יום ראשון, 19 בנובמבר 2017

Pythagorean Number Philosophy by Phineas Mordell

A SOLUTION OF THE PYTHAGOREAN NUMBER PHILOSOPHY SUPPLEMENT TO THE ORIGIN OF LETTERS AND NUMERALS " BY PHINEAS MORDELL.
Copyright 1923 by Phineas Mordell.


The real reason why the Pythagoreans considered numbers to be the elements and origin of everything was that they believed numeral symbols to have preceded all other forms of writing. [1] This is what they really meant when they said: "Numbers by nature are the first and prior to all things." (Aristotle's Metaphysics, Book 1, Chapter 5). The letters of the earliest alphabet having been numeral symbols, all words were originally made up of numeral symbols or numbers.

In the Hebrew language which also resembles the Phoenician language, (and it was anciently believed that Pythagoras himself was a Phoenician) both "words" and "things" are designated by the term דברים. Since "words" is the equivalent of "things" the Pythagoreans said that all things (i.e. words) had their origin and composition in numbers.

Now according to both the Sefer Yetzira and the Pythagoreans, the four elements, fire, air, water, earth, emanated from the first four numbers, I, 11, 111, 1111.
For lamblichus (Life of Pythagoras, transl. by Th. Taylor, London, 1815, p. 332) says: "The fourth tetractys is of the simple bodies, fire, air, water and earth, which have an analogy according to numbers. For what the monad was in the first tetractys, that fire is in this. But the duad is air, the triad is water and the tetrad is earth."[2]. In the Sefer Yetzira however, air is associated with the number one, water with the number two, earth with the number three and fire with the number four. [3] The real four elements of the Pythagoreans were also a series of strokes 1, 11, 111, 1111, amounting to ten which were originally numbers and letters.

The key to the complete identification of the philosophy of Pythagoras with that of the Sefer Yetzira as interpreted in this thesis is the Zero. According to the Sefer Yetzira as explained above
בלימה means Zero. Was the Zero a secret known to the Pythagoreans? A solution to this question will be found in the Pythagorean dualism. The "one" in the Pythagorean dualism is the Symbol, I. Contrary to the prevailing opinion, I believe that the Pythagoreans regarded the Zero, 0, as the second element which was called the infinite, indeterminate duality, infinite binary, etc. In a binary system of notation, the Zero is the second Symbol. We know now that even the decimal system of notation originated from the two symbols the one, I, and the Zero, 0. This is in perfect harmony with the Pythagorean formula that all numbers originated from two elements, the limited (the one, 1,) and the unlimited (the Zero, 0). Therefore, all things according to the Pythagoreans originated from two elements One, 1, and the Zero, 0. [4] Since One, 1 is the finite, the Zero, is the true infinite of the Pythagoreans. The One, 1 was considered the Good, for it represents that which exists, but the Zero, was called the evil, for it represents non-existence.     

The Pythagoreans could not have meant by infinite duality the number two, as many writers believe, for as the one is finite, so the number two and every other number must also be finite. Moreover, in a binary system of notation the symbol one, 1 itself becomes two just as in a decimal system of notation the symbol one, 1 itself, becomes ten.

Boeckh {Philolaos, pp. 53, 140, 148) made it clear that the Pythagorean infinity which is identical with infinite duality is not a number at all. See also A. Heinze, Metaphyische Grundlehren, Leipzig, p. 26. [5]

As Leibnitz has represented God's Creation of the Universe out of chaos or nothing, by means of two digits as symbols, so some Pythagoreans also interpreted the one as God and the second element the infinite binary (the Zero, 0) as the visible world. (Plutarch's Morals translated by W. W. Goodwin, vol. Ill, p. 109. Boston 1870).

"The Deity, it is thought by some, was distinguished by the Pythagoreans as absolute unity, from unity conceived." (A History of Greek Philosophy, Dr. E. Zeller, translated by S. F. Alleyne, London 1881, vol. 1. p. 398.)

Such a view was apparently that of the Sefer Yetzira (above p. 59 para. 15) "Understand wisdom and be wise in understanding, that there is but one Master, and there is no second to him." So in
תיקוני זוהר it is said אנת הוא חד ולא בחושבן Thou art one which is different from the one that is counted. So Solomon ibn Gebiroi in כתר מלכות says אתה אחד ולא כאחד בקנוי והמנוי  Thou art one and not as the one that is created and counted.

