p If one says that physics (in the broadest sense of the term) is a science about the general laws of variation and transformation of realities in inanimate nature or, more definitely, about the laws of variation and transformation of fields and matter, one always implies that the properties of physical realities cannot be separated from their quantitative determinations, i.e. (which is the same thing) that a physical quantity is a kind of synonym of a certain property of a physical reality, and that the regular connections between these properties are expressed as relations between physical quantities (the equations of physics).

p The objects of physics are studied in experiment and in the final analysis should be perceived either directly by the sense organs or in a mediated way, through the readings of the instruments. A necessary premise of the cognition of nature in physics is therefore that, in taking both the quantitative and the qualitative character of physical quantities into account, it finds the correspondence between the empirical data and the quantitative determinations (numbers). The establishing of this correspondence is sometimes called measurement.^^6^^

p There are objections to this definition of measurement: it is said to be, at least, incomplete.^^7^^ And indeed, in our view, it contains too much and at the same time too little for an understanding of measurement. Is it possible, for example, to call determination of the moment of an event measurement? Or does mineralogists’ determination of the hardness of a body by the Mohs scale of hardness in which the hardness of a certain^ten minerals is taken as the standard of hardness (the hardness of talc is 1, of gypsum 2, etc., up to diamond, whose hardness is taken as 10) represent measurement? In other words, can one call the instant of an event or, say, hardness quantities, if it is assumed that a quantity is what can be measured?

p We have arrived, consequently, at the following questions: what is measurement, and what is a physical quantity?

p Measurement differs from so-called arithmetisation. and from what, in our view, should be called quantitative 260 ranking; as will be shown below, they are the conditions of existence of mensuration, although they may also make sense (exist) without it.

p The arithmetisation of a certain class of properties of things is the establishing of rules by which it is possible to determine from the property of a class a number (or certain set of numbers) corresponding to it, and from a certain number (or set of numbers) to determine the property corresponding to it. If, for instance, one ascribes a pair of numbers to each point on a plane, according to a certain rule (each pair of numbers corresponding to one point and one point only on the plane), one thereby arithmetises the plane (since each point on it possesses the property of having a certain position on it).

p Arithmetisation considered by itself is a totally arbitrary process (in the example above the plane can be arithmetised by means of the Cartesian system of coordinates, the system of polar coordinates, or other coordinate systems, and all these methods are equivalent), but in everyday life and scientific research it is governed by quantitative ranking, and through that by measurement, which constitutes its starting points and its peculiar ‘cellule’.

p If the properties of a certain class are such that it is possible to employ the concepts ‘bigger’ or ‘smaller’, they are called intensities (or intensive quantities). If this class is arithmetised in such a way that a higher intensity corresponds to a bigger number, the arithmetisation is quantitative ranking. A liquid B, for instance, is denser than another liquid A if the latter floats on the former, but not conversely (symbolically B > A or A < B). Other examples of intensities, in addition to density, are temperature, the colour of a monochromatic beam (if a certain additional definition is introduced), hardness, and viscosity.

p A set of different intensities A, B, C, D, etc., can be ranked in a sequence of intensities A < B < C < D, and so on, in which A precedes 5, B precedes C, and so on. If the differences between two successive intensities in this sequence are equal (symbolically B > A = C> B = D > C), the intensities are called extensities (or extensive quantities), and the sequence itself becomes a sequence of extensities.

p The arithmetisation of a class of extensities (e.g. a class of lengths, volumes, electrical resistances, masses, etc.) is measurement. The employment of a unit of measurement 261 established arbitrarily is typical of mensuration. When the lengths of things are being measured, for instance, we count how many times a certain rod, selected as the unit of length, can be laid along each of them in a certain way. The situation is analogous to the measuring of, say, the volumes of vessels by means of a measuring vessel, or the masses of bodies by weighing. Such examples are endless, but it is essential to note that such-and-such a type of quantity is measured by means of a method that is specific for it; the quali  tative aspect of the physical quantity finds expression, in particular, in that.

p So, in mensuration, we determine the relation of a ( measured) quantity to another homogeneous quantity (which is taken as the unit of measurement); this relation is expressed by a number (which is called the numerical value of the measured quantity).

p A few words are called for on a fuller and logically closed definition of an extensive quantity, without which mensuration cannot be comprehended. It can be given axiomatically, and there is more than one system of appropriate axioms. Nagel cites Hoelder’s system of axioms characterising the concept of an extensive quantity.^^8^^ This point is also clearly made in Kolmogorov’s article on quantity in the Great Soviet Encyclopedia,^^9^^ which formulates axioms of division, continuity, etc., in addition to the axioms of order and combination. The axiomatic method employed to study the concept of quantity makes it possible to establish all the necessary fundamental characteristics of this concept more fully.^^10^^ If the concept of an extensive quantity is generalised so that the class of these quantities includes negative ones and zero, in addition to positive quantities, then, when an arbitrary positive quantity is selected as the unit of measurement, all the others can be expressed in the form of Q = q [Q] (the basic equation of measurement), where Q is the measured quantity, [Q] is the unit of measurement, q is a real number, and q [Q] is the result of the measurement.

p Extensive quantities are more fundamental in physics than any other kind of quantity. They make it possible to get a quantitative expression of intensive quantities on the basis of established regularities: the value of temperature, for example (as the level of the thermal state), is determined by measuring the temperature interval between the zero value and the determined one. On the other hand, the very 262 definition of an extensive quantity contains an indication towards ’bigger and smaller’. Extensive and intensive quantities are thus two aspects of one and the same concept.

