SPACE, TIME, AND RELATIVITY
(_Paper read to Section A at the Manchester Meeting of the British
Association, 1915, and later before the Aristotelian Society_)
FUNDAMENTAL Problems concerning space and time have been considered
from the standpoints created by many different sciences. The object of
this paper is the humble one of bringing some of these standpoints into
relation with each other. This necessitates a very cursory treatment of
each point of view.
Mathematical physicists have evolved their theory of relativity to
explain the negative results of the Morley-Michelson experiment and of
the Trouton experiment. Experimental psychologists have considered the
evolution of spatial ideas from the crude sense-data of experience.
Metaphysicians have considered the majestic uniformity of space and
time, without beginning and without end, without boundaries, and
without exception in the truths concerning them; all these qualities
the more arresting to our attention from the confused accidental
nature of the empirical universe which is conditioned by them.
Mathematicians have studied the axioms of geometry, and can now deduce
all that is believed to be universally true of space and of time by the
strictest logic from a limited number of assumptions.
These various lines of thought have been evolved with surprisingly
little interconnection. Perhaps it is as well. The results of science
are never quite true. By a healthy independence of thought perhaps we
sometimes avoid adding other people's errors to our own. But there can
be no doubt that the normal method of cross-fertilising thought is by
considering the same, or allied problems to our own, in the form which
they assume in other sciences.
Here I do not propose to enter into a systematic study of these various
chapters of science. I have neither the knowledge nor the time.
First, let us take the ultimate basis of any theory of relativity.
All space measurement is from stuff in space to stuff in space. The
geometrical entities of empty space never appear. The only geometrical
properties of which we have any direct knowledge are properties of
those shifting, changeable appearances which we call things in space.
It is the sun which is distant, and the ball which is round, and the
lamp-posts which are in linear order. Wherever mankind may have got its
idea of an infinite unchangeable space from, it is safe to say that it
is not an immediate deliverance of direct observation.
There are two antagonistic philosophical ways of recognising this
conclusion.
One is to affirm that space and time are conditions for sensible
experience, that without projection into space and time sensible
experience would not exist. Thus, although it may be true to say that
our knowledge of space and time is given in experience, it is not true
to say that it is deduced from experience in the same sense that the
Law of Gravitation is so deduced. It is not deduced, because in the act
of experiencing we are necessarily made aware of space as an infinite
given whole, and of time as an unending uniform succession. This
philosophical position is expressed by saying that space and time are
_a priori_ forms of sensibility.
The opposed philosophical method of dealing with the question is
to affirm that our concepts of time and space are deductions from
experience, in exactly the same way as the Law of Gravitation is such
a deduction. If we form exact concepts of points, lines and surfaces,
and of successive instants of time, and assume them to be related as
expressed by the axioms of geometry and the axioms for time, then we
find that we have framed a concept which, with all the exactness of
which our observations are capable, expresses the facts of experience.
These two philosophic positions are each designed to explain a
certain difficulty. The _a priori_ theory explains the absolute
universality ascribed to the laws of space and time, a universality
not ascribed to any deduction from experience. The experiential theory
explains the derivation of the space-time concepts without introducing
any other factors beyond those which are admittedly present in framing
the other concepts of physical science.
But we have not yet done with the distinctions which in any discussion
of space or time must essentially be kept in mind. Put aside the
above question as to how these space-time concepts are related to
experience--What are they when they are formed?
We may conceive of the points of space as self-subsistent entities
which have the indefinable relation of being occupied by the ultimate
stuff (matter, I will call it) which is there. Thus, to say that the
sun is _there_ (wherever it is) is to affirm the relation of
occupation between the set of positive and negative electrons which
we call the sun and a certain set of points, the points having an
existence essentially independent of the sun. This is the absolute
theory of space. The absolute theory is not popular just now, but it
has very respectable authority on its side--Newton, for one--so treat
it tenderly.
The other theory is associated with Leibniz. Our space concepts are
concepts of relations between things in space. Thus there is no such
entity as a self-subsistent point. A point is merely the name for some
peculiarity of the relations between the matter which is, in common
language, said to be in space.
It follows from the relative theory that a point should be definable in
terms of the relations between material things. So far as I am aware,
this outcome of the theory has escaped the notice of mathematicians,
who have invariably assumed the point as the ultimate starting ground
of their reasoning. Many years ago I explained some types of ways in
which we might achieve such a definition, and more recently have added
some others. Similar explanations apply to time. Before the theories
of space and time have been carried to a satisfactory conclusion on
the relational basis, a long and careful scrutiny of the definitions
of points of space and instants of time will have to be undertaken,
and many ways of effecting these definitions will have to be tried and
compared. This is an unwritten chapter of mathematics, in much the same
state as was the theory of parallels in the eighteenth century.
In this connection I should like to draw attention to the analogy
between time and space. In analysing our experience we distinguish
events, and we also distinguish things whose changing relations form
the events. If I had time it would be interesting to consider more
closely these concepts of events and of things. It must suffice now to
point out that things have certain relations to each other which we
consider as relations between the space extensions of the things; for
example, one space can contain the other, or exclude it, or overlap
it. A point in space is nothing else than a certain set of relations
between spatial extensions.
