Number and Rhythm – Part 1

Chapter VII

The word number comes from numerus (Latin), which derives from nomos (Greek), law, norm. In Greek, its correspondent is the word arithmos that comes from rythmos, from the root rhe, from where rhêo, rhein (verb), means to flow. Therefore, there is a kinship between number and rhythm. There is also an analogy, in which logos both are identified. The flow of creation implies the number.

“Rhythm is perceived periodicity. It acts to the extent to which such a periodicity alters in us the habitual flow of time. Thus, any phenomenon perceptible by our senses contrasts from the set of irregular phenomena (…) to act only upon our senses and impress them in a manner totally disproportionate to the richness of each acting element”, writes Pius Servien. Matila Ghyka synthesizes as follows: “rhythm is the experience of orderly flow of a movement”.

As Warrain asserts, “rhythm is to time as symmetry is to space”, that is to say, spatial harmony (extensist) is symmetric, whilst temporal harmony (intensist) is rhythm. Pythagoras said – and all posterior Pythagoreans confirm such a statement – that arithmos is “posotetos Khyma ex monadon synkeimenon”, i.e., a movable series that emerges (flows) from the Monad.

Thus, arithmos is something of the movable things, of the things that experience mutations of any kind, encompassed within the mutations studied by Aristotle. There is arithmos wherever there is generation and corruption, increase and decrease, alteration, movement (translation[1]). Therefore, all finite things – that constitute the series of created things – are numbers, have numbers.

All finite being is characterized by composition, for the only absolutely simple being – or, of absolute simplicity – is the Supreme Being.

The One (Hen Prote, first one) is not number, as well as the Hen-Dyas aoristos (the indeterminate-dyad-one), since the latter, being generated by the former (generated, not created), is still the former in its procession ad extra. The generation of Hen-Dyas occurs through an ad-intra procession, still within the Supreme Being. The Hen-Dyas (One-multiple) of Plato is a One in its creative activity that creates the indeterminate dyad (determination, which is the formative actuality of Aristotle, and determinability, which is the materiable potentiality). Number arises from the opposition between determination and determinability, since it is the movable series that flows from the Monad, the product of relations between the opposites in the universal substance.

That is what we shall soon demonstrate after outlining a few essential points, such as the following, summarized in our Treatise of Symbolic: “Aristotle defined the number as multiplicity measured by unity. Such a concept, although, indicates that the Aristotelian notion was merely quantitative. In our Theory of Knowledge, we have studied, although in general lines, the Pythagorean concept of number, which is, undoubtedly, the best approach yet.

The Pythagorean sense of the degrees of teleiotes (the degree of perfection for the initiates), number is not merely the measure of the quantitative by unity, but also, the form, as intrinsic proportionality of things, and can be considered – as it really is – under various modalities.

So to synthesize what was stated in our previous books, we can say, about the Pythagorean thought, the following:

Since number is usually an abstractive expression of quantity, scholars have supposed that such was, also, the conception of Pythagoras. Now, he considered, also, as such, but not only so.

The word number comes from the Greek term nomos, which means rule, law, order. He used such term – the word arithmos – as number in a generic sense. Order is a relation between wholeness and its parts and if we consider that where there is such relation, there is certain coherence, the idea of order becomes richer.

For the Master, number is also such order, such coherence that gives physiognomy of tension of wholeness. In current mathematics, we understand that number is not merely quantity, but also relation, and relation of relation, namely, function. For him, number contains, always, the numerous, since demands a relation and any relation demands the existence of another party. One is not number[2]. One is wholeness, the absolute.

“Unity is opposition between limit and unlimited; unity serves as a moment of tension and approximation of two genus of reality”. Such is a Pythagorean postulate.

We can have any definition of essence, but one note is absolutely necessary: in essence, there is always something that is indispensable for the thing to be what it is. For a thing to be what it is, it must have an order, or, better yet, a relation between the parts and the whole, a certain coherence different from others, so it can what it is and not what other things are. Is it not such order of number? So we can state that all things have their number (arithmos) or their order, essence: that is the reason all concept is number.

