Hinton - The Fourth Dimension.pdf

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THE FOURTH DIMENSION
of drawing conclusions and the use of higher space
figures.*
The other instance is chosen on account of the bearing
it has on our fundamental conceptions. In it I try to
discover the real meaning of Kant’s theory of experience.
The investigation of the properties of numbers is much
facilitated by the fact that relations between numbers are
themselves able to be represented as numbers—e.g. 12,
and 3 are both numbers, and the relation between them
is 4, another number. The way is thus opened for a
process of constructive theory, without there being any
necessity for a recourse to another class of concepts
besides that which is given in the phenomena to be
studied.
The discipline of number thus created is of great and
varied applicability, but it is not solely as quantitative
that we learn to understand the phenomena of nature.
It is not possible to explain the properties of matter
by number simply, but all the activities of matter are
energies in space. They are numerically definite and also,
we may say, directedly definite, i.e. definite in direction.
Is there, then, a body of doctrine about space which,
like that of number, is available in science? It is needless
to answer: Yes; geometry. But there is a method
lying alongside the ordinary methods of geometry, which
tacitly used and presenting an analogy to the method
of numerical thought deserves to be brought into greater
prominence than it usually occupies.
The relation of numbers is a number.
Can we say in the same way that the relation of
shapes is a shape?
We can.
* It is suggestive also in another respect, because it shows very
clearly that in our processes of thought there are in play faculties other
than logical; in it the origin of the idea which proves to be justified is
drawn from the consideration of symmetry, a branch of the beautiful.