ARE THERE TRUE SYNTHETIC JUDGMENTS A PRIORI IN THE SENSE OF KANT?
278. The great importance attributed by the German philosopher to
his imaginary discovery, requires us to examine it at length. This
importance may be estimated from what he himself says: "If any of the
ancients had only had the idea of proposing the present question, it
would have been a mighty barrier against all the systems of pure reason
down to our days, and would have saved many vain attempts which _were
blindly made without knowing what was treated of_."[23] This passage
is quite modest and naturally excites our curiosity to know what is
the problem which needed only to be proposed in order to avoid all the
aberrations of pure reason.
[23] _Critik der reinen Vernunft._ Einleitung.
Here are his words: "All empirical judgments, as such, are synthetic.
For it would be absurd to ground an analytic judgment on experience,
since I am not obliged to go out of the conception itself in order to
form the judgment, and therefore can have no need of the testimony of
experience. That a body is extended, is a proposition which stands firm
_a priori_. It is no empirical judgment; for, prior to experience, I
have all the conditions of forming it in the conception of body, from
which I deduce the predicate, extension, according to the principle of
contradiction, by which I at once become conscious of its necessity,
which I could not learn from experience. But, on the other hand, I
do not include, in the primitive conception of body in general, the
predicate, heaviness; yet this conception of body in general indicates,
through experience of a part of it, an object of experience, to which
I may add from experience other parts also belonging to it. I can
attain to the conception of body beforehand, analytically, through
its characteristics extension, impenetrability, form, etc., all of
which are included in the primary conception of body. But I now
_extend_ my cognition, and, as I recur to experience, from which I have
obtained the conception of body in general, I find along with these
characteristics the conception of heaviness. I therefore add this, as a
predicate, to the conception of body. The possibility of this synthesis
therefore rests on experience; for both conceptions, although one does
not contain the other, yet belong as parts to a whole, that is to say,
to experience, which is itself a union of synthetic, though contingent
intuitions. But in the case of synthetic judgments _a priori_ we have
not this assistance. Here we have not the advantage of returning and
supporting ourselves on experience. If I must go out of the conception
A in order to find another conception B, which is to be joined to it,
on what am I to rely? and by what means does the synthesis become
possible?"[24]
[24] Ibid. pp. 9, 10.
279. The reason of this synthesis is found in the faculty of our mind
of forming total conceptions, in which the _relation_ of the partial
conceptions composing it is discovered; and the legitimacy of the
same synthesis is founded on the principles on which the criterion of
evidence is based.
The synthesis of the schoolmen consists in the union of conceptions,
and does not refuse to admit as analytical the total conceptions, from
the decomposition of which results the knowledge of the relations of
the partial conceptions.
If Kant had stopped with the judgments of experience, there would be
no objection to his doctrine. But extended to the purely intellectual
order, it is either inadmissible, or at least expressed without much
exactness.
260. Kant says all mathematical judgments are analytic, and that this
truth which in his opinion "is certainly incontestible and important
on account of its consequences, seems to have hitherto escaped the
sagacity of the analysts of human reason, causing very contrary
opinions." We think it is the sagacity of his Aristarchus, and not that
of the analysts, that is at fault.
"One would certainly think at first sight that the proposition, 7
+ 5 = 12, is a purely analytic proposition, which follows from the
conception of a sum of seven and five, according to the principle of
contradiction. But if we examine it more closely, we find that the
conception of the sum of seven and five contains nothing farther than
the union of both numbers in one, from which it cannot by any means be
inferred what this other number is which contains them both."[25]
[25] Kant, _ubi supra_, § 5.
Were we to say that whoever hears seven plus five, does not always
think of twelve, because he does not see clearly enough that one
conception is the same as the other, although it is under a different
form, it would be true. But from this it does not follow that the
conception is not purely analytic. The mere explanation of both
suffices to show their identity.
That this may be better understood, we will invert the equation thus:
12 = 7 + 5. It is evident that if any one does not know that 7 + 5 =
12, he will not know that 12 = 7 + 5. Now, in examining the conception
12, we certainly see 7 + 5 contained in it. Therefore, the conception
of 12 is identical with the conception of 7 + 5; and just as, because
he who hears 12, does not always think of 7 + 5, we cannot thence infer
that 12 does not contain 7 + 5; so, also, we cannot, because he who
hears 7 + 5, does not always think of 12, thence infer that the first
conception does not contain the second.