To have the experience of such thought, we must divest from the superficial concept that number is the mere indication of quantity. No. Number indicates, aside from quantitative, qualitative, relational, modality, values and other categories.

Therefore, arithmos is quantity, relation, function, tension, law, order, rule. Or, as Philolaus declares, “all things, at least those we know, contain Number; for it is evident that nothing whatever can either be thought or known, without Number” (DK 4).

If we scrutinize the facts that constitute our world – both bodies and psychical facts – we shall realize that they are not static or inert coherence – or, to use the term from Concrete Philosophy, tension – but a dynamic one, which processes, goes from a state to another, takes a direction. Number is, hence, also process, rhythm, vector, flux.

The facts that constitute the world present themselves either similar to each other or different from one another, as well as either completing each other or repealing one another. When two opposite facts are put one in face of the other so to form a correlation, a concordance, a adjustment, as if constituting a new thing, they harmonize.

We all have, through music, an experience of harmony. Pythagoras perceived harmony as the ideal point revealed by nature to all facts, including mankind. Harmony is a result of the adjustment of opposite aspects. Harmony can only occur where there are qualitative oppositions. Two equal beings cannot harmonize, but only “symmetrize”. For harmony to happen it is necessary the existence of additional distinctions other than numerical. Our universe is composed by different unities, and when they adjust with one another, harmony happens.

In aesthetics, advised him, we should not only search for the harmony of symmetry, but for the harmony of opposites, in movement (khiasma). It was abiding by such a great thought that the Greek art was able to create what was, later, the so-called “Greek miracle”.

Pythagoras observed, by studying harmony, that certain relations were verified, once obeyed. Such a relations constitute the so-called “golden numbers” that had in important role in all arts, mainly within superior periods.

Therefore, harmony is the maximum ideal of the Pythagoreans, which consists of adjusting the various elements of nature. Arithmos is also harmony.

He also verified that certain combinations, obedient to certain numbers and in certain circumstances, are more valuable than others. Numbers are also values, since, when actualized, they offer something either beneficial or malefic, revealing values.

The supreme instruction, the supreme knowledge of human and divine things (Mathesis) is an activity; mathema is the study, the knowledge.

One (Hen), which is only (Holos, in Greek, only), is the emanating source of everything. The arithmoi arkhai (from arkhe, supreme, principle) are the supreme principle that arises from One. From the cooperation of such an arithmoi arkhai, only cognizable by initiates, and that are the supreme powers, emerges the organization of Kosmos (universal order). Such a statement demonstrates the influence of the arithmoi arkhai over the platonic forms (eide), which are nothing more than symbols of the Pythagorean arkhai, esoterically exposed by the author of The Republic.

One, as supreme emanating source of the arithmoi arkhai, generates One. One is actuality, pure efficacy, absolute simplicity, and, therefore, pure actuality. Its activity (verbum, in Latin) is its own essence, but represents a role – because, in activity, it is always itself (ipsum esse of the Scholastics), although representing another role (personna – hypostasis), of activity. Such a activity is of the same substance of the Supreme One, which is united, fused by love that unites One to One, forming the first Pythagorean dyad that, once duly analyzed, is little different from the Christian Trinity exposed by St. Thomas Aquinas.

One plus One generated by it plus love that united them forms the Pythagorean triad, symbolized by the equilateral triangle. From emanation (procession ad extra, since the anterior One-One-love is an ad intra procession) emerge Two, the Dyad. The being acquires extreme ways of being that, being inverses, are identified in the being. As Two emerges, which heterogenizes, all numerical combinations (arithmetikai) are possible[3].

Arithmos is also concept, since concept is an arithmos of notes (skhema by aphairesis, i.e., schemes by abstraction). Therefore, we have:

  • quantity (arithmos posótes)
  • quality (arithmos timos)
  • relation (arithmos poia skesin)
  • function (arithmos skesis)
  • law, order, rule (arithmos nomos)
  • process (arithmos proodos, or kethados), which inverse movement is conversion (episthrofe) that does effective return (anados).

Those arithmoi emerge from the arithmoi arkhai and are produced by the emanation of One. And return to One after combining with other arithmoi.

