We have to be on our guard lest we introduce some mystical element
into the idea of necessity. There is nothing mystical about necessity.
Necessary means that a certain operation, if it is exactly the same
operation as another one, will produce exactly the same result. When we
posit two units twice, we shall have the same result as when we posit
one unit four times; and we call this result four. We shall reach the
same product whenever we repeat the same operation. Knowing that we
shall always reach the same result, we can, _a priori_ (or beforehand)
and with certainty, determine the result of certain operations after we
have mentally gone through the same operations for all possible cases
and under any imaginable conditions. That a perfect apriority with an
unfailing certainty is possible only in the domain of formal thought
is natural. The reason is that we know our thought-forms exhaustively.
They contain nothing but that which has been predicated about them. Our
sense-experience however is always piecemeal and never exhaustive.
Comprehension is actually a tracing of form relations and a formulating
them in concise statements. The scientist’s work is based upon the
methods of measuring and counting (i. e. the methods of formal sciences)
and he undertakes to show that certain phenomena, different in some
respects, are the same in other respects, that their sameness can be
stated in a comprehensive and exact formula. In this way he marks out
their determining factors in terms of formal thought (for instance in
numbers), so that we can compute them and predict them according to their
determining factors, so that we can know, according to their conditions,
that they will be always as they are.
The importance of formal thought is paramount in science and the problem
about the necessity which attaches to the laws of formal thought is the
fundamental problem of philosophy.
There are many philosophers, still, troubling themselves to solve the
problem in the fashion of Schopenhauer or of Mill or looking upon the
problem as insolvable. We do not doubt that the solution here presented
is the only possible solution which as soon as it is understood will find
a general acceptance.
Must it be added that the solution of this fundamental problem does
not involve the ready solutions of all other minor problems? Oh no! We
all know that the solution of one problem is only a stepping-stone for
attacking other problems. The possibilities of progress are as unlimited
as the scope of cognition. Light on this general subject gives us hope
that we shall succeed in throwing light also upon other subjects which
are still shrouded to the philosophical inquirer in impenetrable darkness.