An essay on the foundations of geometryRussell, Bertrand
Philosophy
An essay on the foundations of geometry
Russell, Bertrand
Geometry -- Foundations
=133.= The above argument, it may be urged, has overlooked a
possibility. It has used a transcendental argument, so an opponent
may contend, without sufficiently proving that knowledge about
externality must be possible without reference to the matters
external to each other. The definition of a position may be
impossible, so long as we neglect the matter which fills the form,
but may become possible when this matter is taken into account.
Such an objection can, I think, be successfully met, by a reference
to the passivity and homogeneity of our form. For any dependence
of the definition of a position on the particular matter filling
that position, would involve some kind of interaction between the
matter and its position, some effect of the diverse content on
the homogeneous form. But since the form is totally destitute of
thinghood, perfectly impassive, and perfectly void of differences
between its parts, any such effect is inconceivable. An effect on
a position would have to alter it in some way, but how could it be
altered? It has no qualities except those which make it the position
it is, as opposed to other positions; it cannot change, therefore,
without becoming a different position. But such a change contradicts
the law of identity. Hence it is not the position which has changed,
but the content which has moved in the form. Thus it must be
possible, if knowledge of our form can be obtained at all, to obtain
this knowledge in logical independence of the particular matter which
fills it. The above argument, therefore, granted the possibility of
knowledge in the department in question, shows the necessity of a
finite integral number of dimensions.
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