Analogously, there are certain relations between events which we
express by saying that they are relations between the temporal
durations of these events, that is, between the temporal extensions of
the events. [The durations of two events A and B may one precede the
other, or may partially overlap, or may one contain the other, giving
in all six possibilities.] The properties of the extension of an event
in time are largely analogous to the extension of an object in space.
Spatial extensions are expressed by relations between objects, temporal
extensions by relations between events.
The point in time is a set of relations between temporal extensions.
It needs very little reflection to convince us that a point in time
is no direct deliverance of experience. We live in durations, and not
in points. But what community, beyond the mere name, is there between
extension in time and extension in space? In view of the intimate
connection between time and space revealed by the modern theory of
relativity, this question has taken on a new importance.
I have not thought out an answer to this question. I suggest, however,
that time and space embody those relations between objects on which
depends our judgment of their externality to ourselves. Namely,
location in space and location in time both embody and perhaps
necessitate a judgment of externality. This suggestion is very vague,
and I must leave it in this crude form.
_Diverse Euclidean Measure Systems_
Turning now to the mathematical investigations on the axioms of
geometry, the outcome, which is most important for us to remember,
is the great separation which it discloses between non-metrical
projective geometry, and metrical geometry. Non-metrical projective
geometry is by far the more fundamental. Starting with the concepts
of points, straight lines, and planes (of which not all three need
be taken as indefinable), and with certain very simple non-metrical
properties of these entities--such as, for instance, that two points
uniquely determine a straight line--nearly the whole of geometry can
be constructed. Even quantitative coordinates can be introduced, to
facilitate the reasoning. But no mention of distance, area, or volume,
need have been introduced. Points will have an order on the line, but
order does not imply any settled distance.
When we now inquire what measurements of distance are possible, we find
that there are different systems of measurement all equally possible.
There are three main types of system: any system of one type gives
Euclidean geometry, any system of another type gives Hyperbolic (or
Lobatchewskian) geometry, any system of the third type gives Elliptic
geometry. Also different beings, or the same being if he chooses,
may reckon in different systems of the same type, or in systems of
different types. Consider the example which will interest us later.
Two beings, A and B, agree to use the same three intersecting lines
as axes of _x_, _y_, _z_. They both employ a system of measurement of
the Euclidean type, and (what is not necessarily the case) agree as to
the plane at infinity. That is, they agree as to the lines which are
parallel. Then with the usual method of rectangular Cartesian axes,
they agree that the coordinates of P are the lengths ON, NM, MP. So
far all is harmony. A fixes on the segment OU_{1}, on O_x_, as being
the unit length, and B on the segment OV_{1}, on O_x_. A calls his
coordinates (_x_, _y_, _z_), and B calls them (X, Y, Z).
Then it is found [since both systems are Euclidean] that, whatever
point P be taken,
X = β_x_, Y = γ_y_, Z = δ_z_. [β ≠ γ ≠ δ.]
They proceed to adjust their differences, and first take the
_x_-coordinates. Obviously they have taken different units of
length along O_x_. The length OU_{1}, which A calls one unit, B
calls β units. B changes his unit length to OU_{1}, from its original
length OV_{1}, and obtains X = _x_. But now, as he must use the
same unit for all his measurements, his other coordinates are altered
in the same ratio. Thus we now have
X = _x_, Y = γ_y_/β, Z = δ_z_/β.
The fundamental divergence is now evident. A and B agree as to their
units along O_x_. They settled that by taking along that axis a
given segment OU_{1} as having the unit length. But they cannot agree
as to what segment along O_y_ is equal to OU_{1}. A says it is
OU_{2}, and B that it is OU_{2}′. Similarly for lengths along OZ.
The result is that A's spheres
_x_^2 + _y_^2 + _z_^2 = _r_^2,
are B's ellipsoids,
X^2 + β^2Y^2/γ^2 + β^2Z^2/δ^2 = _r_^2,
_i. e._ X^2/β^2 + Y^2/γ^2 + Z^2/δ^2 = _r_^2/β^2.
Thus the measurement of angles by the two is hopelessly at variance.
If β ≠ γ ≠ δ, there is one, and only one, set of common rectangular
axes at O, namely that from which they started. If γ = δ, but β ≠ γ,
then there are a singly infinite number of common rectangular axes
found by rotating the axes round O_x_. This is, for us, the
interesting case. The same phenomena are reproduced by transferring to
any parallel axes.
The root of the difficulty is, that A's measuring rod, which for him is
a rigid invariable body, appears to B as changing in length when turned
in different directions. Similarly all measuring rods, satisfactory
to A, violate B's immediate judgment of invariability, and change
according to the same law. There is no way out of the difficulty.
