Jens Høyrup, Lengths, Widths, Surfaces in Old Babylonian Algebra
Mesopotamian mathematics intimately bound up with state administration and other utilitarian applications..
Jens Høyrup wrote:
Chapter 8
The Historical Framework
So far, we have portrayed the mathematical practice of the Old Babylonian scribes and their teachers; inasmuch as this practice is located differently from ours in historical space, the investigation is certainly a contribution to the history of mathematics — but since it has not approached the development of the practice in question (not to speak of the motive forces of this development) it has not yet approached the history of Babylonian mathematics. The model was that of structural-functional anthropology rather than history.
This history of Babylonian mathematics will be the topic of subsequent chapters; first, however, a brief presentation of the general historical framework may be appropriate. Since the development and orientation of Mesopotamian mathematics was intimately bound up with written administration and the scribal craft, this framework has to include both the development of state and administration and of the scribal profession.
[Footnote extract: [For] a more detailed analysis of the interplay between statal bureaucracy, scribal craft and culture, and the transformations of mathematics from the beginnings through the mid-second millennium, with extensive bibliography [see] Jens Høyrup, 1994. In Measure, Number; and Weight: Studies in Mathematics and Culture.]
Landscape and Periodization
Mesopotamia — "the land between the rivers" Euphrates and Tigris — can roughly be divided into a northern region around Assur and Ninive; a central region, from Eshnunna and Sippar toward the north to Kish toward the south, the centre of which from the second millennium onward was Babylon (somewhat south of present-day Baghdad); and an extreme South, with the cities Ur, Uruk, Shuruppak, Larsa, and others, characterized by irrigation agriculture. Even in the centre, irrigation was practised from an early date, whereas the North depended on rainfall. The political division between the North — Assyria — and the Centre-to-South — Babylonia — thus coincides with an ecological split. To the east of Southern Mesopotamia we find the city Susa as the centre of Elam, a region of river valleys between the Zagros mountains. Since the mid-fourth millennium, strong interactions between Elam and Babylonia were the rule.
Temporally, one may distinguish the protoliterate period (c. 3400 BCE to 3000 BCE according to calibrated radiocarbon datings, subdivided into the early Uruk IV and the late Uruk III phase), in which writing was created in the South, probably in the city Uruk; the Early Dynastic period (c. 3000 BCE to c. 2350 BCE), in which the South was divided between competing city-states; and the Old Akkadian or Sargonic period (c. 2350 BCE to c. 2200 BCE), in which the central and southern region were united in a regional state ruled by an Akkadian dynasty founded by Sargon of Akkad. After an interlude follows the neo-Sumerian or Ur III period (roughly coinciding with the 21st c. BCE), where the South was the core of an extremely bureaucratic state encompassing also the Centre.
After the collapse of the Ur III state follows the Old Babylonian period (c. 2000 BCE to c. 1600 BCE), whose most famous figure is doubtlessly king Hammurapi (1792-1750); in this phase, the Sumerian language disappears as a spoken language, and Akkadian is split into the clearly distinct northern Assyrian and southern Babylonian dialects.
A new societal breakdown followed. At first the masters of the region were the Kassites, a group of warrior tribes. Toward the end of the second millennium BCE, the Assyrian city state expanded and conquered first the whole of Mesopotamia and next the whole Near East. In the final centuries of the Assyrian empire (8th to 7th century BCE), mathematical astronomy may have arisen, but it is only documented directly during the epoch of Persian rule (539 BCE to 331 BCE), which was brought to an end by Alexander's conquest.
After Alexander's death and a brief period of fighting, Mesopotamia fell to one of his generals, Seleucos. The last glow of the ancient Mesopotamian cultural tradition (and its mathematics) falls during the Seleucid period and the early part of the Parthian epoch (312 BCE to first or second c. CE).
Scribes, Administration, and Mathematics
The protoliterate writing system — still only partially deciphered — was created during the Uruk IV phase as a tool for the earliest formation of a bureaucratic state headed by a temple institution.[Footnote: The ‘state’ is understood here as a social system characterized by an at least three-tiered system of control and by extensive specialization of social roles].
