It did not come from Olympus, still inhabited by aging gods, but from the noisy city squares.

What one was convinced of had to be made certain for many. The world of numbers and figures was, according to the initial setting, intelligible by everyone. The authority of the Teacher could indicate the way to such comprehension, but it was not able to replace the work of the mind required to penetrate into its structure. Something else had to replace experience and faith and establish itself as the main organizing principle of the emerging system of knowledge.

And this new thing has come. It did not come from Olympus, still inhabited by aging gods, but from the noisy city squares. VII-VI centuries. BC BC – the era of a great turning point in the life of Greek society, the era of liberation from the power of tribal leaders, the growth of self-governing cities, the intensive development of navigation, trade, crafts, the era of the emergence of the form of government, which was called democracy. The democratic system is the greatest achievement of ancient civilization, the degree of participation of the broad masses of the free population in the decision of state affairs, unprecedented and impossible in the conditions of eastern monarchies.

The coincidence of the appearance on the historical arena of science and democracy is by no means accidental. The activity of the people demanded appropriate forms of expression, and these forms were found: in the agora, the main square of the city, current affairs were discussed, officials were elected, court sessions were held, various interests and opinions clashed and fought among themselves. The consequence of this was the emergence of oratory, which soon reached a high degree of perfection. But the art of the orator is the art of persuasion, moreover, persuasion in conditions of freedom of expression, freedom to ask questions and to doubt – this is what distinguishes him from a preacher, mentor or commander. A priest’s sermon can appeal to a sense of faith, a sense of sacred unity, guarding that which is indisputable and non-controversial. The orator in the agora had to convince people who were not at all eager to believe him in advance. The success of the orator depended on the extent to which he was able to structure his speech in such a way that it gave the impression of a monolith pointing in the direction he needed, a single structure, the links of which are firmly and reliably connected to each other, leaving no room for doubts and objections.

This is how the methods of “correct”, generally accepted, reflecting the general foundations of the psychology of this particular society, of connecting judgments and drawing conclusions, were gradually formed and consolidated in the collective consciousness. So logic arose. And, no less important, it was developed not in abstract discussions, but in real and earthly disputes: about public works, about prices, about the guilt of the defendants, about what constantly constituted the content of people’s lives. This logic, thus, closed itself on universal human experience, selecting from it that universally significant that constituted a single background for the entire life of a person and society.

"The Art of Eudoxus". A portion of a Greek papyrus written in Egypt between 331 and 111. BC, under the general title "Leptin’s Lectures", with a popular presentation of astronomical information. The right column shows a circle with the names of the 12 signs of the zodiac. The three preceding columns contain extracts from the Eudoxus calendar with the addition of information from other calendars (Democritus, Euctemon, Calippus). On the back of the depicted part of the papyrus there are 12 lines of verse, the initial letters of which are the words "The Art of Eudoxus" (Louvre). Image: "Nature"

Of course, the logic developed gradually. It took a long way of spontaneous "testing" of judgments and their connections, "getting used to" them, in order to develop techniques that inspire sufficient confidence in themselves. Later, in the disputes of the sophists, who were involved in the analysis of language a lot, logic was refined, cleansed of accidental layers, understood as a set of universal principles that create the possibility of obtaining equally fair conclusions from fair premises. The logic of the Greeks, therefore, from the very beginning had the character of the logic of dialogue, the logic of dispute, the logic that was isolated as a mechanism that ensures the unity of human communication in conditions when the traditional mythological systems of reference came into sharp conflict with the realities of social life and lost their force. And since it is natural for a person to perceive himself in other people, logical norms were gradually consolidated, automated, made the norms of not only communications, but also the norms of thinking proper.

The process of the separation of logic from its original dialogical beginning went on for many centuries, ultimately leading to the canonization of logical forms as not even universal, but, as it were, superhuman principles of correct thinking. This is how the great systematizer Aristotle interpreted its logic; it is all the more interesting that in this regard he assimilated ideas that had fully developed much earlier, among the Pythagoreans (as well as the Eleatics), and were primarily the result of the use of logical figures in mathematical reasoning.

