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

386 MATHEMATICAL PHILOSOPHY
order by this means to rescue the law of non-contradiction
from the violence of the relations (4), it would be necessary
to suppose our discriminative sensibility to be increased
till the difference between quantities (of magnitude)
corresponding to any two rational fractions, however slight
their difference, would be sensibly discernible, that is,
sensation would have to be a continuous function of
stimulus where continuity signified continuity defined in
the domain of rational numbers. The definition would
be the same as that above recalled except that the function
and its argument would be restricted to rational values.
No such endless refining of sensibility has occurred
nor is it possible, but we know that in the long course of
time the various kinds of sensible magnitude,-the magnitudes revealed by or in sensation,-have gradually been
submitted to a conceptualizing process that is, in a very
important respect, virtually equivalent to the refining
process in question. The various kinds of sensible magnitude,-the various sense " continua " presented respectively in various sense departments of sound, color, weight,
taste, warmth, duration, hardness, spatial extent, velocity,
acceleration, etc.,-have all of them, in the course of the
centuries, got replaced in our thinking with corresponding
insensible or conceptual magnitudes having the structure
of the system of rational numbers. The process has been
partly conscious and partly unconscious. This system,
you know, is what has been sometimes called the mathematical continuum of first order, as by the late Professor
Henri Poincaré, for example, in Science and Hypothesis.
So we may say that one of the great achievements of the
human intellect in its dealing with the magnitudes
revealed in sensation has been the substitution, for such
sense" continua," of conceptual magnitudes or conceptual