CONTINUATION OF THE SAME SUBJECT.
274. We will now explain how the doctrine of identity is applied in
general to all reasoning, whether upon mathematical objects or not:
with this view we will examine some of the dialectical forms in which
the art of reasoning is taught.
Every A is B; M is A: therefore M is B. In the major of this syllogism
we find the identity of every A with B; and in the minor, the identity
of M with B. In each of these propositions there is affirmation, and,
consequently, perception of identity. Let us now see what takes place
in the connection which constitutes the force of the argument.
Why do we say that M is B? Because M is A, and every A is B. M is
one of the As, expressed in the words _every A_; therefore, when we
say, M is A, we say only what we had before said by _every A_. What
difference, then, is there? There is this difference, that in the
expression _every A_, no attention is paid to one of A's contents, M,
of which we had nevertheless affirmed that it was B, in affirming that
every A is B. If, in the expression _every A_, we have distinctly seen
M, the syllogism would not have been necessary, because, in saying
every A is B, we had already understood that M is B.
This observation is so true and exact, that in treating of very
clear relations we suppress the syllogism, and replace it with the
enthymema, which is, it is true, an abbreviation of the syllogism;
but we must see in this abbreviation besides a saving of words, a
saving of conceptions, for the intellect sees one intuitively in the
other, without necessity of decomposition. He is a man, therefore
he is rational; we omit the major, and do not even think of it, for
we intuitively see, in the idea of man, and its application to an
individual, the idea of rational without any gradation of ideas or
succession of conceptions.
Let us suppose that we have to demonstrate that the perimeter of a
polygon inscribed in a circle is less than the circumference, and
that we make the following syllogism: The sum of all the right lines
inscribed in their respective curves is less than the sum of those
curves; but the perimeter of the polygon is the sum of the right lines,
and the circumference is the sum of the arcs or curves; therefore the
inscribed perimeter is less than the circumference. We now ask, will
any one who knows that the sum of the right lines is less than the sum
of the curves, fail to see with equal facility that the perimeter is
less than the circumscribed circumference, provided he understands the
meaning of the words? It is evident that he will not. What necessity,
then, of repeating the general principle? Is it to add any thing to the
particular conception? Certainly not; because nothing can be clearer
than the following propositions: the perimeter of the polygon is a
sum of right lines; the circumference is a sum of arcs or curves;
what the general principle does, is to call attention to a phase of
the particular conception, so that what otherwise could not be seen
in it may be seen on reflection. The certainty of the conclusion does
not depend on the general principle; because, from thinking on the
relations of greater and less only with respect to the right lines of
the perimeter and the arcs, the sum of which forms the circumference,
any one would have inferred the same thing.
This example also tends to prove that the enthymema is not a mere
abbreviation of words; and it shows why we employ it in reasoning upon
matters familiar to the understanding. In any one of the conceptions
we see all that is necessary for the consequence; and, therefore,
one premise suffices, as in it the other is included rather than
understood. A beginner may say: the arc is greater than the chord,
because the curve is greater than the right line; but when familiarized
with geometrical ideas, he will simply say, the arc is greater than
the chord; he will see the idea of the curve in that of the arc,
and the idea of the right line in that of the chord, without need
of decomposition. If the arc is greater than its chord, this is not
because every curve is greater than the corresponding right line. Did
the abstract idea of curve not exist, and were this particular arc of a
circle the only curve thought of; did the abstract idea of right line
not exist, and were this particular chord the only right line thought
of, it would still, as at present, be true that the arc is greater than
the chord.
275. When treating of the _necessary_ relations of things, the general
principles, the middle terms, and all the auxiliaries to reasoning
furnished by logic, are only inventions of art to make us reflect upon
the conception of the thing, and see in it what otherwise we should
not see. Hence our judgments on necessary objects are in some sense
analytical; and Kant equivocates, when he says there are synthetic
judgments not dependent on experience. Without experience we have only
the conception of the thing. We do not pretend that all propositions
express such a relation between the subject and the predicate, that the
conception of the former will always give that of the latter; but we do
hold, that the reason of this insufficiency is the incompleteness of
the conception, either in itself, or in relation to our comprehension.
But if we suppose the conception complete in itself, and a due capacity
in our intellect to understand whatever it contains, we shall find in
the conception all that can be the object of science.
276. An example from mathematics will make this clearer. Large works on
geometry are filled with explanations, demonstrations, and applications
of the properties of the triangle. The conceptions of right lines,
and the angles formed by them, enter into the conception of the
triangle. We ask, can all the explanations and demonstrations of the
properties of triangles in general ever go beyond the ideas of right
lines and angles? No. For the new elements introduced would be foreign
to the triangle, and would consequently change its nature. Necessary
relations neither admit of more nor of less, neither additions nor
subtractions of any sort; what is, is, and nothing more. In passing
from the triangle in general to its different species, such as
equilateral, isosceles, right angled, scalene, it is to be observed
that the demonstration must rigorously attend to what is contained
in the general conception, modified by the determining properties of
the species, that is, the equality of the three sides, of two, the
inequality of all, the supposition of a right angle, and others.
277. What we are now explaining is clearly seen in the application
of algebra to geometry. A curve is expressed by a formula containing
the conception of the curve, or its essence. The geometrician, to
demonstrate the properties of the curve, does not need to go out of
this formula; it is a touch-stone in his hand, and he finds in it all
that he wants. He inscribes triangles, or other figures in the curve,
draws right lines from it to points without, but never goes out of the
conception expressed in the formula; he decomposes it, and finds in it
what before he had not discovered.
In this equation z^2 = (e^2/E^2)(2Ex-x^2), we find the expression of
the relations which constitute the ellipse; E expresses the greater
semi-axis, e the lesser, z the ordinates, and x the abscissas. With
this equation variously developed and transformed, the properties of
the curve are determined; it shows, with the help of constructions,
that the new property is contained in the conception, and to find it,
we have only to analyze it.
If we suppose an intelligence capable of conceiving the essence of the
curve, by an immediate intuition of the law governing the inflection
of points, without the necessity of referring it to any line, whether
one axis instead of two suffices, or in any other manner not even
imaginable by us; this intelligence will not need to follow all the
evolutions which we have made in demonstrating the properties of the
curve; for it will perceive them to be clearly contained in the very
conception of the curve. This supposition is not arbitrary; we see it
realized every day, though on a smaller scale. An ordinary geometrician
conceives a curve as also does Pascal; but while Pascal at a glance
sees the most recondite properties of the curve in this conception,
an ordinary geometrician sees only after long study its most common
properties. Kant made no account of this doctrine, and therefore could
not solve the problem of pure synthetic judgments: had he examined the
subject more profoundly he would have seen that, strictly speaking,
there are no such judgments; and instead of wearing out his genius in
attempting to solve an insolvable problem, he would have abstained from
raising it.(26)