Kant did not call the formal laws _a priori_ in order to characterise
them as innate ideas, but simply to denote that their validity is
necessary and universal. If I have to walk twice a distance of two miles,
I know “beforehand” (i. e. _a priori_) that I shall have to walk four
miles—even before I have made the actual experience. And this wonderful
quality of giving information beforehand is characteristic of all the
laws of formal thought. It is certain that our ability of applying the
laws of formal thought has been acquired by experience in the race as
well as in the individual. But their necessity has to do with experience
in so far only as its recognition is the indispensable condition of all
methodical experience—i. e. of science. The laws of formal thought and
our recognition of their necessity and universality (alias, “apriority”)
are the organ of any and all scientific cognition. The methods of the
sciences are exact measuring and counting based upon the faith that the
laws of measuring and counting are unalterable and unfailingly reliable.
We know beforehand that they will hold good for all possible cases.
Our experience of millenniums suffices to prove that the laws of formal
thought agree with the laws of actual existence, and Kant’s view to
consider them as merely subjective and not objective appears to me
untenable. We may fairly consider Kant’s subjectivism as a thing of the
past. And the agreement of the forms of objectivity with the forms of
subjectivity is easily explained when we bear in mind that the thinking
subject is a part of the objective world. It is but natural that the
forms of existence are impressed upon the thinking subject as forms of
thought.
Yet the question of the necessity and universality of the laws of form
remains. Can we comprehend why the form of objective reality as well as
of subjective thought must be such as they are, and might they not just
as well be different? Is this question legitimate and can it be answered?
We say Yes, the question is legitimate and can be answered.
All the laws of formal thought are products of thought-operations which
are based on no other principle than that of identity (_A_ = _A_). As
soon as thinking beings have developed to that degree of thought-ability
in which they are able to deal with the abstract idea of pure form, they
can make constructions of pure forms. So long as these constructions
of pure forms are made consistently and correctly (i. e. in strict
agreement with the principle of identity), they will be found to hold
good in reality and we can _a priori_—before we have made the actual
experience—rely on their applicability.
The laws of pure forms (forms of thought as well as forms of existence)
can satisfactorily be proved to anyone who acknowledges the principle
of identity. The principle of identity is not an assumption but it is
the generalised statement of the simplest thought-operation, which, if
employed with consistency, can serve as a rule for other and more complex
thought-operations. Consistency is the condition of thought. Consistency
produces order, and order is the most characteristic feature of thought.
We create some pure form in some definite way, for instance in counting
we posit a unit (we call it “one”). Now we create in the same definite
way again a pure form, we again posit a unit (we again call it “one”).
In so far as these two “ones” are the product of the same operation they
will be the same and we express this truth in the sentence 1 = 1 or _A_ =
_A_.
When, for the sake of assistance in the process of abstract thought, we
use real objects to represent our pure forms, similarly as an abacus is
employed for assisting the young mind of a child in learning arithmetic,
the dissimilarity of the objects is of no account. The proposition of
their identity has no reference to the material objects, but to the
operation. Two operations being according to some special and definite
method exactly the same, their products are also exactly the same,
and the rest is not to be minded, because we have in our abstraction
purposely excluded everything else.
Here is not the place to show the palpable advantages of the methods
of abstract thought and especially of the abstract thought-operations
with pure forms. It is sufficient to characterise its main principle of
procedure. We may also parenthetically remark that from our position
we are no longer in need of axioms either in mathematics or in any
other formal science. The data of formal sciences are certain mental
operations, viz. positing pure forms, and combining, separating, and
recombining them. The subject matter of formal sciences consists in the
products of these operations. To formulate some of the simplest products
in axioms is a mistake which has been pointed out by Hermann Grassmann in
his _Ausdehnungslehre_.
We are struck and overawed with the cosmic order of all natural phenomena
which, as science teaches, is produced through the rigidity of the formal
laws of existence. The planetary system with its regularity of motion
which in spite of its many complications has been formulated by Kepler in
most simple laws is an object of wonder to us. This order of nature is
the same order as the grand harmony that prevails in mathematics and all
the other formal sciences. The most complicated laws of both, forms of
nature and forms of thought, are nothing but generalisations of special
applications of the principle of identity in some kind of action that
takes place. While the order of the objective world excites our wonder
we can understand the order of the subjective world of thought-forms
and know that, being the product of certain mental operations according
to the principle of identity, it must be a matter of course. Thus the
intrinsic necessity of the laws of our thought-forms gives us a clue to
the intrinsic necessity of the laws of nature.