Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science. Alexey Stakhov

clarify this contradiction, let’s consider once again the “basic measurement equation” (1.2). The idea of this contradiction is as follows. With the help of the Eudoxus–Archimedes axiom, a certain sequence of the “contractible” segments, which are compared with the measured segment Q, is formed from the measurement unit V according to certain rules, called the measurement algorithm;when this process strives to infinity, on the basis of Cantor’s axiom, for any Q and given V, such segment Q, formed from V, which “absolutely exactly” coincide with Q, will always be found. The most essential part of this proof is a presentation about measurement, as a process that ends in an infinite time (according to Cantor’s axiom). Thus, at the initial stage of the proof of equation (1.2), we use the concept of the potential infinity (Eudoxus–Archimedes axiom), and at the final stage, we “jump over” this concept and use the concept of the actual infinity (Cantor’s axiom). In this regard, it is appropriate to cite the following quotation from the book [16]:

      “It is appropriate to pay attention to the internal contradictoriness (in the dialectical sense) of the set-theoretic theory of measurement (and as a consequence of the theory of real numbers), which allows in its initial positions (continuity axiom) the coexistence of dialectically contradictory ideas about actual infinity: the actual infinityin Cantor’s axiom (and Dedekind’s axiom) and the potential infinity, that is, “becoming”, unfinished in Archimedes’ axiom”.

      The existing mathematical theory of measurement and the theory of real numbers arising from it, based on Cantor’s axiom, are internally contradictory; but such theories, based on the contradictory axioms, cannot be the basis for mathematical reasoning! Otherwise, all mathematics becomes an internally contradictory theory. One wonder what had actually happened in mathematics at the beginning of the 20th century, when the contradictions were found in Cantor’s theory of infinite sets. It is surprising that such a simple idea was not noticed by mathematicians before the book [16].

1.4. Criticism of the Cantor Theory of Infinite Sets

       1.4.1. Infinitum Actu Non Datur

      As it is well known, mathematics became a deductive science in ancient Greece. Already in 6th century BC, Greek philosophers studied the problem of infinity and the continuous and discrete problems related to it. Aristotle paid much attention to the development of this concept. He was the first who categorically began to object against the application of the actual infinity in science, referring to the fact that, despite knowing the methods of counting the finite number of objects, we cannot use the same methods to infinite sets. In his Physics, Aristotle stated as follows:

       “There remains the alternative, according to which the infinite has only potential existence . .. The actual infinite is not exist.”

      According to Aristotle, mathematics does not need actual infinity. Aristotle is the author of the famous thesis Infinitum Actu Non Datur, which translates from Latin as the statement about the impossibility of the existence of the logical or mathematical (that is, only imaginable, but not existing in Nature) actual-infinite objects.

       1.4.2. Criticism of Cantor’s theory of sets in 19th and early 20th centuries

      Cantor’s theory of infinite sets caused a storm of protests already in the 19th century. The detailed analysis of the criticism of this theory was given in Chapter IX “Paradise Barred: A New Crisis of Reason” of the remarkable book by the American historian of mathematics Morris Kline Mathematics. Loss of Certainty [101].

      Many famous mathematicians of the 19th century spoke out sharply negatively about this theory. Leonid Kronecker (1823–1891), who had a personal dislike to Cantor, called him a charlatan. Henri Poincaré (1854–1912) called Cantor’s theory of the infinite sets the serious illness and considered it as a kind of mathematical pathology. In 1908, he declared as follows:

       “The coming generations will regard the theory of sets as the disease, from which they have recovered.”

      Unfortunately, Cantor’s theory had not only opponents but also supporters among famous scientists and thinkers. Russell called Cantor one of the great thinkers of the 19th century. In 1910, Russell wrote: “Solving problems that have long enveloped mystery in mathematical infinity is probably the greatest achievement that our age should be proud of.” In his speech at the First International Congress of Mathematicians in Zurich (1897), the famous mathematician Hadamard (1865–1963) emphasized that the main attractive feature of Cantor’s infinite set theory is that for the first time in mathematical history, the classification of the sets, based on the concept of the cardinal number, was given. In his opinion, the amazing mathematical results, which follow from Cantor’s set theory, should inspire mathematicians to new discoveries. Thus, in Hadamard’s speech, Cantor’s theory of infinite sets was elevated to the level of the main mathematical theory, which can become the foundation of mathematics.

       1.4.3. Research by Alexander Zenkin

      In the recent years, in the works of the outstanding Russian mathematician and philosopher Alexander Zenkin [126], as well as in the works of other authors [127–129], radical attempts have been made to “purify” mathematics from Cantor’s theory of sets based on conception of actual infinity (Fig. 1.3).

Fig. 1.3. Alexey Stakhov and Alexander Zenkin.

      (Moscow University, April 29, 2003: Stakhov’s lecture “A New Type of the Elementary Mathematics and Computer Science Based on the Golden Section”, delivered at the joint meeting of the seminar Geometry and Physics, Department of Theoretical Physics, Moscow University, and Interdisciplinary Seminar Symmetries in Science and Art of the Institute of Mechanical Engineering, Russian Academy of Sciences.)

      The analysis of the Cantor theory of infinite sets, presented in [126], led Alexander Zenkin to the conclusion that the proofs of many of Cantor’s theorems are logically incorrect, and the Cantor theory of the infinite sets is in a certain sense the mathematical hoax of the 19th century. Some famous mathematicians of the 19th century were fascinated by Cantor’s theory and, by accepting his unusual theory without proper critical analysis, elevated it to the rank of the greatest mathematical discovery of the 19th century and laid the foundations of mathematics.

      The discovery of the paradoxes in the Cantor theory of the infinite sets considerably cooled the enthusiasm of mathematicians toward this theory. Alexander Zenkin [126] put the final point in the critical analysis of Cantor’s theory. He showed that the main Cantor error was the adoption of abstraction of the actual infinity, which, according to Aristotle, is unacceptable in mathematics.

      But without the abstraction of actual infinity, Cantor’s theory of infinite sets is untenable! As mentioned above, Aristotle was the first scientist and thinker who drew attention to this problem and warned about the impossibility of using the concept of the actual infinity in mathematics (“Infinitum Actu Non Datur”).

      In Stakhov’s article [129], the following question was posed: “Is modern mathematics not standing on the “pseudoscientific” foundation?” So far, mathematicians have not answered this question concerning the foundations of mathematics.

1.5. Constructive Approach to the Creation of the Mathematical Measurement Theory

      In the framework of the constructive approach to the creation of the mathematical measurement theory, the concept of the actual infinity should be excluded