, formulae completely unknown till the seventeenth
century of our epoch.</p>
<p>The correlation of the facts of sense-experience effected by the
alternative account of nature extends beyond the physical properties of
motion and the properties of congruence. It gives an account of the
meaning of the geometrical entities such as points, straight lines, and
volumes, and connects the kindred<span class="pagenum" title="Page 141"> </span><SPAN name="Page_141" id="Page_141"></SPAN>" ideas of extension in time and
extension in space. The theory satisfies the true purpose of an
intellectual explanation in the sphere of natural philosophy. This
purpose is to exhibit the interconnexions of nature, and to show that
one set of ingredients in nature requires for the exhibition of its
character the presence of the other sets of ingredients.</p>
<p>The false idea which we have to get rid of is that of nature as a mere
aggregate of independent entities, each capable of isolation. According
to this conception these entities, whose characters are capable of
isolated definition, come together and by their accidental relations
form the system of nature. This system is thus thoroughly accidental;
and, even if it be subject to a mechanical fate, it is only accidentally
so subject.</p>
<p>With this theory space might be without time, and time might be without
space. The theory admittedly breaks down when we come to the relations
of matter and space. The relational theory of space is an admission that
we cannot know space without matter or matter without space. But the
seclusion of both from time is still jealously guarded. The relations
between portions of matter in space are accidental facts owing to the
absence of any coherent account of how space springs from matter or how
matter springs from space. Also what we really observe in nature, its
colours and its sounds and its touches are secondary qualities; in other
words, they are not in nature at all but are accidental products of the
relations between nature and mind.</p>
<p>The explanation of nature which I urge as an alternative ideal to this
accidental view of nature, is that nothing in nature could be what it is
except as an<span class="pagenum" title="Page 142"> </span><SPAN name="Page_142" id="Page_142"></SPAN> ingredient in nature as it is. The whole which is present
for discrimination is posited in sense-awareness as necessary for the
discriminated parts. An isolated event is not an event, because every
event is a factor in a larger whole and is significant of that whole.
There can be no time apart from space; and no space apart from time; and
no space and no time apart from the passage of the events of nature. The
isolation of an entity in thought, when we think of it as a bare ‘it,’
has no counterpart in any corresponding isolation in nature. Such
isolation is merely part of the procedure of intellectual knowledge.</p>
<p>The laws of nature are the outcome of the characters of the entities
which we find in nature. The entities being what they are, the laws must
be what they are; and conversely the entities follow from the laws. We
are a long way from the attainment of such an ideal; but it remains as
the abiding goal of theoretical science.</p>
<hr />
<div style="break-after:column;"></div><br />

1 of 2

2 of 2

Previous
Next

Update Required
To play the media you will need to either update your browser to a recent version or update your Flash plugin.