What is a “measurement”? In the Great Soviet Encyclopedia, we find the following definition of this concept:
“Measurement is the operation, by means of which the ratio of one (measurable) magnitude to another homogeneous magnitude (taken as the unit of measurement) is determined; the number, determined with such ratio, is called the numerical value of the measured magnitude”.
Measurement is an important way of the quantitative cognition of the objective world. The Great Russian scientist, Dmitry Mendeleev, the creator of the Periodic Table of Chemical Elements, expressed his opinion about the measurement as follows:
“Science begins from the “measurement”. Exact science is unthinkable without measure”.
The problem of measurement plays the same significant role in mathematics as in other areas of science, in particular, engineering, physics, and other “exact” sciences.
Let’s now trace the evolution of the concept of “measurement” in mathematics [16]. As is known, the first “measurement theory” was the set of rules, which had been used by the ancient Egyptian surveyors. From this set of rules, as the ancient Greeks testify, geometry was obligated with its appearance (and name) to the problem of the “Earth’s measurement”.
However, already in Ancient Greece, the measurement problems had been divided into applied tasks related to logistics and fundamental problems related to geometry; the ancient mathematics focussed on the latter. The science of measurement was developing during this period primarily as a mathematical theory.
The discovery of incommensurable segments, made in the scientific school of Pythagoras, was one of the main mathematical achievements of this period. This discovery caused the first crisis in the foundations of mathematics and led to the introduction of irrational numbers — the second (after natural numbers) fundamental concept of mathematics. This discovery led to the formulation of the Eudoxus method of exhaustion and measurement axioms (see Section 1.2.), to which theory of numbers, integral, and differential calculus go back in their origins.
1.2. Axioms of Eudoxus–Archimedes and Cantor
To overcome the first crisis in the foundations of ancient mathematics, associated with the discovery of “incommensurable segments”, the outstanding geometer Eudoxus proposed the “method of exhaustion”, by which he built the ingenious theory of relations, underlying the ancient theory of the continuum. “Method of exhaustion” played a prominent role in the development of mathematics. Being a prototype of integral calculus, the “exhaustion method” allowed ancient mathematicians to solve the problems of calculating volumes of a pyramid, a cone, a ball. In modern mathematics, the “method of exhaustion” is reflected in the Eudoxus–Archimedes axiom, also called the measurement axiom.
The theory of measuring geometric quantities, which goes back to “incommensurable segments”, is based on a group of axioms, called continuity axioms [16], which include the Eudoxus–Archimedes and Cantor axioms or the Dedekind axioms.
Eudoxus–Archimedes axiom (the “measurement axiom”). For any two segments A and B (Fig. 1.1), we can find the positive integer n such that
Fig. 1.1. Eudoxus–Archimedes axiom.
Cantor’s axiom (about the “contracted segments”). If an infinite sequence of segments A0B0, A1B1, A2B2, …, AnBn, … “nested” into each other is given (Fig. 1.2), that is, when each segment is part of the previous one, then there is at least one point of C, common to all segments.
Fig. 1.2. Cantor’s axiom.
The main result of the theory of the geometric quantities is the proof of the existence and uniqueness of the solution q of the basic measurement equation:
where V is the unit of measurement, Q is the measurable quantity, and q is the result of measurement.
Despite the seeming simplicity of the axioms, formulated above, and the whole mathematical theory of measurement, it is nonetheless the result of more than 2000 years in the development of mathematics and contains a number of in-depth mathematical ideas and concepts.
First of all, it is necessary to emphasize that the Eudoxus exhaustion method and the measurement axiom, resulting from it (Fig. 1.1), are of practical (empirical) origin; they were borrowed by the ancient Greek mathematicians in the practice of measurement. In particular, the “exhaustion method” is the mathematical model for measuring the volumes of liquids and bulk solids by “exhaustion”; the measurement axiom, in turn, concentrates thousands of years of human experience, long before the emergence of the axiomatic method in mathematics, billions of times measuring distances, areas and time intervals, and is a concise formulation of an algorithm of measuring line segment A by using a segment B (Fig. 1.1). The essence of this algorithm consists in successively postponing the segment B on the segment A and counting the number of segments B that fit on the segment A. In modern measurement practice, this measurement algorithm is called the counting algorithm.
1.3. The Problem of Infinity in Mathematics
Cantor’s axiom (Fig. 1.2) contains another amazing creation of mathematical thought, an abstraction of actual infinity. It is this idea of infinity that underlies the Cantor theory of the infinite sets [16].
The idea of actual infinity as the main idea of Cantor’s (set-theoretic) style of mathematical thinking was strongly criticized by the representatives of the so-called completed one it is impossible without gross violence over the mind, which rejects such contradictory fantasies.
Earlier, the famous mathematician David Hilbert (known for his “finite” installations) expressed the same idea in other words and, by discussing the concepts of the finite and infinite, came to the following conclusion [125]:
“From all our reasoning, we want to make some summary on infinity; the general conclusion is the following: the infinite is not realizing anywhere. The infinite does not exist in Nature, and this concept is unacceptable as the basis of our rational thinking — here we have a wonderful harmony between being and thinking … Operation with the infinite can become reliable only through the finite”.
The paradoxes or contradictions in Cantor’s theory of infinite sets, discovered at the beginning of the 20th century, significantly shook the foundations of mathematics. Various attempts have been made to strengthen them. The most radical of them is the constructive direction in the substantiation of mathematics [124], which completely excludes the consideration of the abstraction of actual infinity and uses a much more “modest” abstraction of the infinite called the abstraction of potential feasibility.
The contradiction between potential and actual infinities manifests itself most vividly in the mathematical theory of measurement [16], when we analyze the Eudoxus–Archimedes axioms (Fig. 1.1) and the Cantor axioms (