of the third case as compared with the orthodox for- mulae. But the way is blocked by the ingrained behef 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.
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.
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
Vi] Congruence
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 deter-
minate relations between the spaces at successive in-
stants. It cannot be one timeless space because the
relations between objects change.
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
136 THE CONCEPT OF NATURE Lch.
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 assump- tion. The theory of the relativity of space is incon- sistent with any doctrine of one unique set of points of one timeless space.
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
Vi] Congruence 137
only avoid paradox by taking refuge from the flood of criticism in the comfortable ark of no meaning.
The new theory provides a definition of the con-
gruence of periods of time. The prevalent view pro-
vides 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 any-
thing 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 in-
teresting 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.
138 THE CONCEPT OF NATURE [cn.
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 vaUd 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 centri-
fugal 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 sug-
gestion 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 186 1. Of course anything is behev-
able when a definite physical connexion has been
demonstrated, for example the influence of sunspotSw
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 a. There
are sets of axes at rest in the space of a . These are suitable
dynamical axes. Also a set of axes in this space which
is moving with uniform velocity without rotation is
Vi] Congruence 139
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 a of a set of fixed axes in the space of some other time-system )8. 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.
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 measure-
ment 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 measure-
ment 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,
140 THE CONCEPT OF NATURE [cH.
Newton, Euler and Lagrange. The theory, so far from
founding science on a sound observational basis, forces
everything to conform to a mere mathematical pre-
ference for certain simple formulae.
I do not for a moment believe that this is a true ac-
count 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 funda-
mental physical quantities which we know very well
and wish to correlate.
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-aware- ness. 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.
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
Vi] Congruence 141
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.