Persian astrolabe. Built around 1223. The device was invented by the Greeks, its operation is based on the principle of stereographic projection (Oxford Museum of the History of Science). Image from

Thus, the evidential apparatus of Pythagorean mathematics did not arise from scratch. The principles of discussion, the principles of constructing convincing reasoning have already been more or less identified and assimilated. These principles were transferred by the Pythagoreans to the field of mathematics. Proceeding from some basic, intuitive properties of numbers, they obtained far-reaching consequences using chains of logical reasoning4.

This is how the concept of proof developed, which, according to V.A. Ouspensky, there is reasoning that convinces us so much that with its help we are ready to convince others. This essential characteristic of mathematics, expressed in its attitude towards operating with abstract objects and in substantiating the truth of statements relating to them by means of logical proofs, was fully realized already in early Pythagoreanism. This is exactly how the first great discovery of abstract mathematics was made – the proof of the incommensurability of the diagonal of a square with its side. The proof of this statement, known from the school course of planimetry (and borrowed from Euclid’s "Elements"), is based on the method of reduction to absurdity, that is, it uses the abstraction of logical impossibility; as many historians of mathematics believe, there was also a direct proof using the abstraction of infinity.

Both of these abstractions are possible only upon reaching the appropriate level of logical thinking, they are fundamentally unattainable within the framework of the Babylonian mathematical traditions. The theorem of incommensurability was the greatest triumph of the new style of thinking, and it also became the starting point for deep studies of ancient mathematicians to expand the concept of number, which ended with the logically flawless concept of the number line, developed by the great Eudoxus. It should be noted that the works of mathematicians, of course, in themselves contributed to the explication of the principles of logical reasoning, turning out to be perhaps the most promising field of their application. This is how a new branch of human cognitive activity was formed, which found its inner justification in reasoning and evidence.

Of course, Greek mathematics was in its essence largely an activity of a purely abstract nature. The well-known anecdote about Euclid, who offered a donation to a student ("Give him an obol") 5, who asked about the benefits of geometry, expresses well the ideology of ancient mathematicians. This does not exclude, of course, the importance of the previous stages in the development of mathematical knowledge, as well as the achievements of the ancient Greeks in expanding the use of mathematics for solving practical problems. The work of the great Archimedes can serve as an example here. However, it was in the sphere of abstract activity that a new rationalizing and logical style of thinking was found, and its capabilities, both purely theoretical and applied, were demonstrated.

Ptolemy’s astronomy is already fundamentally different from Babylonian in its model character: the movements of the luminaries here seem to “peel off” from the visible firmament, are placed in absolute space and analyzed in it. In this astronomy, as well as in Archimedean mechanics, one can see not only the conscious use of the theoretical apparatus assumed by mathematics, but also the borrowing of the principles of modeling reality itself imposed by the latter. The foundations of operating with abstract objects, the foundations of separating objects given in direct experience from their idealized counterparts were laid precisely by mathematics, and only later were assimilated by other areas of knowledge.

The only reason mathematics was able to become the universal language of science was that it created general ways of expressing and studying the properties of abstract objects and their relations: such foundations had to be initially understood in isolation from empiricism, in a pure and therefore universal form for all possible applications. Science, in order for it to be able to fertilize technology in many centuries, had to take shape as a fundamentally atechnological activity, fundamentally devoid of direct practical use. Here, in a peculiar form, the general regularity of the cognitive process, which K. Marx called the ascent from the abstract to the concrete, was manifested.

What was stated above is some reconstruction of the process of the origin of ancient mathematics, but not the history of this process. This reconstruction is limited only to a certain line of evolution of scientific thinking and does not pretend to be more. The process of interaction between science and myth, science and technology, science and social structures was extremely complex and multifaceted. It is impossible, for example, to understand the essence of Galileo’s great discoveries without referring to the history of the so-called medieval Renaissance (XII-XIII centuries). Then European culture again took possession of the ideas of Aristotle, and the Greek concept of governing primary principles, combined with the Roman theory of universal law and the Christian idea of ​​the domination of the divine will, led to the first ideas about the existence of intelligible laws of nature, although they have a divine nature, but accessible to the rational thinking of man. But this, however, is a completely different story.

