p Things observed, that are comparable in some common property, may produce an impression of identity or difference in respect to this property on our senses. Quantitatively such identity and difference can be called equality or inequality of the things as regards this property. For example, if a colour studied in a colorimeter proves to be perceptibly 271 identical to a mixture of certain known colours, it is said it and the colour of the mixture coincide. Our sense organs themselves provide] only scanty information about measured quantities expressible by the’equal’_¥or ’not equal’, the more so that their structure puts definite limits on our ability to see, hear, smell, and in general sense (and even within these limits this ability does not provide accurate information about our environment).  [271•* 

p A problem arises of how to go beyond the limits imposed by sense perception and to obtain accurate knowledge about the measured quantities.

p Measuring the length of a field by pacing it, determination of the area of a forest by eye, and determination of the volume of a body by feel give satisfactory information (for certain practical purposes), in spite of their ‘inaccuracy’ (in this case the term ’sensory measurement’ would be legitimate); it was not fortuitous that, in the historically first forms of measurement, when there was not yet science and developed technique, the role of a unit was played by parts of the human body (which is expressed by the terms ‘cubit’, ‘foot’).

p For accurate knowledge such measurements are quite inadequate, of course; and with the development of industry and growth of trade and a private economy more accurate data about quantities were needed. This was solved by the historical development of practice and science, which gave society experimental devices and measuring instruments.

p The first experimental devices (the measuring rod, dividers, and scales), as a matter of fact, simply made more precise what man already knew from simple observation.

p The development of the experimental science and technique in fact also disclosed the imperfection of the sense organs, a fact that stimulated some physicists (including Helmholtz) to doubt the possibility of any exhaustive cognition of the world around us. In reality, however, the fact that we can demonstrate the imperfection of our sense organs and that this proof is based on perceptions coming from these imperfect senses, indicates that human cognition is not only 272 able to surpass the limits imposed by the special structure of the human organs of perception but actually does so in its study of nature.

p Perception is also inevitably involved in accurate measurement by means of instruments. Nobody, for instance, will determine the base line for geodesic measurements by eye; at the same time, however, it is only the eye that detects the coincidence of a hairline and the reference object; or, to take another example, one cannot mark the coincidence of a column of mercury and a division of the scale with one’s eyes closed.

p Perception is thus a necessary component of any accurate measurement, and this implies the coincidence, say, of a needle, a spot of light or the top of a column of mercury with a scale division, or the matching of colours. An instrument reading perceived by the senses underlies a judgment about the result of the measurement, with a more or less long chain of inferences between the reading and the result. For instance, a researcher, observing the displacement of an ammeter needle, records a variation of current; a string of bubbles is perceived visually in a cloud chamber, but the conclusion concerns the trajectory of an alpha-particle.

p Perception is not usually independent in mensuration; its true role in cognition of a quantity can only be understood from the standpoint of the measurement process as a whole, when thought, processing the material of observations into concepts, recreates in the researcher’s mind a quantity that actually exists outside it. If everything occurred differently, it would have been impossible to extract the result of the measurement of a quantity from an instrument reading about the quantity measured.

p Perception is thus only the starting point in the study of quantities. Even direct measurement cannot be reduced to ‘pure’ empirical observation of certain phenomena but is a complex cognitive process in which abstract thought plays .an essential role. In measurements of quantities reducible to .standards the fundamental significance of theoretical thought :for determining the result of the E aasurement is quite clear. Let us take an example first in order to characterise the role <of thought in measurements. In this example, we would note, it is not at all a matter of the method, which is typical of the measurement of length, although it stresses a certain aspect <of mensuration, but rather of a kind of mental experiment.

p It is required to compare the lengths of intervals A and B with each other, adopting the length of B as the unit of measurement. B is laid along A the maximum possible number of times (without division). It may happen that it fits precisely q times (a whole number) into A. Then the relation between the lengths of A and B will be A = Bq.

p It usually happens, however, that an interval R remains over, whose length is smaller than that of B; in symbolic form we have A = Bq -\- R. The same operation can be performed with intervals B and 7?, R and fit, R₁ and /?₂, and so on, and we obtain a series of equations:

p where q, q_lt q₂, . . .q_n are integers, and R, R_lt R_z, ...,/?„ are decreasing lengths of the corresponding remainders.

p In practice, there will be always n for which R_n = 0; then Rn-! will be the common measure of the lengths of two intervals (commensurate intervals the ratio of whose lengths is expressed by a finite continued fraction). But there may also be incommensurate intervals (as is proved in geometry), i.e. which have no common measure of their lengths.

p In the last case the ratio A IB can be expanded into an infinite continued fraction:

12

p

p In

p It can be demonstrated that this infinite continued fraction is a finite irrational number, and A/B can be computed with an arbitrary degree of accuracy by means of rational numbers.

