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has always been a favourite game to prove your axioms absurd.
You will all naturally be very annoyed with me for indulging in
these fatuous pastimes, especially as I started out with a pledge
that I would deal with these subjcts from the hard-headed scientific
point of view. Forgive me if I have toyed with these shining gos-
samers of the thought-web! I have only been trying to break it to
you gently. I proceed to brush away with a sweep of my lily-white
hand all this tenuous, filmy stuff, 'such stuff aØs dreams are made
of.' We will get down to modern science.
24. For general reading there is no better introduction than
'The Bases of Modern Science', by my old and valued friend the late
J. W. N. Sullivan. I do not want to detain you too long with quota-
tions from this admirable book. I would much rather you got it and
read it yourself; you could hardly make better use of your time. But
let us spend a few moments on his remarks about the question of
geometry.
Our conceptions of space as a subjective entity has been com-
pletely upset by the discovery that the equations of Newton based on
Euclidean Geometry are inadequate to explain the phenomena of gravi-
tation. It is instinctive to us to think of a straight line; it is
somehow axiomatic. But we learn that this does not exist in the
objective universe. We have to use another geometry, Riemann's
Geometry, which is one of the curved geometries. (There are, of
course, as many systems of geometry as there are absurd axioms to
build them on. ThÁree lines make one ellipse: any nonsense you like:
you can proceed to construct a geometry which is correct so long as
it is coherent. And there is nothing right or wrong about the
result: the only question is: which is the most convenient system
for the purpose of describing phenomena? We found the idea of
Gravitation awkward: we went to Riemann.)
This means that the phenomena are not taking place against a
background of a flat surface; the surface itself is curved. What we
have thought of as a straight line does not exist at all. And this
is almost impossible to conceive; at least it is quite impossible for
myself to visualise. The nearest one gets to it is by trying to
imagine that you are a reflection on a polished door-knob.
25. I feel almost ashamed of the world that I have to tell you
that in the year 1900, four years before the appearance of Einstein's
world-shaking paper, I described space as 'finite yet boundless,'
which is exactly the description in general terms that he gave in
more mathematical detail.(*) You will see at once that these three
words do describe a curved geometry; a sphere, for instance, is a
finite object, yet you can go over the surface in any direction
without ever coming to an end.
I said above that Riemann's Geometry was not quite sufficient to
explain the phenomena of nature. We have to postulate different
kinds of curvature in different parts of the continuum. And even
then we are not happy!
26. Now for a spot of Sullivan! 'The geometry is so general
that it admits of different degrees of curvature in different parts
of space-time. It is to this curvature that gravitational effects
are due. The curvature of space-time is most prominent, therefore,
around large masses, for here the gravitational effects are most
marked. If we take matter as fundamental, we may say that it is the
presence of matter that causes the curvature of space-time. But
there is a different school of thought that regards matter as due to
the curvatÊure of space-time. That is, we assume as fundamental a
space-time continuum manifest to our senses as what we call matter.
Both points of view have strong arguments to recommend them. But,
whether or not matter may be derived from the geometrical peculiari-
ties of the space-time continuum, we may take it as an established
scientific fact that gravitation has been so derived. This is
obviously a very great achievement, but it leaves quite untouched
another great class of phenomena, namely, electro-magnetic phenomena.
In this space-time continuum of Einstein's the electro-magnetic
forces appear as entirely alien. Gravitation has been absorbed, as
it were, into Riemannian geometry, and the notion of force, so far as
gravitational phenomena are concerned, has been abolished. But the
electro-magnetic forces still flourish undisturbed. There is no hint
that they are manifestations of the geometrical peculiarities of the
space-time continuum. And it can be shown to be impossible tÁo relate
them to anything in Riemann's Geometry. Gravitation can be shown to
correspond to certain geometrical peculiarities of a Riemannian
space-time. But the electro-magnetic forces lie completely outside
this scheme.'
27. Here is the great quag into which mathematical physics has
led its addicts. Here we have two classes of phenomena, all part of
a unity of physics. Yet the equations which describe and explain the
one class are incompatible with those of the other class! This is
not a question of philosophy at all, but a question of fact. It does
not do to consider that the universe is composed of particles. Such
a hypothesis underlies one class of phenomena, but it is nonsense
when applied to the electro-magnetic equations, which insist upon our
abandoning the idea of particles for that of waves.
Here is another Welsh rabbit for supper!
'Einstein's finite universe is such that its radius is dependent
upon the amount of matter in it. Were more matter to be created, the
voÙlume of the universe would increase. Were matter to be annihilat-
ed, the volume of space would decrease. Without matter, space would
not exist. Thus the mere existence of space, besides its metrical
properties, depends upon the existence of matter. With this concep-
tion it becomes possible to regard all motion, including rotation, as
purely relative.'
Where do we go from here, boys?
28. 'The present tendency of physics is towards describing the
universe in terms of mathematical relations between unimaginable
entities.'
We have got a long way from Lord Kelvin's too-often and too-
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