Two rods ρ and σ coincide whenever laid one on the other; ρ is held
still, and both men agree that it does not change. But σ is turned
round. A says it is invariable, B says it changes. To test the matter,
ρ is turned round to measure it, and exactly fits it. But while A is
satisfied, B declares that ρ has changed in exactly the same way as did
σ. Meanwhile B has procured two material rods satisfactory to him as
invariable, and A makes exactly the same objections.
We shall say that A and B employ diverse Euclidean metrical systems.
The most extraordinary fact of human life is that all beings seem to
form their judgments of spatial quantity according to the same metrical
system.
_Relativity in Modern Physics_
Owing to the fact that points of space are incapable of direct
recognition, there is a difficulty--apart from any abstract question
of the nature of space--in deciding on the motion to be ascribed to
any body. Even if there be such a thing as absolute position, it is
impossible in practice to decide directly whether a body's absolute
position has changed. All spatial measurement is relative to matter.
Newton's laws of motion in their modern dress evade this difficulty
by asserting that a framework of axes of coordinates can be defined
by their relations to matter such that, assuming these axes to be at
rest, and all velocities to be measured relatively to them, the laws
hold. The same expedient has to be employed for time, namely, the laws
hold when the measurement of the flow of time is made by the proper
reference to periodic events. Thus the laws assert that the framework
and the natural clock adapted for their use have been successfully
found.
But, if one framework will do, an infinity of others serve equally
well; namely, not only--as is of course the case--all those at rest
relatively to the first framework, but also all those which move
without relative rotation with uniform velocity relatively to the
first. This whole set of frameworks is on a level in respect to
Newton's laws. We will call them Dynamical frameworks.
Now, suppose there are two observers, A and B. They agree in their
non-metrical projective geometry, _e.g._, what A calls a straight
line so does B. They also both apply a Euclidean metrical system of
measurement to this space. Their two metrical systems also agree in
having the same plane at infinity, that is, lines which are parallel
for A are also parallel for B. Furthermore, they have both successfully
applied Newton's laws to the movement of matter, and agree in having
the same sets of dynamical axes. But the framework (among these sets)
which A chooses to regard as at rest is different from the frame (among
the same sets) which B so regards.
Without alteration of their respective judgment of rest, they choose
their co-ordinate axes so that the origins (O for A, and O′ for B) are
in relative motion along OO′, which is the axis of _x_ for both.
Further, since OO′ is the line of symmetry of their diverse Euclidean
systems, we assume that the two measure-systems agree for planes
perpendicular to OO′, _i.e._, we assume a symmetry round OO′.
Then if, for A at O, the distance OO′ be ξ, the relations at any
instant between A's coordinates (_x_, _y_, _z_) and B's
coordinates (_x′_, _y′_, _z′_) for the same point P are
given by
_x′_ = β(_x_ - ξ), _y′_ = _y_, _z′_ = _z_.
Also, according to A's clock, O′ is moving forward with a uniform
velocity _v_, and we measure A's time from the instant of the
coincidence of O and O′.
Thus
ξ = _vt_,
and
_x′_ = β(_x_ - _vt_), _y′_ = _y_, _z′_ = _z_.
We now consider B's clock, and ask for the most general supposition
which is consistent with the fact that their judgments as to the fact
of uniform motion are in agreement.
We do not assume that events in various parts of space which A
considers to be simultaneous are so considered by B. But we assume that
at any point P, with coordinates (_x_, _y_, _z_) for A, there is a
determinate relation between B's time T and _x_, _y_, _z_, _t_.
Put
T = ƒ(_x_, _y_, _z_, _t_).
Write
P = δT/δ_x_, Q = δT/δ_y_, R = δT/δ_z_, S = δT/δ_t_.
Now suppose that the point P is moving, and that (_u_{1},
_u_{2}, _u_{3}) is its set of component velocities along
the axes according to A's "space and clock" system, and (U_{1}, U_{2},
U_{3}) is its set of component velocities according to B's "space and
clock" system. Then by mere differentiation it follows after a short
mathematical deduction that
U_{1} = {(_d_β/_dt_)(_x_ - _vt_) + β(_u_{1} - _v_)}/{P_u_{1} + Q_u_{2} + R_u_{3} + S},
U_{2} = _u_{2}/{P_u_{1} + Q_u_{2} + R_u_{3} + S},
U_{3} = _u_{3}/{P_u_{1} + Q_u_{2} + R_u_{3} + S}.
But we have assumed that, whatever the direction of the resultant
velocity (_u_{1}, _u_{2}, _u_{3}), the velocities (U_{1}, U_{2}, U_{3})
and (_u_{1}, _u_{2}, _u_{3}) are both uniform when either is uniform.
Hence it is easily proved that β, P, Q, R, S are independent of the
coordinates (_x_, _y_, _z_) and of the time _t_. In other words, they
are constant.
Hence we obtain
U_{1} = β(_u_{1} - _v_)/{P_u_{1} + Q_u_{2} + R_u_{3} + S},
and
T = P_x_ + Q_y_ + R_z_ + S_t_.
But we assumed that OO′, _i.e._, O_x_, is an axis of
symmetry. It follows from this assumption that
Q = R = 0.
