CHAPTER XXVI.
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.
CHAPTER XXVII.
CONTINUATION OF THE SAME SUBJECT
269. It is even ridiculous to say that the cognitions of the sublimest philosophers may be reduced to this equation: A is A. This, absolutely speaking, is not only false, but contrary to common sense; but it is neither contrary to common sense nor false to say that all cognitions of mathematicians are perceptions of identity, which, presented under different conceptions, undergoes infinite variations of form, and so fecundates the intellect and constitutes science. For the sake of greater clearness we will take an example, and follow one idea through all its transformations.
270. The equation circle = circle (1) is very true, but not very lucid, since it serves no purpose, because there is identity not only of ideas but likewise of conceptions and expression. To have a true progress in science we must not only change the expression, but also vary in some way the conception under which the identical thing is presented. Thus, if we abbreviate the above equation in this form: C = circle (2), we make no progress, unless with respect to the purely material expression. The only possible advantage of this is to assist the memory, as instead of expressing the circle by a word, we express it by its initial letter, C. Why is this? Because the variety is in the expression, not in the conception. If, instead of considering the identity in all its simplicity in both members of the equation, we give the value of the circle with reference to the circumference, we shall have C = circumference × 1/2 R (3), that is, the value of the circle is equal to the circumference multiplied by one-half the radius. In the equation (3) there is identity as in (1) and (2), because it is affirmed in it that the value expressed by C is the same as that expressed by circumference × 1/2 R; just as in the other two it is expressed that the value of the circle is the value of the circle. But is this equation different from the other two? It is very different. What is the difference? The first two simply express the identity conceived under the same point of view; the circle expressed in the second member excites no idea not already excited by the first; but in the last, the second member expresses the same circle indeed, but in its relations with the circumference and radius; and, consequently, besides containing a sort of analysis of the circle, it records the analysis previously made of the idea of the circumference in relation to the idea of radius. The difference is not, then, solely in the material expression, but in the variety of conceptions under which the same thing is presented.
Calling the value of the relation of the circumference with the diameter N, and the circle C, the equation becomes: C = NR2 (4). Here, also, there is identity of value; but we discover a notable progress in the expression of the second member, in which the value of the circle is given, freed from its relations with the value of the circumference, and dependent solely on a numerical value, N, and a right line, which is the radius. Without losing the identity, and only by a succession of perceptions of identity, we have advanced in science, and starting from so sterile a proposition as circle = circle, we have obtained another, by means of which we may at once determine the value of any circle from its radius.
Leaving elemental geometry, and considering the circle as a curve referred to two axes, with respect to which its points are determined, we shall have Z = 2Bx-x2(5); Z expressing the value of the ordinate; B the constant part of the axis of abscissas; and x the abscissa corresponding to Z. We have here a still more notable progress of ideas: in both members we now express the value, not of the circle, but of lines, by which we may determine all points of the curve; and we easily conceive that this curve, which was contained in the figure whose properties we determined in elemental geometry, may be conceived under such a form as belongs to a genus of curves, whereof it constitutes a species by the particular relations of the quantities 2x and B; thus modifying the expression by adding a new quantity, combined in this or that manner, we may obtain a curve of another species. If, therefore, we would determine the value of the surface contained in this circle, we may consider it, not solely with respect to the radius, but to the areas comprised between the various perpendiculars the extremities of which determine points of the curve and are called ordinates. It results from this, that the same value of the circle may be determined under various conceptions, although this value is at all times identical; the transition from one conception to another is the succession of the perceptions of identity presented under different forms.
Let us now consider the value of the circle dependent on the radius: this will give us C = function x (6). This equation enables us to conceive the circle under the general idea of a function of its radius, or of x, and consequently authorizes us to subject it to all the laws to which a function is subject, and leads us to the properties of their differentials, limits, and relations. By this equation we enter into infinitesimal calculus, the expressions of which present identity under a form which records a series of conceptions of long and profound analysis. Thus, expressing the differential of the circle by dc, and its integral by S. dc, we shall have C = S. dc, (7), an equation in which are expressed the same values as in circle = circle, but with this difference, that the equation (7) records immense analytical labors: it results from a long succession of conceptions of integral calculus, of differentials, and limits of the differentials of the functions, of the application of algebra to geometry, and of a multitude of elementary geometrical notions, algebraical rules and combinations, and of whatever else was needed to arrive at this result. Therefore, when we find the integral of the differential, and obtain by integration the value of the circle, it would clearly be most extravagant to affirm that the integral equation is nothing more than the equation circle = circle; but it is not so to say that at bottom there is identity, and that the diversity of expression to which we have come, is the result of a succession of perceptions of the same identity presented under different aspects. Supposing the conceptions, through which it has been necessary to pass, to be A, B, C, D, E, M, the law of their scientific connection may be thus expressed: A = B, B = C, C = D, D = E, E = M; therefore A = M.
