The Online Library of Liberty

A project of Liberty Fund, Inc.

• Preface.
• Part I.: Special Analysis.
• Chapter I.: A Datum Wanted.
• Chapter II.: The Universal Postulate.
• Chapter III.: Its Corollaries.
• Chapter IV.: Our Present Position.
• Part II.: Special Analysis.
• Chapter I.: Compound Quantitative Reasoning.
• Chapter II.: Compound Quantitative Reasoning (continued).
• Chapter III.: Imperfect and Simple Quantitative Reasoning.
• Chapter IV.: Quantitative Reasoning In General.
• Chapter V.: Perfect Qualitative Reasoning.
• Chapter VI.: Imperfect Qualitative Reasoning.
• Chapter VII.: Reasoning In General.
• Chapter VIII.: Classification, Naming, and Recognition.
• Chapter IX.: The Perception of Special Objects.
• Chapter X.: The Perception of Body As Presenting Dynamical, Statico-dynamical, and Statical Attributes. ∗
• Chapter XI.: The Perception of Body As Presenting Statico-dynamical and Statical Attributes.
• Chapter XII.: The Perception of Body As Presenting Statical Attributes.
• Chapter XIII.: The Perception of Space.
• Chapter XIV.: The Perception of Time.
• Chapter XV.: The Perception of Motion.
• Chapter XVI.: The Perception of Resistance.
• Chapter XVII.: Perception In General.
• Chapter XVIII.: The Relations of Similarity and Dissimilarity.
• Chapter XIX.: The Relations of Cointension and Non-cointension.
• Chapter XX.: The Relations of Coextension and Non-coextension.
• Chapter XXI.: The Relations of Coexistence and Non-coexistence.
• Chapter XXII.: The Relations of Connature and Non-connature.
• Chapter XXIII.: The Relations of Likeness and Unlikeness.
• Chapter XXIV.: The Relation of Sequence.
• Chapter XXV.: Consciousness In General.
• Chapter XXVI.: Results.
• Part III.: General Synthesis.
• Chapter I.: Method.
• Chapter II.: Connexion of Mind and Life.
• Chapter III.: Proximate Definition of Life.
• Chapter IV.: The Correspondence Between Life and Its Circumstances.
• Chapter V.: The Degree of Life Varies As the Degree of Correspondence.
• Chapter VI.: The Correspondence As Direct and Homogeneous.
• Chapter VII.: The Correspondence As Direct But Heterogeneous.
• Chapter VIII.: The Correspondence As Extending In Space.
• Chapter IX.: The Correspondence As Extending In Time.
• Chapter X.: The Correspondence As Increasing In Speciality.
• Chapter XI.: The Correspondence As Increasing In Generality.
• Chapter XII.: The Correspondence As Increasing In Complexity.
• Chapter XIII.: The Co-ordination of Correspondences.
• Chapter XIV.: The Integration of Correspondences.
• Chapter XV.: The Correspondences In Their Totality.
• Part IV.: Special Synthesis.
• Chapter I.: The Nature of Intelligence.
• Chapter II.: The Law of Intelligence.
• Chapter III.: The Growth of Intelligence.
• Chapter IV.: Reflex Action.
• Chapter V.: Instinct.
• Chapter VI.: Memory.
• Chapter VII.: Reason.
• Chapter VIII.: The Feelings.
• Chapter IX.: The Will.

CHAPTER IV.

QUANTITATIVE REASONING IN GENERAL.

