Numerical structuralism and cosmogony in the ancient Near East

Jr. Social Biol. Struct. 1982 5,255-266

Numerical structuralism and cosmogony in the ancient Near East

Robert R. Stieglitz Department of Hebraic Studies, Rutgers University, Newark, NJ 07102, USA

The systematic investigation of sexagesimal numbers, by means of reciprocals and ratios, led to a 'dualism' in the numerical theory developed in ancient Mesopotamia. The historical development of mathematical th inking- the mathopoeic m i n d - resulted in a proto-science, including arithmetic, geometry, astronomy and possibly harmonics, which attempted to formulate a structural theory for the pantheon and cosmos. An exponential matrix, based on sexagesimal considerations, is proposed as a means of exlploring the mathopoeic component in ancient mythology, which often incorporates significant proto-scientific data. Since the mathopoeic mind in Mesopotomian civilization is fully developed by 1750 BC, it is likely that the vast body of data gathered by Near Eastern proto-science served as the historical reference reservoir for the Greek mathopoeic mind - especially for the Pythagoreans. Both of these mathopoeic traditions were, however, de-emphasized in Graeco- Roman Judaism because it appeared that they could divert the believer to pagan 'natural philosophy'.

Mythopoeic and mathopoeic thinking Historians of the ancient Near East, having discovered that speculative thought throughout that region took the form of myth (oral, written and graphic), used the term mythopoeic to characterize ancient thinking, in contrast to modern scientific thought (Frankfort, Wilson & Jacobsen, 1946). Historians of science, however, noted that certain types of myths, especially those narrating some sort of 'astronomical' data and having a wide distribution which appears to be of Near Eastern origin, contain technical data in literary format, which must be deciphered (Santillana, 1961). More recently, McClain has proposed that mathematical harmonics were the common denominator which provided mythopoeic poets in antiquity with the models for their understanding of the cosmos (McClaln, 1976). The aim of this study is to examine the mathematical, or structural, component in ancient Mesopotamian thought as it is revealed in mathematical texts of the Old Babylonian period (ca. 1830- 1530 BC), and in other literary compositions; to propose a mathematical model for a theoretical exploration of numerical structuralism in antiquity; and to assess briefly the impact of this mathopoeic component on the history of structural thinking among other ancient cultures.

The mathopoeic component in Mesopotamian civilization is defined here as that aspect of structural thought which led Mesopotamian mathematicians to the systematic investigation of numbers and their properties; to the development of procedures and operations; and to the formulation of a number and ratio ttieory (Neugebauer & Sachs, 1945).

0140-1750/82/030255+12 $03.00/0 O 1982 Academic Press Inc. (London) Limited

256 R. R. Stieglitz

While it is certain, for example, that the 'Pythagorean theorem' was at best only re- discovered by the Greeks, since it is commonly utilized in mathematical texts a thousand years before Pythagoras, we cannot speak of Old Babylonian 'geometry' in the Greek sense (Neugebauer, 1969). The Mesopotamian mathopoeic mind was different from Greek mathopoeic thinking. The Mesopotamian was laconic, and proclaimed theory by axiomatic analogy. The Greek was dialectical, and proved theory by axiomatic logic.

The dichotomy between the mathopoeic and mythopoeic strands in ancient Mesopotamia is not well det'med as it was in Greek civilization after the impact of Greek philosophy. Indeed, there are indications that the integration of math and myth was a desirable goal in ancient Mesopotamia before the advent of Greek philosophy. If we can speak of Greek 'science' (or 'natural philosophy') in antiquity, then we must regard Mesopotamian structural thinking as part of a 'proto-science', the direct ancestor of Greek, and Western, science.

The foundation of Mesopotamian mathopoeic thinking was its unique mathematical system - a sexagesimal system employing place-value notation (without the use of a zero- sign in the Old Babylonian period). This notation system, whose origins can be traced back to Sumer in the fourth millennium BC, enabled the Babylonian mathematician to perform relatively complex operations with ease. Egyption mathematical procedures, entirely decimal with no place-value notation, and operating with 'unit-fractions', were truly primitive in comparison to their Babylonian counterparts. It is no coincidence that, while both civiliza- tions had a keen interest in observational astronomy, mathematical astronomy developed in Mesopotamia and not in Egypt. Numerical procedures in Egypt could only retard a science so dependent on mathematical descriptions of complex periodic phenomena.