The cause of the equivocation is, that the two identical conceptions
are presented to the intellect under different forms; and until we
have the form, and look to what is under it, we shall not discover the
identity. This is not, strictly speaking, _reasoning_ but _explanation_.
What Kant adds concerning the necessity of recurring, in this case, to
an intuition, with respect to one of the numbers, adding five to seven
on the fingers, is exceedingly futile. First, in whatever way he adds
the five, there will never be anything but the five that is added, and
it will neither give more nor less than 7 + 5. Secondly, the successive
addition on the _fingers_ is equivalent to saying 1 + 1 + 1 + 1 + 1 =
5. This transforms the expression, 7 + 5 = 12, into this other, 7 + 1 +
1 + 1 + 1 + 1 = 12; but the conception, 1 + 1 + 1 + 1 + 1, has the same
relation to 5, as 7 + 5 to 12; therefore, if 7 + 5 are not contained
in 12, neither are 7 + 1 + 1 + 1 + 1 + 1 contained in it. It may be
replied that Kant does not speak of identity, but of intuitions. This
intuition, however, is not the sensation, but the idea; and if the
idea, it is only the conception explained. Thirdly, we know this method
of intuition not to be even necessary for children. Fourthly, this
method is impossible in the case of large numbers.
281. Kant adds that this proposition, "a right line is the shortest
distance between two points," is not purely analytic, because the
idea of _shortest distance_ is not contained in the idea of _right
line_. Waiving the demonstrations which some authors give, or pretend
to give, of this proposition, we shall confine ourselves to Kant's
reasons. He forgets that here the right line is not taken _alone_, but
_compared_ with other lines. The idea of right line alone neither does
nor can contain the ideas of _more_ or _less_; for these ideas suppose
a comparison. But from the moment the right line and the curve are
compared, with respect to _length_, the relation of superiority of the
curve over the right line is seen. The proposition is then the result
of the comparison of two purely analytic conceptions with a third,
which is _length_.
282. If Kant's reasoning were good, even this judgment, "the whole
is greater than its part," would not be analytic; for the idea of
_greater_ enters not into the conception of the _whole_ until the
_whole_ is compared with its _part_. Thus, the judgment, four is
greater than three, would not be analytic, because the idea of four
until compared with three does not include the conception of greater.
The axiom: "things which are equal to the same thing are equal to each
other," would not be analytic, because the conception, _equal to each
other_, does not enter into the conception of _things which are equal
to the same thing_, until we reflect that the equality of the middle
term implies the equality of the extremes.
The x, of which Kant speaks, would be found in almost all judgments,
if we could not form total conceptions involving comparison of partial
conceptions: in this case we should have no analytic judgments except
such as are wholly identical, or directly contained in this formula, A
is A.
283. The comparison of two conceptions with a third, does not take from
the result the character of analytic judgment, as a predicate cannot be
seen in the idea of the subject, without the aid of this comparison.
This comparison is often necessary, because we only confusedly think
of what is contained in the conception which we already have; and
sometimes it even happens that we do not think at all of it. One
often says a thing and then contradicts himself, not observing that
what he adds is opposed to what he had already said. We often ask, in
conversation, do you not see that you suppose the contrary of what you
just said; that the conditions you have just established imply the
contrary of what you now assert?
284. A conception includes not only all that is expressly thought in
it, but all that can be thought. If, on decomposing it, we find in it
other things, it cannot be said that we add them, but that we find
them. It is not a synthesis, but an analysis. Otherwise we must admit
no analytic conceptions, or only such as are purely identical. Except
in this last case, of which the general formula is, A is A, there is
always in the predicate something not thought in the subject, if not
in substance at least in form. The circle is a curve; this undoubtedly
is one of the simplest analytical propositions imaginable; still the
predicate expresses the general conception of curve, which may be
contained in the subject, in a confused manner, with relation to a
particular species of curve. Following a gradation in geometrical
propositions, we may observe that there is nothing in one proposition
not in the preceding, except the greater or less difficulty of
decomposing the conception, so as to see in it what before we had not
seen.
If we say, the circle is a conic section, evidently any one ignorant of
the terms, or who has not reflected on their true sense, will not think
of the attribute in the subject. No addition is made to the conception
of the circle; only a property not before known is discovered, and this
discovery results from comparison with the cone. Is there any synthesis
here? No. There is only an analysis of the two conceptions, the circle
and the cone, compared. As this error destroys the foundation of Kant's
doctrine on this point, we will develop it and place it on a more
solid foundation.
