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

TRANSFORMATION 11t
ôf transformation for an infinite class, by tabulation, ig
not even theoretically possible, but in such cases we must
use an incomplete table or ordinary speech, as iti the
foregoing example where each integer is associated with
(transformed into) its double and as in the example of
Fig. 2z; or, finally, we must express the law, as in Bolzano's example, by means of an equation or system of
equations. Of all methods of expression, the equational
method is the most common, and is usually the most
satisfactory when it is possible, which it is sometimes not.
We have seen thât the meaning of mathematical
transformation has its root in the power we have to associate any idea or thing with any other, however like or
unlike the former. As this power is fundamental and is
continually exercised by all human beings in every kind of
matter, we find, as we should expect to find, not only
that mathematical transformation pervades mathematical
thinking, but that such transformation is only a refinement
of a process present in all our human thinking: a fact
clearly illustrating the general truth that mathematical
activity, instead of being remote from common life,
merely consists in doing, with a peculiar finesse and
ideality, what all human beings, when they think about
the ordinary affairs of life and the world, are doing in a
fashion relatively rough and crude. An ordinary dictionary, for example, is a good illustration of a kind of
transformation that would be genuinely mathematical
were it more precise; for, by definition, the class of words
is transformed into the class of verbal meanings, and,
conversely, the latter class is transformed into the former;
the transformation runs both ways, but it is not one-to-one,
since a given word commonly has tWo or more meanings
and to a given meaning may correspond two or more