The relationship between the mythopoeic and mathopoeic components in Mesopotamia is analogous to the interaction between a 'magical' and 'scientific' tradition which are attested in the field of medicine in Mesopotamia (and in Egypt). The Old Babylonian magician- healer (a~ipu) and physician (asa) were estabilshed and recognized professions, sometimes working together, to heal the sick. They differed in their approach and procedure, but both were honored. Similarly, the Babylonian mathematician was probably a professional scribe and/or priest, whose 'scientific' talents were applied primarily in the service of religion and/or the royal court. His political milieu was, therefore, vastly different from that of his Greek counterpart over a millennium later. The Greek citizen of a polis could choose to apply his talents in a much greater variety of political and social structures.

Number and structure in sexagesimal mathematics The requirements of temple and royal administration led Mesopotamian mathematicians to develop practical devices for use in mathematical operations. Among the most important were the tables of reciprocals, multiplication tables, exponential tables, the 'standard' tables and lists of coefficients. The tables provide us with an insight into the origins of mathopoeic thinking - the realization of a fundamental 'dualism' in the structure of numbers. Numbers were classified into two types: 'regular' and 'irregular'. 'Regular' numbers, as they are termed by modem scholars, were those numbers which were the product of the prime numbers, 2, 3 and 5, raised exponentially to any power (n = 2P3 ~ 5r). The reciprocals of these numbers expressed sexagesimally, it was discovered, were also 'regular' numbers. The 'irregular' numbers, those containing primes above 5 (beginning with 7, 11, 13 etc.), had reciprocals which could not be expressed as a finite series of sexagesimal fractions.

The 'standard' tables were reference lists usually containing 30 pairs of regular numbers and their reciprocals, in the form n::h (i.e. n x ~ = 1). They begin with the pair 2::30 and end with l , 21::44, 26, 40; i.e. they are limited by 2P3 q 5 r ~< 3 ±4 . It is important to remember

Numerical structuralism and cosmogony 257

that because of place-value notation a cuneiform 'number' can, in fact, be interpreted as n x 60*x; thus '3' can stand for 3;0 (= 3), or 0;3 (= 1/20), or 3,0;0 (= 180), etc. The final position of our 'decimal point' in sexagesimal mathamatics is determined only by context.

Of the 30 reciprocal pairs in the 'standard tables, some are redundant pairs since they appear twice (the second time in 'reciprocal' order): e.g. 50:: 1, 12 and 1, 12::50. Indeed, some tables omit several of these redundant pairs (always the second pair). To use the omitted value, the scribe simply went back to the first occurence of the pair. The actual number of unique pairs is not 30, but only 22; and since 1 = 60 in the notation, we are left with only 43 'numerals', These 43 numerals can be arrayed as an exponential matrix generated by the intersection of 2 p and 3 q , up to the limit of the 'standard' table: 2 *6 and 3 *4 . The result is shown as Fig. 1.

o o

o

4 4 , 2 6 , ~ 0

0 x \

o

o o

o

o o

o

0 0

25 56,'-5 o o X o X

5 0 ,15 1,52,30

0 ~X X -- X

/ / \.3,45 /

x X~ 7 £ o

X X X X 0

X X. X 0 0

\ 0 X X X 0

.o/\ / x 1 x o o

X X 0 0

x X o

o

X 0 X 0

o o 0

0 0 0

Fig. 1. Matrix for the sexagesimal 'standard' table o f reciprocals. The 'standard" numbers (= x) are indicated in sexagesimal notation. Numbers not in the table are

represented by (o). The center o f symmetry is I x 60 n

The powers of 5, our third prime number, appear in a sexagesimal base as reciprocals of 2P3 a , since, for example, 5:: 12 and 25::2, 24. All regular multiples of '5 ' , therefore, appear by the sexagesimal generation of the multiples of '2 ' and '3 ' only. The latter two primes, and unity - the prime maker - thus constitute a primordial triad, generating all regular numbers. In mythopoeic language, the poet might say that the 'One' gave birth to a 'Female' (= 2) and a 'Male' (= 3), who in turn mated and thus begot successive generations of 'sons' and 'daughters' (2P3 q 5r), formed in the 'image' of their prototypes.

In Fig. 1, every numeral and its reciprocal in the number field are equidistant from the center of symmetry (= 1 x 60x). Each numeral occupies the same 'reflected' position (mirror-image) with respect to its reciprocal, via unity. This unity, the 'One', was a divinized

258 R. R. Stieglitz

concept throughout the ancient Near East, before and after Hebrew monotheism (Gordon, 1970).

