Front Page Titles (by Subject) CHAPTER I.: COMPOUND QUANTITATIVE REASONING. - The Principles of Psychology
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CHAPTER I.: COMPOUND QUANTITATIVE REASONING. - Herbert Spencer, The Principles of Psychology 
The Principles of Psychology (London: Longman, Brown, Green and Longmans, 1855).
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COMPOUND QUANTITATIVE REASONING.
§ 16. An analysis conducted in a truly systematic manner, must commence with the most complex phenomena of the series to be analysed: must seek to resolve these into the phenomena that stand next in order of complexity: must proceed after like fashion with the less complex phenomena thus disclosed: and so, by successive decompositions, must descend step by step to the simpler and more general phenomena; reaching at last the simplest and most general. As applied to Psychology this mode of procedure, though perhaps, if patiently pursued, the best in its results, is beset with difficulties. The most ordinary operations of consciousness are sufficiently perplexing to those whose thinking powers have not been well disciplined; and its highly involved operations, if dealt with at the outset, may naturally be expected to tax the powers even of the habitual student. Disadvantageous, however, in this respect, as such an arrangement of the subject may be, both to reader and writer, it is so much better fitted than any other for the adequate presentation of the general law which it is the object of this Special Analysis to disclose, that I do not hesitate to adopt it. A little patience only is asked during the perusal of the next few chapters; which will be comparatively abstract and uninteresting. What he finds in them that is not very comprehensible, the reader must pass over until subsequent chapters give the key to it. Should some of the matters discussed seem to him unimportant, perhaps he will suspend his judgment until their bearing upon the doctrine at large becomes visible. And if, as is very possible, he should not perceive the reason for interpreting certain mental phenomena after a particular fashion—for insisting upon a special mode of regarding them and defining them—he is requested to take the analyses upon trust; in the belief that he will presently see them to be the true ones, and eventually see them to be the only possible ones. Thus much premised, let us pass to our immediate topic—Compound Quantitative Reasoning.
§ 17. Of ratiocinative acts exhibiting a high degree of complexity, the following will fitly serve as an example. Suppose an engineer who has constructed a bridge—say an iron tubular bridge—of given span, and who finds that it is just strong enough to bear the strain it is subject to (a strain resulting mainly from its own weight)—suppose such an engineer is required to construct another bridge of like nature, but of double the span. Possibly it will be supposed that for this new bridge he might simply magnify the previous design in all its particulars—simply make the tube double the depth, double the width, and double the thickness, as well as double the length. But, duly acquainted as he is with mechanical principles, he sees that a bridge so proportioned would not support tself—he infers that the depth, or the thickness of the metal, or both, must be more than double. Now by what acts of thought does he reach this conclusion? He knows, in the first place, that the bulks of similar masses of matter are to each other as the cubes of the linear dimensions; and that consequently, when the masses are not only similar in form, but of the same material, the weights also, are as the cubes of the linear dimensions. He knows, too, that in similar masses of matter which are subject to compression or tension, or, as in this case, to the transverse strain, the power of resistance varies as the squares of the linear dimensions. Hence he sees that if another bridge be built proportioned in all respects, exactly like the first, but of double the size, the weight of it—that is, the gravitative force, or force tending to make it bend and break—will have increased as the cubes of the dimensions; while the cohesive force—that is, the sustaining force, or force by which the breaking is resisted—will have increased only as the squares of the dimensions: and that, therefore, the bridge will give way. Or, to present the reasoning in a more formal manner, he sees that the—
whilst at the same time he sees that the—
Whence he infers that as the destroying force has increased in a much greater ratio than the sustaining force, the larger tube cannot sustain itself; seeing that the smaller one has no excess of strength.
But now, leaving out of sight the various acts by which the premisses are reached and by which the final inference is drawn, let us consider the nature of the particular mental process implied by the cognition that the ratio between the sustaining forces in the two tubes, must differ from the ratio between the destroying forces: for this process it is which here concerns us as an example of the most complex ratiocination. There is, be it observed, no direct comparison between these two ratios. How then is it known that they are unlike? It is known by the intermediation of two other ratios, to which they are severally equal.
The ratio between the two sustaining forces equals the ratio 12: 22. The ratio between the two destroying forces equals the ratio 13: 23. And as it is seen that the ratio 12: 22 is unequal to the ratio 13: 23; it is by implication seen, that the ratio between the sustaining forces is unequal to the ratio between the destroying forces. What now is the nature of this implication? or rather—What is the mental act by which this implication is perceived? It is manifestly not decomposable into steps. Though involving many elements, it is a single intuition: and if expressed in an abstract form, amounts to the axiom—Ratios which are severally equal to certain other ratios that are unequal to each other, are themselves unequal: or, reducing it to a still more abstract form—Relations which are severally equal to certain other relations that are unequal to each other, are themselves unequal.