The basis for the invention was an accounting system based on small tokens of burnt clay (cones, spheres, disks, pellets, etc.) which will have stood for various measures of grain, for livestock, etc. The system had been used in the Near East since the eighth millennium. Various transformations and extensions of the system had been introduced in mid-fourth millennium Susa in response to needs created by increasing social complexity. These improvements and extensions were taken over by the even more complex Uruk administration, which soon recast the whole complex into a writing system; apparently this was done in one jump, as the result of genuine invention — no traces of development or gradual extension are known.
[Footnote extract: Obviously, the use of signs for words or concepts and not only for measured quantities constitutes a fundamental change; whether we say that writing was created though a recasting of token accounting system or that this system was part of the inspiration behind the invention is a rather futile [heated] discussion if only we remember that the change was sudden and revolutionary …]
The best evidence for the link between the token system and writing is constituted by the protoliterate metrological notations, which render the traditional tokens — a sphere as an impressed circle, the cone by an oblique impression of the stylus, the disk as a drawn circle. The integral part of the capacity system for grain is likely to continue the pre-literate system (grain accounting will have been old); a system of sub-units may have been a fresh development. The integral units, with relative values, are as follows:
A small cone: 1
A small sphere: 6
A large sphere: 60
A large cone: 180
A large cone, with impressed small circle: Value?
This system fuses quantity and quality; the picture of the small cone, whose original meaning may perhaps have been a standard basket of grain, is still a standard volume of grain; if the grain is malted, the sign itself is modified by addition of small strokes. In the pre-literate period, all metrological symbols are likely to have integrated quantity and quality in this way — a disk marked by a cross may have signified a sheep, etc. With the advent of writing, a separation of quantity and quality became possible, and "2 sheep" would be written as a circle with a cross (a picture of the sheep-token, quality alone) and a numeral meaning 2 (quantity alone). This notation for "almost abstract" number is likely to be a new creation of the protoliterate period, made by adaptation of the grain system to existing spoken numbers (and perhaps extended upwards beyond the range of existing spoken numerals). The signs are indeed the same, but their mutual order and relative values (which in this case are also the absolute values) are different:
A small cone: 1
A small sphere: 10
A large cone: 60
A large cone, with impressed small circle: 600
A large sphere: 3600
A large sphere with impressed circle: 36,000
Systematic mathematical thinking is visible in the construction of the system: 60 is an enhanced 1; 600 is created by impressing the sign for 10 on the sign for 60; 36,000 is composed in analogous way from 3600 and 10.
The characterization of the system as only "almost abstract" refers to its use in the length system: the later usage according to which a number alone would imply the standard unit … goes back to this phase. We may therefore presume that even the length system is a fresh invention; almost certainly, the system of area units based on the length system (but with names that point back to pre-existing "natural" irrigation, ploughing, and seeding measures) is an innovation.
Almost certainly innovative are sub-unit extensions of all metrologies and an administrative calendar decoupled both from the natural month and the liturgical calendar.
In this phase, the stratum of ruling managers of the bureaucratic system seems to have been responsible itself for accounting and written administration (and teaching); no distinct group of scribes can be traced. This identity of the stratum of managers with the numerate and literate class is reflected in a complete integration of mathematics with its bureaucratic applications — school texts are "model documents", distinguishable from real administrative documents only by lacking the name of a responsible official and by the prominence of nice numbers.
But the integration was mutual: bureaucratic procedures, centred on accounting, were mathematically planned, for instance, around the new area metrology and the calendar.
[MGH: ‘metrology’ is the scientific study of measurement e.g. land magnitudes.]
The third millennium continues the mutual fecundation of administrative procedures and the development of mathematics (in a process whose details we are unable to follow). The scope of accounting systems was gradually expanded, and metrologies were modified intentionally so as to facilitate managerial planning and accounting. At the same time, there was a trend toward "sexagesimalization", increasing use of the factor 60.
Around 2600, however, when a distinct scribal profession emerged, numeracy and literacy were emancipated from the full cognitive subservience to accounting and management.
