Zero has become a part of society's present mathematical usage that people are hard to imagine that it was not used in calculation. Zero is perhaps the most sophisticated and historically complex notion in the elementary arithmetic. Recent research has focused on the role of zero as the cross-over between notation and concept, borne from and used within two separate spaces and times—the positional notation system and as a number. [1] The essay charts the historical evolution of zero as a place holder in the sexagesimal system of Babylonia, as a number in classical India, and as a number in medieval and early modern Europe, where it was received with mixed emotions. The proper history of zero is not a discovery story, but a story of the progressive building up of a mathematical object, which depended on varying philosophical commitments.
The earliest known use of zero is in the sexagesimal (base-60) system used by ancient Babylonians. Positional notation was employed by scribes by the Old Babylonian period, similar to the way it is used today in our decimal system. This led to an ambiguity problem: The cuneiform symbol for, say, "2" could be the value of 2, 120, or 1/30 depending on the missing place values. In the mid-first millennium BCE, Babylonian astronomers started to add a symbol (two slanted wedges) to fill in a blank in a numeral, similar to the way that they added a zero to a number like 205. But this placeholder was never used at the end of a numeral and was never used as a numeral. It was not a mathematical object, but a device for bookkeeping. In fact, even in the most complex ancient numeral systems, like the periodical "orders and periods" scheme that Archimedes devised in the 3rd century BCE, there was no zero digit, as Manca demonstrates.
A similar, but independent, development took place in the Maya civilization, in which a shell glyph was used as a zero in the vigesimal (base-20) Long Count calendar system as early as the third century CE. From a contemporary Maya point of view, as Medina (2026) points out, such independent invention calls into question Eurocentric narratives of the history of mathematics as a story of progress emanating from Greece. [2] The Maya zero was completely positional, and, like its Babylonian counterpart, it was used mostly for calendrical and astronomical calculations; however, it was also a part of a complete and economical system of numbers that could be used to represent very large numbers with only a few symbols.
The conceptual breakthrough that made the placeholder into a number was in India. Positional ideas were introduced to India by the mathematicians of the Sanskrit astronomical tradition, who were able to combine positional ideas with their own well-developed arithmetic between about the fifth and seventh centuries CE. The astronomer Brahmagupta formulated rules for working with zero in 628, and he was the first to state that the sum of a number and zero was the number itself, that the product of a number and zero was zero, and--most audaciously--that he had rules for adding and subtracting negative numbers and multiplying and dividing zero. By modern standards his handling of division by zero was not consistent (he proposed that 0/0 equals 0), but the use of zero as an operand in arithmetic was revolutionary. The system was not used in decimal place-value notation in Indian mercantile practice, and was passed on to the Islamic world where the basic form was established by al-Khwarizmi's treatises on calculation with Hindu numerals (c. 825 CE). [3]
There was a slow and hostile response from Europe. Hindus introduced the Arabic numerals to European merchants in Fibonacci's Liber Abaci (1202), but there was some skepticism for centuries. Zero was uncomfortable with the Aristotelian philosophy which denied the reality of a void, and with a culture of numbers in which there was no symbol for nothing. Only with the Scientific Revolution, and in particular the invention of analytic geometry and the calculus, was zero an essential: the concept of a limit (where quantities come closer to zero but never actually reach it) allowed infinitesimal analysis, and the Cartesian coordinate plane fixed zero at the origin of all graphs, without which, Manca (2023) notes, modern algebra and analysis would be impossible.
The story of zero is not a straightforward heroic one, then. To society, it seems self-evident, but at the time, it was limited by philosophical beliefs concerning number, void, and being. Babylonian scribes did not face nothingness, Indian mathematicians made nothingness work, European intellectuals had to overcome metaphysical resistance before zero became commonplace. This multi-layered genealogy is important for the wider methodological claim on the historiography of mathematics: mathematics is a human invention, not an eternal truth that has to be unearthed--one that contemporary scholarship on Indigenous mathematics makes strongly when it shows how easily non-European contributions are excluded from the narrative. [4] The number zero was invented, not discovered: slowly, controversially and more than once.