Our matrix has additional features of interest. The horizontal relationship between two numerals in any row is (2/3) ±x (where x is the number of 'intervals' between numerals, and (+) means movement to the left, (--) to the right (= 3/2)). The vertical relationship, however, is that of (2 x 3) -+x. In a sexagesimal base, where 6 = 0;10, etc., this has the effect of providing us with a vertical multiplication/division table for 10 x , i.e. a decimal substratum within the sexagesimal base. Thus, the mid-field vertical line of the matrix is composed of 10 -2 , 10 -1 , 10 ° , 101 , 102 in sexagesimal notation (it may be useful to regard the sexagesimals in terms of our chronometric system: 1, 40 = 1 h 40 min = 100 min). The historical development of sexagesimal cuneiform notation indicates an unmistakable decimal component, and that phenomenon may be explained by a mathematical model in which both decimal and sexagesimal systems are integrated.

The matrix, of course, may be extended indefinitely along any line of the number-field. Old Babylonian reciprocal tables show that such extensions were carried out routinely. The numeral 2, 5 (= 5±3), for example, is often the starting point for such extention tables. The reason for starting with that particular power of 5 is clear: in the 'standard' tables '5' is developed only to the limit 2P3q5 ±2 . What is significant about the arithmetic extentions is that they are often calcualted to theoretical limits, i.e. involving magnitudes which can only have 'mathematical' or 'scientific' uses, as opposed to practical applications in daily administration. In extreme cases, texts calculate numerals to a limit of 60 -~ is.

The investigations of number/reciprocal was not limited to 'regular' numbers. The 'irregular' numbers were also studied. Text YBC 10529 lists all (regular and irregular) number/ reciprocal pairs from 56 to i , 20 (= 80). The reciprocals of the 'irregular' numbers were simply approximated by rounding off. The prime-triad 7, 11, 13, for example, was the sub- ject of various calculations dealing with prices of goods. But the primes 2, 3, 5 were the basis for all 'standard' calculations and metrology in Old Babylonian times. It seems clear that the sexagesimal base itself (22 x 3 x 5 = 60) was chosen with mathematical operations as a primary consideration.

In the lists of coefficients, prepared for the use of scribes in dealing with various administrative matters (Kilmer, 1960), we Fred the coefficient 1 ;24, 51, 10 (= 1.4142128 +) as the Old Babylonian approximation of the %/'2; it is an extremely go6d calculation. This constant appears written in the middle of text YBC 7289, which presents the geometric solution to an apparently simple problem: what is the diagonal length of a square with a side of 30? McClain (1976) has noted that this problem is of supreme interest in musical theory, and, indeed, that the sexagesimal base is ideally suited for mathematical harmonies. Thus, in the 30-60 octave, the sexagesimal octave, the geometric mean--42 ;25, 35 which is the answer given on the tablet-is the elusive center of symmetry of the octave. But the choice of a square with a side of 30 makes good mathopoeic sense: the sexagesimal number 42, 25, 35 (actually 0;42, 25, 35) is the reciprocal of 1 ;24, 51, 10 (= ~/2). As noted, reciprocals of 'irregular' numbers were also of interest to the mathematicians.

In text YBC 5022, a list of coefficients designated Ud, the text entry in line 65 was read by Neugebauer & Sachs (1945) as 57, 36 Ya-... lu(?)-~-um. The correct reading is likely to be 57, 36 la SAR-a-um '57, 36 :of area'. From a later text the constant 57, 36 is known to be the Old Babylonian estimate of the ratio of the circumference an inscribed hexagon to that of of its circumscribing circle (Neugebauer, 1969). This yields the Old Babylonian approximation of lr as 25/8 (52/23) = 3.125; a better estimate than the Egyption value of 7r ~ 256/81 (2s/34) = 3.1604938. It appears that the constant 57, 36 (actually 0;57, 36) was used as a 'correction' coefficient to determine the area of a circle more precisely. The


usual Old Babylonian method for determining the (approximate) area of a circle (A) from its circumference (c) was: A = 0;5 x c 2 (where 0;5 ~ 1/4rr, and rr ~ 3). A more precise, corrected result (Ao) could be obtained by multiplying the initial estimate by the 'correction' coefficient: Ao = 0;57, 36 x 0;5 x c 2 = 0;4, 48 x c 2. Indeed, the constant 4, 48 is known from the lists of coefficients and from problem texts as a volume-constant in calculations involving cylinders. Evidently the error resulting from rr ~ 3 in volume measurements was unacceptable for practical reasons. In surface measure, the initial approximation could be corrected, when necessary, by using the '57, 36' of area'.

The colophon in the coefficient list YBC 5022, and in other mathematical texts, reads: 'the deity Nidaba', implying, as noted by Neugebauer & Sachs (1945), 'that the whole text belongs to the so-called "wisdom" literature'. As we shall see below, 'numerals' in Meso- potamian civilization were indeed representatives of deities.