George Boole inThe Laws of Thought, London 1854, p. 48, also says: "The respective interpretations of the symbols 0 and 1 in the system of Logic are Nothing and Universe."
In ancient times various interpretations were given to the two symbols 1 and 0, as the Platonic antithesis of "being" and "non-being." (Boole, ibid, p. 414,) and Leucippus's "Plenum" and "vacuum," into which all existence was resolved (Boole, ibid, p. 413). The Philosophy and religion of Dualism apparently arose from an interpretation of the symbols one, 1 and zero, 0.

Various views prevailed about the origin of these two symbols. Some believed that the Zero, preceded the One, 1. Perhaps the Chinese cosmogony is based on this view. [6] The author of the Sefer Yetzira and the Pythagoreans believed that the one preceded the zero. Therefore, the zero was second.
The view held by some Kabbalists that the nine Sefiroth or numerals (1-9) have emanated from infinity אין סוף implies that the Zero 0, has preceded the One, 1. But according to the author of the Sefer Yetzira and the Pythagoreans even infinity emanated from the One, 1.
After the invention of the symbols 11, 111, 1111, when the binary system of notation developed into a decimal, the zero was regarded as the fifth symbol and hence the fifth element. The five symbols or elements 1, 11, 111, 1111, 0, were supposed to symbolize the whole cosmos which was believed to be in harmony with a decimal system of notation expressible by these symbols. [7]

The four elements, air, water, earth and fire were also symbolized by ten dots * ** *** **** arranged in a triangular form and the sphere of void encompassing them was symbolized by a circle. Hence the void and the universe were symbolized by 


and apparently was simplified in a

 which according to Boethius, the Neo-Pythagoreans used as a symbol for zero.
The void and the universe may also be symbolized by a circle containing the nine numeral figures 1-9. The nine numerals may be designated by Kabbalistic names as
1 – חכמה, 2-בינה, 3-דעת, 4-חסד, 5- תפארת, 6- גבורה, 7 – נצח, 8 – הוד, 9 - יסוד
and the zero circle embracing them may be designated by אין סוף or כתר מלכות.



About the Hebrew Kabbalistic origin of our numerals see A. J. H. Vincent in Journal de Mathematiques Par Joseph Liouville Paris 1839 pp. 261-280 and in Revue Archaologique 18J5-1845) pp. 601-621.)

According to the Sefer Yetzira the Hebrew alphabet consisting of ten double letters and twelve simple letters, representing the vigesimal and duodecimal systems of notation, still more fully symbolizes the universe.

If we consider every stroke as a separate symbol the zero is the eleventh as 1,1,1,1,1,1,1,1,1,1,0, or the tenth as 1,1,1,1,1,1,1,1,1,0

From a Pythagorean point of view the fifth element "aether" like infinity is only another name for vacuum. [8] Therefore, aether also is logically identical with zero (Com. above p. 30). Now "modern physics following Einstein asserts that . . . since the 'aether' as a substance obstinately evades all our attempts at observing it, and all phenomena occur as if it did not exist, the word 'aether' lacks physical meaning, and therefore aether does not exist" (M. Schlick, Space and Time, p. 12). "Since we are free to use words at pleasure there is no objection to using the word 'aether' in the future to represent the vacuum . . . we must be very cautious, however, not to picture it as matter" (ibid p. 20).
In the above explained system of notation from which our system of notation originated, the principle of position (place value) plays an important role. It is this principle that makes it possible to express all numbers, by only two symbols the One, 1 and the Zero, 0. The principle of position is the "miracle" that makes one, many and many, one. In the Sefer Yetzira (above p. 59 para. 15) the principle of position is indicated b}' the words "The ten digits and zero their end is joined with their beginning, as a flame is attached to the coal." Now if the Pythagorean number philosophy was based on such a system of notation as that on which the philosophy of the Sefer Yetzira is based, the principle of position, (place value) must have been known to the Pythagoreans and must have played an important role in their philosophy. But how did they express it? Did their "harmony" mean the principle of position? [9]