285. Synthesis, properly so called, requires something to be added
to the conception, which in nowise belongs to it, as the example
brought by Kant shows. The conception, extension, is contained in
the conception, body; but heaviness is an entirely foreign idea,
which we can unite to the conception, body, only because experience
authorizes it. Only with this addition is there properly synthesis.
The union of ideas which results from the conception of the thing,
although comparison may be necessary in order to fecundate them, does
not make a synthesis. The conceptions are not wholly absolute, they
contain relations, and the discovery of these relations does not give
a synthesis, but a more complete analysis. If it be said that in this
case there is something more than the primitive conception, we answer
that the same thing happens in all not purely identical. We may also
add that by the comparison a new total conception is formed resulting
from the primitive conceptions; and the properties of the relations
are then seen, not by synthesis, but by the analysis of the total
conception.
According to Kant, true synthesis requires the union of things so
different from one another, that the bond uniting them is a sort of
mystery, an x, whose determination is a great philosophical problem. If
this x is found in the essential relation of the partial conceptions
constituting the total conception, the problem is resolved by a simple
analysis, or, to speak more exactly, it is shown that the problem did
not exist, because the x was a known quantity.
We know of no judgment more analytical than that in which we see the
parts in the whole, since the whole is only the parts united. If we
say, one and one are two, or, two is equal to one plus one; it cannot
be denied that we have a total conception, two, in the decomposition
of which, we find one plus one. If this be not an analytic conception,
that is to say, if the predicate be not here contained in the idea
of the subject, it will be hard to tell what is. But even here there
are different conceptions, one plus one; unite them, and they form
the total conception. The relation, although most simple, exists; and
whether it be more or less, simple or complicated, and, consequently,
seen with more or less facility, does not alter the character of the
judgments, or from synthetic convert them into analytic.
286. We will complete this explanation with an example from elementary
geometry. "The surface of a rhomboid is equal to the surface of a
rectangle having the same base and altitude." First: in the idea of the
rhomboid, we do not see the idea of its equality with the rectangle;
and this we cannot see, because the relation does not exist when there
is no other term to which it may relate. The idea of the parallelogram
does not contain that of the rectangle, and consequently not that of
equality. Second: the relation results from the comparison of the
rhomboid with the rectangle; and, consequently, it must be found in a
total conception containing them both. It cannot, therefore, be said
that we add any thing to the conception of the parallelogram which
does not belong to it. On the contrary, we see this equality flow from
the conception of the rhomboid and that of the rectangle, as partial
conceptions of the total conception, formed by the combination of them
both. The analysis of this total conception opens to us the relation we
are now in quest of; for it must be observed that when the simple union
of the conceptions compared does not suffice, we make use of another
including them, and also something more; and from the new conception,
duly analyzed, we deduce the relation of the parts compared.
287. In the geometrical construction, that serves for the
demonstration of the above theorem, which we have used as an example,
may be seen what we have just explained with regard to total
conceptions containing other conceptions besides those compared. If we
place the rectangle and the rhomboid upon the same base, we at once see
that there is something common to both, namely, the triangle formed by
the base, a part of one side of the rhomboid, and a part of one side of
the rectangle. Neither synthesis nor analysis is here required, because
there is perfect coincidence, and this in geometry is equivalent to
perfect equality. The difficulty is in the two remaining parts, that
is, in the trapezoids to which the parallelograms are reduced by the
subtraction of the common triangle. The mere sight of the figures
teaches nothing concerning the equivalence of the two surfaces; we see
only that the two sides of the rhomboidal surface go on extending,
but including a less distance in proportion as the angle becomes more
oblique, under these two conditions: length of sides, and diminution of
distances between two limits, of which one is infinity, and the other
the rectangle. The relation of the equivalence of the surfaces may be
demonstrated by prolonging the parallel opposite the base, and thus
forming a quadrilateral of which the trapezoids are parts; to discover
the equality of these trapezoids, it is only necessary to decompose
the quadrilateral, attending to the equality of two triangles, each
respectively formed by one of the trapezoids and a common triangle.
