(1) An intuitive, “practical” idea about the finiteness of the measurement process, according to which every measurement is completed in the finite number of steps.
(2) Constructive idea of the potential feasibility, in accordance to which we ignore the limitations of our possibilities in choosing the measuring means and the number of measurement steps (i.e., the number of measurement steps is always finite and can be choosen to be arbitrarily large and there exists always the potential possibility to do the next measurement step).
Such a seemingly insignificant change in the approach to the measurement leads us to rethink many problems of the mathematical theory of measurement. With the set-theoretic approach, the measurement is carried out to the “point”, i.e., to the absolutely exact coincidence of the measurable and measuring segments (a possibility of such an absolutely accurate measurement follows from Cantor’s axiom).
With the constructive approach, the measurement never reaches the “point”, and the measurement result reduces always to certain uncertainty interval regarding the true value of the measurable quantity. By increasing the number of measurement steps, this interval narrows and can be made arbitrarily small, but this interval never turns into the “point”.
In his famous work On the Philosophy of Mathematics [130], Hermann Weil pays attention to the following distinction between the classical and constructive definitions of the continuum concept:
“In modern analysis, the continuum is considered as the set of its points; in the continuum, it sees only a special case of the basic logical relationship between an element and a set. But it is amazing that the same fundamental relationship between the whole and its part had not yet found a place in mathematics! Meanwhile, the possession of parts is the basic property of the continuum and Brauer’s theory puts this relationship in the basis of the mathematical study of the continuum. This is actually the basis of the above attempt to proceed not from points, but from intervals, as from the primary elements of the continuum”.
One of the important moments in the set-theoretic theory of measurement, based on the Eudoxus–Archimedes axiom and the Cantor axiom, is the choice of the measurement algorithm, which is defined by the numeral system, in which the measurement result is represented.
With the infinite (in terms of the actual infinity)number of measurement steps, i.e., when we are measuring up to the “point”, the measurement algorithm does not affect the final measurement result and therefore the problem of choosing measurement algorithms as a serious mathematical problem does not arise here. The choice of the measurement algorithm has an arbitrary character, and, as a rule, it reduces to the “decimal” or “binary” algorithms.
With the infinite (in terms of the potential infinity)number of measurement steps, i.e., when we are measuring up to the “interval” [130], between the measurement algorithms, there arises the difference in the measurement “accuracy”, achieved with the help of the given measurement algorithm. Let’s recall that the measurement “accuracy” is equal to the ratio of the initial uncertainty interval to the uncertainty interval on the final step of measurement. For such conditions, the constructive idea about the “efficiency” of the measurement algorithms [131] comes into play, and the problem of the synthesis of efficient or optimal measurement algorithms is put forward as the central problem of the constructive (algorithmic) theory of measurement [16].
Thus, the constructive approach to the theory of measurement leads us to the formulation of the problem, which, in essence, has never been considered in the mathematical theory of measurement as a serious mathematical problem, namely the problem of finding the “optimal” measurement algorithms. The solution of such problem led us to the creation of the algorithmic measurement theory [16], which can be considered as the constructive direction in the mathematical theory of measurement.
1.6. The “Indicatory” Model of Measurement
1.6.1. The conceptions of the “indicatory” element (IE) and the “indicatory” model of measurement
When we set forth the task of creating the algorithmic measurement theory, it is necessary to clarify once again what the measurement is, what its purpose is, what the measurement algorithm is and what tools we use to implement the measurement.
First of all, we note that if we want to measure something, we must know some source data concerning the measurement, in particular, the range of measurable values. It is one thing to measure the cosmic distances, for example, the distance from the Earth to the Sun and quite another to measure the atomic distances.
However, when we turn to the mathematical measurement, we are distracted from the physical character of the measured quantities;in this case, for all the cases of measurement, we can represent the measurement range in the form of the geometric segment AB.
It is clear that the measurable value is one of the possible values, belonging to this range, that is, before the beginning of the measurement, there is some uncertainty regarding the measurable value; otherwise, the measurement would simply be meaningless. This situation of the “uncertainty” can be depicted by using the “unknown” point X located on the segment AB (Fig. 1.4).
Fig. 1.4. Constructive (“indicatory”) model of measurement.
Now, we can formulate the purpose of measurement. The purpose of measurement consists in the determination of the length of the segment AX (Fig. 1.4). In practice, this purpose is realized with the help of special devices, for example, the “lever scales” or “comparators”.
The “comparators” carry out the comparison of measurable quantity with certain “standard quantities” or “measures”, formed from the “measurement unit” V and, depending on the result of the comparison, they give us information about the position of the measured segment AX on the initial segment AB.Thus, the essence of the measurement reduces to the successive comparisons of the measurable segment AX with some “measures”, which are formed at each measurement step from the unit V according to the measurement algorithm.
In order to model the process of the comparison of the measurable segment AX to the “measures”, the important concept of “indicatory element” (IE), which is a peculiar model of the “comparator” or the “lever scales”, the basic means of any measurement, was introduced in Ref. [16].
We assume that each “indicatory element” (IE) can be enclosed to some “known” point C of the segment AB, according to the measurement algorithm, that is, we can form for every “indicatory element” some “measure” on each step of the measurement.
The “indicatory element” gives us information about the relative position of the “unknown” point X and the “known” point C. If the IE is to the right of the point X, it “indicates” the binary signal 0; otherwise, the binary signal 1.
1.7. The Concept of the Optimal Measurement Algorithm
In Section 1.6, we construct the following mathematical model of measurement (see Fig.