al.

For contemporary thought concerning the mathematical sciences this
altered point of view generates peculiarly pressing problems.
Mathematicians have weighed the old logic and found it wanting. They
have builded themselves a new logic more adequate to their ends. But
they have not whole-heartedly recognized the change that has come about
in psychology; hence they have retained the faculty of intelligence
knit into certain indefinables such as implication, relation, class,
term, and the like, and have transported the faculty from the human soul
to a mysterious realm of subsistence whence it radiates its ghostly
light upon the realm of existence below. But while they reproach the old
logic, often bitterly, their new logic merely furnishes a more adequate
show-case in which already attained knowledge may be arranged to set off
its charms for the observer in the same way that specimens in a museum
are displayed before an admiring world. This statement is not a sweeping
condemnation, however, for such a setting forth is not useless. It
resembles the classificatory stage of science which, although not itself
in the highest sense creative, often leads to higher stages by bringing
under observation relations and facts that might otherwise have escaped
notice. And in the realm of pure mathematics, the new logic has
undoubtedly contributed in this manner to such discoveries. Danger
appears when the logician attains Cartesian intoxication with the beauty
of logico-mathematical form and tries to infer from the form itself the
real nature of the formed material. The realm of subsistence too often
has armed Indefinables with metaphysical myths whose attack is valiant
when the doors of reflection are opened. It may be possible, however, to
arrive at an understanding of mathematics without entering the kingdom
of these warriors.

It is the essence of science to make prediction possible. The value of
prediction lies in the fact that through this function man can control
his environment, or, at worst, fortify himself to meet its vagaries. To
attain such predictions, however, the world need not be grasped in its
full concreteness. Hence arise processes of abstraction. While all other
symptoms remain unnoticed, the temperature and pulse may mark a disease,
or a barometer-reading the weather. The physicist may work only in terms
of quantity in a world which is equally truly qualitative. All that is
necessary is to select the elements w