re abstract
mechanical characteristic of direction of movement;--although the
persistency with which intuitionally minded geometers have sought to
adapt such illustrations to their needs has somewhat obscured this fact.
However, even a cursory examination of a modern treatise on geometry
makes clear what has taken place. For example, Professor Hilbert begins
his _Grundlagen der Geometrie_, not with definition of points, lines,
and planes, but with the assumption of three different systems of things
(Dinge) of which the first, called points, are denoted A, B, C, etc.,
second, called straight lines (Gerade), are denoted a, b, c, etc., and
the third, called planes, are denoted by [Greek: alpha], [Greek: beta],
[Greek: gamma], etc. The relations between these things then receive
"genaue und vollstaendige Beschreibung" through the axioms of the
geometry. And the fact that these "things" are called points, lines, and
planes is not to give to them any of the connotations ordinarily
associated with these words further than are determined by the axiom
groups that follow. Indeed, other geometers are even more explicit on
this point. Thus for Peano (_I Principii di Geometria_, 1889) the line
is a mere class of entities, the relations amongst which are no longer
concrete relations but types of relations. The plane is a class of
classes of entities, etc. And an almost unlimited number of examples,
about which the theorems of the geometry will express truths, can be
exhibited, not one of which has any close resemblance to spatial facts
in the ordinary sense.

Philosophers, it seems to me, have been slow to recognize the
significance of the step involved in this last phase of mathematical
thought. We have been so schooled in an arbitrary distinction between
relations and concepts, that while long familiar with general ideas of
concepts, we are not familiar with generalized ideas of relations. Yet
this is exactly what mathematics is everywhere presenting. A transition
has been made from relations to types of relations, so that instead of
speaking in terms of quantitative, spatial and temporal relations,
mathematicians can now talk in terms of symmetrical, asymmetrical,
transitive, intransitive relational types and the like. These present,
however, nothing but the empirical character that is common to such
relations as that of father and son; debtor and creditor; master and
servant; a is to the left of b, b of c; c of d; a is older than