– Fluxes (arithmoi khyma), by which the Pythagorean mathematized the studies about emanations and fluxes of any specie (of light, for instance).

– Number rhythm (arithmos rythmos), periodical number;

– Sets are numbers (arithmos plethos)

– And when they become tensions, are arithmoi tonoi.

Pythagoras was also concerned with the conjunction of numbers that produced transitional qualitative aspects, different from composing elements, such as the percussion of different notes forming a new qualitative aspect. Thence, the symphonic numbers (arithmoi symphonikoi), which, by their turn, form the numbers of harmony (harmonikoi arithmoi).

Proportions of all sort lead to the construction of analogical numbers (analogikos arithmos).

There were still other types of numbers within Pythagorean mathematics. There were the numbers of punctual growth, which were nothing but the segment numbers of Dedekind, the dynamei symetroi (numbers potentially commensurable), and others, such as the sympathetikoi arithmoi and antipathetikoi arithmoi, different from episthemikos arithmos, the scientific number, of profane mathematics.

Only by considering number in its true Pythagorean sense, one can comprehend its symbolic, which is, actually, object of Arithmosophy, the study of significability of numbers. However, we cannot forget that, within the various religious myths, number can appear as having a value in itself, when, in fact, as we shall observe, it is not a power per se, but an indication of such a power, a reference to the arithmoi arkhai, the archetypical numbers.

Natural phenomena and their laws lead to coefficients that are numbers, and all things of the cosmic world are arithmetically realities that imitate numbers. Crystals, plants, humans, stars, sounds, chemical spectra, etc., all reveals numbers and numerical laws. Mathematics shows us how number is an extraordinary instrument of knowledge, inasmuch as we cannot apprehend a phenomenon when we are not able to reduce it to numbers.

As asserted by Pascal, “there are properties common to all things, and the knowledge of them opens the mind to the greatest wonders of nature”. And such “common properties” analogizes one fact to another, allowing the apprehension of references to numbers, thus indicating us the symbolic that emerges throughout time.

Leibnitz acknowledged that “mathematical language” could communicate the secrets of nature and, many times, it was repeated amongst philosophers that mathematics is the language of God and that the divinity built the universe as a perfect mathematician, symbolized through various artistic and religious manifestations, such as in Christianity.

Numbers were studied since earliest times, with references amongst the Vedas, the Egyptians, the Chaldeans, the Babylonians, the Greeks and the Early Church Fathers. Saint Augustine was keen to emphasize that “the unintelligibility of numbers prevents the comprehension of many figurative and mystical passages from Scripture”.

In general, for the Pythagoreans, numbers were entities intermediary between the Supreme Being – the One, which is not number – and the other beings, in which, because created and thus finites, number is a negative limit, since indicates where such entity is what it is, and a positive one, indicating what it is, its quid, for form, as morphe or eidos or schema, in the Aristotelian sense, is number, as Aristotle partially comprehended.

The Aristotelian form corresponds to the Pythagorean form, which is the law of intrinsic proportionality of beings, because if the being is this and not that, it is as it has a certain intrinsic proportionality, which is its arithmos.

For a genuine Pythagoreanism, we must consider the set of created beings according to two triads, superior and inferior, that offers a clear view of reality. Considering the sensible things as the beings more directly in contact with our senses, it is unproblematic, from the outset, to realize that they are constituted by a geometric structure revealed by two dimensions.

Such structures can be reduced to mathematical numbers (arithmoi mathematikoi), as, for instance, done by algebra, algebraic geometry, etc. Therefore, the inferior triad is formed by the following, which can be schematized by mathematics:

  • mathematical numbers,
  • geometrical structures,
  • sensible things

However, the schematical possibilities of knowledge are absolutely not depleted if considered only under such triad. And such soon becomes clear as things reveal an intrinsic proportionality, a scheme that makes it be what it is and not something else, namely: its form.

Such forms (commonly called platonic ideas) constitute the connection point with the inferior triad. Forms are no longer object of sensible knowledge but of an intellectual one, for they demand abstractive activity of the spirit that separates from the phantasm (phantasma, what appears, emerges, comes to light; phaos, light) the eidetic scheme (eidos, morphe) of the thing, that of which (quo) the thing is what it is and not something else, that intrinsic proportionality, that arithmos plethos (number of proportional set), that reveals aa arithmos tonos (a tension, a coherence of its parts with the whole).