We thus obtain the simplified results
T = P_x_ + S_t_, }
U_{1} = β(_u_{1} - _v_)/(P_u_{1} + S), } (I)
U_{2} = _u_{2}/(P_u_{1} + S), }
U_{3} = _u_{3}/(P_u_{1} + S). }
Here we remember that (_u_{1}, _u_{2}, _u_{3}) are
the velocities of any particle according to A's "space and clock"
system, and that (U_{1}, U_{2}, U_{3}) are the velocities of the same
point according to B's "space and clock" system. We have obtained the
most general relations consistent with the facts that (1) they both
employ Euclidean systems, related as described above, and (2) they
agree in their judgments on the uniformity of velocity.
We now compare their judgments on the magnitudes of velocities.
Let the magnitude of the velocity of P be V according to A's judgment,
and V′ according to B's’ judgment.
Then
V^2 = _u_{1}^2 + _u_{2}^2 + _u_{3}^2,
V′^2 = U_{1}^2 + U_{2}^2 + U_{3}^2.
Also we can put
_u_{1} = _l_V, _u_{2} = _m_V, _u_{3} = _n_V,
where (_l_, _m_, _n_) have nothing to do with the magnitude V, but
simply depend on the direction of motion. In fact (_l_, _m_, _n_) are
the "direction cosines" of the velocity according to A's judgment. By
substituting in the above equation for V^2 we see that
_l_^2 + _m_^2 + _n_^2 = 1.
Now, substituting for (_u_{1}, _u_{2}, _u_{3}) in
the equations (I) above, and squaring and adding, and eliminating
_m_^2 + _n_^2 by the relation just found, we at once find
V′^2 = ((β^2 - 1)V^2_l_^2 - 2β^2V_vl_ + β^2_v_^2 + V^2)/(PV_l_ + S)^2.
It is thus seen that in general the relation of V′ to V depends on the
direction cosine _l_. Now _l_ is the cosine of the angle
which the direction of the velocity V makes with O_x_, according
to A's judgment.
The meaning of this relation is, that if A discharges, from guns at
the point P, shells with a given muzzle velocity V according to his
judgment, B will consider that their muzzle velocities are different
from each other, except in the case of pairs of guns equally inclined
to the axis OO′. Instances of this type of diversity of judgment can
be noted any day by any one who looks out of the window of a railway
carriage, and forgets that he is travelling.
Now, suppose the velocity V′ bears a relation to the velocity V, which
is independent of _l_. Then _l_ must disappear from the above
formula. There are two conditions to be satisfied
One condition is
V^2 = β^2_v_^2/(β^2 - 1),
or in a more convenient form
β^2 = 1/(1 - _v_^2/V^2).
The meaning of this condition is, that there is one, and only one,
muzzle velocity V (according to A's judgment), namely, the muzzle
velocity given by the above formula, which can have the property that
B will judge that all the guns are firing in their diverse directions
with one common muzzle velocity.
Let us now suppose that V has this peculiar value: that is, if we look
on this value V as known, we must suppose that β is given by the second
of the above formulæ.
The other condition allows P and S to be put in the forms
P = -β_v_/λV^2, S = β/λ,
where
V′ = λV.
Thus we have the bundle of formulæ
β^2 = 1/(1 - _v_^2/V^2),
T = β{_t_ - _vx_/V^2}/λ,
V′ = λV.
The value which we give to λ is purely a matter for the adjustment of
units. If we want A and B to agree in their judgments of the magnitude
of this peculiar muzzle-velocity, we put λ = 1.
We then get the formulæ usually adopted, namely
β^2 = 1/(1 - _v_^2/V^2), }
T = β{_t_ - _vx_/V^2}, } (II)
V′ = V. }
But if we prefer that A and B should reckon (according to A's judgment)
in the same units of time, we put λ = β, and obtain
β^2 = 1/(1 - _v_^2/V^2),
T = (_t_ - _vx_)/V^2,
V′ = βV.
But A and B are in any case in such hopeless difficulties over their
comparisons of time-judgments that the detail of using the same units
does not help them much. Accordingly the formulæ marked (II) are those
used. Thus A and B agree in their judgments as to the magnitude of one
special velocity V, whatever may be the direction in which the entity
possessing it is moving.
In order to reach this measure of agreement, they have to disagree as
to their space judgments and their time judgments. The root cause of
their disagreement is their diverse judgment as to which axis system is
to be taken at rest for the purpose of measuring velocities.
Before discussing the nature of the disagreement disclosed in formula
(II), let us ask why we should bring these difficulties on our heads by
supposing that two people in relative motion, who both (for the purpose
of measuring velocities) assume that they are at rest, should agree in
their judgments in respect to this special velocity V.
Such an agreement has no counterpart in any of our obvious judgments
made from railway carriages. Surely we can wait till the contingency
occurs before discussing the confusion which it creates.