In conclusion, the following should be emphasized. Science is probably not one of those cultural phenomena that arise with automatic regularity in various societies, leading to the admiration of later historians. Science arose only once, and subsequently this "act of creation" has never been repeated. The importance of this fact and the need for its comprehensive understanding do not diminish in the least from the fact that it happened twenty-five centuries ago.

1 Art. Lem defines technology as "the means of achieving the goals set by society, conditioned by the state of knowledge and social efficiency." See: Lem St. The amount of technology. M., 1968, p. 23.

2 Thales is considered by tradition to be a great geometer. Although it is hardly possible now to determine with certainty what he did himself and what he borrowed from the East, undoubtedly, in the style of his thinking, he did not go beyond the technological, instrumental-empirical relation to numbers and figures.

3 For more details see: P.P. Gaidenko. How science came into being. – Nature, 1977, no.

4 For example, contained in Euclid, but found much earlier, proof of the unboundedness of the set of primes.

5 Obol is a small coin.

Fig. 1. The Perseus cluster of galaxies in the optical (left) and X-ray (right) ranges. The scale of the images is different: the optical image corresponds to the most central part of the X-ray. X-rays do not come from the galaxies themselves, but from the entire cloud of hot intergalactic gas in which these galaxies are immersed. Spectroscopic measurements of this glow make it possible to find out the physical conditions in the intergalactic environment. Optical photo taken from, X-ray photo from

Two groups of researchers immediately reported that a new emission line with an energy of 3.57 keV was found in the X-ray spectra of galaxy clusters. This radiation should come from hot intergalactic gas filling a cluster of galaxies, but, unlike other identified emission lines, this cannot be attributed to any atomic transition. If the non-standard origin of this line is confirmed, it may indicate the decay of dark matter particles with a mass of 7.1 keV.

Intergalactic environment in galaxy clusters

Clusters of galaxies are the largest gravitationally bound objects in the universe. They contain hundreds, sometimes thousands of galaxies immersed in a common huge cloud of dark matter. The intergalactic space in the cluster may appear completely empty, judging by only optical observations, but in fact it is filled with very hot rarefied plasma with a temperature of tens of millions of degrees (Fig. 1). There is a lot of this plasma in the cluster; its total mass is an order of magnitude larger than the mass of stars in all galaxies of the cluster. This plasma contains not only hydrogen and helium, but also a variety of heavy elements that were synthesized during the burning of stars and supernova explosions, and then accumulated in the intergalactic environment. Drawing an analogy with geology, we can say that the isotopic composition of intergalactic gas is an ancient astrophysical "layer" of matter, in which the chronicle of stellar evolution in galaxies for billions of years is recorded.

Due to its high temperature, intergalactic plasma in clusters glows in the X-ray range. This radiation is well recorded by satellite observatories that observe the sky in X-rays, and from it it is possible to reconstruct the isotopic composition and physical conditions in this environment. At such temperatures, all atoms are highly ionized, and the spectrum of this radiation shows numerous lines corresponding to transitions between different electronic levels in various ions. By registering X-ray photons and measuring their energy, one can construct an X-ray spectrum from a cluster and register individual emission lines in it. By comparing these lines with the known transition lines of highly charged ions, as well as measuring the intensity of these lines, one can find out the composition and conditions in the intergalactic medium inside the cluster.

To avoid misunderstandings, we should immediately mention the redshift. Distant space objects are moving away from us at a significant speed due to the expansion of the universe. As a result, the spectrum recorded by us turns out to be shifted to the region of longer wavelengths (into the "red region") in comparison with the original emitted spectrum. When astronomers talk about X-ray spectra of galactic clusters, they mean redshift spectra, that is, the spectra converted to the source’s reference frame. It is these spectra that can be compared with tabular values ​​and with each other.

There are several satellite observatories capable of taking X-ray spectra in the energy range of several keV.

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