Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science. Alexey Stakhov
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      Pythagoras (570–500 BC) is perhaps one of the most famous scientists in the history of science. He is revered by every person who studies geometry and is familiar with the “Pythagoras theorem”, one of the most famous theorems of geometry. In ancient literature, Pythagoras has been described by his contemporaries as a well-known philosopher and scholar, a religious and ethical reformer, an influential politician, a demigod in the eyes of his disciples and a charlatan. His popularity was such that during his lifetime, coins with his image were issued in 430–420 BC. For the fifth century BC, this was an unprecedented case! Pythagoras was the first Greek philosopher awarded to a special assay (Fig. 1.1).

      Fig. 1.1. Pythagoras.

      The important role of Pythagoras in the development of Greek science consists in the fact that he fulfilled a historical mission in transferring the knowledge of the Egyptian and Babylonian priests into the culture of Ancient Greece. It was thanks to Pythagoras, who undoubtedly was one of the most educated thinkers of his time, that Greek science received a huge amount of knowledge in the fields of philosophy, mathematics and natural sciences, which, by getting into the favorable environment of ancient Greek culture, contributed to its rapid development.

      Pythagoreans created the doctrine of the creative essence of the number. Aristotle in “Metaphysics” notes this particular feature of the Pythagorean doctrine:

      “The so-called Pythagoreans, having engaged in mathematical sciences, first had put forth them forward and after their study began to consider them the beginnings of all things . . . Since, therefore, everything else was explicitly compared to numbers throughout their essence, and numbers took first place in the whole of nature, they had recognized harmony and number as the basis of all things and all Universe.”

       1.1.3. The contribution of Heraclitus to the development of the doctrine of Harmony

      Starting from antiquity to the present day, Heraclitus remains one of the most popular philosophers in the history of philosophy. In 1961, on the recommendation of the World Peace Council, the 2500th anniversary of the birth of Heraclitus was celebrated. Such an anniversary is usually celebrated to commemmorate the history of some world-famous ancient cities or countries, but to do so for a person is rare and unusual.

      Heraclitus believed that everything is constantly changing. The idea of the eternal motion was presented by Heraclitus in the bright image of the ever-flowing river (Fig. 1.2). The postulate on the universal variability of the world, one of the cornerstones of all dialectics, is compressed by Heraclitus in the famous formula: “It is impossible to enter twice into the same river.”

      Fig. 1.2. Heraclitus.

      As Shestakov points out [10], “in the aesthetics of Heraclitus, ontological understanding of harmony is at the forefront. Harmony is inherent, above all, the objective world of things, the cosmos itself what is inherent to the nature of art. It is characteristic that when Heraclitus wants to reveal the nature of harmony most clearly, he turns to fine arts. Best of all, Heraclitus illustrated the harmony of the Cosmos by the image of the lyre, in which the differently strained strings create a perfect harmony.”

      But, in the aesthetics of Heraclitus, there is also a moment of evaluation. This is especially pronounced in the doctrine of two kinds of harmony: “hidden” and “obvious”. Heraclitus prefers the “hidden” Harmony. Widely known is the following saying of Heraclitus: “The hidden harmony is stronger than the obvious.”

      Cosmos, as the highest and most perfect beauty, is an example of the Hidden Harmony. Only at first glance the cosmos seems to be a chaos. In fact, a play of elements and events conceals “the most beautiful harmony”.

       1.1.4. The musical harmony of Pythagoras and the music of the spheres

      Pythagoreans made wonderful discoveries in music. Pythagoras found that the most pleasant to ear consonances are obtained only when the lengths of the strings, that produce these consonances, have ratios as the first natural numbers 1, 2, 3, 4, 5, 6, that is, 1:1, 1:2 (unison and octave), 2:3, 3:4 (quint and quart), 4:5, 5:6 (thirds), etc. The discovery he made (the law of consonances) shocked Pythagoras. It was this discovery that first pointed out the existence of numerical patterns in Nature, and it was this that served as a starting point in the development of Pythagorean philosophy and in the formation of their basic thesis: “Everything is a Number.” Therefore, the day when Pythagoras discovered the law of consonances, was declared by the German physicist A. Sommerfeld the birthday of theoretical physics.

      The discovery of mathematical regularities in musical consonances was the first “experimental” confirmation of the Pythagorean doctrine of Number. From this moment, the music and the related doctrine of Harmony began to occupy a central place in the Pythagorean system of knowledge. The idea of musical relations soon acquired the “cosmic scales” among the Pythagoreans and grew into the idea of Universal Harmony.

      The Pythagoreans began to assert that the entire Universe is organized on the basis of simple numerical relationships and that the moving planets demonstrate “the music of the heavenly spheres” and ordinary music is merely a reflection of Universal Harmony prevailing everywhere. Thus, music and astronomy had been reduced by the Pythagoreans to the analysis of numerical relations, that is, to arithmetics and geometry. All four MATEMs (arithmetics, geometry, harmonics and spherics) began to be considered mathematical and called by one word — “mathematics.”

       1.1.5. Once again about the term of the Mathematics of Harmony

      As mentioned in the preface, for the first time, the term Mathematics of Harmony was introduced in a short article “Harmony of spheres”, published in The Oxford Dictionary of Philosophy [103]. This doctrine, often attributed to Pythagoras, leads to the unification of mathematics, music, and astronomy. Its essence is the fact that the celestial solids, being huge objects, must produce music during their movement. The perfection of the heavenly world requires that this music must be harmonious; it is hidden from our ears only because it is always present. The Mathematics of Harmony was a Central Discovery of Immense Importance for the Pythagoreans.

      Thus, the concept of the Mathematics of Harmony in [103] is associated with the “Harmony of spheres”, which was also called the Harmony of the World or world music. The “Harmony of spheres” is an ancient and medieval doctrine about the musical and mathematical structure of the Cosmos, which goes back to the Pythagoras and Plato philosophical tradition.

      Another mention about the Mathematics of Harmony, applied to ancient Greek mathematics, is found in the book by Vladimir Dimitrov A New Kind of Social Science. The Study of Self-Organization of Human Dynamics, published in 2005 [54]. СКАЧАТЬ