It does not matter the plane in which it is considered. For instance, a painting of a portrait, if seen through microscope, would represent only granules of diverse colours over the canvas, and, albeit in such state the same overall view (or the apprehension of its arithmoi plethos) is spoiled, it would not prevent a viewer from seen the portrait through such set of coordinates. Its form, in such relation, is such and, in other, it would present heterogeneity of form. If in one position we see it as a whole (plethos), in other we would see it as a heterogeneous being of other totalities, without excluding the fact that, in such set of coordination, it constitutes a coherent whole, a tension that is different from the tensions of its composing elements, which, on its turn, can form other tensions with heterogeneous elements, and so forth.

Such point, of capital importance in our General Theory of Tensions, reveals that substantial forms, particularly, are the arithmos of the tension, which, by its turn, is a coherent scheme that implies the heterogeneous, since, as tension (tonos), is one and homogeneous, but heterogeneous in its parts that are transcended by the whole, which forms an unity qualitatively different from the composing parts, which, in totality, can only be considered quantitatively.

Therefore, form is not a sensible being, is not a thing persistent of per se, but that occurs within the thing, since the thing is what it is by the form it has, i.e., by the schematic that present an intrinsic proportionality of its parts.

Hitherto Plato reached it in the dialogues, since hitherto it was the exoterical field of the Pythagorean thought. Things imitate forms, since they are of this or of that. Therefore, in a triangle made of wood or of steel, for instance, triangularity is the scheme of intrinsic proportions of such wood triangle, which is a triangle not because it is made of wood, but for participating in the proportionality of triangles that constitute its essence.

Thus, the eidectic scheme of the triangle is the law of intrinsic proportionality of the triangle, imitated (in a Pythagorean sense) by such and such object, or participated (in the Platonic sense) by them. However, such and such object are not triangularity itself, but only triangles, since participate in triangularity,

Triangularity is not a being per se subsistent as something that occupies a stance. It has no “where” or “when”. It does not happen here or there. It is and subsists in the being; or, better yet, in the infinite power of the being. Triangularity is a “can be” that sensible things imitate, triangularizing by the intrinsic proportionality they have. And, thus, the noetic eidetic scheme – build in the spirit – is the enunciation of the law of intrinsic proportionality of the triangle in intentional terms, in noetical terms, according to our spirit and its capacity of assimilation and construction of schemes that apprehend, within the facts, triangularity. Therefore, to Platonism as well as to Pythagoreanism, the eidetic scheme of a thing belongs to the omnipotence of the being; it is, therefore, ante rem. In the thing, we have the concrete scheme by imitation (mimesis), i.e., in re, and, in the human mind, we have the noetic eidetic scheme, post rem (after the thing)[4]. The Pythagorean and the Platonic (since Plato was a Pythagorean) thoughts cannot be fully understood if the terms are disposed differently.

We have, then, the forms as the first element of the superior triad. Forms, however, reveal an ontological structure that corresponds, in the eidetic field, to the geometrical structures in the field of inferior triad (of the sensible things). The intrinsic proportionality of things, the arithmos eidetikos, presents an ontological structure, whilst sensible things present an ontic, singular structure.

Such ontological structure reveals the arithmoi arkhai (the archetypical numbers) that are immediately inferior to One, the Supreme Being, the Divinity, which is not number, since number belongs to multiplicity, to what is dual, to dyad, as in the esoterical thought of Pythagoreanism, which we shall discuss further on.

We have, now, the two triads, disposed as follow:

Superior Triad Inferior Triad
arithmos arkhai (archetypical numbers) arithmos mathematikoi (mathematical numbers)
Ontological structures Geometrical structures
Forms (arithmoi eidetikoi) Sensible things

In the field of Symbology, we could say that the sensible things participate in the geometrical structures, in figures, in mathematical numbers, in forms, etc. Thus, things symbolize the higher until reaching arithmoi arkhai.