But the contingency has occurred. It occurs when we consider the
velocity of light. Perhaps I may venture to remind a philosophical
society that light moves so very quickly that it is difficult to
consider its velocity at all. So we need not be surprised that this
peculiar fact concerning its velocity is not more obvious.
Now V being the velocity of light, unless _v_ is large, _v_/V
(and still more _v_^2/V^2) will be quite inappreciable. The only
velocity ready to hand which is big enough to give _v_/V an
appreciable value is the velocity of the earth in its orbit.
Many diverse experiments have been made, and they all agree in
concluding that a man who assumes the earth to be at rest will find by
measurement that the velocity of light is the same in all directions.
Furthermore, when the same man turns his attention to interstellar or
interplanetary phenomena, and assumes the sun to be at rest, he will
again find the velocity of light to be the same in all directions.
These are well-attested experiments made at long intervals of time.
This is the exact contingency contemplated above.
Again the velocity of light _in vacuo_ has recently taken on a
new dignity. It used to be one among other wave velocities such as
the velocity of sound in air, or in water, or the velocity of surface
waves in water. But Clerk Maxwell discovered that all electromagnetic
influences are propagated with the velocity of light, and now modern
physical science half suspects that electromagnetic influences are
the only physical influences which relate the changes in the physical
world. Accordingly the velocity of light becomes the fundamental
natural velocity, and experiment shows that our judgment of its
magnitude is not affected by our choice of the framework at rest, so
long as we keep to a set of dynamical axes. These experiments on light
have been confirmed by other electromagnetic experiments not involving
light.
Thus we are driven to equations (II), where V is the velocity of light.
The first conclusion to be drawn from equations (II) is that two people
who make different choices of bodies at rest will disagree as to
their measuring rods in the way described above. There is no peculiar
difficulty about that. The only wonder is that all people agree so well
in their judgments as to metrical systems. A mathematical angel would
naturally expect incarnate men to be in violent disagreement on this
subject.
But the case of time is different. For simplicity of statement we speak
of A as at O, and B as at O′. We remember that O′ is moving relatively
to O with velocity _v_ in direction OO′. Suppose A and B are
looking in this direction; and they both measure their time from the
instant when they met, as O′ passed over O. Then we have
T = ((_t_ - _vx_)/V^2)/((1 - _v_^2)/V^2).
Now, suppose we consider all the events all over space which A
considers to have happened simultaneously at the time _t_. The
events of this set which occurred anywhere on a plane perpendicular to
OO′ at a distance _x_ in front of O (according to A's reckoning),
will have occurred according to B's reckoning at the time T as given
above. Let us fix our attention on the fact that B does not consider
all these events to be simultaneous. For let T_{1}, and T_{2} be B's
times for such events on planes _x__{1}, and _x__{2}. Then
T_{1} - T_{2} = _v_(_x__{2} - _x__{1})/(V^2 - _v_^2).
Thus if _x__{2} be greater than _x__{1}, T_{2} is less than
T_{1}. Thus B judges the more distant events in front of him to have
happened earlier than the nearer events in front of him, and _vice
versa_ for the events behind. This disturbance of the judgment of
simultaneity is the fundamental fact. Obviously the measurement of time
intervals is a detail compared to simultaneity. A may think a sermon
long, and B may think it short, but at least they should both agree
that it stopped when the clock hand pointed at the hour. The worst of
the matter is that so far as any test can be applied there is no method
of discriminating between the validities of their judgments.
Thus we are confronted with two distinct concepts of the common world,
A's space-time concept, and B's space-time concept. Who is right? It is
no use staying for an answer. We must follow the example of the wise
old Roman, and pass on to other things.
Thus estimates of quantity in space and time, and, to some extent, even
estimates of order, depend on the individual observer. But what are
the crude deliverances of sensible experience, apart from that world
of imaginative reconstruction which for each of us has the best claim
to be called our real world? Here the experimental psychologist steps
in. We cannot get away from him. I wish we could, for he is frightfully
difficult to understand. Also, sometimes his knowledge of the
principles of mathematics is rather weak, and I sometimes suspect----
No, I will not say what I sometimes think: probably he, with equal
reason, is thinking the same sort of thing of us.
I will, however, venture to summarise conclusions, which are, I
believe, in harmony with the experimental evidence, both physical and
psychological, and which are certainly suggested by the materials for
that unwritten chapter in mathematical logic which I have already
commended to your notice. The concepts of space and time and of
quantity are capable of analysis into bundles of simpler concepts. In
any given sensible experience it is not necessary, or even usual, that
the whole complete bundle of such concepts apply. For example, the
concept of externality may apply without that of linear order, and the
concept of linear order may apply without that of linear distance.
Again, the abstract mathematical concept of a space-relation
may confuse together distinct concepts which apply to the given
perceptions. For example, linear order in the sense of a linear
projection from the observer is distinct from linear order in the sense
of a row of objects stretching across the line of sight.
Mathematical physics assumes a given world of definitely related
objects, and the various space-time systems are alternative ways of
expressing those relations as concepts in a form which also applies to
the immediate experience of observers.
