Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

330 MATHEMATICAL PHILOSOPHY
sometimes felt that no student is intellectually fit to be
graduated from college who does not easily and habitually
recognize the immense and fundamental difference between
those domains. Such a student is as meagrely disciplined
as one who believes that, if two things or persons be each
of them indispensable, they are therefore of equal importance-a rank fallacy vitiating o9 per cent of current social
philosophy throughout the world. I once heard a railway
section-hand -argue that, because his work was indispensable, he was just as important as the railway president.
Have you observed that among the hyperspaces which
we have so far taken occasion to notice there is no hyperspace of points? The examples have been hyperspaces of
point-pairs or of point-triads,.. or of n-sets (of points),.. or of lines or of circles or of line-pencils or of spheres,
and you are now doubtless prepared to extend the list of
such examples indefinitely, for " the clue, grown familiar
to the hand, lengthens as we go and never breaks."
But it is not this kind of hyperspace that mystifies the
layman. What he desires to have you make clear to him,
-though he may not be able to say so very clearly,-is the
conception of a hyperspace of points. When he asks you
to " explain the fourth dimension," he is really asking
you to explain the idea of a space that is 4-dimensional
in points in the sense in which ordinary space is 3-dimensional in points, a plane 2-dimensional and a line I-dimensional. And so we see that our further task is thus
defined. I have said that hyperspace has three intimately
related meanings. One of them has been explained.
The other two attach to the term, hyperspace of points,
or-wbhat is tantamount-point-space of four or more
dimensions. And we are now to see what the meanings
are. They are not hard to see if we but look attentively.