One of the most intriguing Old Babylonian mathematical texts is known as Plimpton 322. The text investigates secant 2 values in a near-linear descending order of 15 different right triangles, varying by about 1 °, whose dimensions have been calculated in terms of 'Pythago- rean triplets'. McLain (1978) saw in this text a prototype for Plato's musical trigonometry. We should note, however, that the principal column of the text was the listing of the secant 2 values within the significant interval < 2--< x/2. Again, a mathematical and/or musical aim may underline this remarkable text.

Mesopotamian mathematical texts indicate, as noted by Neugebauer (1969), that mastery of numerical procedures, ratio theory, geometrical operations and even a numerical theory was essentially completed by 1750 BC. The developments that took place in later periods were dominated by interest in astronomy: the 'linear zig-zag' functions, ephimerides for lunar and planetary theory, and a 360 ° zodiac for mathematical calculations. McClain (1976) stressed the dominant role played by musical metaphors in Greek philosophy and suggested that mathematical harmonics can be traced back to Old Babylonian times and even earlier.

Although we still lack the harmonizing evidence, we do have Old Babylonian textual evidence of a well developed musical terminology and tuning system. There are also some indications of attempts to integrate music and cosmology by means of mythology. The missing link, in a sense, is the direct correlation between numeral and tone in Old Baby- lonian texts.

An interesting example of musical mythopoeics is provided by the names of individual strings in a nine-string lyre. Text Nabnitu XXXII (Kilmer, 1965) lists these names in Sumerian, the classical language of Mesopotamian civilization, and in Akkadian, the dialect of its Semitic peoples. The nine Sumerian names are listed as follows: fore-string; next; third- string, thin; fourth-string, small; fifth; fourth from behind; third from behind; second from behind; behind-string. The Akkadian names, as expected, are the translations of the Sumerian names with a notable exception. The fourth string, called in Sumarian sa.4. tur 'fourth-string, small' was named in Akkadian A-bana 'Ea-creator', i.e. in Akkadian it was named after the deity Ea (= Enki in Sumerian), patron of arts and creative crafts, while the Sumerian string- name merely gives its numerical position in the tuning order. What is implied here is that sometimes after the establishment of the Semitic Dynasty of Akkad (ca. 2360-2180 BC) cosmology and musical theory were in the process of being harmonized. For the priest- musicologist, the preeminent role of this fourth string (and its tone) in the tuning order merited its identification with the god Ea, who first structured the universe. The numeral '40', i.e. 40•60 = 2/3, was also identified with Ea, as part of a system to be discussed below. Does this indicate that the Ea-creator string (2/3) was taken as the 'creator' from which the other tones (in a cycle of fifths) were generated?

260 R. R. Stieglitz

Ea, the Creator, is, in any event, the key figure in Mesopotamian cosmology. In the Babylonian creation epic, the Enuma Elish, which cannot antedate the 24th century BC and may have been introduced by the Western Semites who founded the First Dynasty of Babylon ca. 1830 BC (Jacobsen, 1968), Ea is called the father of Marduk. The latter was the patron god of Babylon, who restructured the cosmos in mathopoeic fashion.

The diffusion of Sumerian cultural elements to Syria, including the cuneiform script and its sexagesimal notation, is now attested in the middle of the third millennium BC at Ebla. It is not surprising, therefore, to f'md that the Old Babylonian musical tuning (a Sumero- Akkadian heritage) was current at the Syrian coastal city of Ugarit a millennium later (G6terbuck, 1970). The agents of this musical transmission were the Hurrians, a people who also played a major role in the diffusion of Oriental cults and literary traditions to Anatolia.

In the Aegean, the numerical procedures in the Minoan administrative texts (Linear A), which are contemporary with the Old Babylonian period, show that they were primarily influenced by Egyptian decimal practices. But the later Mycenaean Greek texts (Linear B) indicate that the fractional system bears a closer resemblence to Mesopotamian cuneiform practices than to its Minoan predecessor (Stieglitz, 1978). It is likely that any stimulus diffusion of Mesopotamian practices to the Aegean was transmitted via Syria by Canaanites (Phoenicians) and/or Hurrians, along the same routes by which the Canaanite alphabet reached the Greeks.

Mesopotamian mathematicians also had some interest in non-sexagesimal procedures. Text MLC 1731, dated to ca. 1700 BC, is concerned with the conversion of sexagesimals to unit-fractions, but not according to Egyptian (nor Greek) methods (Sachs, 1946). Although contacts between Mesopotamia and Egypt are known in Pre-Dynastic times, we still know very little of any 'scientific' exchanges between these two centres of civilization during the second millennium BC. Such 'scientific' contacts between Mesopotamia and Egypt probably took place intermittently during that period, and must have been quite intense after the Assyrian conquest of Egypt in the seventh century BC. By the Hellenistic period, for example, the Graeco-Babylonian zodiac is prominent in Egypt, and it is most interesting, therefore, that the Alexandrian scientist Ptolemy recorded (Neugebauer, 1969) that he had access to Mesopotamian eclipse records dating to 747 BC onwards.