Yet there must be one way of expressing the relations between objects
in a common external world. Alternative methods can only arise as the
result of alternative standpoints; that is to say, as the result of
leaving something added by the observer sticking (as it were) in the
universe.
But this way of conceiving the world of physical science, as composed
of hypothetical objects, leaves it as a mere fairy tale. What is really
actual are the immediate experiences. The task of deductive science
is to consider the concepts which apply to these data of experience,
and then to consider the concepts relating to these concepts, and
so on to any necessary degree of refinement. As our concepts become
more abstract, their logical relations become more general, and less
liable to exception. By this logical construction we finally arrive
at conceptions, (i) which have determinate exemplifications in the
experience of the individuals, and (ii) whose logical relations have
a peculiar smoothness. For example, conceptions of mathematical time,
of mathematical space, are such smooth conceptions. No one lives in
"an infinite given whole," but in a set of fragmentary experiences.
The problem is to exhibit the concepts of mathematical space and time
as the necessary outcome of these fragments by a process of logical
building up. Similarly for the other physical concepts. This process
builds a common world of conceptions out of fragmentary worlds of
experience. The material pyramids of Egypt are a conception, what is
actual are the fragmentary experiences of the races who have gazed on
them.
So far as science seeks to rid itself of hypothesis, it cannot go
beyond these general logical constructions. For science, as thus
conceived, the divergent time orders considered above present no
difficulty. The different time systems simply register the different
relations of the mathematical construct to those individual
experiences (actual or hypothetical) which could exist as the crude
material from which the construct is elaborated.
But after all it should be possible so to elaborate the mathematical
construct so as to eliminate specific reference to particular
experiences. Whatever be the data of experience, there must be
something which can be said of them as a whole, and that something is
a statement of the general properties of the common world. It is hard
to believe that with proper generalisation time and space will not be
found among such properties.
_Commentary added on reading the Paper before the Aristotelian
Society_
The first six pages of the paper consist of a summary of ideas which
ought to be in our minds while considering problems of time and space.
The ideas are mostly philosophical, and the summary has been made by
an amateur in that science; so there is no reason to ascribe to it any
importance except that of a modest reminder. There are only two points
in this summary to which I would draw attention.
On pp. 192 and 193 there occurs--
"Wherever mankind ... unending uniform succession."
If I understand Kant rightly--which I admit to be very
problematical--he holds that in the act of experience we are aware
of space and time as ingredients necessary for the occurrence of
experience. I would suggest--rather timidly--that this doctrine should
be given a different twist, which in fact turns it in the opposite
direction--namely, that in the act of experience we perceive a whole
formed of related differentiated parts. The relations between these
parts possess certain characteristics, and time and space are the
expressions of some of the characteristics of these relations. Then the
generality and uniformity which are ascribed to time and space express
what may be termed the uniformity of the texture of experience.
The success of mankind--modest though it is--in deducing uniform laws
of nature is, so far as it goes, a testimony that this uniformity of
texture goes beyond those characteristics of the data of experience
which are expressed as time and space. Time and space are necessary
to experience in the sense that they are characteristics of our
experience; and, of course, no one can have our experience without
running into them. I cannot see that Kant's deduction amounts to much
more than saying that "what is, is"--true enough, but not very helpful.
But I admit that what I have termed the "uniformity of the texture of
experience" is a most curious and arresting fact. I am quite ready to
believe that it is a mere illusion; and later on in the paper I suggest
that this uniformity does not belong to the immediate relations of the
crude data of experience, but is the result of substituting for them
more refined logical entities, such as relations between relations, or
classes of relations, or classes of classes of relations. By this means
it can be demonstrated--I think--that the uniformity which must be
ascribed to experience is of a much more abstract attenuated character
than is usually allowed. This process of lifting the uniform time and
space of the physical world into the status of logical abstractions has
also the advantage of recognising another fact, namely, the extremely
fragmentary nature of all direct individual experience.
My point in this respect is that fragmentary individual experiences
are all that we know, and that all speculation must start from these
_disjecta membra_ as its sole datum. It is not true that we are
directly aware of a smooth running world, which in our speculations we
are to conceive as given. In my view the creation of the world is the
first unconscious act of speculative thought; and the first task of a
self-conscious philosophy is to explain how it has been done.
There are roughly two rival explanations. One is to assert the world
as a postulate. The other way is to obtain it as a deduction, not a
deduction through a chain of reasoning, but a deduction through a chain
of definitions which, in fact, lifts thought on to a more abstract
level in which the logical ideas are more complex, and their relations
are more universal. In this way the broken limited experiences sustain
that connected infinite world in which in our thoughts we live. There
are three more remarks while on this point I wish to make--
(i) The fact that immediate experience is capable of this deductive
superstructive must mean that it itself has a certain uniformity of
texture. So this great fact still remains.
(ii) I do not wish to deny the world as a postulate. Speaking without
prejudice, I do not see how in our present elementary state of
philosophical advance we can get on without middle axioms, which, in
fact, we habitually assume.
