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The Fables of Ishango, Or the Irresistible Temptation of Mathematical Fiction

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The Fables of Ishango, Or the Irresistible Temptation of Mathematical Fiction The fables of Ishango, or the irresistible temptation of mathematical fiction by Olivier Keller, agrégé in mathematics, PhD, EHESS Figure 1: Map of modern-day Uganda. Ishango is located in the Democratic Republic of Congo (a former colony of the Kingdom of Belgium), close to the border with Uganda and situated on the northern shores of Lake Edward (its approximate position is shown by the red square). Prehistoric artefacts are silent by definition, imbued with a delectable air of mystery. They are captivating and, we feel, rich in meaning. We naturally try to “make them talk”, but it is hard to know if they have something to say and, if so, to make them speak the truth. All manner of fictionalised anachronisms lie poised for ambush, and one can very easily get caught out, especially when they are camouflaged … as mathematical truths. Mathematics, it is true, naturally look like definitive and timeless definitive truths. To top it off, almost any marking can be described or interpreted in mathematical language. The researcher acting in good faith can get carried away despite himself, yielding to the siren song of anachronism with the firm conviction that he is speaking the truth, for “this is mathematics”! The two perpendicular and symmetrical planes of a biface (hand axe),1 it will be argued, show that the maker of the tool had mastered the 1. A tool that was mass produced by Homo erectus and Homo ergaster during the Lower Palaeolithic, starting from 1.5 million years ago. 1 Euclidian conception of space. Similarly, a given group of n strokes or notches will invariably be taken as number n. We offer the reader a typical case study of this phenomenon, and one that is rather unusual on BibNum. Instead of analysing a noteworthy scientific text, we will show how, through mathematical illusion, a prehistoric artefact – the first of the two Ishango bones – rose to fame by being presented as a noteworthy scientific text. Discovered in 1957 by the Belgian archaeologist Jean de Heinzelin, the object’s fame is now assured. It is on display at the Royal Belgian Institute of Natural Sciences, in Brussels, and presented to visitors as “humanity’s oldest calculator”. To mark the International Year of Mathematics in 2000, the Belgian Post Office published a stamp alluding, among other things, to this famous bone. The stamp shows a series of three then six vertical strokes at its base. Figure 2: Stamp issued in 2000 by the Belgian Post Office to mark the International Year of Mathematics. The vertical strokes are an allusion to the notches on the Ishango bone. In 2007, an international conference entitled “Ishango, 22000 and 50 Years Later: The Cradle of Mathematics?”2 was held in Brussels, and on 28 February, under the title “The incised bones of Ishango and the birth of numeration in Africa” [Les os incisés d’Ishango font naître la numération en Afrique], Le Monde reported that: […] they could represent the oldest evidence of humanity’s mathematical capacities, fifteen millennia before numeration emerged at the same time as writing in Mesopotamia (modern-day Iraq). 2. “Ishango, 22000 and 50 Years Later: The Cradle of Mathematics?” 2 Figure 3: The first Ishango bone seen from two sides. On the left: the so-called “middle column” containing, from top to bottom and depending on whose interpretation it is, groups of 3, 6, 4, 8, 9 (or 10), 5 and 7 notches. On the right: part of the so-called “right column” (11 notches above and 9 below) and the “left column” containing, from top to bottom, groups of 11, 13, 17 and 19 notches. (Photographs by the Royal Belgian Institute of Natural Sciences) The Ishango bone, it seems, is now a permanent fixture in both mathematical history books and popular science magazines, which venerate it as the oldest, or at least one of the oldest, records of human scientific endeavour. It can even inspire truly cosmic lyricism, for, as speakers at the 2007 conference explain: […] in Belgium, media coverage began in 1996 when Dirk Huylebrouck wrote The Bone that Began the Space Odyssey in The Mathematical Tourist, and continued with attempts to send the bone into space in homage to the Central African contribution to the development of technology.3 3. Els Cornelissen, Ivan Jadin and Patrick Semal. “Ishango, a history of discoveries in the Democratic Republic of Congo (DRC) and in Belgium”, in Dirk Huylebrouck (ed.), Ishango, 22000 and 50 Years Later: The Cradle of Mathematics?, 28/02–02/03 2007, pp. 23–39. Brussels, Koninklijke Vlaamse Academie Van Belgie Voor Wetenschappen En Kunsten. 