p Let us return to measurement. From the methodological standpoint, or rather from the angle of the general methods of obtaining the results of measurement, the division of measurements into direct and indirect ones is of very great interest. In direct measurement, the result is obtained from the act of measuring the quantity itself independently of measuring other quantities. In indirect measurement the result is obtained in terms of direct measurements of quantities that are connected with the measured one by a certain mathematically expressed dependence. In this section we shall discuss only direct measurement.

p Measurement in science, unlike measurement in everyday life, is above all accurate measurement. There is a well- developed classification of ‘accuracies’, including the ’highest accuracies’ in the theory and practice of measurements.¹¹ The concept ’metrological accuracy’ is essential for our theme, for this reason: metrological accuracy is the highest accuracy that can be attained in the measurement of a given quantity in certain established units.^^18^^ Allowing for the fact that measurement results are no more accurate than the standards are,  [262•*  one can say that measurements made with metrological accuracy are those that are reducible to standards.

p We now have to consider on the logical plane: how did measurement with metrological accuracy arise, or how did that form of measurement which we are entitled to call its standard form arise?

p The standard^ form of measurement developed from simpler ones. The initial form is the random or individual form of measurement, whose specific feature is that a certain kind of quantity, characterising one thing, is measured by means of any other single thing characterised by the same kind of quantity.

p Thing A = b things B (the equals sign means ’equal with respect to such-and-such a property’).^^13^^

p This form already reveals the special features of measurement as a cognitive process providing information about the measured quantity. The measured property of object A is expressed qualitatively through the capacity of another object B to be comparable with respect to this property; its quantitative expression is that B appears as a numerically determined property of A. The property of an object (in its quantitative determination) does not exist simply in its expression by means of another object, but independently of any such expression, i.e. it is not the result of measurement that determines the quantity but the quantity itself that determines the result. On the other hand, within the limits of the relation between object A and object B in terms of a common property, B is not expressed in any way, i.e. it functions simply as a measure.

p Furthermore, a feature of mensuration is that the individual becomes the representative of its own opposite, the general, in measurement. We would stress yet again that here the individual represents the general only within the limits of the relation of the things in terms of this general: a definite quantity of iron represents only heaviness with respect to a sugar-loaf whose weight is being measured by it; iron fulfils this role, however, only within the context of the weight relation (into which it enters with sugar), and sugar enters this relation only because both iron and sugar possess weight.^^14^^ Finally, it must be considered a specific feature of measurement that a thing functions in it as a property.

p The individual form of measurement is only met in the early historical stages of the development of production and human culture. In Babylonia, for example, there were three separate, unconnected groups of measures of length, which had arisen independently of one another: one based on the ‘cubit’ (a finger’s breadth [digit], span, and cubit), which measured short intervals; one’based on the gar ( approximately equal to six metres); and a third group—the ‘mile’ and ’hour’s walk’—measures for long distances.^^15^^

p The individual form of measurement is quite unsatisfactory for the tasks of measurement. In it a thing expresses the properties of only one object; all other objects possessing the same property are not involved in the expression. At the same time the individual form of measurement passes by itself as it were to developed or expanded form. By the first 264 form the property of an object A is only measured in one thing B, regardless of what this thing is (a span, cubit or arshin, or metre if the matter in hand is, say, the length oj object A). To the extent that one and the same thing enters the relation with respect to one and the same property sometimes with one and sometimes with another thing, various individual expressions of measurement result. The individual expression of measurement thus turns into a number of various individual expressions, and we obtain the developed, or complete form of measurement:

p Object A^=b objects B = c objects C

p Leaving aside analysis of the developed form of measurement,  [264•*  let us simply point out its drawbacks:

p (1) the series of expressions is not finished; (2) they are not connected with each other; (3) if the properties of all the objects that constitute a given series are measured in this form, a vast set of series is obtained, extraneous to one another.

p Each of the equations involved in the complete form of measurement (and, therefore, the whole series) can be reversed.  [264•**  In this case we obtain the universal f orm of measurement

p { Object B =— of object A

p Object C =—of object A

p In this form the properties (of one and the same kind) of things are measured by one and the same thing singled out from the aggregate of these things; for example, the lengths of solids are measured by the metre and so represent their lengths through their relation to the metre. In this case the 265 metre, in realising length, differs from itself as an individual element and from all other solid bodies as individual elements, and thereby expresses what it has in common with other bodies, i.e. expresses length. In their equating with the metre the solids possessing length prove to be not only qualitatively equal, i.e. lengths in general, but at the same time quantitatively comparable magnitudes of length.

p Reasoning abstractly it can be said that every thing in a set of things characterised by the same property can be the universal measure of this property. Historical practice and science, however, necessarily choose one definite object as the universal measure which is therefore singled out from a set of things (it is not necessarily a natural thing but can be made artificially).

p As a universal measure a thing has only one specific property with respect to other things from which it has been singled out, namely, to be their universal measure. When the singling out proves to be the final lot of one definite thing, it begins to function as a standard. In general, we call the standard of a quantity that object whose physical properties coincide with the ‘property’ of expressing this quantity or, in short, a standard is an objectivised ( embodied) universal measure. That is how the standard metre, represented by a platinum-iridium bar with a certain crosssection, got the privileged position of the standard of length.

With the transition from the individual form of mensuration to the developed one, and from the developed form to the universal, essential changes take place from the angle of the accuracy of measurements. In the transition from the universal form to the standard, on the contrary, progress consists solely in the ‘property’ to be a universal measure now finally merging, by virtue of circumstances we shall discuss in the next section, with the physical properties of the standard as a definite body.