CHAPTER XXVIII.
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 z2 = (e2/E2)(2Ex-x2), 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)
CHAPTER XXIX.
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.
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
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
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.