Mesopotamian mathematics from the Old Babylonian period is principally from the second half, roughly 1800-1600 BC. When the first archeological discoveries were made the evidence suggested that it appeared fully-formed out of nowhere, flourished briefly, and then disappeared again for a thousand years. But this could merely be a happenstance of discovery. After all, examples from much later Babylonian history, in the Seleucid era, shows mathematical expression similar to the earlier period. The interregnum material, and the expansion of the knowledge that we find in later Babylonian evidence, merely became lost to human record. We cannot now trace that evolution. Should we then extrapolate that experience of loss back into the past, prior to 1800 BC? Is it possible the use of clay to record intellectual activities was part of the world environmental changes, and that earlier human records were made on media that have now totally disappeared?
We possess several hundred Old Babylonian mathematical tablets. They are conventionally divided into table-texts and problem texts. The difficulty for scholars is rather as if you were to try to reconstruct modern mathematics armed only with the exercises from a text-book and a few worked examples. Much speculation, deduction, and extrapolation have occupied OB scholars for many years.
The Babylonians did not produce texts stating general procedures. They gave worked examples that show they knew general principles for tackling problems, as though following general procedures. Where this knowledge derived from or how it originated is unknown. Perhaps statements lie moldering in some undiscovered tell but the impression we get from the body of their work is that they never felt a need for such expression. Perhaps they were so preoccupied with the basic problems of survival they did not have time for such developments.
One of the most striking features of both OB and Egyptian mathematics, from that time period, in the texts that have survived, is the manner in which they present problems. They give highly practical examples, utilitarian problems. As I studied the evidence I realized that we have tutorials, in both Egypt and Mesopotamia. We have only tutorials; we have no theoretical math. Was there no theoretical understanding? Did the procedures derive strictly out of practical discoveries by trial and error? Modern scholarship has looked upon this phenomenon in just such manner. But did not individuals arise here and there who were curious? Did they have a psychology that never reached curiosity? Were their minds not similar to modern minds? Or did they neglect such activities because they remembered an earlier inspiration that had given them the general rules and they had faith in such methods.
Plimpton 322 is so important because it offers insight into the social evolution that led to OB mathematics. That bothersome fact has no easy formulation in our modern ideas about the past. The tablet is not a mundane artifact, for us to scoff at, or reduce its significance.
Robson must reduce the significance of Plimpton 322 because of the implications for social evolution. She makes quite an issue about fitting the tablet into the cultural context of its supposed origin. She emphasizes that we should not take modern understanding or expression of mathematics and turn it back to that era. According to her, if we are to properly understand Plimpton 322 we should regard it the way the originators understood it. On page 176 Robson offers a list of criteria for judging the merit of the ancient mathematical tablets, with special emphasis on Plimpton 322. While her list is sound, it is incomplete in concept, and conditioned by her notions. I would elevate her criteria to a different view of the past.
Historical Sensitivity should be recognized in a more ancient contribution to OB mathematics. I have demonstrated this vividly in Egypt. See:
Robson would limit our understanding to the known culture of Old Babylon. But the Semitic Akkadians developed from an earlier culture. A wider approach would not be conceptually anachronistic, but a more comprehensive regard for that earlier contribution. We just simply do not know what mathematics may have existed beyond the evidence available to us. (Note how this early knowledge was expressed in Egypt, and remained obscure from all eyes for five thousand years.) The fact that we are limited in the scientific sense, and then regard that as the only evidence, does not deny that earlier contribution. In fact, going back to the beginnings of known Akkadian writing, we see a high level of administrative culture. Social contracts and other social documents exhibit a sophistication that is not far from the modern world. But modern evolutionary views, with their confining notions, prohibit us from gaining insight into that past. I personally do not feel constrained by such rules. We should not form conclusions based on lack of evidence. Plimpton 322 fits within a much larger context.
Robson's Cultural Consistency may not offer a full understanding of Plimpton 322 if its ultimate origins are beyond what is known of OB mathematics. Such suggestion takes it outside the context offered by her, but we should not deny that possibility merely because we assume that no other cultural contribution was possible. Our recent notions of the evolution of man may be highly inaccurate and should not precondition us.
Calculational Plausibility confined to OB methods and techniques may not offer an explanation of Plimpton 322 if the tablet were based on some other, earlier knowledge of mathematics, the path of which is now lost to us.
Physical Reality offered by Robson does, indeed, show a plausible size and proportion to the (broken, with pieces now lost) tablet.
Textual Completeness was a sound rule followed by Robson. Her speculative prior columns do offer an understanding of the remainder of the extant tablet.
Tabular Order was another unique insight offered by Robson. We must credit her with the ability to form the extant table into a meaningful structure, but we should be mindful that Neugebauer was there before her.