We can symbolize through figures (which are geometrical structures) a sensible being. For instance, in a cubist expression of Napoleon, we would have an apparent inversion, since the participant would be symbolized by the participate. But that is not the case. There is the association, through the figure of Napoleon, when symbolizing him by a cubist figure, reduced to a figurative scheme. It is not a complete symbolization, but a copy, an imitation of such geometrical structure. Symbol includes more in its language, since moves towards the eidetical (for instance, when symbolizing Napoleon by an eagle).

Symbol contains something of imitative, for there is no assimilation without corresponding accommodation, which implies imitation. But if imitation is a co-principle of symbol, it is not, of per se, sufficient to indicate its essence, for, on the contrary, we would have to include all imitations in the specie of symbol.

It the figurative can symbolize, there is no, herein, the revelation of occult, which is also a characteristic of the symbol that points to it. Such does imply that the figurative cannot symbolize, but that it only does it partially, for it points to the figure of the immediate symbolized. It symbolizes when points to the symbolized or turn present notes of it that is not contained in the symbol that belongs to the symbolized. Symbol points out, through the imitative, to the symbolized, but it does not intend only that, but what it is of the symbolized, not contained in the symbol. Therefore, the symbol is, hierarchically, always less than the symbolized, for the symbol participates in something of the symbolized, which is the participated, and participates in a lower degree in what the latter has in plenitude.

Symbol is a way to make present what is absent. Therefore, it is not only the imitative that has to be considered, but what is more in the symbolized. Such similarity implies something that differentiates. The aesthetical delight provoked by an artistic symbol comes from referred aspect. A work of art per se expresses the figurative aspect, but, as it points to beyond and allows the delight in a plenitude, it offers a aesthetical delight beyond mere sensible apprehension, since, on the contrary, art would be considered only under the angle of esthesia, senses, regardless of the angle of the spirit.

The aesthetical sensation is made not only of immediate intuition of exterior expression, but of apophantic – therefore mystical – intuition that allows a penetration within the intrinsic of the work of art, which is lived in different levels according to the spectator’s ability. That is the reason art can never be exclusively realist, in the abstratist sense as a copy of reality. In any way, such “reality” speaks a symbolical language, reason why the realists are actually “impossible realists”, since, whether they like it or not, their work goes beyond their intentions. Therefore, all work of art is, in its way of expression, realist, but symbolically transcendent, despite the artist’s intentionality. It always allows a symbolic interpretation, sometimes in disagreement with “the first impressions” of the artist.

We can classify the numbers (arithmoi) according to the sciences that include them as material object. Therefore, we have:

  • Pure numbers: Arithmology
  • Scientifical numbers: epistemikós arithmos
  • Sensible numbers: arithmos logistikós (calculation numbers of vulgar mathematics)

Scientifical numbers, according to Nicomachus of Gerasa, are:

  • Limited multitude (posotes): Quantitative number, abstraction of quantity.
  • Monads composition (plethos, tonós): Classes of classes.
  • Flux (khyma).

Nicomachus defined the second type as follows: “The Pythagoreans considered all terms of a natural series of numbers as principles, so that, amongst sensible objects, the triad is the principle of three, tetrad is the principle of four, etc.” Such definition is similar to what some modern logicians offer to the concept of numbers as classes of classes.

Pure numbers, matter of Arithmology, are defined by Nicomachus as “principles (arkhai), as origins of Number and all and everything, are ‘the Same and the Other’, or the quality of being the same thing and of being something else”.

The relation between two objects or quantities is arithmos skesis. And harmony, according to Philolaus, is the “unity of multiplicity, and the agreement of discordances” (DK 10).

Therefore, the essence of things, the Forms, are also numbers. Some identify form as essence, and the latter with numbers. However, it is important to distinguish them. Eidetic form, as exemplary in the order of Supreme Being, is ante rem. Forms, within things, the concrete forms, in re, are laws of intrinsic proportionality that constitute the formal structure of sensible things, the eidola (little forms) of Plato.