For the first time, writing served to record literary texts (proverbs, hymns, and epics); and we find the first instances of supra-utilitarian mathematics — mathematics starting from applicable mathematics but going beyond its usual limits.
[MGH: the term ‘supra-utilitarian’ is discussed earlier in the book but is self-explanatory here.]
It seems as if the new class of professional intellectuals had set out to test the potentialities of the professional tools — the favourite problem was the division of very large round numbers by divisors that were more difficult than those handled in normal practice.
The language of the protoliterate texts is unidentified, whereas the language of the third-millennium southern city states was Sumerian.
[Footnote: The protoliterate script had been logographic, deprived of phonetic and grammatical elements; documents were organized as schemes that corresponded to bureaucratic routines, and did not attempt to render sentences; the grammatical elements and incipient use of phonetic principles that begin to turn up around 2700 are indubitably Sumerian. If we dismiss fanciful ideas about foreign conquerors from Caucasus, Tibet, Thailand, etc. (of which there have been many, but never supported by the least evidence), it seems a natural assumption that the language spoken in the southern region remained unchanged, and that the language of protoliterate Uruk was thus an early Sumerian. Much in the structure of Sumerian suggests, however, that it may have developed locally from a creole spoken by enslaved populations in the late fourth millennium and then taken over by the minority of masters (as often happens in such situations when no metropolis can protect the original language of these).]
Toward 2300, however, the Akkadian-speaking Sargon dynasty conquered the whole Sumerian region, and soon the entire Syro-Iraqian area. Sumerian remained the administrative language (and hence the language of scribal education) during this "Sargonic" or "Old Akkadian" phase, but new problem types suggest inspiration from a "lay" (that is, non-school), possibly non-Sumerian surveyors' tradition — area computations that are very tedious unless one knows that [MGH: un-transcribable equation is omitted here] and the bisection of a trapezium by means of a parallel transversal [all this is discussed earlier in the book].
The 21st century is of particular importance. After the breakdown of the Old Akkadian empire, the new "neo-Sumerian" territorial state ruled by the "Third Dynasty of Ur" (whence the other name, Ur III) established itself in 2112 ("middle chronology"). A military reform under king Sulgi in 2074 was followed immediately by an administrative reform, in which scribal overseers were made accountable for the outcome of every 1\60 of a working day of the labour force allotted to them according to fixed norms; at least in the South, the majority of the working population was subjected to this regime, probably the most meticulous large-scale bureaucracy that ever existed.
Several mathematical tools seem to have been developed in connection with the implementation of the reform (all evidence is indirect): A new book keeping system — not double-entry book-keeping, but provided with analogous built-in controls; the sexagesimal place-value system used in intermediate calculations; and the various mathematical and technical tables needed in order to make the place-value system useful [their use is described earlier in the book].
No space seems to have been left [yet] to autonomous interest in mathematics [i.e. non-utilitarian]; once again, the only mathematical school texts we know are "model documents". As we shall see, this absence of problem texts will not be due to the (bad) luck of excavations.
For several reasons (among which probably the exorbitant costs of the administration) even the Ur-III state collapsed around 2000. A number of smaller states arose in the beginning of the succeeding "Old Babylonian" period (2000 to 1600), all to be conquered by Hammurapi around 1760. Without being a genuine market economy, the new social system left much space to individualism, both on the socio-economic and the ideological level.
In the domain of scribal culture, this individualism expressed itself in the ideal of ‘humanism’ ([omitted] Sumerian for "being human"): scribal virtuosity beyond what was needed in practice. This involved the ability to read and speak Sumerian, now a dead language known only by scribes, as well as supra-utilitarian mathematical competence.
