72 THE FOURTH DIMENSION. 



III. 



THE INTRODUCTION OF THE NOTION OF FOUR-DIMENSIONAL 

 POINT-AGGREGATES, PERMISSIBLE. 



In the preceding section it was shown that we can conceive not 

 only of manifoldnesses of one, two, and three dimensions, but also 

 of manifoldnesses of any number of dimensions. But it was at the 

 same time indicated that our world-space, that is, the totality of all 

 conceivable points that differ only in respect of position, cannot in 

 agreement with our notions of things possess more than three di- 

 mensions/ But the question now arises, whether, if the progress of 

 science tends in such a direction, it is permissible to extend the no 

 tion of space by the introduction of point-aggregates of more than 

 three dimensions, and to engage in the study of the properties of 

 such creations, although we know that notwithstanding the fact 

 that we may conceptually establish and explore such aggregates of 

 points, yet we cannot picture to ourselves these creations as we do 

 the spatial magnitudes which surround us, that is, the regular three- 

 dimensional aggregates of points. 



To show the reader clearly that this question must be answered 

 in the affirmative, that the extension of our notion of space is per- 

 missible, although it leads to things which we cannot perceive by 

 our senses, I may call the reader's attention to the fact that in arith- 

 metic we are accustomed from our youth upwards to extensions of 

 ideas, which, accurately viewed, as little admit of graphic concep- 

 tion as a four-dimensional space, that is, a point-aggregate of four 

 dimensions. By his senses man first reaches only the idea of whole 

 numbers the results of counting. The observation of primitive 

 peoples* and of children clearly proves that the essential decisive 

 factors of counting are these three : First, we abstract, in the count- 

 ing of things, completely from the individual and characteristic at- 

 tributes of these things, that is, we consider them as homogeneous. 

 Second, we associate individually with the things which we count 



* See the essay Notion and Definition of Number in this collection. 



