Here the defender of the traditional multiplication table of our textbooks has a lot to learn. Each number is different from another number, such as 1 from 2, 2 from 3, 3 from 4, etc. But it is also true that between every two adjacent numbers there is a whole abyss of transitional fractional numbers; and this abyss is so incalculable that it is never possible to go from 1 to 2, or from 2 to 3, or from 3 to 4. We also need to learn to understand how we can suddenly go from 1 to 2 and not go from 2 at once, go from 2 to 3 at once and at the same time and not go over. This is a remarkable dialectic, and in this treatise it is beautifully represented by the theory of one and two.

Fourthly, the treatise not only establishes these two dialectical principles, but gives a consistent development of everything that is formed by these two principles.

It is clear, first of all, what a trinity is. If neither the one nor the binary spoke of any form or structure, then the trinity is a symbol of this first structure, where there is not only the indivisibility of the unit and the divisibility of the dual, but also their formation into a single figure. And then the quaternary is that which is the carrier of the structure, that is, the body, which in the quintuple is treated as a living body, and in the sixfold as an organism. Already at the stage of the six, thought encounters what is usually called the cosmos, since the cosmos is an organically living body, a mental-bodily structure. But this is still the beginning of the dialectical depiction of the cosmos. In the septenary, the cosmos is enriched by the presence in it of a ubiquitous and equally rhythmic well-being, which at the stage of the eightfold reaches cosmic panharmony, and at the stage of the ninefold - to the actively arranged sphericity of the cosmos.

This, in fact, is the end of the picture of the cosmos, thought out by the methods of dialectical arrhythmology. Only one question remains: what is the cosmos taken as a whole, if nothing else exists besides it? Obviously, it can now only be said that it is precisely the cosmos, and not something else. And this question did not arise before, because at the previous stages we were inside the cosmos and talked about those structures that are inside the cosmos, and only at the ninefold stage did we start talking about the cosmos as a whole as an actively landscaped sphericity. Now, after all these internal and external definitions of the cosmos, the question is raised about what the cosmos itself is in general. And as soon as we said that the cosmos is precisely the cosmos, this meant that from the cosmos itself we passed to the idea of the cosmos, that is, to its paradigm, by virtue of which it received its eternal arrangement. The decimal character characterizes the cosmos as the complete identity of the prototype embedded within it and the material corporeality of the cosmos. And only now can the task of dialectical arrhythmology be considered complete.

It has become a habit to present all these Pythagorean-Platonic numbers as a continuous and incoherent confusion, as incredible fantasy, or even simply as a mixture of childish naïve stupidity and mystical delirium. We allow ourselves to hope that the study we have proposed above for the first time destroys this age-old injustice, which almost all previous investigators showed in relation to ancient Pythagorean Platonism. It seems that in this treatise this is not a mess at all, but a consistently developed line from chaos to the cosmos, a line of primordial antiquity. Characteristically, among the definitions of unity, "chaos" directly appears, both in essence and even terminologically, in comparison with which the ten is a complete cosmos in itself in the fullness of its ideal predetermined and material realization. Whoever does not see the sequential arrangement of the philosophical material in this treatise is not only deprived of the opportunity to analyze this treatise, but also understands nothing of the thousand-year history of Pythagorean Platonism. True, the material of the treatise, as we have often seen above, is given both compilatively and ontologically, and often contradictory. But science exists to sort out any confusion and to bring the factual confusion of historical materials to complete clarity and intelligibility.

Fifthly, just as the term "arithmetic" appearing in the title of a treatise is usually misunderstood and misunderstood, it is also common and useless to imagine what "theology" is, which is also present in the title of the treatise as a grammatical participle from the corresponding verb "to theologize". It would be inappropriate here to cite and analyze the innumerable tastes of what theology is. We are not interested here in the ideas and tastes of the contemporary reader, but in the tastes and ideas that are present in the treatise itself. And in the treatise itself, "god" or "deity" and any myth in general is understood only as the ultimate generalization of various areas of the cosmos and the cosmos itself as a whole.

Let us imagine the sea as a whole, with all its geographical, physical, biological, and socially significant features, and not only now, but always, throughout all eternity. This is Poseidon as the ultimate generalization of the sea and everything marine. The same must be said of Demeter as the ultimate generalization of the entire agricultural region. And about Athena in connection with the generalization of honest and just war or heroism, and in another sense - as a symbol of wisdom. The same is thought of Ares, who is the ultimate generalization of a villainous, treacherous, and unjust war, or a war for war's sake.