Numbers as gods

Perhaps the most fascinating aspect of Mesopotamian mathopoeic thinking is the 'deification' of a carefully selected group of numbers. In this system, the 'name' of the deity, preceded by the determinative-sign for 'deity', is written with a numeral. The numerals representing the deities are-with one exception-regular numbers, found in the 'standard' tables. The main seven deities/numerals,in addition to 1 = 60 = Anu, are listed together with the 'standard' numbers and reciprocals in continuous arithmetic order in Fig. 2. Our 'standard' 22 pairs are now grouped into ' 1 ' + 21 reciprocal pairs-Anu presiding over a divine assembly- constituting a numerical 'continuum' with Unity as its beginning and end. It is significant that the 'numeral' 21 was utilized as a cryptogram for the god Anu (Labat, 1965).

In light of this arrangement it seems hardly coincidental that Egyptian mythology presents the next-world judgement-chamber in the very same format (1 + 21 'pairs'): Osiris presiding over two rows of 21 judges each, assembled in the 'Broad-Hall of Double-Truth'. Was the Egyptian mythopoeic image based on a similar mathematical model?

McClain (1976) has proposed a 3 q x 5 r matrix, with the limits of 320 x 514 , as the fundamental 'holy mountain' to correlate number-tone with mythopoeie imagery in a wide geo- graphical range. The essence of this admirable model, however, cannot be divorced from


I = 60 Anu

1,0, 56,15

I ,6,40 54

I, 12 50 Enlll

1,15 tff~

1,20 45

1,21 44,26,/1-0

1,30 40 Ea-Enkl

1,40 36

1,52,30 32

2 30 Sln

2,13,20 27

2,24 25

2,30 24

3 20 Shamash

3,20 18

3,/~5 16

4 1 5 Ishtar

5 12 Nergal

6 i0 B~l-i:arduk

6,40 9

7,30 8

Fig. 2. The22 'standard'reciprocals and ma]or Mesopotamian deities. The 'standard" numerals are arranged in an arithmetic 'continuum" with 1 = 60 as the beginning

and end

sexagesimal mathopoeic thinking, since it involves only 'regular' numbers in the form 2 p 3 q 5 r (the powers of 2 are often omitted by McClain, in various versions of the 'mountain', because 2 p is always assumed, and it merely produced 'octave equivalences'). Furthermore, in order to explain the Egyptian mythological image, and others, a reciprocal 'mountain' must be imposed on the original pattern.

Since our sexagesimal matrix, as well as McClain's 'holy mountain', is based on Meso- potamian mathematical considerations, this again raises the question of possible 'proto- scientific' data diffusion from Mesopotamia long before 1750 BC. Present archaeological evidence indicates that Mesopotamia was the source of artifacts and artistic motifs imported into Egypt at the end of the fourth millennium BC, but no Egyptian influences on Meso- potamia are apparent during that era. It is difficult to see Egyptian mathematical practices, which handled fractions by means of unit-fraction sums (e.g. 5/6 was written as 1/2 (+) 1/3), as the source for such mathematical models. But they could have been developed in Egypt under Mesopotamian stimulus diffusion. A clue may be provided by the Egyptian 'exception' ratio 2•3, which was not written as a unit-fraction. The ratio 2•3, the essence of our matrix, is also the key ratio in Mesopotamian mathopoeic cosmogony, and was therefore deified as the god Ea (= 40), the Creator (who corresponds to Ptah in Egyptian mythology).

Another reason to believe that sexagesimal, not decimal, investigation of numbers first led to the development of attempts to view cosmic phenomena in conjunction with number

262 R. R. Stieglitz

theory is the case of the prime number 7. For it is only in Mesopotamian sexagesimal mathopoeic thought that this number has a special significance.

The exception to the use of 'regular' numbers as deities, was the deified 'Seven'. This 'irregular' number received special attention and treatment because it was the f'trst cardinal, ordinal and prime number which confronted the Mesopotamian mathematician with the problem of the 'irrational': '7 does not divide' is the laconic comment in the mathematical texts. What is meant, is that the reciprocal of 7 cannot be expressed as a finite sum of sexagesimal fractions (ff = 0; 8, 34, 17, 8, 34, 17, 8, 34, 1 7 . . . ) . The reciprocal of 7, an infinite cyclic repetition of fractions, has no ratio. This in itself, in mythopoeic thinking, would be sufficient to associate it with the gods. Indeed 'Seven' plays a unique religious role, as the first symbol of the 'irregular', the 'mystical', sacred world.

