5,000 Years Old Sumerian Computers — And They Have No Explantion

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[00:00:00]I want to start with a tablet. Not the famous ones, not the King list, not the Epic of Gilgamesh, not the Venus observations.

[00:00:09]a category of tablet that sits in collections across the world. In the British Museum, in the Oriental Institute in Chicago, in the Louver, in the Istanbul Archaeological Museum, that has been studied for over a century, that has been translated, published, and discussed in academic literature, and that contains something the academic literature consistently notes and then declines to pursue. These tablets perform calculations, not record calculations that a human performed. Perform them. The tablets contain mathematical tables, premputed results for multiplication, division, square roots, cube roots, and reciprocal values organized in a format specifically designed to allow a user to look up answers rather than derive them.

[00:00:57]You bring your numbers to the table. You find the relevant entry. You read the answer. This is what a calculation device does. This is this is the functional definition of a computational tool. A system that performs mathematical operations and makes the results accessible to a user without requiring the user to perform the underlying operations themselves. The Samrians had them 5,000 years ago.

[00:01:22]The YBC7289 tablet held at Yale University dates to approximately 1,800 BCE and contains a calculation of the square root of two accurate to six decimal places. The number 1 moing 41421356, the irrational number that describes the relationship between the side of a square and its diagonal, is inscribed in Babylonian sexesimal notation on a clay tablet that is over 3,000 years old with a precision that was not exceeded in the western mathematical tradition until well into the modern era. But I want to go further back than 1800 B.CE. The Yale tablet, impressive as it is, is Babylonian, a later development of a tradition that begins considerably earlier. The Sumerian mathematical tablets, predating the Babylonian period by centuries, contain the same structural approach, precomputed tables organized for lookup, covering the mathematical operations that the civilization's administrative and engineering needs required. The UR3 period from 2,112 to 2004 B.CEE produced an extraordinary archive of administrative documents. The scribes of this period were operating a complex bureaucratic system managing labor, agriculture, livestock, and construction across multiple cities simultaneously. The mathematical demands of this administration were significant.

[00:02:54]Rations had to be calculated for thousands of workers.

[00:02:57]Construction projects required volume and area calculations for materials.

[00:03:02]Agricultural yields had to be projected and recorded. The mathematical tables that survive from this period are not academic exercises. They are working tools for the administration.

[00:03:14]And the sophistication of those tools has never been fully explained. The standard explanation goes like this. The Samrians developed their mathematical knowledge gradually driven by practical necessity, refining their techniques across generations as administrative complexity increased.

[00:03:32]The mathematical tables are the product of accumulated knowledge produced by intelligent people working on difficult problems over a long period. That explanation is coherent and it has a specific problem that the academic literature acknowledges and then moves past without adequate engagement. The sexesimal system base 60, the number base in which all Sumerian mathematics is conducted, is present in the earliest written records, not in a primitive form that developed toward a sophisticated form, rather in a sophisticated form from the beginning. The earliest protouniform accounting tablets from the late 4th millennium B.CE before Samrian writing had fully developed already use sex aesimal notation. The number base, the most mathematically sophisticated aspect of the entire system, was there before the writing system that should have been required to develop and transmit it. I discussed this in the context of timekeeping and the 60-second minute. The point I want to make here is different.

[00:04:33]What I want to focus on is not the choice of base 60, but what the Samrians built with it.

[00:04:40]Because what they built was not just a recording system. They built lookup tables. Precomputed answer repositories.

[00:04:49]A system specifically designed to remove the computational burden from the individual user by making the answers available in advance. This is the conceptual architecture of a computer, not the physical architecture. I'm not claiming there were electronic devices in ancient Mesopotamia.

[00:05:07]I'm claiming something more specific and more interesting.

[00:05:11]The Samrians understood and implemented the fundamental principle that underlies all computational technology that computation can be separated from use.

[00:05:21]That answers can be produced in advance, stored in accessible form, and retrieved by users who need them without those users having to perform the computation themselves.

[00:05:31]The multiplication table is the simplest implementation of this principle.

[00:05:35]Every school child who ever memorized a times table was using a lookup system, a precomputed answer repository that removes the need for calculation at the point of use. The Samrian mathematical tablets extend this principle across a remarkable range of operations.

[00:05:53]Reciprocal tables allow division to be performed as multiplication.

[00:05:57]Square root tables allow geometric problems to be solved without iterative approximation. Cube root tables extend this to three-dimensional calculations.

[00:06:08]Tables of squares and cubes allow rapid calculation of areas and volumes.

[00:06:14]Together, these tables constitute a complete toolkit for the mathematical operations required by a sophisticated administrative civilization.

[00:06:24]And they were produced in systematic and organized form at the beginning of that civilization, not at the end of a long developmental arc, not as the culmination of centuries of mathematical refinement, but at the start.

[00:06:38]The Plimpmpton 322 tablet is the most discussed example of this phenomenon held at Columbia University and dating to approximately 1800 B.CE. It is a table of pythagorean triples sets of three numbers that satisfy the relationship a squared plus b^2= c^² which describes the relationship between the sides of a right triangle. The table contains 15 rows of such triples organized in a way that suggests systematic generation rather than accidental discovery and it covers a range of values that goes significantly beyond what would be required for practical construction problems. The Pythagorean theorem was not formally proved in the western mathematical tradition until the Greek period approximately a thousand years after this tablet was made. The Babylonians were not just using it. They were systematically tabulating its implications across a range of values.

