fibonacci, form, Laplace, mathematics, nothing, number, origin, sign, value, void, zero

A Brief History of Nothing

The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought.

Alfred North Whitehead

Leibniz called zero “a fine and wondrous refuge of the divine spirit.” But where does the idea come from? The history of the word may afford us a clue to this mystery. We receive the English word ‘zero’ from the French zéro which comes (along with ‘cipher’) from the Italian zefiro. The latter originates in turn from the Arabic sifr (from safira = “it was empty,” a translation of the Sankskrit sunya = “void” or “un-reality.”)

There are at least two distinct ideas of “zero,” each with more or less unique histories and origins. First, there is the modern, everyday notion of zero as itself a number (the additive identity, “nothing”) which is closest to the original meanings of the Arabic and Indian words. However, zero has a secondary meaning as well: there is a less “formal” idea of zero as simply a kind of positional place-holder (for example, to distinguish 4053 and 4503 from 453, where “0” stands for “none” in the tens’ or hundreds’ place.) Thus zero denotes both a digit and a value, but these two ideas are not completely separate. Neither zero has a simple, common or easily-described history.

What is certain is that, for over a thousand years, the Babylonians used a positional numbering system without zero. (Original texts from that era simply depend on context to resolve ambiguity.) There is certainly no evidence Babylonians felt any problems with the ambiguities which existed. Indeed, until around 400 BC, we do not find even spaces between numerals to denote positional difference (the earliest but debatable use of zeroes in Babylonia is in 700 BC, falling quite close to earliest use in Indian writing around 650 BC.)

When zeroes do arrive in the cuneiform script (in the form of double wedges, or triple hook symbols) they are almost only used between numerals — that is, never at the end of a number (for example, 2015 but not 2150, which would often still be written 215.) Some researchers have suggested that these symbols were likely not intended by the Babylonian mathematicians to indicate zeroes as numbers, but were probably marks intending to clarify the interpretation of a number.

The length of time that mathematics was actively studied and investigated prior to the invention of zero suggests the discovery (or invention?) of “zero” is by no means obvious or natural. The late Hellene astronomer Ptolemy was one of the first Greeks to use “true” zeros (and these resulted from empirical measurements.) Even Fibonacci himself, who introduces Arabic and Indian numbering systems into Europe, still doesn’t treat zero as a “number” but as a “sign.”

Zero continued to be staunchly resisted by thinkers outside of the Muslim world until hundreds of years after Fibonacci. Still by the time of Cardan (who was solving cubic and quadratic equations without zeros) the value was not really a part of European mathematics. Only about 400 years ago does zero really come into widespread usage as a number in its own right, as a concept representing the amount “before” counting starts.

Does ‘zero’ somehow provide the basis for algebraic or “meta-structural” re-evaluation of the properties of numbers themselves? Laplace called zero “ …a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it lent to all computations put our arithmetic in the first rank of useful inventions.” The role of zero in the development of mathematics is incontestable. It is more than possible that Western society depends in many ways upon the uncanny and mysterious power of “nothing.”

While the origins of zero may remain clouded in mystery, a final question about the future of mathematics may perhaps be posed. What does this idea of zero still hide? What does this question of “nothing” evade?

What power could still lie stored within zero — waiting within this mysterious void at the heart of number — or rupture in the essence of value? What unpredictable and undreamt-of mysteries are waiting for mathematicians of the future?

Further Reading:

A. Aaboe, Episodes from the Early History of Mathematics (1964).
G. Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
B. L. van der Waerden, Geometry and Algebra in Ancient Civilizations (New York, 1983).

Articles and other resources:

S. K. Adhikari, “Babylonian mathematics,” Indian J. Hist. Sci. 33 (1) (1998), 1-23.
S. Gandz, “A few notes on Egyptian and Babylonian mathematics,” in Studies and Essays in the History of Science and Learning Offered in Homage to George Sarton on the Occasion of his Sixtieth Birthday, 31 August 1944 (New York, 1947), 449-462.
K. Muroi, “Babylonian mathematics – ancient mathematics written in cuneiform writing” (trans. from Japanese), in Studies on the history of mathematics (Kyoto, 1998), 160-171.
G. Sarton, “Remarks on the study of Babylonian mathematics,” Isis 31 (1940), 398-404.

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8 thoughts on “A Brief History of Nothing

  1. Pingback: History

  2. Haven’t, but it sounds pretty interesting — math as experimental science. I wonder how it fits in with Badiou’s conception of math as fundamental ontology. Are our ontologies imaginary, and if so, what does that mean about possible worlds…? If mathematics is a kind of imagination-machine, ultimately at the service of writers… but then math would lose its status as revealing something essential about our knowledge, unless we recognize that truth is not an impersonal ontology, but already a language — or, at least, theoretically expressible in an imaginary language.🙂

  3. I met Brian a few weeks ago at a conference and his opinion of ‘Being and Event’ wasn’t favorable. The mathematics was out-of-date and influenced too much by the Bourbaki school — which was in vogue back when Badiou was still an active mathematician — and less by Grothendieckian category theory. As you probably know, category theory is exactly what Badiou used with his later work, although Brian wasn’t aware of that.

    And I wonder if Brian’s ideas can be attributed to the fact that he did a lot of research in combinatorics. I forget the details but i think Badiou, in some essay, said something to the effect that his philosophy in ‘Being and Event’ goes against the Hilbertian understanding mathematics as a game of formal languages.

  4. Thank you so much for this comment. Remember that a lot of Badiou’s work is about chance too — this is, strangely enough, where Badiou reaches out to poetry and discourse as well. There is an uncanny kind of eloquence about the axioms of probability, isn’t there? I think it’s because they “decide” an undecidable — they emerge late (relatively speaking, of course) and as the discovery of a certain kind of “internal” limitation on the extent of the validity of deduction. A transcendental limitation by which we are made free — precisely by limiting the scope of our axioms, we open up completely new generative fields, radical sources of creative exploration and experimentation. Imaginary machines are quite real: I am reminded of Deleuze and Guattari’s description of modern society, oscillating between a capitalistic “internal” limit and a schizophrenic “external” limit. Does the inventor-discoverer of a novel system of mathematics always run a double risk — that of over-extending an axiomatic, over-generalizing and hence discounting the role of human ingenuity; as well as that of becoming totally absorbed in details and thus becoming completely oblivious to the power of imagination? Perhaps the constitutive “break” within mathematics itself — Badiou’s “meaningless” primacy of truth — is no more real than the rupture between the human and natural sciences, or even science and literature.

  5. This is a very interesting topic, and I don’t think there has been any direct research – at least on the side of theoretical mathematics – on the position of the event in mathematical research.

    What’s for sure is that axiomatics is pretty much dead. Mathematics no longer just about reduction, derivations and deductions. Gian-Carlo Rota, in ‘Indiscrete Thoughts’, wrote something about what he called ‘mathematicial enlightenment’. And I remember that Alain Connes, in ‘Triangle of Thoughts’, said something about what he called ‘primordial mathematicial reality’, which might be close to Deleuze’s transcendental field. Perhaps the theory of computation, complexity or heuristics might have something important to say. It is a popular opionion by some computer scientist that the mathematical event can be understood by looking into the gap between P and NP complexity classes.

    • Hey there,

      This article is great. Very well written. I’m writing an essay myself on “nothing” and I was wondering what your source was for the history of the word “nothing”. I saw your work cited, but wasn’t sure which went with which.

      -Parker

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