I do not propose here to enter upon an analysis of this highly complex intuition; but simply present it as an example of the more intricate acts of thought which occur in Compound Quantitative Reasoning—an example to which the reader may presently recur if he pleases. A nearly allied but somewhat simpler intuition will better serve to initiate our analysis.
§ 18. This intuition is embodied in an axiom which has not, so far as I am aware, been specifically stated; though it is taken for granted in Proposition XI. of the fifth book of Euclid; in which, as we shall presently see, the wider of two assumptions is assigned in proof of the narrower. This proposition, which is to the effect that “Ratios which are equal to the same ratio are equal to one another,” it will be needful to quote in full.∗ It is as follows:—
Take of A, C, E, any equimultiples whatever G, H, K; and of B, D, F, any equimultiples whatever L, M, N.† Therefore since A is to B as C to D, and G, H, are taken equimultiples of A, C, and L, M, of B, D; if G be greater than L, H is greater than M; and if equal, equal; and if less, less. Again, because C is to D as E to F, and H, K, are equimultiples of C, E; and M, N, of D, F; if H be greater than M, K is greater than N; and if equal, equal; and if less, less. But if G be greater than L, it has been shown that H is greater than M; and if equal, equal; and if less, less: therefore, if G be greater than L, K is greater than N; and if equal, equal; and if less, less. And G, K are any equimultiple whatever of A, E; and L, N, any whatever of B, F; therefore as A is to B so is E to F.”
Let us now, for the sake of simplicity, neglect all such parts of this demonstration as consist in taking equimultiples and drawing the immediate inferences; and inquire by what process is established that final relation amongst these equimultiples which serves as the premiss for the desired conclusion. And to make the matter the clearer, let us here separate these equimultiples from the original magnitudes; and consider by itself the argument concerning them.
From the hypothesis and the construction, it is proved that if G be greater than L, H is greater than M; and if equal, equal; and if less, less: and, similarly, that if H be greater than M, K is greater than N; and if equal, equal; and if less, less. Whence it is inferred (and here comes the petitio principii) that if G be greater than L, K is greater than N; and if equal, equal; and if less, less. That this is an assumption, under a less definite form, of the very thing to be proved, will readily be seen on simplifying the verbiage. For what, in general language, is the fact established when it is shown that if G be greater than L, H is greater than M; and if equal, equal; and if less, less? The fact established is, that whatever relation subsists between G and L, the same relation subsists between H and M: whether it be a relation of superiority, of equality, or of inferiority: in other words, that so far as they are defined, the relations G to L and H to M are equal. So, too, with the relations H to M and K to N, which are proved to be equal in respect to the characteristics predicated of them. And then, when it has been shown that the relation G to L equals the relation H to M; and that the relation K to N also equals it; it is said that therefore the relation G to L equals the relation K to N. Which therefore, involves the assumption that relations which are equal to the same relation, are equal to each other—an assumption differing only in its higher generality from the proposition that “Ratios which are equal to the same ratio, are equal to each other,”—an assumption which itself needs proof, if the proposition to be established by it needs proof.
The only rejoinder which it seems possible to make to this criticism is, that in asserting that if G be greater than L, H is greater than M; and if equal, equal; and if less, less; it is not asserted that the relation G to L equals the relation H to M: for that, without negativing the assertion, G may be supposed to exceed L in a greater proportion than H exceeds M; and that, in this case, the relations will not be equal. One reply is, that the possibility of this supposition arises from the extreme vagueness of the definition of proportional magnitudes; and that it needs only to seize the true meaning of that definition, to see that no such assumption is permissible. Not to dwell upon this, however, it is a sufficient answer to the objection, that though the relations G to L, and H to M, are left to some extent indeterminate, and cannot therefore be called equal in an absolute sense, yet, so far as they are determinate, they are equal; and that if it be allowable to assume of indeterminate relations, that in the respects in which they are equal to the same, they are equal to each other, it must be allowable to assume as much of determinate relations. This will be clearly perceived on considering the matter under any one of its concrete aspects. Suppose it to have been shown that if G be greater than L, H is greater than M; and that if H be greater than M, K is greater than N; then it is said that if G be greater than L, K is greater than N. What now are here the premisses and inference? It is argued that the first relation being like the second in a certain particular (the superiority of its first magnitude); and the third relation being also like the second in this particular; the first relation must be like the third in this particular. If now it be allowable to assume that two relations which are severally like a third in any particular, are like each other in that particular; it is allowable to assume as much when they are like in all particulars, or are equal. The one truth is not more self-evident than the other. The act of thought is the same in each case; and is valid either in both or in neither. Evidently, then, the reasoning involves a disguised petitio principii.