My position is, that by careful scrutiny we should extrude such
postulates from every part of our organised knowledge in which it is
possible to do without them.
Now, physical science organises our knowledge of the relations between
the deliverances of our various senses. I hold that in this department
of knowledge such postulates, though not entirely to be extruded, can
be reduced to a minimum in the way which I have described.
I have not the slightest knowledge of theories respecting our emotions,
affections, and moral sentiments, and I can well believe that in
dealing with them further postulates are required. And in practice I
recognise that we all make such postulates, uncritically.
(iii) The next paragraphs on pp. 193 and 194 are as follows--
"The opposed philosophical method ... physical science."
It will be noted that, in the light of what has just been stated, the
first of these paragraphs (which, I hope, faithfully expresses the
experimental way of approaching the problem) really obscures the point
which I have been endeavouring to make. The phrase, "If we form the
exact concepts of points, etc.," is fatally ambiguous as between the
method of postulating entities with assigned relations, and the method
of forming logical constructions, and thus reaching points, etc., as
the result of a chain of definitions.
Turning now to pp. 194-195, we come to the following paragraphs--
"The other theory ... eighteenth century."
We note again that the relational theory of space from another point
of view brings us back to the idea of the fundamental space-entities
as being logical constructs from the relations between things. The
difference is, that this paragraph is written from a more developed
point of view, as it implicitly assumes the things in space, and
conceives space as an expression of certain of their relations.
Combining this paragraph with what has gone before, we see that the
suggested procedure is first to define "things" in terms of the data of
experience, and then to define space in terms of the relations between
things.
This procedure is explicitly assumed in the next short paragraph: "In
this connection ... from the events."
The gist of the remaining paragraphs of this section is contained in
the paragraph at the bottom of p. 196: "The point in time ... new
importance."
The sentence, "We live in durations, and not in points," can be
amplified by the addition, "We live in space-extensions and not in
space-points."
It must be noted that "whole and part" as applied to extensions in
space or time must be different from the "all and some" of logic,
unless we admit points to be the fundamental entities. For "spatial
whole and spatial part" can only mean "all and some" if they really
mean "all the points and some of the points." But if extensions and
their relations are more fundamental than points, this interpretation
is precluded. I suggest that "spatial whole and spatial part" is
intimately connected with the fundamental relation between things from
which our space ideas spring.
The relation of space whole to space part has many formal properties
which are identical with the properties of "all and some." Also when
points have been defined, we can replace it by the conception of "all
the points and some of the points." But the confusion between the two
relations is fatal to sound views on the subject.
_Diverse Euclidean Measure Systems_
The next section deals with the measure systems applicable to space.
A measure system is a group of congruent transformations of space
into itself. Consider a rigid body occupying all space. Let this body
be moved in any way so that the particles of the body which occupied
points P_{1}, P_{2}, P_{3}, etc., now occupy points Q_{1}, Q_{2},
Q_{3}, etc. Then any point P_{1} in space is uniquely related to the
corresponding point Q_{1} in space by a one-to-one transformation with
certain characteristics. By the aid of these transformations we can
achieve the definition of distance in a way which definitely determines
the distance between any two points, provided that we can define what
we mean by a congruent transformation without introducing the idea of
distance. If we introduced the idea of distance, we should simply say
that a congruent transformation is one which leaves all distances
unchanged, _i. e._, if P_{1}, P_{2} are transformed into Q_{1},
Q_{2} then the distance P_{1}P_{2} is equal to the distance Q_{1}Q_{2}.
But mathematicians have succeeded in defining congruent transformations
without any reference to distance.
There are alternative groups of such congruent transformations, and
each group gives a different measure system for space. The distance
P_{1}P_{2} may equal the distance Q_{1}Q_{2} for one measure system,
and will not equal it for another measure system. All these different
measure systems are on the same level, equally applicable. A being
with a strong enough head could think of them all at once as applying
to space. The result so far as it interests us in respect to the
theory of relativity is explained on pp. 197-200, ending with "The
most extraordinary fact ... same metrical system." This final sentence
bears on Poincaré's assertion that the measure system adopted is
purely "conventional." I presume that by "conventional" a certain
arbitrariness of choice is meant; and in that case, I must express
entire dissent. It is true that within the circle of geometrical ideas
there is no means of giving any preference to any one measure system,
and any one is as good as any other. But it is not true that if we
look at a normal carriage wheel, and at an oval curve one foot broad
and ten feet long, we experience any arbitrariness of judgment in
deciding which has approximately the form of a circle. Accordingly to
Poincaré the choice between them, as representing a circle, is entirely
conventional.
Again, we equally form immediate judgments as to whether a body is
approximately rigid. We know that a paving stone is rigid, and that a
concertina is not rigid. This again necessitates a determinate measure
system, selected from among the others.
Accordingly we conclude that (i) each being does, in fact, employ a
determinate measure system, which remains the same, except possibly
for very small variations, and (ii) the measure systems of different
human beings agree, to within the limits of our observations. These
conclusions are not the less extraordinary because no plain man has
ever doubted them.
