ON THE ACCURACY AND TERMINOLOGY OF MEASUREMENTS

V. Ya. STEPANOV

It is impossible to overestimate the importance and significance of measurements for modern science and technology. Measurements of physical quantities constitute the starting point of all our knowledge and are the final authority that passes judgment on any modern theory. “There is no more important subject,” wrote the well-known French mathematician A. Lebesgue [1] about measurements, dedicating to this subject the book On the Measurement of Quantities.

The problem of increasing the accuracy of measurements is closely connected with the scientific and technical level of society and with the progress of that level. On the one hand, a high level of measuring technology is the result of scientific and technical progress; on the other hand, accuracy is not only the basis of the reliability of technology, but an increase in accuracy opens up new possibilities for the development of science.

In light of this, the need for methodologically grounded, precise terminology concerning measurements is understandable; this is important for all branches of science and technology.

In modern systems of units of measurement, in order to measure a quantity it is necessary to compare it with a quantity of the same kind adopted as the unit. The result of measurement is expressed by a number. We shall show that there exist fundamentally irremovable limitations on the accuracy of measurements.

There is an opinion that the process of increasing measurement accuracy can continue indefinitely and that the measured values of quantities can asymptotically approach the true ones. Thus, for example, the well-known American scientist and metrologist D. DuMond writes in the article “Precise Measurements of Universal Physical Constants” [2].

We believe that ideal measurement accuracy not only cannot be attained, and not only is it unattainable (in the sense of a potential infinity associated with asymptotic approximation), but that an unlimited approach to the true value is in principle unattainable because of fundamentally irremovable errors that arise at some stage of asymptotic approximation and interrupt it. Let us consider some of these errors.

1. Errors of the subject, arising as a consequence of the obligatory participation of a human being at some stage of measurement. Every person has individual psychophysiological characteristics (different habits, speed of reaction, visual acuity, etc.), which influence measurement. There is a known experiment involving repeated measurement of parts

by three persons: an experienced mechanic, a young engineer, and a novice fitter [3]. Experience showed the indisputable influence of the “characteristics of the subject” on measurement.

1. Errors introduced by the instrument, owing to the impossibility of creating ideal instruments, without friction, with ideal balancing, etc.

2. Error arising from the practically ineradicable influence of external conditions: electric, magnetic, and gravitational fields, temperature, pressure, and other external parameters subject to the same changes.

3. Error associated with the finite accuracy of any standard.

4. Error of incommensurability (the discovery of incommensurability belongs already to Plato), owing to the fundamental incommensurability of quantities. Therefore the result of any measurement is expressed by an irrational number.

5. Error due to the absence of absolutely rigid parameters and properties in time, which leads to the uniqueness of the value at each given moment and ultimately forces us to use statistical methods.

6. Error due to the discrete character of quantities; for example, the dimensions of real bodies cannot be conceived as being measured with an accuracy exceeding the dimensions of the atom. This list of fundamentally ineradicable errors limiting the accuracy of measurements can be continued. Annoying errors arise in measurements connected with quantum mechanics. At present many special works are devoted to this question.

In view of the foregoing, certain points connected with the definition of errors and accuracy of measurement in the draft recommendations of VNIIM, “Basic Metrological Terms and Definitions” [4], do not appear successful.

In particular, the definition of measurement error as the algebraic difference between the value obtained in measurement and the correct (true) value of the measured quantity (item 74) appears methodologically unsuccessful.

Earlier, a definition is given of the actual value of a measured quantity (item 73). In the note to item 74, however, it is stated that, since “the correct (true) value of the measured quantity remains unknown, in practice the measurement error is taken to mean its approximate value, found from experimental data. In this sense the term ‘error’ is also used in other places of this recommendation.”

Why, one may ask, give definitions that cannot be applied and that have to be understood in a sense different from that in which they are defined?

It seems more logical and consistent to define error as the difference between the measured value and the actual one.

The true value will be understood as the limit approached by the actual value, but which it cannot attain, since we would have to go beyond the abstractions (such as plane, straight line, etc.) laid at the foundation of measurements.

P. K. Rashevskii wrote about this in the preface to Hilbert’s Foundations of Geometry: “Moreover, not only is it impossible to attain an ideally plane form, but, owing to the atomic structure of matter, it is not even possible to approach it without limit. Indeed, when we increase the required accuracy, the metal plate breaks up into individual atoms, and then it no longer makes any sense at all to speak of its surface.”

The definitions indicated are free of inconsistencies and, besides the methodological value of correctly understanding one’s actions and possibilities, are also of definite philosophical and worldview interest, since they are a successful application of the basic conception of the Marxist-Leninist theory of knowledge concerning the fundamental knowability of reality (the actual value) and the impossibility of its complete knowledge (the true value). In addition, it must be remembered that the process by which the actual value approaches the true value cannot proceed along a smooth asymptotic curve, owing to fundamental limitations on the accuracy of measurements, which restricts the accuracy of measurements at every level of development of science and technology, but is not an obstacle limiting cognition. It also seems illogical to confuse the concepts of accuracy of measurements and error of measurements (note to item 78), since these are reciprocal quantities. It would be more reasonable to retain Gauss’s definition of accuracy as the reciprocal of the relative error. The quantitative assessment of accuracy, associated with the accuracy class by a directly proportional dependence, should be called, as is customary, the “accuracy class.” Then such contradictions as in the note to item 78 will not arise, where it is said that the “quantitative assessment of the accuracy of measurements” is replaced by an indication of the error, while the smaller the error, the greater the accuracy. In conclusion, one remark. The so-called manifolds—sets of enumerated elements—can be measured absolutely exactly. For example, the charge m. b. is measured exactly if the charge of the electron, which is still the smallest, is taken as the fundamental unit. This emphasizes one interesting point in the choice of the basic units of a system.

REFERENCES

1. A. Lebeg. On the Measurement of Quantities. Moscow, Publishing House of the Ministry of Education of the RSFSR, 1960.
2. Science and Humanity. Moscow, “Znanie,” 1964.
3. S. F. Malikov, N. I. Tyurin. Introduction to Metrology. Moscow, Publishing House of Standards, 1966.
4. Proceedings of the Institute of the Committee of Standards, Measures, and Measuring Instruments. Issue 57 (117), M—L, Standartgiz, 1962.