§ 27. Leaving details, and considering the facts under their most general aspect, it is to be remarked that Quantitative Reasoning involves, with more or less constancy, the three ideas, coextension, coexistence and connature:∗ or to speak less accurately, but more comprehensibly—sameness in the quantity of space occupied; sameness in the time of presentation to consciousness; and sameness in kind. The germ out of which Quantitative Reasoning grows—the simple intuition of the equality of two magnitudes, necessarily involves all these: seeing that there can be no comparison between them unless they are of the same kind; and their coextension cannot be perceived unless they are coexistent. So too with geometry, throughout its entire range. Each of its propositions predicates the coextension or non-coextension of two or more connatural things which coexist: and its demonstrations proceed by asserting that certain coexistent, connatural things are invariably coextensive, or the reverse; or that certain connatural and coextensive things invariably coexist with certain other things. When the propositions are numerical, and when, as frequently happens in Algebra and the calculus generally duration is one of the elements dealt with, it would appear that coexistence is not involved; and further, that when force and value are the other elements of the question, there is not even any implication of coextension. These, however, are illusions resulting from the abstract character of numerical symbols. Simply representing as these do, equal units, and groups of equal units, of any order whatever; and being, as it were, created at any moment for the purposes of calculation; numerical symbols seem at first sight, independent alike of Space and Time; and able to establish quantitative relations between magnitudes that are not homogeneous. The fact, however, is exactly the reverse. On tracing them back to their origins, we find that the units of Time, Force, Value, Velocity, &c., which figures may indiscriminately represent, were at first measured by equal units of Space. The equality of times, becomes known either by means of the equal spaces traversed by an index, or the descent of equal quantities (space-fulls) of sand or water. Equal units of weight, were obtained through the aid of a lever having equal arms (scales); and were obtainable in no other way. The problems of Statics and Dynamics are primarily soluble, only by putting lines to represent forces. Mercantile values are expressed in units, which were at first, and indeed are still, definite weights of metal; and are therefore, in common with units of weight, referable to units of linear extension. Temperature is measured by the equal lengths marked alongside a mercurial column. And similarly, all the definitely quantitative observations of science, are made by means of subdivisions of linear space. Thus, abstract as they have now become, the units of calculation, applied to whatever species of magnitudes, do really represent equal units of linear extension; and the idea of coextension underlies every process of mathematical analysis. Similarly with coexistence. Numerical symbols are, it is true, purely representative; and hence may be regarded as having nothing but a fictitious existence. But one of two things must be admitted respecting the reasoning processes carried on by means of them. Either these processes imply a conscious reference to the things symbolized—in which case the equalities predicated in them are really those which were previously observed between coexistent things; or else the things symbolized cease to be thought of, and the relations among the symbols are alone considered—in which case these symbols require to be made coexistent to consciousness before their relations can be determined. In fact, the phenomena of motion and sequence can be treated quantitatively, only by putting coexistent magnitudes to represent magnitudes that do not coexist. The relative lengths of two times, not being ascertainable directly, has to be indirectly ascertained, by comparing the spaces which a clock-finger traverses during the two times; that is, by comparing coexistent magnitudes. In brief, regarding it in the abstract, we may say that the Calculus in general is a means of dealing with magnitudes that do not coexist, or are not homogeneous, or both, by first translating them into magnitudes that do coexist and are homogeneous, and afterwards reducing them back to their original form.

But, perhaps, the fact that perfect quantitative reasoning deals exclusively with intuitions of the coextension of coexistent magnitudes that are connatural, will be most clearly seen when it is remarked that the intuitions of coextension, of coexistence, and of connature, are the sole perfectly definite intuitions of which we are capable. Whilst, on applying two equal lines together, we can perceive with precision that they are equal; we cannot, if one is greater than the other, perceive, with like precision, how much greater it is: and our only mode of precisely determining this, is to divide both into small equal divisions, of which the greater contains so many, and the less so many: that is—we have to fall back upon the intuition of coextension. Again, whilst we can perceive with the greatest exactness that two things coexist, we cannot, when one thing follows another, perceive with like exactness the interval of time between them: and our only way of definitely ascertaining this, is by means of a scale of time made up of coextensive units of space. Once more, we can recognize with perfect definiteness, the equality of nature of those things which admit of quantitative comparison. That straight lines are homogeneous, and can stand to each other in relations of greater and less, though they cannot so stand to areas or cubic spaces; that areas are connatural with areas, and cubic spaces with cubic spaces; that such and such are magnitudes of force, and such and such are magnitudes of time—these are intuitions that have as high a degree of accuracy as the foregoing ones—a degree of accuracy which our intelligence cannot exceed. Beyond these three orders of intuitions, however, we have none but what are more or less indefinite. All our perceptions of degree and quality in sound, colour, taste, smell; of amount in weight and heat; of duration; of velocity; are in themselves inexact. Now, as we know that by quantitative reasoning of the higher orders, perfectly definite results are reached; it follows that the intuitions out of which it is built must be exclusively those of coexistence, connature and coextension: an inference which will be confirmed on calling to mind that in any case of imperfect quantitative reasoning, some other species of intuition is palpably involved.

