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the earliest alphabet having been numeral symbols, all words were originally
made up of numeral symbols or numbers.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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