Thus the general truth that relations which are equal to the same relation are equal to each other—a truth of which the foregoing proposition concerning ratios is simply one of the more concrete forms—must be regarded as an axiom. Like its prototype—things that are equal to the same thing are equal to each other—it is incapable of proof. Seeing how closely, indeed, the two are connected both in nature and origin, perhaps some will contend that the one is but a particular form of the other, and should be included under it—that a relation is simply one species of thing; and that what is true of all things is, by implication, true of relations. Much as may be said in support of this position, it is, however, necessary, as will presently be seen, to specifically enunciate this general law in respect to relations, even if it be held derivative. At the same time the criticism serves to bring into yet clearer view the axiomatic nature of the law. For whether it be or be not true that a relation must be regarded as a thing, it is unquestionably true that in any intellectual process serving to establish the general fact—Relations that are equal to the same relation are equal to each other—the concepts dealt with are the relations, and not the objects between which the relations subsist; that the equality of these relations can be perceived only by making them the objects of thought, and not by thinking of the related objects; and that hence the axiom, being established by the comparison of three concepts, is established by just the same species of mental act as though it referred to substantive things instead of relations.
The truth—Relations that are equal to the same relation are equal to each other—which we thus find is known by an intuition,∗ and can only so be known, underlies many important geometrical truths. An examination of the first proposition in the sixth book of Euclid, and of the deductions made from it in succeeding propositions, will show that there is a large class of theorems having this axiom for their basis—theorems which are at present ostensibly based upon the demonstration above shown to be fallacious.
§ 19. But this axiom has far wider and far more important applications. It is the foundation of all Mathematical Analysis. Alike in working out the simplest algebraic equation, and in performing those higher analytical processes of which algebra is the root, it is the one thing perpetually taken for granted. Whilst other axioms are specifically stated, this axiom is tacitly assumed at every step. It is true that the assumption is limited to that particular case of the axiom in which its necessity is so self-evident as to be almost unconsciously recognized; but it is not the less true that this assumption cannot be made without involving the axiom in its entire extent. The successive transformations of an equation we shall find to be linked together by acts of thought, of which this axiom expresses the most general form. Let us take an example and analyse it.
Now it may seem that the only assumptions involved in these three steps are—first, that if equals be added to equals, the sums are equal; second, that the square roots of equals are equals; and third, that if equals be taken from equals, the remainders are equal. But a little reflection will show that the several results reached in virtue of these assumptions lead to no conclusion if they stand alone: and they cannot be co-ordinated to any purpose without some further assumption being made. What is that assumption? As at present written, there is nothing to mark any connexion between the first form of the equation and the last. Manifestly, however, the validity of the inference x = 2, depends upon there being some perfectly specific connection between it and the original premiss x2 + 2x = 8; and this connection implies connections between the intermediate steps. This premised, the real process of thought involved will be at once recognized on inserting the required symbols, thus:—
That only in virtue of the successive cognitions thus represented does the conclusion legitimately follow from the original premiss, cannot fail to be seen, on considering that the argument is worthless unless the value of x in the last form of the equation, is the same as its value in the first; and that this implies the preservation throughout of a constant relation between the function of x and the function of its value under all their transformations—a constancy which is more strictly expressed by saying that their successive relations are equal. But now arises the question—In virtue of what assumption is it that the final relation subsisting between the two sides of the equation is asserted to be equal to the initial one? On this assumption it is that the worth of the conclusion ultimately depends; and for this assumption no warrant is assigned. I answer, the warrant for this assumption is the axiom—Relations that are equal to the same relation are equal to each other. Probably, at first sight, it will not be altogether manifest that this axiom is involved. It needs but to simplify the consideration of the matter, however, to render the fact apparent. Suppose that we represent the successive forms of the equation by the letters A, B, C, D. If now A, B, C, D had represented substantive things; and if, when it had been shown that A was equal to B, and B was equal to C, and C was equal to D, it had been concluded that A was equal to D; what would have been assumed? There would have been two assumptions of the axiom—Things that are equal to the same thing are equal to each other: one to establish the equality of A and C by the intermediation of B; and one to establish the equality of A and D by the intermediation of C. Now, the fact that A, B, C, D do not represent things, but represent relations between things, cannot be supposed fundamentally to alter the intellectual process by which the equality of the first and last is recognized. If, when A, B, C, D represent things, the equality of the first and last can be shown only by means of the axiom—Things that are equal to the same thing are equal to each other; then, manifestly, when A, B, C, D represent relations, the equality of the first and last can be shown only by means of the axiom—Relations that are equal to the same relation are equal to each other.
