CAN ALL COGNITIONS BE REDUCED TO THE PERCEPTION OF IDENTITY?
264. Immediate evidence has for its objects those truths which the
intellect sees with all clearness, and to which it assents without the
intervention of any _medium_, as its name denotes. These truths are
enunciated in propositions called _per se notæ_, first principles, or
axioms, in which it is sufficient to know the meaning of the terms
to see that the predicate is contained in the idea of the subject.
Propositions of this class are few in all sciences; the greater
part of our cognitions are the fruit of reasoning which proceeds
by mediate evidence. In geometry the number of truths that do not
require demonstration, but only explanation, is very limited. The
body of geometrical science, with its present colossal dimensions,
has proceeded from reasoning: even in the most comprehensive works
the axioms occupy but a few pages; the rest is composed of theorems,
propositions not of themselves evident, but requiring demonstration.
The same is true of all other sciences.
265. Since in axioms the intellect perceives the identity of the
subject with the predicate, intuitively seeing that the idea of the
latter is contained in that of the former, there arises a very grave
philosophical question which may prove very difficult, and cause
strange controversies, if care be not taken to place it upon its true
ground. Is every human cognition reduced to the simple perception of
identity? and can its general formula be this: A is A, or: a thing is
itself? Some philosophers of note maintain the affirmative; others
the contrary. We think there is a confusion of ideas not so much as
to the question itself as to its state. Clear and exact ideas of what
judgment is, and of the relation affirmed or denied by it, will greatly
facilitate the accurate solution of the question.
266. There is in every judgment perception of identity or non-identity,
accordingly as it is affirmative or negative. The verb _is_ does not
express the union, but the identity of the predicate with the subject;
and when accompanied with the negation _not_, it simply expresses
non-identity, abstracting union or separation. This is so true and
so exact, that in things really united an affirmative judgment is
impossible, because they have no identity. We must, then, in such
cases, if we would be enabled to make an affirmation, express the
predicate in the concrete, that is, in some sense involving the idea
of the subject itself in it; for the same property affirmed in the
concrete cannot be in the abstract, but must rather be denied. Thus
we may say, man is rational; but not, man is rationality: a body is
extended; but not, a body is extension: paper is white; but not, paper
is whiteness. Why is this? Is it that rationality is not in man,
extension is not united to body, nor whiteness to paper? Certainly
not; but if rationality be in man, extension in body, and whiteness in
paper, we have only not to perceive identity between the predicates and
subjects, to render affirmation impossible; on the contrary, despite
the union, we have negation: thus we may say, man is not rationality; a
body is not extension; paper is not whiteness.
We have said that, in order to save the expression of identity, we used
the concrete instead of the abstract term, and involved in the former
the idea of the subject. It cannot be said that paper is whiteness, but
it may be said that paper is white; for this last proposition means
that paper is a white thing; that is, we make the general idea of a
_thing_, or the idea of a modifiable subject, enter into the predicate
while in the concrete; and this subject is identical with the paper
modified by whiteness.
267. Thus it is easy to see, that the expression, _union of the
predicate with the subject_, is, at the best, inexact. Every
affirmative proposition expresses the identity of the predicate with
the subject. Use authorizes these modes of speaking, which still
produce some confusion when we endeavor perfectly to understand these
matters. And it must be observed, that ordinary language here, as often
elsewhere, is admirably exact and appropriate. Nobody says, paper
is whiteness, but, paper is white. It is only when we would greatly
heighten the degree, to which a subject possesses a quality, that
we express it in the abstract, and then we join with it the pronoun
_itself_. Thus, speaking hyperbolically, we say a thing is beauty
itself, whiteness itself, goodness itself.
268. Even what in mathematics is called equality, also means identity.
Thus in this class of judgments, besides what we have observed of
general in them all, to wit: the identity saved by expressing the
predicate in the concrete, the very relation of equality denotes
identity. This needs explanation.
Whoever says 6 + 3 = 9, expresses the same as he who says 6 + 3 are
identical with 9. Clearly in the affirmation of equality, no attention
is paid to the form in which the quantities are expressed, but to the
quantities themselves alone; otherwise we should be unable to affirm
not only identity, but also equality; for it is evident that 6 + 3,
as to their form, neither written, spoken, nor thought, are identical
with, or equal to, 9. The equality is in the values expressed, and
these are not only equal but identical; 6 + 3 are the same as 9. The
whole is not distinguished from its united part; 9 is the whole, 6 + 3
its united parts.
The different manner of conceiving 6 + 3 and 9 does not exclude the
identity. The difference is in the intellectual form, and occurs not
only here but also in the perceptions of the simplest things; there
is nothing which we do not conceive under different aspects, and
whose conception we may not decompose in various ways; but we do not
therefore say that the thing ceases to be simple and identical with
itself.
What we have said of an arithmetical equation may be extended to
algebraical and geometrical equations. If we have an equation whereof
the first member is very simple, as Z, and the second very complicated,
as the development of a series, we cannot say that the first expression
is equal to the second; the equality is not in the expression but in
the thing expressed, in the value designated by the letters; in this
sense it is true, in the former it is evidently false.
Two circumferences having the same radius are equal. Here we seem to
treat solely of equality, since there are two distinct objects, the
two circumferences, which may be traced on paper or represented in the
imagination; yet not even in this case is the distinction true, it is
only apparent, for here, as in algebraical and arithmetical equations,
there is distinction and even diversity in form with identity at
bottom. The principal argument, on which the distinction is founded,
may be combatted by observing that the circumferences which may be
traced or represented, are only forms of the idea, not the idea itself.
Whether traced or represented they have a determinate size and a
certain position on the planes seen or imagined; in the idea, and in
the proposition containing it, there is nothing of this; we abstract
all size, all position, and speak in a general and absolute sense.
True, the representations may be infinite either externally or in the
imagination; but this, so far from proving them identical, shows their
diversity, since the idea is one and they are infinite; the idea is
constant, they are variable; the idea is independent of them, they
are dependent on the idea, and have the character and denomination of
circumferences, inasmuch as they approach it by representing what it
contains.
What, then, is expressed in the proposition: two circumferences, having
the same radius, are equal? The fundamental idea is, that the value
of the circumference depends upon the radius, and the proposition
here enunciated is simply an application of this property to the
case of the equality of radii. The circumferences, then, conceived
by us as distinct, are only examples which we inwardly consider in
order to render the truth of the application apparent; but in what
is purely intellectual, we find only the decomposition of the idea
of circumference, or its relation to the radius applied to the case
of equality. Then there are not two circumferences in the purely
ideal order, but one only, whose properties we know under different
conceptions, and express in various ways.
If in all judgments there is affirmation of identity, or non-identity,
and all our cognitions either begin or end in a judgment, it would seem
that they all ought to be reduced to a simple perception of identity.
The general formula of our cognitions will then be: _A is A_, or, _a
thing is itself_. This result strikes one as an extravagant paradox,
and is so, or not, according to the sense in which it is understood;
but if rightly explained, it may be admitted as a truth, and a very
simple one. From what has been said in the preceding paragraphs, the
meaning of this opinion may be discerned: but the importance of the
present matter requires still further explanation.