The number 7, and its multiples, also play a prominent role in mythopoeic thought beyond the borders of Mesopotamia, leading us to believe that Mesopotamia was indeed the focal point of its original diffusion. The Mesopotamian deified '7' is usually identified astronomically with the Pleides, but this is not certain. As a sacred numeral it is attested in Canaanite religious literature in the second millennium BC, and is especially prominent in the Bible. In the Hebrew creation epic, for example, only the Seventh Day, later called the Sabbath, was both blessed and sanctified by the Creator (Genesis 2 : 2). Multiples of 7, such as 14, 42,49 and 70, figure prominently in both Judaism and Christianity. By late Antiquity, Mesopotamian astrology popularized the cosmic counterparts of the deified '7' in the form of the '7 planets', each identified with a major deity.

The very deification of selected 'regular' numbers, and the 'irregular' prime 7, indicates that the Mesopotamian mathopoeic mind saw in numerals a means of expression transcend- ing the requirements of practical operations. Numerals as gods were thus 'cosmic coefficients'. Since numbers were utilized as ordinary coefficients in secular operations, were utilized in puns (!), and in sacred matters such as omens (Gadd, 1967), they were also used metaphoric- ally as metaphysical symbols. The gods were related by their geneology, and so were the numerals. Divine Unity was Anu because, as is already stated in an ancient Mesopotamian exegesis, Anu was 'the primordial deity, father of the gods' (Labat, 1965).

But Anu himself, in epic and in our mathematical matrix, was already static. He was removed from the fore-front of action by his very deffmition (1 x 60 x 'divided' by 1 x 60x). The driving forces of the universe were in the hands of a younger generation of gods. The most important of these was the 'son' of Anu, Ea = 40 (2 'divided' by 3), representing creativity, the horizontal vector on our matrix. Enlil = 50 (5 'divided' by 2 x 3) was the deity of dynamic force, the vertical vector on our matrix. At a later stage in history, Marduk = 10 (1 'divided' by 2 x 3), the 'grandson' of Anu and 'son' of Ea, endowed with the 50 names, is elevated to rule the universe which he has restructured.

In the Enuma Elish, the divine assembly of 50 gods (43 divine 'numerals' (= 1 + 2 x 3 x 7) and the deified '7'?) elects Marduk to battle, and eventually to defeat, the 'older orde¢ led by the goddess Tiamat. After his victory, Marduk 'divides' Tiamat in order to re-form the (present) world. In his re-organization, he establishes three constellations for each of the 12 months in order to 'determine the year'. The Babylonian month consisted of 30 days (in spite of observational astronomy, a mean synodic month of 30 'days' was utilized for mathematical reasons), so that each constellation was therefore allotted 10 days. Now this system is suspiciously similar to the 36 'decans', and the 360-day year in Egyptian astronomy of the second millennium BC, and merits further investigation. Marduk then establishes three bands in heaven: his own Nebiru-line (the planet Jupiter), above it the Enlil-line, and below it the Ea-line.

In Fig. 3, we have outlined the locations of the eight major deities of Mesopotamia


0 0 0 0 X 0 X

.............................................. ~ : / ~ , .......... / _ ~,,-,,0,__

dA\ \ / 0 X X X X X 0

\ //' \"t, / o x x x o

10 ~ / Neblru-llne

. . . . . . . . o . . . . . 2 .......... i .... .... ;i x 2 .......... 2 .... o . . . . . . . . . . . . .

. . . . . . . o . . . . . . . . . . . . . . . . . ~:_// . . . . , d _ . . . . . _ _ / _ _ . . . . o . . . . . . . . . . . ~ - ' - - ' - " - , - ~ . r ~ , ,

o x 2 x o o

\ o o x o o

x

0 X X X ' X ~ X 0

/ \ / / , \ / o o o

X 0 X 0 0 0 0

Heaven

Netherworld

lezen~ l=Anu

50=Enlll 4O=Ea )O=S1n 20=She~esh

15=Ish~r 12=Ner~al lO=karduk

Fig. 3. Ma/or deities and the bands o f heaven. The eight major Mesopotamian deities, connected by double-lines, are plotted to indicate their relationship to the limits o f the matrix (Fig. 1). The division of heaven is that performed by Marduk (= 10) in the Enuma Elish. The deities are arranged, and divided, symmetrically