It is true that in this case the relations dealt with are relations of equality; and the great simplification hence resulting may produce some hesitation as to whether the process of thought really is the one described. Perhaps it will be argued that the successive forms of the equation being all, in virtue of their essential nature, relations of equality, it is known by an act of direct intuition that any one of them is equal to any other; or that if an axiom be appealed to, it is the axiom—All relations of equality are equal to each other. It must, without doubt, be conceded, that relations of equality, unlike all other relations and unlike all magnitudes, are in their very expression so defined as that the equality of any one of them to any other may be foreknown. But admitting this, the objection may be met in two ways. In the first place, it may be replied that every relation of equality can be known to equal every other relation of equality only through the cognition—Relations that are equal to the same relation are equal to each other. For like all general truths it must be originally derived from particular experiences: the particular experience forming the first step to it must be a perception of the equality of some two relations of equality: further progress towards the general truth requires a perception of the equality of one of these to some third relation of equality: and now be it observed that any further carrying out of this process to a fourth and a fifth, cannot lead to the generalization that all relations of equality are equal, until they have been compared in some other than their serial order. As in the case of magnitudes that have been recognized as successively equal, each to the next, the assertion that they are all equal implies an act of thought in which some two that are not adjacent have been perceived to be equal in virtue of their common equality to an intermediate third; so, in the case of relations, however obviously they are all equal, a like act of thought must be gone through. Yet a simpler proof is assignable. As the truth—All relations of equality are equal to each other, is more general than the truth—Relations of equality, that are equal to the same relation of equality are equal to each other; it must include this last; and cannot be reached without presupposing it. If this reply be considered inconclusive—as it will possibly be by those who contend for innate forms of thought—the second reply may be given; namely, that the relation subsisting between the two sides of an equation when reduced to its final form, is known to be a relation of equality only in virtue of its affiliation upon the original relation of equality, by means of all the intermediate relations. Strike out in the foregoing case, the several transformations which link the first and last forms of the equation together, and it cannot be logically known that x is equal to 2. If then this ultimate relation can be known to equal the first, only because it is known to equal the penultimate relation, and the penultimate relation to equal the antepenultimate, and so on; it is manifest that the affiliation of the last relation upon the first, unavoidably involves the axiom—Relations that are equal to the same relation are equal to each other.
It must be admitted that in cases like these in which this general axiom is applied to relations of equality, it seems very much like a superfluity—a formula that is more circuitous than the intuition it represents. And it is doubtless true that in such cases the cognition seems to merge into a simpler order of cognitions, from which it is with difficulty distinguishable. Nevertheless, I think the arguments adduced warrant the belief that the mental process described is gone through; though perhaps almost automatically: and indeed, if, when the relations are not relations of equality, the intuition expressed by this axiom is consciously achieved, it seems unavoidably to follow, that when the relations are those of equality, it is also achieved, even if unconsciously. And for this belief yet further warrant will be found, when, under another head, we come to consider the case of inequations—a case in which no such source of difficulty exists, and yet in which the process of thought is of like nature.
§ 20. Leaving here its several applications, and turning to consider the axiom itself, as being predicable alike of all relations, whether of equality or any degree of inequality, we have now to inquire by what process of thought it is known that relations which are equal to the same relation are equal to each other. We have seen that the fact is not demonstrable, but can be reached only by direct intuition. What is the character of this intuition?