Eidetic noetic forms that fit logical definitions are in intellectu post rem; they are built according to human intentionality, which are nothing but concepts. They can be logically devised, when depleted of all pragmatic content and considered only on its logical structure, according to Aristotelian norms that fit the definition, which is equal to the next genre and the specific difference, and the social-historic concept, noetic form, in which there is a contribution to human experiences with immense variations, object of the science of Schematology.

Once number (arithmos) is placed in its genuine Pythagorean sense, all false interpretations immediately crumble: the true thought of the Samian becomes transparent. One then understands that Mathematics in the Pythagorean sense is not common mathematics, as studied in schools. The latter is included within the former, but it does not encompass the totality of the mathematical thought. So, in order to avoid common mistakes, we shall call such Pythagorean theorizing Metamathematics, since the term mathematics is impaired by such vulgar misconception.

In our commentaries to Metaphysics of Aristotle, we shall have the opportunity to examine the mistakes of the Aristotelian analyses, emerged from an unfamiliarity of the Stagirite about the genuine Pythagoreanism – which is acceptable, since the Pythagoreanism was a prohibited thought that became deliberately disfigured, distorted and falsified. On the other hand, many philosophers are Pythagoreans without even knowing it. And that is the reason we shall further on demonstrate that Pythagoras fecundated, more than any other, western thought, and the presence of his thesis can be identified within most of the speculative works.

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[1] The conversion of something from one form or medium into another.

[2] It cannot be mistaken with the arithmetical one.

[3] One generates One, in in intra procession of the Pythagorean trinity, similarly to the Christian concept. In ad extra procession, it generates an one (universal substance) that is dyad (two) in its operation.

[4] For not having a here nor a where, the eide has no figure (qualitative determination of quantity) nor limitative determination of any kind. Therefore, in order to better understand them, we cannot reduce them to the schematic of our sensible intuitions (phantasmata), as intended by those without adequate mens philosophica.

Mathematics and Pythagoreanism

Book: Pythagoras and the Theme of Number

Chapter XVIII

The great improvement of mathematics within modern science, mainly in the extraordinary results of Physics, authorizes the statement – already widespread – that modern science is under the aegis of Pythagoras. As medieval science was predominantly Aristotelian and the pre-relativistic, Democritean, modern science is Pythagorean.

For obvious reason, such classifications can be exaggerated, mostly when, instead of indicating a prevalence of those ideas, one considers that those sciences are totally dominated by the respective school of thought.

We have demonstrated, in Aristotle and Changes[1], as groundless the statement that modern physics is predominantly Democritean and distant from Aristotelian postulates, since, as it was verified, modern atomic theory is more Aristotelian than Democritean. Nevertheless, one can never doubt that science, in general, has a stronger inclination towards Pythagorean postulates than usually believed, once one has a clearer and truer view of the thought of the great Samian philosopher.

Modern mathematics establishes the undisputed triumph of abstraction. The Set Theory, deriving from Function Theory, is an achievement of modern mathematics that is nigh to the Pythagorean thought, mainly the arithmoi plethoi, the arithmoi tónoi and also the arithmoi khymai – well-known by the initiates of second degree onwards – with so extraordinary achievements that would amaze the modern spirit.

In his book El Nombre d’Or, vol. II, Matila Ghyka – one of the most enlightened Pythagoreans of our time – writes the following words we shall quote and comment:

“By the operation of our last mathematical symbols, we emphasized an image of the physical world in which only the structure is considered a philosophy of Pure Form, Form and Rhythm, or at least periodicity, since, in this world of physical phenomena (what formerly was called world or material “plane”) we shall see that, following the words of Nicomachus, knowledge cannot encompass but relations and structures; Number, not substance, is the only, eternal reality”.

This is, undeniably, the inclination observed in modern science. Atomic theory can be reduced to a mathematical formulation, ascribable to which Physics experienced a progress that astonishes the human spirit.