Is any thing here added to the conception of each trapezoid? No. We
only compare them. They could not be compared directly, and therefore
we included them in a total conception, the mere analysis of which
enabled us to discover the relation sought for. The conception does not
give this relation; it only shows it; for if the conception of the two
figures compared were more perfect, so that we might intuitively behold
the relation existing between the increment of the sides and the
decrement of their distance from each other, we should see that there
is here a constant law, which supplies on one side what is lost on the
other; and consequently we should discover, in the very conception
of the rhomboid, the fundamental reason of the equality, that is,
the permanent value of the surface, notwithstanding the greater or
less obliquity of the angles; thus obtaining what we deduced from the
above comparison, and generalize with reference to two constant lineal
values, base and altitude. The same would happen with respect to the
equivalence of all variable quantities differently expressed, could we
reduce their conceptions to such clear and simple formulas as those of
apparent functions; for example, nx/mx, from which, whatever the value
of the variable, there always results the same value of the expression,
which is constant, to wit, n/m.
288. Let not these investigations be imagined useless. In this, as
in many other questions, it happens that most important truths are
the result of a philosophical problem which, in appearance, is merely
speculative. Thus, in the present case, we observe Kant explaining the
principle of causality, in an inexact, and, as we understand him, in an
altogether false sense; but, perhaps, the origin of his equivocation
lies in his considering the principle of causality as synthetic,
although _a priori_, whereas it must be regarded as analytic, as we
shall show when treating of the idea of cause.
In consideration of the great importance of clear and distinct ideas
on the present subject, we will in a few words, sum up the doctrine we
have explained concerning mediate and immediate evidence.
There is immediate evidence when, in the conception of the subject, we
see its agreement or disagreement with the predicate, without requiring
any other means than mere reflection on the meaning of the terms.
Judgments of this class are with propriety called analytic, because we
have only to analyze the conception of the subject to find therein its
agreement or disagreement with the predicate.
There is mediate evidence when, in the conception of the subject, we do
not immediately see its agreement or disagreement with the predicate,
and therefore have to call in a middle term to make it manifest.
290. Here arises the question whether judgments of mediate evidence
are analytic. It is clear that if we mean by analytic only those in
which we have solely to understand the meaning of the terms in order to
see the agreement or disagreement of the predicate, the judgments of
mediate evidence cannot be called analytic; but if by analytic judgment
we mean a judgment in which it is only necessary to _decompose_ the
conception of the subject in order to find therein its agreement or
disagreement with the predicate, we must say that the judgments of
mediate evidence are analytic, and the means employed is only the
formation of a total conception containing the partial conceptions, the
relation of which we seek to discover. In the union of these partial
conceptions there is a synthesis, it is true; but there is none in the
discovery of their relation, for this is done by analysis.
A judgment is not the less analytic because formed by the union of
different conceptions; for then no judgment would be analytic. When we
say, man is rational, the two conceptions of _animal_ and _rational_
enter into the conception of _man_, but do not take from it its
analytical character; for this, as its very name imports, consists
in the analysis of a conception, being sufficient to show certain
predicates in it, without reference to the manner of this conception's
formation, whether two or more conceptions are united in it, or not.
291. This clearly shows in what mediate evidence consists. The
predicate is indeed contained in the idea of the subject; but, owing
to the limitation of our intellect, either these ideas are incomplete,
or we do not see them in all their extension, or else we do not well
distinguish what we in a confused manner perceive in them; and hence,
to know the meaning of the terms does not enable us immediately to
see that the predicate is contained in the idea of the subject.
Moreover, the objects, even such as are purely ideal, are presented
to us separately; and hence, not knowing the sum of them all, we pass
successively from one to another, discovering their mutual relations in
proportion as we approach them.
292. It may, from what we have said, be inferred that all judgments
in the purely ideal order are analytic, since every cognition of
this order is obtained by the intuition of whatever is more or less
complicated in the conception, and there is no more synthesis than is
necessary to bring the objects together, by uniting their conceptions
in one total conception, which serves for the discovery of the relation
of the partial conceptions.
293. The x, therefore, of which Kant speaks, and the removal of which
is one of the most important problems of philosophy, is nothing more
than the faculty possessed by the soul to unite the conceptions of
different things in one total conception, and to discover in it their
mutual relations. This faculty is no new discovery, for the schools
have all recognized it under one name or another. No one ever denied
to the intellect the faculty of comparing; and comparison is the act
whereby the intellect places two or more objects before its sight so
as to perceive their mutual relations. In this act the intellect forms
a total conception, of which the conceptions compared are a part. Thus
we have seen that in geometry to verify the mutual relation of certain
figures, we construct a new figure which includes them all, and is a
sort of field whereon the comparison is made.
This exposition of analytic and synthetic judgments will suffice for
the present; as we proposed to treat of them here only in general, and
as related to certainty; consequently we will not descend to their
particular application to various ideas, the analysis of which belongs
to other parts of this work.