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

THE MATHEMATICS OF PSYCHOLOGY 887
"continua" of the type of the mathematical "continuum " of the first order. As already pointed out, the
great replacement or substitution has been logically
coerced by the mentioned contradiction that inheres in
the " continua " of sense. The aim, which has been conscious or unconscious, and the issue have been emancipation, intellectual harmony, increase of freedom from
logical discord: we know that, if qi, q2, q3 be quantities of
a conceptual magnitude or " continuum " of the type in
question, and if qi =q2, and q2 =q3, then ql =q3 without
exception.
The Need of Further Emancipation.-But the replacement mentioned is not enough. For starting with the
kind of conceptual magnitudes in question, we are destined
to encounter contradictions or discords of another sort.
This fact may be shown as follows: Consider a sensible
line. It is for sensibility a thing like a rope or a cable
or a chalk mark. In it inheres the old contradiction (4).
Now suppose it replaced by a corresponding conceptual
magnitude of the type of the first order " continuum."
The new thing can be made as thin and as narrow as the
difference between any two rational numbers. But this
difference can be made to approach as near as we please
to zero. Taking the limit, we have a line having only
length, the thickness and the width being zero. Such
a line is, we say, geometric; it is not sensible, but is purely
conceptual. Let us now take two such sensible lines
having a (sensible) bulk or piece in common, as indicated
in Fig. 31, and let us suppose them submitted to the conceptualizing process above indicated. The result, we say,
is two geometric lines and a geometric point common to
them, the point being, we say, the limit of the bulk that
the lines have in common as sensible lines and also as