And here, with a view of showing the various combinations into which these intuitions enter, and also with a view of exhibiting sundry facts not yet noticed, it will be well to group, in their ascending order, the successive forms which quantitative reasoning assumes: such repetition as will be unavoidable, being, I think, justified by the completer comprehension to be given, by presenting the phenomena in their genesis and their totality.

§ 28. The intuition underlying all quantitative reasoning is that of the equality of two magnitudes. Now, the immediate consciousness that— implies—first, that A and B shall be coexistent; for otherwise, they cannot be so presented to consciousness as to allow of a direct recognition of their equality—second, that they shall be magnitudes of like kind, that is, connatural or homogeneous; for if one be a length and the other an area, no quantitative relation can exist between them—third, that they shall not be any homogeneous magnitudes, but they shall be magnitudes of linear extension; seeing that these alone admit of that perfect juxtaposition by which exact equality must be determined—these alone permit their equality to be tested by seeing whether it will merge into identity, as two equal mathematical lines placed one upon the other do—these alone exhibit that species of coexistence which can lapse into single existence: and thus the primordial quantitative idea, unites the intuitions of coextension and coexistence in their most perfect forms.

To recognize the negation of this equality—to perceive that A is unequal to B—or, more explicitly, to perceive either that— involves no such stringent conditions. It is true that, as before, A and B must be connatural magnitudes. But it is no longer necessary that they should be coexistent; nor that they should be magnitudes of linear extension. Provided the superiority or inferiority of A to B is considerable, it can be known in the absence of one or both; and can be known when they are magnitudes of bulk, weight, area, time, velocity, &c.

The simplest act of quantitative reasoning, which neither of these intuitions exhibits when standing alone, arises when the two are co-ordinated in a compound intuition; or when either of them is so co-ordinated with another of its own kind. When, by uniting two of the first intuitions thus—

we recognize the equality of A and C; it is requisite, as before, that if the two equalities are to be known immediately, the magnitudes shall be those of linear extension, though, if the equalities have been mediately determined, the magnitudes may be any other that are homogeneous; but it is no longer necessary that all of them shall coexist. At one time A must have coexisted with B; and at one time B must have coexisted with C; but the intuitions of their equalities having once been achieved, either at the same or separate times, it results from the ability which we have to remember a specific relation with perfect exactness, that we can, at any subsequent time, recognize, the equality of the relations A to B and B to C, and the consequent equality of A and C; though part, or even all, of the magnitudes have ceased to exist.

By uniting the first and second intuitions, and by uniting the second with another of its own kind, we obtain the two compound intuitions, formulated as follows:—

In the first of these cases it is requisite, when the relations are immediately established, that the magnitudes be linear; but not so if the equality of A and B has been indirectly established: and whilst A and B must have coexisted, it is not necessary that B and C should have done so. In the second case the magnitudes need not be linear; but, if the inequalities are considerable, may be of any order. Further, it would at first sight appear that they need none of them be coexistent. But this is not true; for if the superiority or inferiority of A to B and of B to C be so great that it can be perceived by comparing the remembrances of them, then the superiority or inferiority of A to C can be similarly perceived, without the intermediation of B; and the reasoning is superfluous. The only cases to which this formula applies, are those in which the inequalities are so moderate, that direct comparison is required for the discernment of them: whence it follows that, as in the third formula, each pair of magnitudes must have been at one time coexistent. And in strictness this consideration applies also to the fourth formula.

