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

106 MATHEMATICAL PHILOSOPHY
geometry of the Euclidean plane has its birth in a certain
transformation-a point-to-dyad transformation-which
consists in the fact that a one-to-one correspondence subsists between the points of the plane and the number dyads
(x, y) of N. By virtue of this transformation, to any
given relation among points in doctrine Di there corresponds a definite relation among dyads in doctrine D4;
and conversely, for the correspondence runs both ways.
Do not fail to note now very carefully, for this is the crux
of the matter, that, owing to the mentioned correspondence, we can translate a problem respecting points into a
problem respecting dyads, then solve the latter (algebraically) and finally translate the result in terms of points,
thus getting a proposition in Di; and, conversely, we can
translate a problem respecting dyads into a problem
respecting points, then solve the latter (geometrically)
and finally translate the result in terms of dyads, thus
getting a proposition in D4: in other words, we can investigate algebraically the point relations making up D1
and, conversely, we can investigate geometrically the
dyad relations making up D4. It is now obvious that,
instead of calling both of these converse procedures
analytic, or algebraic, geometry, the former ought to be
called analytic, or algebraic, geometry; and the latter
geometric analysis or geometric algebra. Observe that
neither of them yields a new doctrine; each of them is
simply a new method of establishing an old doctrine;
and the fundamental distinction between the two doctrines, D1 and D4, remains in undisturbed serenity.
You perceive at once that the foregoing discussion
applies, mutatis mutandis, to D1' and D4'.
The Possibility of Yet Other Interpretations of HAF and
HaF'.-To each of these doctrinal functions have now