“The paradoxical subtlety of Cantor’s theory of transfinite numbers (foundation of the Set Theory) had already inflicted disruptions amongst some mathematicians, and the controversies between finitistics and infinitistics about the logical possibilities of an actual infinite, of a mathematically achievable (in thought) infinite, not only the never reaching limit, such as the exasperating one of classical algebra, Cantor develops, manipulates, numbers, portions in achievable (conceptually) infinite processions of different orders; one does not perceive at first sight the impression of a disruptive fantasy, but of a discipline worthy of integrating the classical temple of Mathesis. The audacity of conceptions, this symbolics in which the Hebrew Aleph of Zohar and Torah becomes the symbol of the cardinals, and the Gnostic Omega, of the transfinite ordinals, refers us to kabalistic lucubration, sefirotic pyramid, white magic tower, Golem of Symbols of dreadful growth, of Rabi Loew’s some disciple over the diapers of Hradschin…”

To the Pythagoreans, it is the sacred decad, the link of everything, from where the numbers and the rhythms sprung.

Rhythm is, according to him (in the best definition we know), the experience of orderly flow of a movement. In rhythm, there is the timing and the intensity. It is the connection of the intensist to the extensist, of the qualitative to the quantitative. The pythagorean arithmoi rythmoi coalesced[2] with the arithmoi posotes, the quantitative numbers – which Aristotle considered to be the only Pythagorean numbers.

Modern mathematics is at the verge of qualitative. It is not only an abstractive realization of third level, but is also “concretioning”, since modern relativity coalesces the coordinates into a set and the calculation of sets encompasses “concretion”. Modern mathematics is not a higher level of abstraction, but an activity that, from abstracting to the extreme, becomes concretioning, since connects, “concretionizes”, assemble in sets or groups, what the analyses hitherto had separated. Modern mathematics, in its “critics”[3] – in the sense we use this word – is more sincritics than diacritics, therefore defining a new vector, which final outcomes are still to be totally discovered.

He proceeds in referring to Cantor: “But the creations of this disquieting wizard of the transfinite has incorporated into our mathematics and logic as the pageant armor of the Set Theory, which another cabalistic, tamer and enchanter of symbols, making use of the Group Theory, took us in three transcendental jumps, from paradox to paradox, to the ultra-Pythagorean synthesis, enunciated above the physical universe in number-ideas”. A clear reference to Einstein, the Pythagorean of our time, who undoubtedly gave science new directions by synthesizing in his magnificent work so much of the current achievements of mathematics and physics.

If the ‘raw material’ has been finally discovered, so was found that all so-called material bodies, so-called solids, including our live bodies, due to the enormous spacing between the constituting molecules and despite of the apparent framework, are, in reality, gaseous (the only relatively solid bodies known in the Universe are three recently discovered stars, including the white dwarf or Sirius B, with a compressed matter of 60.000 times the density of water, so much that a ton of it could be contained within a matchbox, and the nucleus of its atoms should probably be quite nigh to suppress, to the most part, the zone of electric orbits and even the planetary electrons). Moreover, those molecules and atoms, undecipherable for the past forty years, are finally known as small solar systems, almost empty zones, in which, in turn, from relatively astronomic distances comparing to their dimensions, gravitate around one another, last particles of ‘substance’ (not of matter, since they have lost the only material quality of matter, i.e., constant mass), pure electricity particles, negative of positive (electrons or protons).”

One cannot confuse the philosophical concept of matter, such as demonstrated in Concrete Philosophy, with the one ascribed by Physics, which still follows the prejudices of the 19th century, after all, mass is not the essence of matter. Nevertheless, it is without doubt that modern physics withdraws from 19th century’s prejudices, the vulgar materialism of that time. Matter becomes pure numbers to physicists such as Heisenberg, mere mathematical structures, consistent with the famous sentence by Pythagoras, “All things are numbers”.

Those words of Ghyka was never so meaningful as today: “And if accidentally the Knowledge also regards the bodies, material suppositum[4] of incorporeal things, it specially connects, nonetheless, to the latter, since those immaterial and eternal things constitute the genuine reality. But what is subject of creation and destruction (matter and bodies) is not actually real, by essence. One can then recognize inasmuch as this worldview is similar to the one of modern mathematical physics, where only the invariable, the structure counts”.