p This well-known Euclidean algorithm interests us in many respects. (1) It call’ be shown directly sensually that the lengths of intervals are commensurate and only commensurate, which is due to the existence of a threshold of sensitivity in our sensory organs. When they are equipped with the appropriate devices, this only increases the number of steps in the measurement; from the aspect of 274 perception, however, the proposition above is still valid. The numerical result of measurement is therefore always directly represented by a rational number. (2) It can be shown, in terms of geometrical theory, that there are both incommensurate and commensurate ratios of quantities. Hence, (3) the perceptible result of the comparison of quantities (without the appropriate theoretical correction) is not yet an accurate result. Generally speaking, exact ratios between (uniform) quantities cannot be established by their directly perceptible comparison: such a comparison yields only the preliminary material for determining the exact ratio of quantities. (4) Accuracy of measurement is intimately related to the concept of infinity, and the incommensurate ratios only witness to this in their own way.

p Geometrical measurements are not the only ones that need thought. Theoretical thought is an element of the measurement of any physical quantity. In the next section we shall consider the role that physical laws (the ratios of quantities) play in obtaining exact measurement results. Discovery of a law necessarily implies mental activity. As we shall see later, the finding of exact expressions for the results of measuring quantities, the simple form of which is the A/B considered above, coincides with the discovery of laws of nature.

p Physics is thus not satisfied with individual empirical measurements; it uses them to move towards exact knowledge, generalising the empirical material and ridding it of haphazard elements.

p Since physics became established as a science (Galileo, Kepler, Newton), its systematising factor and the most important source of its concepts (together with experience) has been mathematics; conversely, mathematics has grown from physics. Mathematical ideas shape the notions and principles of physics, and in modern physics they play a tremendous heuristic role on their own. But in relation to physics mathematical abstractions acquire physical flesh, so to say, only through measurement; on the other hand, experimental observations are only raised to the level of theoretical generalisation through measurement.

p From this it will be clear that the concept of the connection of mathematical abstractions (which figure in physical equations) with experimental observations, or the ’measurement recipes’ (as Mandelstam put it^^19^^), are extremely important for interpreting physical concepts. Each period in the 275 development of physics and mathematics has made its contribution to analysis of this concept. Any logically formed physical theory of broad scope has its own mathematical apparatus or formalism (e.g. classical formalism has numbers and vectors, the formalism of quantum mechanics—linear operators) which corresponds to its own specific rules of the relation between its mathematical abstractions and experimental observations.

p On that plane a physical concept is a kind of synthetic result of perception and abstract cognition, the physical concept itself being interpreted according to the specific features of the formalism of a certain physical theory. In this interpretation, the point of view of Niels Bohr is of fundamental significance. He never tired of explaining that it would have been impossible to describe real experiments without employing concepts of classical physics that represent a generalisation of everyday experience.

p According to him, the question of the physical meaning of the abstractions of classical mechanics (which expresses most clearly the epistemological and methodological features of classical physics) did not lead to any special difficulties in it (the values of the variables of its mathematical apparatus are numerical values of physical quantities mathematically expressed by these variables). In non-classical theories matters have become more complicated. In quantum mechanics, for instance, solution of the problem of how to express the physical meaning of the concepts of its formalism, considering observation data described in classical concepts, has proved far from trifling. It is not the purpose of this chapter to analyse this solution; matters relating to it were discussed in Chapter III, but we would like to make a comment relevant to the theme of this section.

p In quantum formalism the eigenvalues of its operators correspond to the numerical values of physical quantities that are represented mathematically by operators. The specific nature of quantum operators and relations between them reflects the specific nature of quantum quantities.  [275•*  In order 276 to infer the position of an electron, say, from the observed distribution of specks on a photographic plate, a system of definite principles and concepts that are ‘odd’ from the standpoint of classical physics (e.g., ’relativity with respect to the means of observation’, ’probability as the numerical measure of the potentially possible’, ’the difference between the potentially possible and the realised’) is required.  [276•* 

p On this basis appropriate conclusions about the physical quantities relating to micro-particles not directly perceivable are drawn from the observation data. If, for instance, the isolated concepts of a particle’s velocity and position employed in classical theory reflect the fact that the latter studies the motion of macroscopic bodies, in quantum mechanics the situation is quite different. The electrons in the atom do not behave either as particles or as waves but possess particle and wave properties simultaneously: it is then already impossible to speak of an electron’s isolated position and velocity; it is necessary to employ new concepts that are remote from the usual classical ones and yet connected with them.

p The roles of perception and abstract thought in a physical theory are thus equally important in their own way, and this comes out quite definitely in the measurement of physical quantities.

In summing up, we would like to stress that both sensory perception and abstract thought have a place in mensuration, or rather the dialectical unity of the two.