Herbart is said to have given the deathblow to faculty psychology. Man
no longer appears endowed with volition, passion, desire, and reason;
and logic, deprived of its hereditary right to elucidate the operations
of inherent intelligence, has the new problem of investigating forms of
intelligence in the making. This is no inconsequential task. "If man
originally possesses only capacities which after a given amount of
education will produce ideas and judgments" (Thorndike, _Educational
Psychology_, Vol. I, p. 198), and if these ideas and judgments are to be
substituted for a mythical intelligence it follows that tracing their
development and observing their functioning renders clearer our
conception of their nature and value and brings us nearer that exact
knowledge of what we are talking about in which the philosopher at least
aspires to equal the scientist, however much he may fall below his
ideal.
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 which are most effective for
prediction and control. Such selection gives the principle that
dominates all abstractions. Progress is movement from the less abstract
to the more abstract, but it is progress only because the more abstract
is as genuinely an aspect of the concrete starting-point as anything is.
Moreover, the outcome of progress of this sort cannot be definitely
foreseen at the beginnings. The simple activities of primitive men have
to be spontaneously performed before their value becomes evident. Only
afterwards can they be cultivated for the sake of their value, and then
only can the self-conscious cultivation of a science begin. The process
remains full not only of perplexities, but of surprises; men's
activities lead to goals far other than those which appear at the start.
These goals, however, never deny the method by which the start is made.
Developed intelligence is nothing but skill in using a set of concepts
generated in this manner. In this sense the histories of all human
endeavors run parallel.
Where the empirical bases of a science are continually in the
foreground, as in physics or chemistry, the foregoing formulation of
procedure is intelligible and acceptable to most men. Mathematics seem,
however, to stand peculiarly apart. Many, with Descartes, have delighted
in them "on account of the certitude and evidence of their reasonings"
and recognized their contribution to the advancement of mechanical arts.
But since the days of Kant even this value has become a problem, and
many a young philosophic student has the question laid before him as to
why it is that mathematics, "a purely conceptual science," can tell us
anything about the character of a world which is, apparently at least,
free from the idiosyncrasies of individual mind. It may be that
mathematics began in empirical practice, such philosophers admit, but
they add that, somehow, in its later career, it has escaped its lowly
origin. Now it moves in the higher circles of postulated relations and
arbitrarily defined entities to which its humble progenitors and
relatives are denied the entree. Parvenus, however, usually bear with
them the mark of history, and in the case of this one, at least, we may
hope that the history will be sufficient to drag it from the
affectations of its newly acquired set and reinstate it in its proper
place in the workaday world. For the sake of this hope, we shall take
the risk of being tedious by citing certain striking moments of
mathematical progress; and then we shall try to interpret its genuine
status in the world of working truths.
I