(connected by double lines) in order to indicate their positional relationship to each other and to the limits of our 'standard' matrix (solid line). The horizontal mid-field line of the matrix-representing the earth-is the Ea-line as established by Marduk. In this zone, Ea-Enki (his Sumerian name Enki means 'Earth-Lord') is aligned with his father Anu. The upper-half of our number-field above the earth represents heaven. Marduk's planet, Jupiter, is aligned with Ishtar (Venus), so that the Nebiru-line (whose nightly pass is very close to the ecliptic) now divides the heavens according to a 2/3 ratio. Directly below Marduk and Ishtar are the Sun (= 20) and the Moon (= 30). Above Marduk's planet is Enlil (Saturn?). His infernal counterpart is the god Nergal (= 12), ruler of the Netherwofld, which is represented by the lower-half of the number-field.

Our matrix, then, not only offers a structural explanation for the relationship among the deified 'regular' numerals, but also correlates with significant mythopoeic elements in the so- called 'scientific' section of the Enuma Elish (IV: 141-V:22). Such mathematical models can thus offer new dimensions for the investigation of mythopoeic images and numerical elements in mythology, since these are usually neglected in traditional literary and philo- logical studies.

The Pythagorean heritage and post-exilic Judaism The historical outline of Western science in the wake of Greek science in antiquity is well known. What is less clear is the relationship, if any, between Oriental 'proto-seience' and the Greek mathopoeic achievement. In any such linkage, the Pythagorean heritage would be of central importance.

The influence of Mesopotamian mathopoeic thought on the birth and evolution of

264 R. R. Stieglitz

Pythagorean religion is still not known. What characterized the Pythagorean teaching, was a curious mixture of science and magic, of mathematical harmonics mixed with various super- stitions. A major tenet of their beliefs was the identification of numerals, and geometrical figures, with major deities. Plutarch (The E at Delphi, 20; On 1sis and Osiris, 10, 75) tells us that the Pythagoreans identified Unity and the numeral '1' with Apollo; Duality with Artemis; the first Cube with Poseidon; the Hebdomad and equilateral triangle, with Athena, and so on. This synthesis of mathopoeic and mythopoec thinking, coupled with the deification of numerals, is precisely what was characteristic of Old Babylonian religion. It is prob- able that the vast body of proto-scientific data gathered by the mathopoeic tradition in the anient Near East served as the historical reference reservoir, and played some role in the shaping of the Pythagorean philosophy.

We know few details about the life of Pythagoras. The traditions about his extensive sojourn in the Orient make it likely that he was acquainted with some of the mathematical lore accumulated in Babylon and Egypt. Pythagoras and his followers saw in these mathematical studies the means by which to integrate astronomy, geometry, music and arithmetic. The Pythagoreans, like the Hebrews before them, found that God is a nomad. Unlike the Hebrews, they found their god in the study of nature and number.

Pythagorean mathematical harmonics reached new heights in the philosophy of Plato 0YlcClain, 1978), and enjoyed considerable success in the Graeco-Roman world. Plato's motto for his academy was 'Let no one ignorant of Geometry enter• For Geometry is the knowledge of the Eternal Being'. Greek mathopoeic thinking, unlike that of the Mesopo- tamians, was willing and able to separate math from myth. In the Graeco-Roman period it was common, in certain circles, to view classical mythology as mostly allegorical (at the same time, Babylonian astrology, a 'scientific' religion, was immensely popular). The old myths were seen through the eyes of the new math, that of Greek philosophy. The rationali- zation process is quite explicit in Plutarch (On Isis and Osiris, 55):

• . . they relate a legend that Hermes cut sinews of Typhon, and used them as strings for his lyre, thereby instructing us that Reason adjusts the Universe and creates concord out of dis- cordant elements, and that it does not destroy but only cripples the destructive forces•

The Pythagorean school, therefore, with its emphasis on the integration of mathematics, astronomy and music, perpetuated an older tradition of mathopoeic thinking-albeit along the path of Greek philosophy-that saw in 'ratio' the basis of attaining true understanding of the cosmos.

The prophet Ezekiel, an older contemporary of Pythagoras in Babylonian captivity, was also very much preoccupied with numerical formulae and visions of the divine. But Ezekiel reaffirmed the unity of God, as Moses did before him, by personal theophany, and not by 'natural philosophy'. The attitude of monotheism towards the phenomenon of divine numbers was unequivocal-God is One (Deuteronomy 6:4;Zachariah 14:9).

Psalm 82 describesElohim (the Creator in Genesis 1-2:4) as still presiding over the divine assembly, but they turn out to be false gods, who perish like ordinary mortals (Psalms 82 : 7). They can all be 'divided' until only One is left. Hebrew monotheism thus rejected the mathopoeic notion of deified numerals, except for the '1', since only it was the beginning and end (lsaiah 44: 6). God, indeed, 'counts the number of stars, to all of them he gives their names. Great is our Lord, his understanding has no number' (Psalms 147:4-5).

