Quantification

“ Therefore, one has much to say for the thesis that the whole logical value of the quantitative conceptions in science is due, not to any peculiar advantage of the concept of quantity, as such, but to the exactness of the forms of serial order which are discoverable in case of any quantitative realm of facts. ”

“ And yet, were we to take members of the Mathematical Society, who, having severally devoted themselves to the laws of quantitative relations, know that, simple as these are intrinsically, a life's study is required for the full comprehension of them—were we to ask each of these his opinion on some point of social policy, the readiness with which he answered would seem to imply that in these cases, where the factors of the phenomenon are so numerous and so much involved, a general survey of men and things gives data for trustworthy judgment. ”

“ Apart from quantitative verification, I doubt whether it would be possible in the whole range of science to find a case where we can be more confident that one phenomenon has been conditioned by another. ”

“ As it is impossible to comprehend aright the value of quantitative laws without constantly bearing in mind the degree of quantitative approximation to the truth probably attained, I have devoted a special chapter to the theory of approximation, and, however imperfectly I may have treated this subject, I must look upon it as a very essential part of a work on scientific method.
"It then remains to illustrate the sound use of hypothesis, to distinguish between the portions of knowledge which we owe to empirical observation, to accidental discovery, or to scientific prediction. ”

“ But how are measures of light to give us fractions of a magnitude? How can the vague, qualitative relation, the brighter the light the higher the magnitude, become an exact and quantitative one? The discussion of this question may be of use by showing that, even with matters of so uncertain a nature, science does not proceed by guesswork. It is a general law, that the human senses measure ratios and not differences. ”

“ There is an interesting class of experiments which enable us to observe a number of quantitative results in one act. Generally speaking, each experiment yields us but one number, and before we can approach the real processes of reasoning we must laboriously repeat measurement after measurement, until we can lay out a curve of the variation of one quantity as depending on another. We can sometimes abbreviate this labour, by making a quantity vary in different parts of the same apparatus through every required amount. ”

“ Similarly, in chemistry, we try to ascertain the quantity of the matters brought into combination, in order to find out the measures or proportions conditioning such combinations, that is to say, those quantities which give rise to definite qualities. In statistics, too, the numbers with which the study is engaged are important only from the qualitative results conditioned by them. Mere collection of numerical facts, prosecuted without regard to the ends here noted, is justly called an exercise of idle curiosity, of neither theoretical nor practical interest. ”

“ We can not, therefore, reason respecting causation, without introducing considerations of quantity and extension at every step; and if the nature of the phenomena admits of our obtaining numerical data of sufficient accuracy, the laws of quantity become the grand instrument for calculating forward to an effect, or backward to a cause. That in all other sciences, as well as in geometry, questions of quality are scarcely ever independent of questions of quantity, may be seen from the most familiar phenomena. ”

“ The necessity of qualitative measurement of statements not only per se, but with reference to the sources from which they are drawn, is generally recognized: but historians as yet seldom heed the twin requirement of quantitative measurement of our data in comparison with the sources whence they come. ”

“ A man with a head of hair or a heap of grain are what they are, so long as nothing in them is changed. The change of quantity is translated into change of quality, not because the first concept is constitutive of the second, but, on the contrary, because the second is constitutive of the first. Quantity has been obtained, measurement has been effected, by starting from quality, determined in the pure individual judgment and made homogeneous in the empirical judgment, which is the basis of the judgment of enumeration and of measurement. ”

“ It may be asked, as Mill asks, Why spend so much trouble in calculating from imperfect data, when a little trouble would enable us to render a conclusion certain by actual trial? Why calculate the probability of a measurement being correct, when we can try whether it is correct? But I shall fully point out in later parts of this work that in measurement we never can attain perfect coincidence. Two measurements of the same base line in a survey may show a difference of some inches, and there may be no means of knowing which is the better result. ”

“ Measure is the qualitative quantum, in the first place as immediate,—a quantum, to which a determinate being or a quality is attached.
Measure, where quality and quantity are in one, is thus the completion of Being. Being, as we first apprehend it, is something utterly abstract and characterless: but it is the very essence of Being to characterise itself, and its complete characterisation is reached in Measure. Measure, like the other stages of Being, may serve as a definition of the Absolute: God, it has been said, is the Measure of all things. ”

“ But we must emphasize the fact once more that in deciding matters in this way we deal solely with questions of expediency, not with questions of the nature of our mental capacity.
36. Measurement. Measuring is in a certain way the opposite of counting. While, in counting, the things are regarded in advance as individual, and the group, therefore, is a body compounded of discontinuous elements, measuring, on the other hand, consists in co-ordinating numbers with continuous things, that is, in applying to continuous things a concept formed upon the hypothesis of discontinuity. ”

“ Such is the character of the judgments which the calculus enables us to form with respect to the occurrence of a certain difference between the real value of any quantity under measurement and the value assigned to it by the measurement. The confidence that the constants which we have determined are accurate within certain limits is a subjective feeling which cannot be dislodged from an important part of probabilities. ”