 The Pythagorean philosophy is the greatest enigma of all philosophical systems of antiquity. Various sources such as Egyptian, Indian, Greek, Phoenician, Hebrew, etc., have been advanced as the origin of the Pythagorean philosophy. Many years of study have convinced me that the Pythagorean philosophy is identical with the philosophy of the Sefer Yetzira and both are of Hebrew origin. The philosophy of the Sefer Yetzira apparently emanated from the Hebrew prophetic guild or school as conjectured by Joseph Molitor. That which really was the philosophy of the Sefer Yetzira, the Greeks designated as Pythagorean philosophy and transmitted this to posterity in such a mutilated form that in spite of everything that has been written on the Pythagorean philosophy since Aristotle, it has never yet been really understood. The entire Greek account of Pythagoras and his philosophy is unreliable and most likely is a forgery. However, that may be, I hope I have at least succeeded in removing some obstacles to a better understanding of the Sefer Yetzira and the Pythafrorean philosophy and their relation to each other.

Notes

[1] '"The invention of Signs to represent numbers is doubtless much older than any form of writing" (Chambers Encyclopaedia, 1893, Vol. VII, p. 548). "Numeral characters, were first invented because they were first necessary to mankind." (Th. Astle — The Origin and Progress of Writing, London 1784).

[2] "Die Pythagoreer fuehrten daheralles auf die Zahl und nicht auf die mathematischen Figuren zurueck" (S. A. Byk. Die Vorsokratische Philosophie 1. Leipzig 1876, p. 116).

[3] It is the "central fire" (fiery angels the throne of glory) that is associated in the Sefer Yetzira (above p. 61 para. 21) with the number four or ten which the principle of place value makes a harmonious one (see below about harmony). The harmonious one (zusammengefuegtes Eins) was confounded with the plain one and thus fire came to be regarded as the Pythagorean first material element.

[4] The Chinese Philosophers even actually said that the circle 0 and the line ------- are the first elements from which all writing and everything originated. (Thimus Harmonikale Symbolik Koeln 1876, vol. 1, pp. 79-83).
By the "bounded line" and "unbounded line" from which according to the Pythagoreans everything originated (Diels H. Die Fragmente der Vorsokraliker p. 250) they surely meant the line and the circle the symbols for one and zero.

[5] After having written this supplement I lound that Mr. S. Klyce (Universe 1921 para. 43 f) advances the view that zero and infinity are logically identical.

[6] There was first ot all a period when Nothing existed . . . Grad- ually Nothing took upon itself the form and limitation of Unity, represented by a point at the centre ot a circle" (H. A. Giles, A History of Chinese Lit- erature, New York, 1901, p. 3).

[7] '"Die Weltkugel besteht aus fuenf Koerpern: dies sind innerhalb der Kugel Feuer, Wasser, Erde, Luft und ausserdem fuenftens das Gehaeuse der Kugel selbst." (Nestle, Die Vorsokratiker. Jena 1908, p. 162).

[8] "'Das fuenfte Element ist also nicht aus der fuenfren mathematischen Figur geworden, da dasselbe als Urgrund ganz formlos ist. Deswegen spricht auch Stobaeus von bios vier Elementen, . . . weil das fuenfte Element als reiner Urgrund noch keine Zahl und daher im Kosmos nicht voshanden ist." (S. A. Byk, Die I'orsokratische Philosophic 1. Leipzig 1876, p. 94).
"Infinity is not the affirmation of space but its disappearance" (H. Waldon Carr, The General Principles of Relativity 1920, p. 152).
Perhaps it is more correct to read (above p. 60 para. 16) ותכליתן אין סוף "and their termination is infinity" instead of אין להם סוף "to their aim there is no limit."


[9] '"Das absolute Princip dieser Harmonic sahen die Pythagoraeer in der Monas, sie ist ilinen die nur im Gedanken vorhandene mit sich scibst noch unentzweite Einheit, die das viele zu ihrem Gegensatze hat. In diesem Stad- ium ist sie nur Monas. Zur Harmonic wird sie erst, wenn sie aus sich selbsi ztim Vielen wird." (S. A. Byk, ibid, p. 108).

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