[00:07:33]The table is not a proof. It is a lookup resource. Precomputed answers to problems involving right triangles organized for practical use. Academic debate about Plimpmpton 322 has focused on whether it was a teaching tool, a reference table for practical geometry, or something with a more theoretical mathematical purpose. What that debate has not seriously addressed is the question of what kind of mathematical knowledge was present in the civilization that produced it and where that knowledge came from. Now I want to take you to something that has received almost no attention in the popular discussion of ancient mathematics and that I find more significant than any individual tablet.

[00:08:17]The Samrian scribal schools, the Iduba, trained a professional class of mathematicians and administrators whose technical knowledge was transmitted through a curriculum that has been partially reconstructed from the tablets they produced. The curriculum included the production of mathematical tables.

[00:08:32]Students copied tables, check tables, and eventually produced new tables. The training was standardized across multiple schools in multiple cities, producing a consistent mathematical tradition that operated with remarkable uniformity across the Sumerian world.

[00:08:49]This is an institutional infrastructure for computational knowledge, not individual mathematicians producing isolated insights. a system for training, standardizing, and transmitting a body of precomputed mathematical knowledge across a civilization and across time. The Adoba was not just a school. It was a knowledge preservation and distribution system, a way of ensuring that the computational tools the civilization needed were produced consistently, maintained accurately, and made available to the administrative class that use them. The infrastructure preceded the knowledge it was preserving. The Adoba as an institution appears at the beginning of the literate period fully formed with a curriculum that includes advanced mathematics from its earliest documented phase. You do not build an institutional knowledge preservation system before you have the knowledge you are preserving.

[00:09:45]The infrastructure implies that the knowledge was already there needing to be institutionalized rather than being developed incrementally within the institution.

[00:09:55]Where was the knowledge before the institution was built to preserve it?

[00:09:59]The Samrian answer to that question is consistent across multiple independent texts. The knowledge came from the Appcallu.

[00:10:07]The seven beings who emerged from the sea before the flood taught humanity everything that civilization requires.

[00:10:12]Among the things they taught, listed as gifts transmitted, were the mathematical arts, not as a vague cultural achievement, but as a specific name component of the package of knowledge transmitted before the flood, preserved through the flood, and institutionalized after it in the scribal school system.

[00:10:33]The Aduba was preserving something it received. The tablets are copies of a tradition, not the origin of one. Now, let me show you where the sophistication of the Samrian mathematical tradition leads when you follow it far enough. The Antiqua mechanism is a Greek device from approximately 100 BCE, a bronze mechanical calculator that could predict astronomical positions and eclipse cycles.

[00:10:58]It is considered one of the most remarkable technological objects from the ancient world and the sophistication of its gearing is such that nothing comparable appears in the archaeological record for over a thousand years after it was made. When researchers first described it in the early 20th century, they assumed it was a later medieval device misdated because a machine of that level of complexity should not have existed in the ancient world. The antiiththerra mechanism was built by Greeks who inherited their astronomical and mathematical tradition from Babylonia who inherited it from Sumer.

[00:11:32]The precomputed astronomical tables that the mechanisms gearing implemented in physical form had existed in clay tablet form for over a thousand years before the mechanism was built. The sorrow cycle that the mechanism used to predict eclipses was in the Samrian astronomical record. The mathematical relationships encoded in its gears were the same relationships encoded in the sexesimal tablets which shows a direct ancestry of ideas.

[00:12:02]The antither mechanism is what happens when you take the Samrian lookup table and build it into a machine. The conceptual architecture is identical.

[00:12:11]The implementation differs only in the medium bronze gears rather than clay tablets. The principle is the same.

[00:12:19]Computation separated from use, answers produced in advance, and results accessible to a user without performing the underlying calculations.

[00:12:28]The Samrians implemented this principle in clay because clay was their medium.

[00:12:33]Given a different medium, given different materials and different manufacturing capabilities, the same conceptual architecture produces different physical implementations.

[00:12:43]The anti-therra mechanism is the Samrian mathematical table made mechanical.

[00:12:49]What would the Samrian mathematical table look like implemented in a medium more sophisticated than clay? We cannot answer that question from the physical record. But we can observe that the conceptual framework precomputed answers lookup architecture computation separated from use is the same framework that underlies every computational device from the clay tablet to the bronze gear mechanism to the electronic computer. The Samrians were there first 5,000 years ago. They had a system of precomputed mathematical tables covering multiplication, division, square roots, cube roots, reciprocals, Pythagorean triples, and astronomical calculations organized in a standardized institutional framework for preservation and distribution. They said they received it from beings who came from the sea and taught them everything before the flood. The academic literature notes the sophistication of the tablets. It notes the absence of a developmental sequence that should have preceded them. It notes the institutional infrastructure that appears fully formed at the beginning of the literate period. And then it moves on without asking the question that all three of those observations together are clearly raising. Who computed the first tables?

[00:14:04]Before the Aduba institutionalized the tradition, before the literate period began, before the writing system that should have been required to develop and transmit this knowledge existed, someone worked out the square root of 2 to six decimal places. Someone systematically generated Pythagorean triples across a range of values. Someone organized the reciprocal relationships of the sexed giimal system into a complete and internally consistent lookup architecture before the institutions that preserve these things existed. The tablets say who did it? The answer is in every text that addresses the origin of Samrian knowledge. We just keep deciding not to take it seriously.