Clearly if the equality of the first and third relations cannot be established by an act decomposable into steps, but can be established only by a single act, that single act must be one in which the first and third relations are brought into immediate relation before the consciousness. Yet any direct comparison of the first and third without the intermediation of the second would avail nothing; and any intermediation of the second would seem to involve a thinking of the three in their serial order—first, second, third; third, second, first—which, even could it be called a single act, would not bring the first and third into the immediate relation required. Hence, as neither a direct comparison of the first and third, nor a serial comparison of the three, can fulfil the requirement, it follows as the only remaining alternative, that they must be compared in couples. And this is what is really done. By the premisses it is known that the first and second relations are equal; and that the second and third relations are equal. There are, therefore, presented to consciousness, two relations of equality between relations. The direct intuition is that these two relations of equality are themselves equal. And as these two relations of equality possess a common term, the intuition that they are equal, involves the equality of the remaining terms. The nature of this mental process will, however, be best expressed by symbols. Suppose the several relations to stand thus:—A : B = C : D = E : F, then the act of thought by which the equality of the first and third relations is recognized may be symbolized thus:—∗
Careful introspection will, I think, confirm the inference that this represents the mental process gone through—that the first and second relations, contemplated as equal, form together one concept; that the third and second, similarly contemplated, form together another concept; and that, in the intuition of the equality of these concepts, the equality of the terminal relations is implied: or that to define its nature abstractedly—the axiom expresses an intuition of the equality of two relations between relations.
Probably to the minds of some readers, this analysis will not at once commend itself. Indeed, as at first remarked, it is an inconvenience attendant on commencing with the most complex intellectual processes, that the propriety of formulating them after a certain manner cannot be clearly perceived until the analysis of the simpler intellectual processes has shown why they must be thus formulated. After reading the next few chapters, the truth of the above conclusion will become manifest. In the meantime, though it may not be positively recognized as true by its perceivable correspondence with the facts of consciousness, it may yet be negatively recognized as true by contemplating the impossibility, lately shown, of establishing the equality of the first and last relations by any other intellectual act.
Before ending the chapter it should be observed, that the relations thus far dealt with are relations of magnitudes; and, properly speaking, relations of homogeneous magnitudes; or, in other words, ratios. In the case of the geometrical reasoning quoted from the fifth book of Euclid, this fact is definitely expressed; and though in the case of the algebraical reasoning it may at first be thought that the magnitudes dealt with are not homogeneous—seeing that the same equation often includes at once magnitudes of space, time, force, value,—yet it needs but to consider that these magnitudes can be treated algebraically only by reducing them to the common denomination of number—only by considering them as abstract magnitudes of the same order, to at once see that the relations dealt with are really those subsisting between homogeneous magnitudes—are really ratios; and might have been so named throughout. The motive for constantly speaking of them under the general name, relations, of which ratios are but one species, will be understood when it is seen, as it presently will be, that only when regarded under this most general form do they permit the intellectual processes by which they are co-ordinated to be brought under the same category with other acts of reasoning.
[∗]In some editions the enunciation runs,—“Ratios which are the same to the same ratio are the same to each other;” but the above is much the better.
[†]For the aid of those who have not lately looked into Euclid, it will be well to append the definition of proportionals, which is as follows:—“If there be four magnitudes, and if any equimultiples whatsoever be taken of the first and third, an any equimultiples whatsoever of the second and fourth, and if, according as the multiple of the first is greater than the multiple of the second, equal to it, or less, the multiple of the third is also greater than the multiple of the fourth, equal to it or less; then, the first of the magnitudes is said to have to the second the same ratio that the third has to the fourth.“
[∗]Here, and throughout, I use this word in its ordinary acceptation as meaning any cognition reached by an undecomposable mental act; whether the terms of that cognition be presented or represented to consciousness. Sir William Hamilton, in classing knowledge as representative and presentative or intuitive, restricts the meaning of intuition to that which is known by external perception. If, when a dog and a horse are looked at it is seen that one is less than the other, the cognition is intuitive; but if a dog and a horse are imagined, and the inferior size of the dog perceived in thought, the cognition is not intuitive in Sir William Hamilton's sense of the word. As, however, the act by which the relation of inferiority is established in consciousness, is alike in the two cases, the same term may properly be applied to it. And I draw further reason for using the word in its common acceptation, from the fact that the line of demarcation between presentative and representative knowledge cannot be maintained. Though there is much knowledge that is purely representative, there is none that is purely presentative. Every perception whatever involves more or less of representation. And this is asserted by Sir William Hamilton himself, when, in opposition to Royer Collard's doctrine, that perception excludes memory, he writes, “On the contrary, I hold, that as memory, or a certain continuous representation, is a condition of consciousness, it is a condition of perception.”
[∗]The sign (:) used in mathematics to express a ratio, is, in this formula, as in many that follow, placed somewhat unusually in respect to the letters it connects, with a view to convenience of reading. And it may here be explained in preparation for subsequent chapters, that this sign, though here marking, as it commonly does, a ratio, or quantitative relation, will hereafter be employed to mark any relation.