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

ESSENTIAL DISCRIMINATIONS 107
been given three geometric interpretations but only one
non-geometric one, and the latter is algebraic. It is
natural to ask: are there other algebraic interpretations?
The answer is, there are. I shall not tarry to present
them, for we have many other things to consider, but we
may pause a moment to convince ourselves of their
existence. Let us recall our third interpretation of
HAF, for example, giving rise to the geometric doctrine
D3. It is plain that in it we may replace point by dyad
and pathocircle by an equation determining a perfectly
corresponding system of dyads, and thus obtain a new
algebraic interpretation of HaF and therewith a new twodimensional theory of dyads and dyad systems. And
so on-an algebraic interpretation for each geometric one
and conversely.
How many geometric and how many algebraic interpretations of HaF or of HàF' are possible? Is the number
finite or infinite? I will state-without giving the proof
-hat each of the two functions admits of an infinitude
of interpretations of either sort. And I may add,-again
omitting the proof, which is easy,-that from any given
interpretation, whether geometric or algebraic, one can
derive an endless series of different interpretations,
correspondingly geometric or algebraic, drawing them,
each out of its predecessor, unceasingly as the successive
joints of an infinitely-many-jointed telescope. Most of
the interpretations thus obtainable and the corresponding
doctrines are devoid of interest for us human beings, but
that statement is a commentary upon our supersimian
curiosity and not upon the intrinsic merits of the doctrines.
Do HAF and HAF' admit of interpretations that are
both non-geometric and non-algebraic? Yes: each of the
functions admits of an infinite variety of such interpreta