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

164 MATHE3MATICAL PHILOSOPHY
compared with one another, element for element and
proposition for proposition, giving rise in this way to
what might be appropriately called the Comparative
Anatomy of Doctrines. These examples of the use of
transformation, though they must suffice for the present,
are only as pebbles picked up at random on the ocean
beach.
"The hour contracts " and we have yet to speak of
The Connection of the Concept of Transformation with
that of Relation and that of Function.-We shall see that
the three concepts are very similar-but three aspects
indeed of one and the same thing seen from different points
of view. Let us, in the first place, try to understand
clearly what a relatio' is, This is necessary because,
though countless hosts of relations are present everywhere
in the world and are used by everybody all the time, even
in their dreams, yet the scientific conception of what a
relation precisely is, is not familiar; even the great major.
ity of logicians and mathematicians are not familiar with
it; it seems a little strange that such is the fact, for the
logical theory of relations,-the logical theory having the
nature and the properties of relations as subject.matter,goes back to the logical work of J. H. Lambert (I728-77),
-mathematician, physicist, astronomer,-and especially
to that of Augustus DeMorgan (18o6-78),-mathematician, logician and wit; was advanced by important
researches of our fellow countryman, the late C. S. Peirce,
who called it the logic of relatives; and has now reached
a high state of development in the Principia of Whitehead
and Russell, who have made the theory of abstract relations supreme in logical doctrine. This great theory —
fupdamental alike in philosophy and in mathematicshas riot yet become in most universities a subject of