Nevertheless, the Bible contains much mathopoeic data• The Hebrew mathopoeic component, like its Oriental counterpart, did weave significant numeral data into the sacred texts. Prominent mathopoeic elements are included in such narratives as Noah and the Flood, The Ages of Abraham's Ancestors, The Tabernacle Complex, The Numbers of Israel,


and Ezekiels's Temple Complex. While it is outside the scope of this paper to analyse Bibli- cal numerology, it must be noted that non-numerical images in the Bible may also allude to mathematical models.

As an example of the latter, let us look at the Biblical guardians of the 'Way of the Tree of Life' (Genesis 3:24). They are called the cherubim (27), and lahat, the rotating sword. In later Israelite tradition, the two cherubim were the winged guardians of the Ark of the Covenant (Exodus 25 : 18-22, etc.), and a standard epithet of God was 'Dweller of the Cherubim' (1 Samuel 4:4; lsaiah 37: 16, etc.). The concept of the supreme deity flanked symmetrically by lesser divinities has numerous pagan parallels in art and epic; and, of course, it brings to mind our mathematical model (Fig. 1). The model also shows the supreme deity (= 1) dwelling between symmetrical 'Cherubim'. The latter could be divinized in any polythestic system, and they were, but not in monotheism. Lahat, the rotating sward ('a flaming sward which turned every way' in the RSV translation), may also be seen as an apt reference to such a mathematical model, whose center represents the concept of invariance ('the Tree of Life ')- the only property of God not attained by man in the Garden of Eden (Genesis 3 : 22).

When Judaism was confronted with Greek 'natural philosophy' in the Hellenistic period, its reaction was to de-emphasize both the Greek and the older Oriental mathopoeic component (with which it was well acquainted), i.e. any Judaic mathopoeic thinking was set aside. The Rabbinic attitude, that followed by mainstream Judaism, was crystallized by the view that 'astronomy and mathematics are the after-courses of Wisdom' (Mishnah, Pirkei- Avot 3 : 23).

Rabbinic Judaism elected to concentrate on the exegesis of the revealed divine law, the Torah. Mathopoeic studies, such as astronomy, were held in high esteem by some Rabbis but were in general considered as secondary disciplines in education. The mathopoeic tradition was not forgotten in Judaism, it was merely put aside because it might distract the believer, and could even divert one to pagan 'natural philosophy'. Only the mystical tradition in Judaism, the Kabbalah-which the Rabbis warned was not for general consumption-perpetuated some sort of a mathopoeic heritage. The Kabbalistic creation epic, the Sefer Yetsirah 2), states that the 22 Hebrew letters (= numbers) and sounds ( ' 1 ' + 21 pairs?) comprise the foundations of the universe.

References Frankfort, H. & H. A., Wilson, J. A. & Jacobsen, T. (1946). The Intellectual Adventure of

Ancient Man. Chicago: University of Chicago Press. Gadd, C. J. (1967). J. cuneiform Stud. 21, 52-63. Gordon, C. H. (1970). J. NearEast. Stud. 29,198-199. Gfiterbock, H. G. (1970). Revue Assyriol. Archdol. orient. 64, 45-52. Jacobsen, T. (1968). J. Am. orient. Soc. 88, 104-108. Kilmer, A. D. (1960). Orientalia 29,273-308. Kilmer, A. D. (1965). In Studies in Honor ofBenno Landsberger (Assyriological Studies No.

16). Chicago: University of Chicago Press. Labat, R. (1965). In Studies in Honor ofBenno Landsberger (Assyriologicai Studies No. 16).

Chicago: University of Chicago Press. MeClain, E. G. (1976). The Myth oflnvariance. New York: Nicholas Hays. McClain, E. G. (1978). The Pythagorean Plato. New York: Nicholas Hays. Neugebauer, O. & Sachs, A. (1945). Mathematical Cuneiform Texts. Hew Haven: American

Oriental Society. Neugebauer, O. (1969). The Exact Sciences in Antiquity, 2nd Ed. New York: Dover. Sachs, A. (1946). J. NearEast. Stud. 5,203-214.

266 R. R. Stieglitz

Santillana, G. de (1961). The Origins of Scientific Thought. Chicago: University of Chicago Press.

Stieglitz, R. R. (1978). Bull. Am. Soc. Papyrolo~sts 15, 127-132.

Documents

Numerical structuralism and cosmogony in the ancient Near East