On page 175 Robson offers a line-by-line explanation of all the errors found in the tablet. For reasons I shall show later I accept all of her corrections.
According to her analysis three are simple copying errors; four are mathematical errors. If the tablet was intended as a pedagogical tool, as Robson insists, why does it have so many errors? If it was designed to generate problems involving right triangles and reciprocal pairs, as a teaching tool, why would a teacher permit errors to creep into his examples? If I were a teacher I certainly would not permit such errors; I would be very careful to make sure my work was accurate. I would not want to propagate sloppiness to my students. Was it done in haste? Did the teacher wish to illustrate examples, but was not careful to ensure accuracy? Would that attitude not defeat the supposed purpose of the tablet?
The tablet has the form of a work that was unique, not something casually borrowed from a general catalog of examples. As others have pointed out, it is not a common school text, even to the point of carrying a sophistication beyond the repertoire of OB mathematics. It has the appearance of preserving knowledge for purposes that were not tutorial.
But here Robson objects; she is unwilling to admit a sophistication beyond her notions of the cultural evolution of 2,000 BC. She wishes to reduce it to an ordinary text, not something exceptional. We strongly differ from Robson. She sees it as commonplace, even though no other tablets like it have been found. She must give it a profane explanation. Everything must have an ordinary "evolutionary" origin, evolutionary according to her view of the past history of mankind. In that world view there are no extraordinary events, such as borrowing knowledge from an earlier past.
The tablet has the form of something copied, without recognition of its errors. If the scribe had recognized that he made mistakes would he have permitted them to remain? Would he not start all over again, the next time careful to not make that same mistake? Or do we see errors as borrowed from another source along with the content of the tablet, and the scribe did not know he had errors? Beyond mere copy mistakes, the scribe evidently did not possess the analytical skills to deduce the cause of the errors he left uncorrected on the tablet. The scribe did not know he had errors!
We can ask the further question: Did the errors require dynamic participation on the part of the scribe? Or was he a blind copyist who did not participate in creating the errors?
We can gain some answers by examining the errors. I shall use Robson's designation of Column # and Line #. I shall work from the most simple to the more complex.
II, 9 is easy to understand; it could be caused by merely adding one punch mark. As Robson noted, 8 and 9 are easily confused in cuneiform.
I, 13: -- Robson sees a missing place in the sexagesimal notation that may not be missing. She notes that the empty sexagesimal place should be marked with a blank space. In all other cases on the tablet the scribe leaves a space whenever he encounters a "zero" (empty sexagesimal space) either before or after a mating number. Where he encounters a 00 he leaves an extra wide space. This extra wide space suggests room for the "zero-zero" double empty space. Some of the difficulties of judging the "value" of these spaces are discussed in my analysis. See:
I, 2 and I, 8: -- Robson sees the first as the conflation of two numbers, 56 for 50 06, and the second as an arithmetical error, 59 for 45 14. The are both the same type of error, the adding together of two sequential sexagesimal digits. In context of the lines the digits are all tightly spaced except where the scribe gave space for 03 in line 8.
If the scribe were using this tablet as a teaching tool he here makes major mistakes. These errors would produce numbers that would violate the Pythagorean triples. These type errors suggest that the scribe was unaware of his copy mistakes, or what they did to the mathematical integrity of the tablet.
II, 13: -- Here the (correct) value of 2 41 is replaced with its square, 7 12 01. Did the scribe engage in this calculation, for reasons that are now unknown to us? He now raises his interference in the integrity of the tablet to a higher level, performing an act which is much more than the simple conflating of two numbers. In such act he truly throws off the understanding of a student. He again violates the pattern of the Pythagorean triplets, as though he did not recognize them. Or, if he were a mere copyist, this error was introduced in an earlier (now lost) version of the tablet by someone else who also did not recognize the significance of the error. We should keep in mind that if he had created the rough draft of this tablet, he would have known the proper values of the Pythagorean triplets.
II or III 15: -- 56 is double the value of 28, which would violate the Pythagorean triplet, or 53 for 1 46 with the same result. However, the alternate Pythagoreans are 28-53-45, (the same digits in sexagesimal), or 56-106-90 (56, 1 46, 1 30). The second series of numbers are double of the first (2 multiplier). Since these numbers are proportional to one another they produce the same D/L ratio, 1.1777 . . .
In summary, such mistakes show that the scribe did not truly understand the nature of the text. He was copying from an earlier version which included the mistakes, or he was reworking the values to his idea of what was correct, without proper understanding. If this tablet served as a pedagogical tool the OB scribal schools were surely degraded. Highly unlikely.
The implications of Pythagorean triples may be lost on Robson but they were not lost on Neugebauer or Sachs, nor untold thousands who followed.
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