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

THE MATHEMATICS OF PSYCHOLOGY $97
second by conceptual continua of the type of the mathematical continuum of second order. Of course, I do not
mean to imply that the first replacement was completed
before the second began. I mean merely that the order
indicated is, roughly speaking, correct. Neither do I
intend to imply that the process of substitution has been
always a conscious one nor that it has been always accompanied by a realizing sense of its actuating motive.
Much of our intellectual life is not attended by consciousness either that it is going on or why it proceeds in
this direction rather than that. That the replacements
have been actually made, however, is sufficiently evident
in the fact that students of natural science,-physicists,
for example, or astronomers or chemists,-habitually and
freely employ the real numbers, whether rational or
irrational, algebraic or transcendental, to express quantities or amounts of the various kinds of physical magnitude. Are these students aware that they are thus dealing with purely conceptual continua and not with such
as are revealed in sense? It must be said that, for the
most part, they are not. For the most part these students
are not indeed aware that there are such continua even
in mathematics; much less are they informed regarding
the inner structure or constitution of them; they employ
them naïvely, as children may handle tools which they
have not yet analyzed. This is not said in any spirit of
reproach or derogation, for it is only in very recent times
that even mathematicians themselves have made the constitution of continua a subject of deliberate investigation, though the matter figured itself vaguely in the background of their thought for more than two thousand
years. Even Aristotle in his Physics made a stab at the
question.