The vast majority of Mesopotamian mathematical texts come from the Old Babylonian school (almost exclusively teacher's texts or copies from these, not student exercises as the third millennium specimens). They are invariably in Akkadian, another indication that the whole genre of "humanist" mathematics had no Ur-III antecedents. …
[Footnote extract: … The use of Sumerograms is no more proof of a Sumerian origin than the Vatican Latin dictionary is proof that the ancient Romans knew about railways, trade unions, and nuclear fission. In some but not all cases, the denial of Akkadian mathematical innovations has certainly been an expression of anti-Semitism …]
… Inner weakening (which had already led to the loss of the Southern provinces) followed by a Hittite raid put an end to the Old Babylonian state in 1600.
Then Kassite warriors subdued the Babylonian area, for the first time rejecting that managerial-functional legitimization of the state which, irrespective of suppressive realities, had survived since the protoliterate phase and made mathematical-administrative activity an important ingredient of scribal professional pride.
The school institution disappeared, and scribes were trained henceforth as apprentices within scribal "families". Together, these events had the effect that mathematics vanishes almost completely from the archaeological horizon for a millennium or more (one Kassite problem text and one table text have been found; the problem text seems to derive from the style of the Old Babylonian periphery); metrologies were modified in a way that would fit practical computation in a mathematically less sophisticated environment (making use of normalized seed measures or of systems based on broad lines in area mensuration; though no longer an object of pride, mathematical administration did not disappear).
Around the "Neo-Babylonian" mid-first millennium, mathematical texts tum up again, for instance, concerned with area mensuration, the conversion between various seed measures, and some supra-utilitarian problems of the kind that had once inspired Old Babylonian algebra. This and other features may reflect renewed interactions between the scribal and the lay traditions, which so far cannot be traced more precisely.
One of the Neo-Babylonian texts starts by listing the sacred numbers of the gods before going on with genuinely mathematical topics. This break-down of cognitive autonomy corresponds to what the texts tell us about their owners and producers (such information is absent from the Old Babylonian tablets); they identify themselves as "exorcists" or "omen priests" (another reason to believe that their practical geometry was borrowed from lay surveyors).
A final development took place in the Seleucid era (311 onwards). Even this phase is only documented by utterly few texts: some multi-place tables of reciprocals probably connected to astronomical computation; one theme text; an anthology text focusing on practical geometry; and an unfocused anthology text. The unfocused anthology text … seems to be a list of new problem types, either borrowed from elsewhere or fresh inventions in the area. The Seleucid texts make heavy use of Sumerian word signs, but in a way that sometimes directly contradicts earlier uses. At least to some extent they represent a new translation into Sumerian of a tradition that must have been transmitted outside an erudite scribal environment.
Chapter 10
The Origin and Transformations of Old Babylonian Algebra
Practitioners' Knowledge and Specialists' Riddles
Several offhand references were made in the preceding chapters to "riddles", to "practitioners", and to "lay surveyors". In order to make these consider ations meaningful and fruitful, a somewhat more systematic discussion may be needed. The following applies to the situation as it looked (with a few exceptions, mostly regarding the Islamic world) until the seventeenth century, that is, to a situation where the knowledge system of a practical profession or craft was much more autonomous than today. As is well known, the relation between theoreticians' and practitioners' knowledge began to change in the late Renaissance, and was wholly transformed by the nineteenth-century advent of the modern engineers' professions; in consequence, today a large part of the practitioner's knowledge (though still far from all of it) is applied theory, which complicates the relation between the two types of knowledge without wholly abolishing their distinctive characteristics.
The gist of practitioners' knowledge is, by definition, know-how, not know-why — in Aristotle's terminology “productive knowledge”, not “theoretical knowledge” [in Metaphysica]. This certainly does not mean that practitioners can have no knowledge “of principles” (Aristotle again) but that the orientation of the practitioners' knowledge system differs from that of the theoretical system. The distinction between these two orientations is of general validity, but has particular implications for mathematics; what follows is therefore specifically adjusted not only to the pre-Modern period but also to the domain of mathematics, whose "practitioners" in pre-Modern times were surveyors, master builders, calculators — and, with mathematical practice as only part of the professional duties of the craft, "scribes" and “clerks”. …
The Source:
Jens Høyrup, Lengths, Widths, Surfaces A Portrait of Old Babylonian Algebra and Its Kin Springer-Verlag New York 2002
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