A Placeholder Promoted to a Number

Zero is usually presented as one invention. It is two, and they are centuries apart in difficulty.

The first job is positional. In a place-value system you need a way to say that a column is empty, or 205 and 25 become the same marks on the page. Several civilisations worked this out independently: Babylonian scribes used a pair of slanted wedges, Mesoamerican calendars had a shell glyph, and the Chinese counting-rod system left a space. These are features of a notation. A scribe can use a placeholder perfectly for a lifetime without ever treating the thing it marks as a quantity.

The second job is being a number. Something you can add, subtract, multiply by, and reason about on the same footing as three and seventeen. That is a conceptual commitment rather than a convenience, and the earliest surviving text that makes it is Brahmagupta's Brahmasphutasiddhanta, written in 628.

What Brahmagupta does there is write rules. A fortune minus zero is a fortune. A debt minus zero is a debt. Zero minus a debt is a fortune. He is treating zero and the negatives as members of the system and stating how they behave, which is exactly what it means to let something into arithmetic.

He also tried division by zero and got it wrong, concluding at one point that zero divided by zero is zero. Later Indian mathematicians, Bhaskara II in particular, worked on it further, and the settled modern answer is that the operation is undefined rather than large or small. That is worth noticing: the person who first let zero into arithmetic also produced its first mistake, and the mistake is in precisely the place where the subject still puts a warning sign.

Now the manuscript, and the dispute, which is a better lesson in how to read scientific news than it is in the history of numbers.

The Bakhshali manuscript was found in a field near Mardan, in what is now Pakistan, in 1881. It is written on birch bark and contains the earliest known Indian use of a zero symbol. In 2017 the Bodleian Library in Oxford had portions of it radiocarbon-dated and announced, by press release, dates as early as the third century, which would have made it dramatically older than anyone had thought.

Specialists objected, and the objection was not to the chemistry.

A birch-bark manuscript is a physical object that gets copied, repaired, supplemented and rebound over centuries. Radiocarbon dating returns the age of the material of a particular folio. So the oldest folio dates the oldest surviving page, and the youngest dateable folio is evidence about when somebody was still writing in the book. Dating one leaf does not date the composition of the text, and the leaves in this manuscript turned out to span something like seven hundred years.

The scholars who responded, Plofker and Wujastyk among them, also objected to the announcement route: a result presented to the press before peer review, where the inference drawn from it was the contestable part.

Keep the shape of that. A measurement can be sound while the conclusion drawn from it is not, and the two fail separately. It is the same distinction the base-rate lesson in the probability book is about, arriving from a completely different direction.