The next complication, and the one which characterizes all quantitative reasonings save these simplest and least important kinds just exemplified, is that which arises when, in place of conjoined relations, we have to deal with disjoined relations—when the compared relations instead of having one term in common have no term in common. Wherever this happens—wherever we have four magnitudes instead of three, sundry new laws come into force: the most important of which is, that the magnitudes need no longer be all of the same order. In every one of the foregoing cases, it will be observed that while the intuition of coexistence is sometimes not immediately involved but only mediately so, even where the judgment reached is perfectly quantitative—while, where the judgment is imperfectly quantitative, the intuition of coextension is not involved, save as the correlative of non-coextension—the intuition that is uniformly involved is that of the connature of the magnitudes, their homogeneity, their sameness in kind. Without this, no one of the judgments given is possible. But with disjoined relations it is otherwise. The four magnitudes may be all homogeneous; or they may be homogeneous only in pairs, either as taken in succession or alternately. Let us consider the resulting formulæ.

When all the magnitudes are homogeneous we have for the first group of cases the symbol

in which each of the disjoined relations is one of equality, and the second is some transformation of the first. This, as before shown, represents the mental act taken in every step of an equation; and stands for the several axioms—When equals are added to, subtracted from, multiplied by, or divided by, equals, the results are equal. For the second group of cases we have the symbol

in which each of the relations is one of inequality. This comprehends all the cases of proportion: whether they be the numerical ones in which the degrees of inequality are definitely expressed; or the geometrical ones (as those subsisting between the sides of similar triangles) in which the degrees of inequality, though known to be alike, are not definitely expressed. For the third group of cases, forming the antithesis to the two preceding groups, and being but imperfectly quantitative, we have the symbol

which represents such general truths as that if equals be taken from unequals the remainders are more unequal; that if to equals unequals be added, the sums are unequal; and so forth: and which also stands for the instances in which two ratios differ so widely, that their inequality is at once recognized. It needs only to be further remarked respecting these three groups of cases in which the magnitudes are all homogeneous, that the equality or inequality predicated between the two pairs, always refers directly or indirectly to the space-relations of their components, and not to their time-relations.

Passing to the other disjunctive class, in which the several magnitudes are not all homogeneous, we find that the equality predicated between the relations may refer either to comparative extension or comparative existence. The first group of them, which may be symbolized thus:—

so as to indicate the fact that the magnitudes of the first relation are of one species, whilst those of the second relation are of another species, comprehends cases in which one line is to another line as one area to another area; or a bulk to a bulk, as a weight to a weight—cases like those in which it is seen that triangles of the same altitude are to one another as their bases; or that the amounts of two attractions are to each other inversely as the squares of the distances from the attracting body. Here it is manifest that though the first pair of magnitudes differs in kind from the second pair, yet the antecedent and consequent of the one, bear to each other the same quantitative relation as those of the other; and hence the possibility of ratiocination. The second group of cases is that in which each relation consists of two heterogeneous magnitudes, as a line and an angle; but in which the two antecedents are of the same nature, and the two consequents are of the same nature. It may be formulated thus:—

Here, neither of the compared relations can be a quantitative one: seeing that in neither do the components possess that connature without which relative magnitude cannot be predicated. Hence the two relations can be equal only in respect of the coexistence of their elements; and, as it would seem, considerations of quantity are no longer involved. But though, under the conditions here stated, the reasoning merges into that inferior species remaining to be treated of in the next chapter; there are other conditions under which this form represents reasoning that is truly quantitative: namely, when the coexistence holds only in virtue of certain defined quantitative relations, by which the heterogeneous magnitudes are indirectly bound together. Thus, when the theorem—“The greater side of every triangle has the greater angle opposite to it,” is quoted in the proof of a subsequent theorem, the act of thought implied is of the kind above symbolized. The greater side (A) of a triangle, has been found to stand in a relation of coexistence with the greater angle (b); and in some other triangle the greater side (C) and greater angle (d) are perceived to stand in the same or an equal relation: but this relation is not simply that of coexistence; it is coexistence in certain respective positions: and though there can be no direct quantitative relation between a side and an angle, yet, by being contained between the two lesser sides, the greater angle is put in indirect quantitative relation with the greater side. It may be questioned, however, whether in this, as in the innumerable like cases that occur in geometrical reasoning, A, b, C, and d should not be severally regarded rather as relations between magnitudes, than as magnitudes themselves. To elucidate this question, let us consider the theorem—“The angle in a semicircle is a right angle.” Here the word “semicircle” denotes definitely quantitative relations—a curve, all parts of which are equidistant from a given point, and whose extremities are joined by a straight line passing through that point: the words “angle in a semicircle” denote further quantitative relations: and the thing asserted is, that along with this group of quantitative relations coexists that other quantitative relation which the term “right angle” denotes between two lines containing it. Taking this view, the reasoning will stand thus:—

