The fourth dimension

86 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 fandamental 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, 7.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.