Already at the Renaissance, Leonardo da Vinci said that “there is no certainty in sciences where one of the mathematical sciences cannot be applied, or which are not in relation with these mathematics”. One can comprehend the extent of accuracy of da Vinci’s sentence once undertaking the concept of mathematics in the genuine Pythagorean sense, distinct from the strictu sensu predominant concept.

Our – man built – mathematics proportionally repeats, to the human intentionality, the mathematical interpretation man is able to build of the great mathematics of the universe, the great content of Mathesis Megisthe (mathema, mathematos, mathematics). Our mathematical language expresses something of the one that presided the great realization of creation. Reason why Sir James Jeans does not hesitate in saying: “From the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician”.

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Commenting the Greek contributions to our culture, Matila Ghyka acknowledges as the two most important, one of Egyptian origin, represented by the Pythagorean thought and its influence, such as the roughly holy aspect of geometry, the perfection of form, the importance of secret, the magical value of the Logos, the magical value of sign, symbols, rhythm, etc., and other of hyperborean origin, such as referred by Herodotus and Heraclides Ponticus.

From the Greek spirit, the following are genuine Pythagorean contributions to our time:

– Prevalence of synthesis and clarity of synthesis;

– Perfect formal beauty, in the work of art;

– Geometry as an ideal model of a synthesis founded on axioms and on catenation of unassailable logical deductions;

– Establishment of a theory of numbers, in which the universe is ruled by or arranged according numbers;

– Concepts of proportion and rhythm, derived from the two mentioned theories (theory of forms and theory of numbers) and applied to the research of Beauty;

– Theory of musical harmony;

– Harmonic model of Cosmos;

We deem as the most valuable – and forgotten – of them all, the axiomatization of philosophy. There is a place in philosophy for an axiomatics to the same extent as for mathematics. An axiomatization would allow a metamathematization of philosophy, in the eminent Pythagorean sense. That was exactly what we have done in our book Concrete Philosophy, which, by gathering the apodictically demonstrated “positivities”, remains under the auspices of the Samian master.

Pythagoreanism nourished countless sects, due more to the deficiency of its disciples than to the prominence of the great initiate that was Pythagoras. Thereby, the Gnostic sects such as Ophites, Essenians, Manicheans, Paulicians, Bogomils, Albigensians, Cabalistics, Rosecrucians, Masons, and innumerable other imbibed from the Pythagorean source and their heterogeneity accrues from heterogeneous ways of interpreting the philosophy of the Samian master.

Our intent in this work is not to evaluate those sects as true or false, right or wrong, but, to the resemblance of Cuvier’s method, by using the steps of our decadialectics and our concrete dialectics, which we consider to be the most expedient means for examining an philosophical thought, to reconstruct the Pythagorean doctrine from unequivocally valid postulates. And, from those postulates, through rigorous ontological derivations, similarly to what was done in Concrete Philosophy, restore the true doctrine of Pythagoras. Starting from those restorations, with self-providing value, since they are proportionally worthy the provided demonstrations – as in the words of St. Thomas Aquinas, “the only authority in Philosophy is demonstration” – we can, therefore, appreciate the value of the various positions and discern those that can be considered legitimate inheritors of the philosophy of Pythagoras.

So shall be our endeavor.

[1] Commentaries on Aristotle’s work On Generation and Corruption.

[2] Mario Ferreira had created an unique vocabulary to treat some of the aspects of his Concrete Philosophy. Some words, he created. Others, he used an archaic version of Portuguese, to emphasized their Latin roots. One of Mario’s neologisms were “concrecionar”, which could be translated into “concretionize”, with a similar meaning as the word coalesce, i.e., to grow together into a single body. “Concretionize”, however, is preferable since it has the same root as “concrete”, as in Concrete Philosophy.

[3] Another of Mario’s neologism, related to the Latin word krisis, separation, abyss, as referring to a separation in between things that propitiates the analytical critics, the exam of parts of a whole so to more concretely apprehend the latter. It has a vectorial dualism: diacrisis, the constant separation, and sincrisis, the reassembly of the formerly scattered elements. He develops this subject on his book Philosophy of Crisis.

[4] Translator’s note: Suppositum is a generic word including all individual substances, applied equally to rational and irrational, living and non-living individuals.