Candidate #2. The Ishango Bones
The Lembobo Bone is dated at 35000 BC, but before we go backwards in time let’s step forward to 20000 BC to look at another pair of bones. The first is extremely famous and is the most common inclusion in any math history book that wants to name check prehistory; the second is (as of the time of this writing) wildly obscure.
The first Ishango bone was found in 1960 by Belgian geologist Jean de Heinzelin, working near Lake Edward on the border of the Congo and Uganda.
[Image credit: Science Museum of Brussels.]
There are three rows around the bone containing sets of tally marks:
First row: 19, 17, 13, 11
Second row: 7, 5, 5, 10, 8, 4, 6, 3
Third row: 9, 19, 21, 11
(A more detailed diagram is here at the Wikipedia article.)
The presence of so many numbers on such an early artifact has sparked all sorts of commentary. It has lead some to speculate the first row is a table of prime numbers, or that the bone as a whole represents another lunar calendar (spanning six months):
When I examined this tiny petrified bone in the Musée d’Historie Naturelle in Brussels, I found that the engraving, as nearly as microscopic examination could differentiate the deteriorated markings, was made by thirty-nine different points and was notational. It seemed, more clearly than before, to be lunar.
— Alexander Marshack, The Roots of Civilization
Jean de Heinzelin, the discoverer of the original bone, on his deathbed disclosed the existence of a second bone. Scholarly work followed, and the bone was only revealed to the public in 2007. Here it is:
[Image credit: Royal Belgian Institute of Natural Sciences.]
There are six sets of marks:
14 long marks, 6 short marks
6 long marks
18 long marks
6 long marks
20 long marks (with complex secondary marks)
6 long marks, 2 short marks
Detailed work on the second bone is still a wide open topic. The link above suggests the marks reflect a multi-base system, a suggestion I find implausible for several reasons, most of all because no number system in history used 18 tally marks or more in a row to represent a single digit.
Now, you may be puzzled why I’ve mentioned these artifacts in the first place, given the Lebombo Bone comes roughly 15000 years earlier. However, some still call the first Ishango Bone (no math text I know of has mentioned the second) the oldest mathematical artifact because they have a particular definition of mathematical:
…most scholars do not consider recording dates to be proper mathematics.
— Simon Singh, The Ishango Bone – Is This The World’s Oldest Mathematical Artefact?
That is, mere tally marks on the Lembobo Bone are not enough to qualify as mathematical.
From an ethnomathematical standpoint, I find this absurd. Being able to count to 29 is an accomplishment not every culture has made, and even what is a simple act for us now required in history a leap of mathematical imagination. Additionally, there is a difference between counting by 1-1 correspondence and counting by sequence, and in the case of the (likely lunar calendar) Lembobo Bone, being able to match the two types of counting required an even greater mathematical intuition than, say, making tally marks to correspond with the number of one’s sheep.
Still, the Ishango Bones deserve some sort of title, perhaps with an appropriate adjective; I’ve read of it as “the earliest complex mathematics” or “the earliest logical mathematics” but I believe most fitting would be to call it “the earliest substantial mathematics”.
To keep a running time line, then:
35000 BC Lembobo Bone: earliest mathematics where counting is used for practical purposes
20000 BC Ishango Bones: earliest substantial mathematics
In parts 3 and 4 of this series I’m going to look at some artifacts that are even older, but (as far as I know) not yet written about by mathematicians.