It is an interesting subject to investigate exactly what are the
fundamental uniformities of experience which necessitate this
conclusion. It is not so easy as it looks, since we have to divest
ourselves of all aid of scientific hypothesis if our conclusions are to
be demonstrative.
_Relativity in Modern Physics_
Pp. 201-202, "Owing to the fact ... which B so regards."
The fundamental formulæ for the theory of relativity are the relations
between diverse co-ordinate systems given on p. 203, and formulæ II
at the bottom of p. 207. The general explanation of one method in
which these formulæ arise--namely, Einstein's method--is given on
pp. 201-211. Namely, we seek the condition that for all dynamical
axes the velocity of light should be the same, and the same in all
directions. It should be noted that the experiments which, so far as
they go, confirm these formulæ, can also be explained in another way
which makes the theory of relativity unnecessary. We need only ascribe
to the ether a certain property of contraction in the direction of
motion, and the thing is done. So no one need be bludgeoned into
accepting the rather bizarre doctrine of relativity, nor indeed any
other scientific generalisation. The good old homely ether, which we
all know, can in this case serve the purpose. Just as an author of
genius, if he lives long enough, survives the inevitable accusation
of immorality, so the ether by dint of persistence has outlived all
reputation of extravagance. But if we detach ourselves from the glib
phraseology concerning it, the scientific ether is uncommonly like the
primitive explanation of the soul, as a little man inside us, which can
sometimes be caught escaping in the form of a butterfly. As soon as the
ether has to be patched up with special properties to explain special
experiments, its scientific use is problematical, and its philosophic
use is _nil_.
Philosophically the ether seems to me to be an ambitious attempt
to give a complete explanation of the physical universe by making
an elephant stand on a tortoise. Scientifically it has a perfectly
adequate use by veiling the extremely abstract character of scientific
generalisations under a myth, which enables our imaginations to work
more freely. I am not advocating the extrusion of ether from our
scientific phraseology, even though at special points we have to
abandon it.
But the key to the reasons why it is worth while to consider seriously
the doctrine of relativity is to be found on pp. 209, 210: "Again the
velocity of light ... not involving light." Namely, we have begun to
suspect that all physical influences require time for their propagation
in space. This generalisation is a long way from being proved.
Gravitation stands like a lion in the path. But if it be the case, then
all idea of an immediate presentation to us of an aspect of the world
as it in fact is, must necessarily be abandoned. What we perceive at
any instant is already ancient history, with the dates of the various
parts hopelessly mixed.
We must add to this the difficulty of determining what is at rest
and what is in motion, and the further difficulty of determining a
definite uniform flow of time. It is no use discussing this matter
as though, but for the silly extravagant doctrine of relativity,
everything would be plain sailing. It isn't. You may be quite sure
that when, after prolonged study, you endeavour to give the simplest
explanation of a grave difficulty, you will be accused of extravagance.
I have no responsibility for the doctrine of relativity, and hold no
brief for it, but it has some claim to be considered as a comparatively
simple way out of a scientific maze.
In the first place, we use the Newtonian dynamical sets of axes,
and the Newtonian clock to extricate ourselves partially from the
difficulties of rest, motion, and time. These have proved capable
of scientific determination within the limits of our experimental
accuracy. Thus the only thing left over is the choice of the axes
at rest, which is a completely indeterminate problem on Newtonian
principles.
Again, so far as we can at present guess by adopting the theory
that all metrical influence is electromagnetic, all influences
are propagated with the velocity of light _in vacuo_. This
electromagnetic hypothesis is by no means established, but it gives
the simplest of all possible results in respect to the propagation of
influence, which we therefore adopt.
But what dynamical axes are we taking as at rest? Now our practical
choice gives a range of relative velocities small compared to that of
light. So except for certain refined experiments it does not matter.
There are two possibilities--
(i) We may assume that one set of axes are at rest, and that the others
will show traces of motion in respect to the velocity of light; or--
(ii) That the velocity of light is the same in all directions whichever
be the dynamical axes assumed.
The first supposition is negatived by experiment, and hence we are
driven to the second supposition; which immediately lands us in the
whole theory of relativity.
But if we will not have this theory we must reject the earlier
supposition that the velocity of light _in vacuo_ is the same in
all directions. This we do, in fact, by assuming an ether, and assuming
a certain law for its modification. Then we, in fact, adopt the first
supposition so far as to hold that there are dynamical axes specially
at rest, namely, at rest relatively to the undisturbed ether. Then an
assumed law for the modification of the ether so alters the velocity of
light that we explain why no dynamical axes show traces of motion.
I wish now to go back to the point which I made a few minutes ago,
that what we perceive at any instant is ancient history with its dates
hopelessly mixed. In the earlier part of my comments I emphasised the
point that our only data as to the physical world are our sensible
perceptions. We must not slip into the fallacy of assuming that we are
comparing a given world with given perceptions of it. The physical
world is in some general sense of the term a deduced concept.
Our problem is, in fact, to fit the world to our perceptions, and not
our perceptions to the world.
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