Book: Pythagoras and the Theme of Number
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, 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 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” – 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 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”.
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.
 Commentaries on Aristotle’s work On Generation and Corruption.
 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.
 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.
 Translator’s note: Suppositum is a generic word including all individual substances, applied equally to rational and irrational, living and non-living individuals.