271. What we have just explained cannot be well understood unless we recall some characteristics of our intellect, in which is found the reason of so great anomalies. Our intellect is so weak as to perceive things only successively: only after much study does it see what is contained in the clearest ideas. Hence a necessity, to which corresponds with admirable harmony a faculty that satisfies it: the necessity is of conceiving under various, and different, as well as distinct, forms, even the simplest things: the faculty is that of decomposing the conception into many parts, and multiplying in the order of ideas what in that of reality is only one. This faculty of decomposition would be useless were not the intellect, in passing through the succession of conceptions, to find means of connecting and retaining them: otherwise it would continually lose the fruit of its labors; it would slip from its hands as fast as it grasped it. Happily it has this means in signs either written, spoken, or thought; those mysterious expressions which at times not only designate an idea, but also are the compendium of the labors of a whole life, and perhaps of a long series of ages. When the sign is presented to us, we do not see certainly and with full clearness all that it expresses, nor why the expression is legitimate; but we know confusedly the meaning therein contained; we know that in case of necessity, it is enough for us to follow the thread of the perceptions through which we have passed, thus going back even to the simplest elements of science. In making calculations, the most eminent mathematician does not clearly see the meaning of the expressions he uses, except as they relate to the object before him; but he is certain that they do not deceive him, that the rules by which he is guided are sure; because he knows that at another time he established them by incontestible demonstrations. The progress of a science may be compared to a series of posts on which the distances of a road are marked: he who marked the numbers on the posts uses them without necessity of recalling the operations which led him to mark the quantity before him; he is satisfied with knowing that the operations were well made, and that he wrote the result correctly.
272. The proof of this necessity of decomposition, besides being fully established by the above example, is found in the elements of all instruction, where, under a form of demonstration, it is necessary to explain propositions which express simply the definitions or axioms that have been before established. For example: we find in the elementary works on geometry this theorem: all the diameters of a circle are equal; and we must, if we would have beginners understand it, give a demonstrative form to that which neither is nor can be any thing more than an explanation, and is almost a repetition of the idea of the circle. When we describe a circle, we fix a point around which we revolve a line called the radius; since then the diameter is nothing more than the sum of two radii continued in the same right line, the mere enunciation of the theorem would seem sufficient to show that it is evidently contained in the idea of the circle, and is as a sort of repetition of the postulate, on which the construction of the curve is founded: still it is not so, and it must be explained as if it were a proof; we must show the diameter to be equal to two radii, these radii to be equal, and at times repeat that this is supposed in its construction: in a word, it is necessary to employ many conceptions to show a truth, which ought to have been known by the simple intuition of one alone, as is the case when the geometrical powers of the intellect have acquired a certain strength and robustness.
273. We may now appreciate at its just value, the opinion of Dugald Stewart, who, in his Elements of the Philosophy of the Human Mind, says: "It may be fairly questioned, too, whether it can, with strict correctness, be said of the simple arithmetical equation, 2 plus 2 = 4, that it may be represented by the formula A = A. The one is a proposition asserting the equivalence of two different expressions; to ascertain which equivalence may, in numberless cases, be an object of the highest importance. The other is altogether unmeaning and nugatory, and cannot, by any possible supposition, admit of the slightest application of a practical nature. What opinion then shall we form of the proposition A = A, when considered as the representative of such a formula as the binomial theorem of Sir Isaac Newton? When applied to the equation 2 plus 2 = 4, (which in its extreme simplicity and familiarity is apt to be regarded in the light of an axiom:) the paradox does not appear to be so manifestly extravagant; but, in the other case, it seems quite impossible to annex to it any meaning whatever."22 This philosopher does not observe that the pretended extravagance arises from his wrong interpretation of his adversaries' opinion. No one ever thought of denying the importance of the discoveries which prove different expressions equivalent: no one doubts that Newton's formula of the binomial is a great advance upon the formula A = A: but the question consists not in this, but in seeing whether Newton's formula of the binomial is any thing more than the expression of identical things; and whether even the merit of the expression is or is not the fruit of a series of perceptions of identity. Were the question presented under Dugald Stewart's point of view, it would be unworthy of discussion: for philosophy should not dispute upon things that are ridiculous as well as absurd.