An essay on the foundations of geometryRussell, Bertrand
Philosophy
An essay on the foundations of geometry
Russell, Bertrand
Geometry -- Foundations
=206.= But whence arises, on this view, the paradox that we cannot
but regard space as having more or less thinghood, and as divisible
_ad infinitum_? This must be explained, I think, as a psychological
illusion, unavoidably arising from the fact that spatial relations
are immediately presented. They thus have a peculiar psychical
quality, as immediate experiences, by which quality they can be
distinguished from time-relations or any other order in which things
may be arranged. To Stumpf, whose problem is psychological, such a
psychical quality would constitute an absolute underlying content,
and would fully justify his thesis; to us, however, whose problem is
epistemological, it would not do so, but would leave the _meaning_ of
the spatial element in sense-perception free from any implication
of an absolute or empty space[201]. May we not, then, abandon empty
space, and say: Spatial order consists of _felt_ relations, and _quâ_
felt has, for Psychology, an existence not wholly resolvable into
relations, and unavoidably _seeming_ to be more than mere relations.
But when we examine the information, as to space, which we derive
from sense-perception, we find ourselves plunged in contradictions,
as soon as we allow this information to consist of more than
relations. This leaves spatial order alone in the field, and reduces
empty space to a mere name for the logical possibility of spatial
relations.
=207.= The apparent divisibility of the relations which constitute
spatial order, then, may be explained in two ways, though these are
at bottom equivalent. We may take the relation as considered in
connection with empty space, in which case it becomes more than a
relation; but being falsely hypostatized, it appears as a complex
thing, necessarily composed of elements, which elements, however,
nowhere emerge until we analyze the pseudo-thing down to nothing,
and arrive at the point. In this sense, the divisibility of spatial
relations is an unavoidable illusion. Or again, we may take the
relation in connection with the material atoms it relates. In this
case, other atoms may be imagined, differently localized by different
spatial relations. If they are localized on the straight line joining
two of the original atoms, this straight line appears as divided by
them. But the original relation is not really divided: all that has
happened is, that two or more equivalent relations have replaced
it, as two compounded relations of father and son may replace the
equivalent relation of grandfather and grandson. These two ways of
viewing the apparent divisibility are equivalent: for empty space, in
so far as it is not illusion, is a name for the aggregate of possible
space-relations. To regard a figure in empty space as divided,
therefore, means, if it means anything, to regard two or more other
possible relations as substituted for it, which gives the second way
of viewing the question.
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