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

328 MATHEMATICAL PHILOSOPHY
regarded as a plenum or space of point-pairs, a line has
two, a plane four, and ordinary space six, dimensions.
You see that the dimensionality of a space depends, not
only upon the space itself, but also upon the entity
employed as element. You see easily that, in respect to
point-triads, the dimensionality of a line is three, that of
a plane is six, and that of ordinary space is nine; and
you see that, if we take for element the point-set containing n points, then a line is an n-dimensional space, a plane
has 2n dimensions, and the dimensionality of ordinary
space is 3n. There is no sense in simply saying that a
plane, for example, or that ordinary space has such-andsuch a dimensionality (or number of dimensions); what
we have to say is that it has such-and-such a dimensionality when it is conceived as a space or plenum of
elements of such-and-such a kind. When people say
simply, as they often do, that space (meaning ordinary
space) has three dimensions, they mean-though they
do not know well what they mean-that it has three
dimensions as a space of points. If you think they know
what they mean, ask them what they mean. For additional examples showing that space dimensionality depends upon space element, consider the following. In
a previous lecture we saw that in a plane a line has two
coordinates, two degrees of freedom-a plane being precisely as rich in lines as in points; and so a plane is a twodimensional space of lines, as it is, we have seen, of points.
What is the line dimensionality of ordinary space? It is
easily seen to be four. To see it, reflect that a line is
determined by two points, say a point in the plane of the
floor of this room and a point in the plane of the ceiling;
each of the points (kept in its plane) has two coordinates,
two degrees of freedom, and so, you see, the line has four.