And this seems to be the more correct analysis of those kinds of quantitative reasoning, in which the antecedents are not homogeneous with the consequents.

The only further complication needing consideration here, is the one arising when, instead of two equal relations, we have to deal with three. As, from that first simple intuition in which two magnitudes are recognized as equal, we passed to the union of two such intuitions into a compound one involving three magnitudes; so again, from the foregoing cases in which two relations are recognized as equal, we now pass, by a similar duplication, to the still more complex case in which three relations are involved. This brings us to the axiom—“Relations that are equal to the same relation, are equal to each other;” formulated, as we before saw after this fashion:—

In which symbol it will be seen that each pair of relations is united in thought, after the same general manner as any of the pairs lately treated of. The various modifications of this form which result when the relations are unequal, it is unnecessary here to detail. And it is also unnecessary to go at length into those yet more complicated forms which result when this conjunctive arrangement is replaced by a disjunctive one—when, in place of three relations, we have to deal with four; as in the case of the axiom given at the outset (§ 17)—“Relations which are severally equal to certain other relations that are unequal to each other, are themselves unequal.” The laws of the evolution have been sufficiently exemplified to render this, and the allied intuitions, readily comprehensible. All that needs further be done, is to point out how, by successive developments, we have progressed from a simple intuition of the equality or inequality of two magnitudes, to a highly complex intuition of the equality or inequality of relations between relations.

§ 29. And, now, having examined quantitative reasoning in its genesis, and found that, either mediately or immediately, it always involves, in their positive or negative forms, some or all of the ideas—sameness in the nature of its magnitudes; sameness in their quantity; sameness in their time of presentation to consciousness; and sameness in degree between relations of the same nature subsisting among them; it will be well, finally, to observe that we may recognise, à priori, the impossibility of carrying on any quantitative reasoning, save by intuitions of the equality or inequality of relations. It is the purpose of a quantitative argument to determine with definiteness the relative magnitudes of things. If these things stand to each other in such wise that their relative magnitudes are known by simple intuition, argument is not involved. There can be argument, therefore, only when they are so circumstanced as not to be directly comparable: whence it follows that their relative magnitudes, if determined at all, must be determined by the intermediation of magnitudes to which they are comparable. The unknown quantitative relation between A and E, can be ascertained only by means of some known quantitative relations between each of them and B, C, D; and it is the aim of every mathematical process to find such intermediate known relations, as will bring A and E into quantitative comparison. Now, no contemplation of magnitudes alone can do this. We might go on for ever considering B, C, and D, in their individual capacities, without making a step towards the desired end. Only by observing their modes of dependence can any progress be made. If A and E are in an unknown quantitative relation, which we desire to determine, we can determine it only as being equal or unequal to certain other relations, which we know mediately or immediately. There is no way, even of specifically expressing the relation, save by this means. The ascertaining what a thing is or is not, signifies the ascertaining what things it is like or not like—what class it belongs to. And when, not having previously known the relation of A to E, we say we have determined it, our meaning is, that we find it to be the same, or not the same, as some relation which is known. Hence it results, à priori, that the process of quantitative reasoning, must consist in the establishment of the equality or inequality of relations.

[∗]I coin this word partly to avoid an awkward periphrasis; and partly to indicate the kinship of the idea signified, to the ideas of coexistence and coextension. As we have already in use the words connate and connatural, the innovation is but small; and will, I think, be sufficiently justified by the requirement.