3 It is acknowledged that such a promising artefact, and one that arouses so much enthusiasm, deserves its place in an institution devoted to the history of science. Let’s therefore consider what it is exactly, and examine from a technical, historical and methodological point of view the mathematical interpretations4 that have made it famous. THE ISHANGO BONES AND THEIR NUMERICAL INTERPRETATIONS The first Ishango bone (Fig. 3) was discovered in the area after which it is named, in the Democratic Republic of Congo. A fragment of quartz affixed to one end shows that it was a tool handle. It is usually dated to 20,000 years BCE. Placed lengthwise (10 cm), it shows three rows of more or less parallel but asymmetrically grouped notches (Fig. 4). Several of the notches are worn way or barely visible, which immediately makes any numeric interpretation suspect. Figure 4: Full view of the first Ishango bone, with its three columns (“left”, “middle” and “right”). Source: Dirk Huylebrouck, “L’Afrique, berceau des mathématiques”, in Mathématiques exotiques, Pour la science dossier, April/June 2005. Yet its inventor, Jean de Heinzelin, nevertheless took this risk and published his results in Scientific American in June 1962: Take the first column, for example: 11, 13, 17 and 19 are all prime numbers (divisible only by themselves and by one) in ascending order, 4. We will not consider Alexander Marshack’s interpretation of the bone as a lunar calendar, which is now largely discredited. 4 and they are the only prime numbers between 10 and 20. Or consider the third: 11, 21, 19 and 9 represent the digits 10 plus one, 20 plus one, 20 minus one and 10 minus one. The middle column shows a less cohesive set of relations. Nevertheless, it too follows a pattern of a sort. The groups of three and six notches are fairly close together. Then there is a space, after which the four and eight appear – also close together. Then, again after a space, comes the 10, after which are the two fives, quite close. This arrangement strongly suggests appreciation of the concept of duplication, or multiplying by two. It is of course possible that all the patterns are fortuitous. But it seems probable that they were deliberately planned. If so, they may represent an arithmetical game of some sort, devised by a people who had a number system based on 10 as well as a knowledge of duplication and of prime numbers.5 Let’s take the middle column: according to the author, 3 is doubled to 6, 4 to 8 and 5 to 10. But the 5 and the 10 are doubtful: one of the sets of 5 is genuinely illegible, and in reality the 10 could be a 9. In addition, in the case of a duplication of 5, 3 and 4, there is to explanation as to why the set of five notches is shown twice, whereas the group of three and five are shown only once. And what is the role of 7, which is neither involved in duplication nor doubled? Unless the bottom of the middle column reads 10, 4, 5 and 7 (and not 10, 5, 5 and 7), which would give us 7 doubled, with 10+4, and 5 doubled, with 10. In the left column, Heinzelin sees a list of prime numbers. If this is the case, the people of Ishango must have had a detailed understanding not just of simple duplication, but of the far more complex matter of multiplication in general, which would have made the two-times table on the side rather ridiculous, a bit like placing an addition table alongside a table of antiderivatives in a contemporary publication. And if the right column represents 10+1, 20+1, 20-1 and 10-1, why isn’t the left column, for example, 15-4, 15-2, 15+2, 15+4 (instead of a list of prime numbers), in other words a kind of “arithmetical game” based on the average of the 10s and 20s in the right column? And here’s another potential “game”: on the left, the two end numbers total 30, as do the two middle numbers, whereas on the right the first and the third numbers total 30, and likewise the second and the fourth. Once it has been decided that the sets of notches are numbers, it’s easy – given a few arrangements here and there – to load the bone with meaning, or 5. Jean de Heinzelin, “Ishango”, Scientific American 206, 105–116 (1962), doi: 10.1038/scientificamerican0662-105. 5 even, if one pursues the argument a little further, as above, to make it say contradictory things. @@@@@@@ The Ishango bone owes its fame not to the calculations of Jean de Heinzelin, but to the dedication and new calculations of two Belgian scientists, the mathematician Dirk Huylebrouck, and Vladimir Pletser of the European Space Agency. Though they reject Heinzelin’s specific conclusions, they accept his premises, namely a purely mathematical interpretation.
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