Constructive approach to measurement theory

The famous mathematician Georg Cantor introduced Cantor's axiom in 1872. This one underlies in the base of the classical measurement theory together with Eudoxus-Archimede axiom. The latter involves in itself yet one more unusual phenomenon of mathematical thinking, i.e. the abstraction of the "actual", or "final" infinity. To clear up this concept we have to compare Cantor's axiom with Eudoxus-Archimede's axiom. Both in the former and in the latter axioms we use the concept of "infinity". However there exists a huge difference between the "infinity" concepts used in these axioms. As for Cantor's axiom, the infinite set of the "contracting" line segments together with the point C joining them is considered to be given by all the objects simultaneously. The "final", "actual" infinite sets represent the most distinctive feature of Cantor's theoretical-set mathematical style.

Due to its empirical origin Eudoxus-Archimede's axiom is constructive and contains in itself a more "simple" abstraction of the infinity, which is called the abstraction of potential feasibility.

According to Eudoxus-Archimede's axiom, a number of measurement steps in this axiom is always finite, but it is "potentially unlimited". It is this consideration that makes the sense of constructive approach to the concept of the infinity.

Thus, considering the classical measurement theory based on Eudoxus-Archimede's and Cantor's axioms we touched upon two the profoundest mathematical abstractions of the infinity, the abstraction of "actual" or "final" infinity and the abstraction of "potential" infinity. These two abstractions divide mathematics into two parts: (1) mathematics based on the "actual" or "final" infinity and (2) mathematics based on the "potential" infinity and refuted the "actual" infinity.

A struggle between these two approaches to mathematics is continuing until now and is the essence of the new crisis in modern mathematics.

The idea of measurement as the process completed during the infinite time period finds, on the one hand, a deep gap between the experimental data of the natural research, and on the other hand, according to the Russian mathematician A. A. Markov, "to think of the infinite, i.e. uncompleted process as of the final one is impossible without a rough violence upon the mind rejecting such contradictory fantasy".

Like to the "problem of incommensurability" in the ancient mathematics the "problem of infinity" is the topic of the deep modern philosophical research. The Russian philosopher Prof. G.V. Chefranov (Taganrog, Taganrog State University of Radio Engineering) is one of the best World philosophers contributed essentially to this problem.

Prof. A.P. Stakhov and Prof. G.V. Chefranov (Taganrog, 1974)
Prof. .P. Stakhov and Prof. G.V. Chefranov (Taganrog, 1974)

Chefranov's book "The Infinity and Intellect" (1971) devoted to the problem of infinity simulation using mathematical and general languages is one of the most deep philosophical works on this problem.

Chefranov's book 'The Infinity and Intellect' (1971)
Chefranov's book 'The Infinity and Intellect' (1971)

What will happen to the classical mathematical measurement theory if the abstraction of the actual infinity would be excluded? It means that we exclude Cantor's axiom from the measurement theory. First of all it would mean that the measurement theory should be constructed on the constructive idea of the finitness of measurement. According to this idea, any measurement should be performed in the finite number of steps. But according to the constructive concept of the potential infinity the number of measurement steps can be chosen as big as is wished and we can do next step after the preceding step.

The stated methodological basis results in the fact that any measurement has the principal measurement error called the quantization error. As this takes place the new interpretation of the basic problem of the measurement theory comes into being. For the given number n there appears a difference between the different n-step measurement algorithms, which reach the different "measurement exactness" for the given number n, the number of measurement steps. So, the constructive approach to the measurement theory brings forth into the problem of effectiveness of measurement algorithms as the principal problem of the constructive or algorithmic measurement theory. We should search the "optimal" measurement algorithms and this problem is the main problem of the "constructive" or "algorithmic" measurement theory.

    "Problem of measurement" became the topic of the fruitful philosophical discussions between Prof. Stakhov and Prof. Chefranov in period since 1974 to 1977. The results of these discussions are the following:
  1. The modern crisis in mathematics is determined mainly by crisis in interpretation of the "infinity" concept.
  2. The difference in approach to the "infinity" concept reveals itself the most brightly in the mathematical measurement theory based on Eudoxus-Archimede's and Cantor's axioms Because each of these axioms uses the "infinity" concepts, which are in contradiction and are incompatible one to another (the concept of the "actual" infinity in Cantor's axiom and the concept of the "potential" infinity in Eudoxus-Archimede's axiom) it follows from here the conclusion that the classical measurement theory and all the mathematical theories based on it are contradictory.
  3. The elimination of the "actual" infinity abstraction from the mathematical measurement theory brings automatically into the elimination of Cantor's axiom from it and put forward the problem of development of the constructive measurement theory based on the "potential" infinity abstraction.
  4. The consideration of measurement as the process completed for the "finite" but "potentially unlimited" number of steps brings into appearing of the quantization error, which is principally irremovable in the framework of constructive approach to measurement. This brings into moving forward of the problem to search the "optimal" measurement algorithms as the main problem of the "constructive" or "algorithmic" measurement theory.

But what was happen from these deep philosophical reasoning? We will tell about this at the next pages of our Museum. Follow us!