How Many Authenticated Ancient Mathematical Artifacts are Known?

“How many authenticated ancient mathematical artifacts are known?”

I recently asked myself this question while researching the history of mathematical proof. Ultimately, all historical theories must answer to the evidence. For mathematics, this means studying surviving parchment documents, cuneiform tablets, bamboo strips, bone markings, Stella inscriptions, calculating boards, and other objects, that inform our mathematical origin stories. And, when it comes to the ancient history of mathematics, that’s all we have – stories. Stories that are spun from a very small number of authentic artifacts.

“How bad is it?”

When I started tracking down references to ancient mathematical artifacts, I quickly discovered that everyone keeps referring to the same short list of objects. Instead of going down a rabbit hole, you find yourself wandering in a circular maze littered with objects like the Ishango Bone, Plimpton 322, the Rhind Papyrus, Yale YBC 7289, the Suàn shù shū, oracle bones, the Bakhshali manuscript, W 19408,76, the Khmer Zero, and a few surviving medieval versions of Euclid’s Elements. Surely the dense convoluted tomes going on about the history of mathematics cannot — when traced back to source materials — depend on such a meager collection: a collection so small it would fit in the bed of your pickup with enough space left over for your dog!

Ancient authenticated mathematical artifacts are exceedingly rare. They are among the rarest of all ancient artifacts. The following table was taken, from The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook; it lists all the known major Egyptian mathematical artifacts. There are fifteen objects in this table spanning some two thousand years!

I think we can safely infer that our knowledge of ancient Egyptian mathematics is woefully incomplete. The ancient Greeks, still widely lauded for their many mathematical achievements, have left no original documents. Euclid’s famous Elements, a book we still esteem and admire today, can only be traced to the Middle Ages – almost a thousand years after its avowed creation date. A thousand years is plenty of rewrite time. How much of The Elements dates to 300 BCE and how much derives from later amendments is still hotly contested.

Some cultures are better represented in the archeological record than others. The Mesopotamians, actually a series of cultures spanning thousands of years, that lived in the fertile crescent, the land between the Tigris and Euphrates rivers in modern Iraq, wrote on cuneiform clay tablets. When dried or baked cuneiform is very stable. Even better the Mesopotamians recycled their tablets by using them as filling for walls. Many thousands of these tablets have survived and are still readable today. Because so many cuneiform tablets have survived it’s possible to apply rudimentary statistics. Of the 5000 tablets found in the late fourth-millennium city of Urak about 10% were “school” exercises. Most of these exercises are word lists and other writing exercises but a few exhibit calculations. We can’t be certain if these tablets are representative but they do show that ancient Urakians had the same aversion to mathematics that’s found in modern cultures. Math nerds have never been the cool kids! In any case, the moral of the story is clear; if you want your stuff read thousands of years from now, write on durable materials.

For cultures that wrote on perishable materials like papyrus (Egyptians, Ptolemaic Greeks), bamboo strips (China, Japan, Korea), tree bark (India), string (Inca and Pre-Inca quipus), and fig-tree fibers (Mayan codices), the record is extremely sparse. For example, the famous ancient library of Alexandria supposedly held hundreds of thousands of papyrus documents, but not a single one has survived! History is a cruel castrating bitch! Lack of material is a common theme: there are no Indian manuscripts older than 200 AD. The famous Bakhshali manuscript hints at older materials but none of its predecessors have been found. Modern Indians make bold claims about the antiquity of their mathematics but when it comes to hard artifacts their situation is comparable to the ancient Greeks, another culture that left us few originals to study. The oldest Chinese bamboo strips date to around 200 BCE. There are older “oracle bone” markings that date to Bronze Age 1200-1000 BCE that show the ancient Chinese had developed base 10 numerals but sadly, the bone scripts do not contain any known mathematical texts or cheat sheets. Too bad!

The oldest known unambiguously mathematical artifact is a cuneiform tablet labeled W 19408,76 that dates to 3320 BCE. Five thousand years ago somebody was calculating areas. There are even older objects that hint at paleolithic mathematics. Tally sticks — mostly notched bones like the Ishango and Lebombo bones —suggest people were “counting” tens of thousands of years ago. You can’t infer much from tally sticks but that hasn’t stopped so-called ethnomathematicians from making extravagant claims. In the wilder versions, paleolithic number theorists were contemplating prime numbers twenty thousand years ago. It’s not impossible, but as Carl Sagan famously said, “Extraordinary Claims Require Extraordinary Evidence,” and right now (2023), the data does not align with the drivel.

And speaking of “data”, The Mathematical Association of America maintains one of the best historical mathematical object lists: The Index to Mathematical Treasures. This list contains roughly 1300 objects. I screen-scraped the HTML, extracted the list, and charitably assigned hard dates to ambivalently dated objects. For example, the entry for the Zhoubi suanjing refers to a book printed in 1603 that claims descent from a 100 BCE original. So, I take 100 BCE as the date of this object. This charitable date assignment will tend to make objects older than they probably are. After assigning dates I computed the following half-millennia histogram.

Negative numbers refer to BCE dates, and yes, I know the Year Zero does not exist in our calendar. It should but that’s a rant for another day. The second column is a cumulative object count. From -25000 to 0 we have 21 objects! Remember, I am assigning charitable dates. The actual count is even lower. The MAA treasures list is not comprehensive but the trends it exhibits are likely to be stable as the object count increases. The vast majority of entries date from the year 1500 to the present and this corpus constitutes our actual history of mathematics. Before 1500 and movable type printing, the object count is profoundly underrepresented and it only gets worse the further back you go. We just don’t have a lot of germane material to study. Keep this in mind the next time some blowhard starts opining on the history of mathematics and its subdisciplines like the history of proof.

So, the answer to my question, “How many authenticated ancient mathematical artifacts are known?” is simply, not many! People are still looking and perhaps in the years to come more objects will be found that will render our alleged mathematical histories less mythical and more mathematical.