“ As science has been developed, the domain of quantity has everywhere encroached on that of quality, till the process of scientific inquiry seems to have become simply the measurement and registration of quantities, combined with a mathematical discussion of the numbers thus obtained. It is this scientific method of directing our attention to those features of phenomena which may be regarded as quantities which brings physical research under the influence of mathematical reasoning. ”

“ The process of measure, instead of being only the wrong infinite of an endless progression, in the shape of an ever-recurrent recoil from quality to quantity, and from quantity to quality, is also the true infinity of coincidence with self in another. In measure, quality and quantity originally confront each other, like some and other. But quality is implicitly quantity, and conversely quantity is implicitly quality. ”

“ There are no "quantitative aspects" in the world, besides these similarities; the adjective "quantitative" can be applied to the method only, not to nature itself; science does not "pick out" the quantitative sides of things, and it does not think of overlooking qualities as if they did not exist. ”

“ But if it comes to be discovered that variations of quality in any class of phenomena, correspond regularly to variations of quantity either in those same or in some other phenomena; [Pg 249] every formula of mathematics applicable to quantities which vary in that particular manner, becomes a mark of a corresponding general truth respecting the variations in quality which accompany them: and the science of quantity being (as far as any science can be) altogether deductive, the theory of that particular kind of qualities becomes, to this extent, deductive likewise. ”

“ We say that a man is tall, or heavy, or that the room is hot—qualitative statements; or we may say exactly how tall, or how heavy, or how hot—quantitative statements. The distinction between qualitative analysis and quantitative analysis in chemistry implies the same idea. Thus we may speak of a piece of wealth as having a definite quantity of value, or say that the value of the piece of wealth is a definite quantity. ”

“ Measurement owes its existence to Earth; Estimation of quantity to Measurement; Calculation to Estimation of quantity; Balancing of chances to Calculation; and Victory to Balancing of chances. ”

“ So that in this simplest, and perhaps earliest case of quantitative prevision, the phenomena are not only thrust daily upon men's notice, but Nature is, as it were, perpetually repeating that process of measurement by observing which the prevision is effected. ”

“ In statistics, it is evident that empirical laws may sometimes be traced; and the tracing them forms an important part of that system of indirect observation on which we must often rely for the data of the Deductive Science. ”

“ Again, within the aspects selected for attention, all that experiment can establish is that the deviation from uniform law, if there is any deviation, is not sufficiently great to affect our measurements and calculations. But how far our standards of measurement are from rigid precision may be readily learned from the chapter on physical standards in any good work on the logic of the inductive sciences. ”

“ We want to account for things, which means that we want to know to which of the various accounts opened in our mental ledger we ought to carry them—and how can we do this if we admit a phenomenon to be neither one thing nor the other, but to belong to half-a-dozen different accounts in proportions which often cannot even approximately be determined? ”

“ In the foregoing chapters it is only by way of exception that there has been made any formal use of statistical data, or any reference to scientific authorities;—in fact, there has been a studied avoidance of the sympathetic literature of the subject. But it seems wise and, above all, just to the reader, to guard well every salient position, to throw round every argumentative assertion a bulwark of mathematical evidence—a task that presents little difficulty, since in general the facts in the case are well ascertained and the testimony unanimous. ”

“ Quantity, on the contrary, is the character external to being, and does not affect the being at all. Thus e.g. a house remains what it is, whether it be greater or smaller; and red remains red, whether it be brighter or darker. Measure, the third grade of being, which is the unity of the first two, is a qualitative quantity. All things have their measure: i.e. the quantitative terms of their existence, their being so or so great, does not matter within certain limits ”

“ The discovery of quantitative laws is especially satisfactory, because it not only explains why an event happens at all, but why it happens just in this direction, degree, or amount; and not only is the given relation of cause and effect definitely assimilated to other causal instances, but the effect is identified with the cause as the same matter and energy redistributed; wherefore, whether the conservation of matter and energy be universally true or not, it must still be an universal postulate of scientific explanation. ”

“ On account, then, of the difficulties of the subject and of the profound importance of results which can be reached by no other known method, it is of the utmost consequence that the principles and processes of statistical mechanics should be put upon a firm and certain foundation. ”

“ This charge against science has no more justification than all the others. The quantitative method, characterised by the use of numbers in the representation of logical structure, is for practical as well as theoretical reasons, the very best instrument of knowledge, and science should be praised for using it as much as possible. The theoretical reasons, if I may just hint at them, lie in the fact that counting and numbering is made possible by the repeated occurence of similar or equal events in experience ”

“ Statistical facts are usually numerical or quantitative in character, so that, if many causal agencies are in operation at the same time, their effects are simply added together algebraically, and are inextricably merged into a general total. ”

“ Size, on the other hand, and its alteration regarded merely as size, is for the qualitative determination, when it is unable to assert itself in terms of measure, an indifferent determination. That is to say, measure is quantity, precisely in so far as it can give to itself an aspect which qualifies it, and thereby a qualitative determination is united to the purely quantitative one. ”

“ Knowledge being scientific, as it can apply number and measure, an attempt is in order to see if a scientific order of values cannot be built "on a number and measure scale of force," ascent in the scale signifying increase of value, descent diminution of value—all other estimations being prejudices, naïvetés, misunderstandings. ”