Therefore, when reading the term "theologoumena" in the title of this treatise, one should not faint, formalistically and literally translating the Greek term "theology" as "theology", but one should understand this subject historically. Historically, ancient theology was simply a teaching about the ultimate generalizations of the cosmos and everything cosmic. Therefore, the title "Theologumena of Arithmetic", which is terrible for many, if we treat this subject strictly philologically and strictly historically and philosophically, should be translated as follows: arrhythmology as a study of extremely generalized cosmic structures. It goes without saying that our proposal to understand the title of the treatise in this way, and not otherwise, is dictated by the desire to formulate its rational kernel. This latter acquires its concrete historical form only in connection with all the other content of the treatise, which is least amenable to rational understanding. Historically, all this rational and all this irrational (we have also talked enough about the latter) must still be able to unite.

Sixthly, the philosophical terminology used in the treatise against the background of very contradictory and confused characteristics is in any case distinguished by great historical and philosophical accuracy, and there is absolutely no arbitrary confusion in it. Such, for example, are the terms "potency" and "energy." Both of them are by no means used in some philistine and disorderly sense. They testify that we are not talking here about any other, but purely semantic becoming. Since the time of Aristotle, this understanding of these two terms has remained in ancient philosophy forever. Such are the terms "logos" and "eidos". These terms are so specific to the whole of ancient philosophy that we did not even dare to translate them into any modern language. Both of these terms also belong to a purely semantic area, and the first indicates more to the semantic method, and the second to the semantic picture and the visual result of the logical method. Such are the terms "paradigm", "demiurge" or "mind". In general, these are purely Platonic terms with their very meticulous development in subsequent antiquity. Finally, the terms "cosmos" or "harmony" are of the same kind. In all ancient philosophy, these terms are necessarily understood materially and corporeally, with an ideal treatment of everything material and corporeal, reaching sculptural clarity and distinctness. In this respect, the treatise we are analysing is absolutely typical.

Отдельно стоит сказать о стоическом термине "семя", или "семенной", и тоже о стоическом понимании первоогня как "художественного". Подобного рода стоицизм нисколько не противоречит основной пифагорейско-платонической традиции трактата.

Наконец, в трактате заметно использование тоже достаточно древнего геометрического понимания элементов. Огонь, например, понимается как пирамида. Все это трактуется вполне в рамках исходной пифагорейско-платонической традиции.

В-седьмых, вопрос, который не может не ставиться в нашем труде, именно вопрос историко-эстетический, если принять во внимание предложенный у нас выше анализ, решается и просто и легко. Кто же будет отрицать, что в античной философии структурные проблемы представлены ярче всего? А вот в данном трактате как раз они-то и ставятся в первую очередь. Или кто может отрицать постоянную склонность античной мысли идти от хаоса к космосу и отчетливо формулировать все подобные этапы этого перехода? А вот это как раз и есть проблематика данного трактата. Также невозможно отрицать огромное значение в античной эстетике таких проблем, как единичность, свертывание и развертывание действительности, ноуменальное осмысление и душевная самодвижность космоса. Все это не только типичная античная философия, но и типичная эстетика античности. Если твердо стоять на той позиции, что эстетика есть учение о выразительных формах, то какую же еще более значительную выразительность можно находить в античной мысли, если не в ее космологизме, то есть если не в ее идеально-гармонически сформированном космосе? Трактат "Теологумены арифметики" есть целиком трактат из истории античной эстетики, конечно, если эту эстетику понимать не как-нибудь вообще, а только так, как ее понимала сама античность.

Наконец, в-восьмых, правомерно поставить также и вопрос о новаторстве философского учения трактата. Было ли здесь что-нибудь новое в сравнении с многовековой пифагорейско-платонической традицией? Что автор везде следует этой традиции - с этим трудно спорить. Однако в двух отношениях трактат, несомненно, отличается некоторой новизной.

Прежде всего аритмология поставлена здесь в подчиненное положение к философии. Здесь очень трудно указать, где кончается аритмология и где начинается вообще диалектика бытия и всей действительности. Это, например, не так заметно хотя бы у Филолая, о котором в трактате имеется один небольшой пассаж (74, 10-15). В трактате имеется также большой экскурс о Спевсиппе (82, 10-85, 23). В этом экскурсе, в котором, кстати сказать, Спевсипп объединяется с Филолаем, тоже не заметно сколько-нибудь яркого синтеза аритмологии и платонизма. Так, у Спевсиппа, как он изложен в нашем трактате, выдвигается на первый план мало разработанное и не очень понятное учение о совершенстве декады. Для нашего же трактата нет никаких сомнений в тождестве пифагорейства и платонизма.