A quantity, a placeholder, a point of reference: new concepts of zero are constantly being developed in mathematics. | Grafik: Bodara

Firstly, zero can signify the absence of quantity, like the number of apples left after giving away all we had. This concept took time to find its place in science: “The idea of representing the absence of something by using a symbol with the same nature as we use to signify presence is not obvious”, says Roy Wagner, a professor of history and philosophy of mathematics at ETH Zurich.
In the 7th century, the Indian mathematician Brahmagupta treated zero as a whole number, on which one can perform usual arithmetic operations. This is ‘shunya’ in Sanskrit (meaning nothing), which became ‘sefr’ in Persian, notably in the works of scholar Al-Khwarizmi, and – probably – ‘zefiro’ in ancient Italian, which gave rise to the word ‘zero’.

Another major use of zero is found in positional notation, like when it indicates the absence of tens in the number 1203. This zero of position has some relation to that of quantity, but limited: subtraction in columns with remainders gives it temporarily the value ten, something impossible for the zero of quantity.
Four millennia before the advent of Roman numerals and their additive numbering (MCCIII), positional systems appeared in Babylonian mathematics, built around base 60, as well as with Archimedes. Numbers were initially represented in tabular form (e.g., with an abacus), which allowed expressing the absence of a certain factor by a simple blank space. Writing numbers without a table would later require introducing a dedicated symbol, which is not always the same as that for the zero of quantity.

Zero has other uses too. It’s found in counting, for example, when time starts at midnight (00:00) rather than an hour. Ancient Egyptian architects used the hieroglyph of zero to indicate geometric reference points in constructions. Zero also appears in digital technologies, which represent information using 1 and 0. The use of binary in computing comes from Boolean algebra, which set logical formality in stone. By associating one and zero with true and false statements, it takes advantage of a similarity between logical operations (‘and’, ‘or’) and calculations made on numbers 0 and 1.
All these zeros represent distinct mathematical concepts, with similarities but also differences. “Understanding the idea of zero means being able to navigate between these different contexts”, says Wagner. “Moreover, new concepts of zero emerge all the time; some endure, others don’t”.
Are zero and infinity just mathematical objects or are they part of the world? “Separating signifier from signified doesn’t seem useful to me”, he says. “Each sign has real materiality, and each object has symbolic significance. After all, the concept of a hammer is just as approximate as that of zero”.