This month's topics:

Much ado about the history of zero

The press release was dated September 14, 2014: "Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol 'zero'." The sub-head: "Ancient Indian mathematical text at Oxford's Bodleian Libraries revealed to be centuries older than previously thought." The manuscript, discovered near Peshawar in 1881, has recently been carbon-dated: "The surprising results of the first ever radiocarbon dating conducted on the Bakhshali manuscript, a seminal mathematical text which contains hundreds of zeroes, reveal that it dates from as early as the 3rd or 4th century--approximately five centuries older than scholars previously believed. This means that the manuscript in fact predates a 9th-century inscription of zero on the wall of a temple in Gwalior, Madhya Pradesh, which was previously considered to be the oldest recorded example of a zero used as a placeholder in India. The findings are highly significant for the study of the early history of mathematics."

A detail of the Bakhshali manuscript, folio 46 recto. The numerical part is transcribed by G. R. Kaye (The Bakhshali Manuscript, Government of India Central Publication Branch, Calcutta 1927) as the fraction $\displaystyle{\frac{110895125022}{1554615000}}$. Image from Kaye, courtesy of Bill Casselman. Full image of folio.

The release links to a video in which the Oxford mathematics professor Marcus du Sautoy recounts that he expected the dating to be around mid-ninth century, and "I'm actually staggered to find that this is way earlier." "This manuscript is between 200 and 400 AD."

Hannah Devlin's report that day in The Guardian is more specific, and politely sceptical: "In the latest study, three samples were extracted from the manuscript and analysed at the Oxford Radiocarbon Accelerator Unit. The results revealed that the three samples tested date from three different centuries, one from 224-383 AD, another from 680-779 AD and another from 885-993 AD, raising further questions about how the manuscript came to be packaged together as a single document."

Relying on the prestige of Oxford's faculty, libraries and laboratories, the media took details of the press release and the video as statements of fact.

It was left to a team of science historians (Kim Plofker et al., including the Bakhshali expert Takao Hayashi) to point out ("The Bakhshali Manuscript: A Response to the Bodleian Library's Radiocarbon Dating," History of Science in South Asia, 5 134-150) that the folios dated 224-383 AD and 680-779 AD were in fact consecutive leaves in the manuscript: written in what was clearly the same hand, and containing consecutive parts of the same calculation. Something is wrong with that dating. Their last sentence: "It should not be hastily assumed that the apparent implications of results from physical tests must be valid even if the conclusions they suggest appear historically absurd."

"Pariah moonshine,"

the title of an article published September 22, 2017 in Nature Communications, requires some decoding. When the classification of finite simple groups was completed in 2002, besides infinite families with natural geometric definitions (e.g. symmetries of regular prime-number-sided polygons) there were 26 "sporadic" groups, the largest of which is called "the Fischer-Griess monster" (it has about 1000 times as many elements as there are atoms in the Earth). As the story goes, the idea that the monster could be related to the study of complex functions (and thence to theoretical physics, or "nature") seemed at the time like lunacy, and a series of conjectures made in that direction were termed "monster moonshine." But in fact such connections do exist. According to the article, "The generalised moonshine conjectures were recently proven by [Scott] Carnahan, so moonshine illuminates a physical origin for the monster, and for the 19 other sporadic groups that are involved in the monster. Therefore, 20 of the sporadic groups do indeed occur in nature." For the sporadics that did not fit in, the "pariahs," the question had been posed but was open until this announcement, which describes a connection for two of them. As the authors, John F. R. Duncan (Emory), Michael H. Mertens (Köln) and Ken Ono (Emory) state, "pariah groups of O'Nan and Janko do play a role in nature."

It is significant that the authors chose to place this announcement of their results not in a mathematics journal but in one aimed at a much larger scientific audience; their writing is appropriately expository, especially in the introduction. Nature itself ran an assessment of the paper in their "News and Views" section, October 4: "Mathematics: A pariah finds a home," by Terry Gannon. Gannon sets the stage, again in terms suitable for wide consumption, and sketches out the story. He ends "It is always difficult to gauge the importance of a mathematical result without the hindsight that many years brings. Nevertheless, Duncan et al. have shown us a door. Whether it is to a new closet, house or world, we cannot yet say, but the results are certainly unexpected, and no one will think of the pariahs in the same way again."

Talk of numbers on Connecticut NPR

On October 12, 2017 the Colin McEnroe show on Connecticut NPR broadcast a segment, "The Secret Lives Of Numbers," about "the anthropological, psychological, and linguistical ramifications of the concept of numbers." McEnroe's guests were Caleb Everett (Anthropology, Miami), who grew up among the Pirahã tribe in the Amazon basin; and Brian Clegg, a physicist by training and the author of Are Numbers Real? The Uncanny Relationship of Mathematics and the Physical World. We learn from Everett that members of a non-numerical tribe like the Pirahã have no problem with small sets of elements, but have trouble differentiating one cardinality from another when they are close and bigger than, say, 7. (Nevertheless, he adds later in the program, "they're pretty well adapted to their environment and they're leading pretty successful lives, by their standards.") Clegg has this nice perspective on the relation between mathematics and the physical world: "Mathematics exists separately to the universe. What we've done is find a way of bringing the two together; so we've almost acted as a translator between the universe and parts of mathematics."

Tony Phillips
Stony Brook University
tony at