The Invention of Zero

So opens the account of Creation in the King James Bible, a poetic evocation of the obscure relationship between absence and presence, nothing and everything. This is difficult, slippery territory. In working to understand the nature of this relationship, we are drawn to the very limits of language and conceptual thought.

The Hebrew word tohu—here translated as “without form”—occurs nineteen times in the Bible, rendered in varying contexts as wasteland, wilderness, an empty place, nothingness, vanity, and confusion. Bohu (“void”) appears only three times: once in Genesis 1:2 (above), once again in Isaiah 34:11 (there translated as “emptiness”), and a third time in Jeremiah 4:23, in direct reference to Genesis and so preserving the translation as “void.”

These are rarely used, fearful words. According to a note in the Oxford Annotated Bible, the formless void that somehow precedes or underlies Creation is associated with an ancient belief that the world originated from “a watery chaos, personified as a dragon in the Babylonian creation epic.” This “dragon” is the Leviathan referred to in Job and the Psalms, a fire-spitting, serpentine creature with fangs and claws—a graphic embodiment of our primal terror in the face of an abyss of nothingness undefined by time or space, an impenetrable, ungraspable, fathomless black hole out of which we emerge and into which we inevitably return.

This ancient story of the universe emerging ex nihilo (“from nothing”) resonates with a mythopoetic authority that has captivated the imagination of generations of commentators who struggled to come to terms with their own fears of what cannot, by its nature, be known. Like his Hebrew predecessors, the Greek philosopher Aristotle (384–322 BCE) found any reference to absolute nothingness troubling. For Aristotle, the problem was not so much poetic or psychological as rational. In his Physics he maintained that matter could have no beginning, as a beginning would itself have to begin, and that second beginning would in turn have to begin, implying an infinite regress and therefore a logical fallacy. One might wonder what exactly is the difference between an endless regress of beginnings and no beginning at all.

Aristotle’s opinion was enormously influential and more or less prevailed for centuries (even though it clearly rejected the biblical account of the creation) until Sa’adiah ben Yosef Gaon (882–942)—a Jewish theologian writing in Arabic—composed his masterwork, The Book of Beliefs and Opinions, in which he countered Aristotle’s fear of infinite regress with a rigorous, systematic defense of the idea of creation from nothing. Over time, his arguments were refined by Muslim scholars, who carried them across the Middle East into North Africa and from there into Moorish Spain, where they caught the attention of the 12th-century Sephardic philosopher Maimonides and influenced his efforts to harmonize Aristotelian reason with the teachings of the Bible.

Carried forward by Jewish philosopher-mystics and Christian apophatic theologians, the doctrine of creation from nothing increasingly gained prestige until, at long last, it became respectable for Christian intellectuals to conceive of a nothing out of which everything is born. In 1215, at the fourth Lateran Council, Pope Innocent III certified creatio ex nihilo as the official doctrine of the Roman Catholic Church. After a thousand years the conflict between Aristotle and the Leviathan was resolved. The serpent prevailed.

But how is it that Sa’adiah ben Yosef Gaon’s ideas gained so much traction among European intellectuals, both Jewish and Christian? Even the most creative mind does not operate in a cultural vacuum. Where did he derive the inspiration for his radically new understanding of the relationship between absence and presence? While there may well be no single answer to these questions, history provides some tantalizing clues.

Rabbi Sa’adiah Gaon was born in Egypt, but he lived and worked for eleven years in Baghdad under the Abbasid Caliphate, during what is often considered the Golden Age of Islam. Arabs had conquered Persia in 651 CE, incorporating it into a vast Islamic empire that stretched from Spain to the frontiers of South Asia. During the 7th and 8th centuries, foreign ideas flowing west out of India into Persia exerted a profound influence on Muslim intellectuals,who in turn passed these ideas along to Europe. Among them was the ten-digit Indian numerical system, incorporating the concept of a number that is, in itself, nothing. This system was described in 825 CE by the Persian mathematician Muhammad Ibn Musa al-Khwarizmi (ca. 780–850) in a work synthesizing Greek and Indian thought. The Sanskrit name for this mysterious number was shunya. The phonetically similar Arabic word sifr (“empty”) was adopted as a translation for Sanskrit shunya, which was represented by a small, open circle.

By the early 9th century the Moors had conquered Spain and Sicily, bringing with them this revolutionary mathematical concept; al-Khwarizmi’s book was translated into Latin in 1145 and was, for the next four centuries, the principal mathematical textbook in European universities (the English word algorithm is derived from his name). In Italy, sifr became zefiro, zefro, or zevero, corresponding to the French zéro, which—minus the accent—made its way into English.

Though Sa’adiah Gaon was born some thirty years after al-Khwarismi’s death and does not appear to have directly referred in his writing to the concept of zero, it is difficult to believe he would not have been familiar with al-Khwarizmi’s work. In any case, Sa’adiah Gaon’s defense of creation ex nihilo and al-Khwarizmi’s explication of the mathematical concept of zero moved together from Persia through the Middle East, across North Africa and into Moorish Spain, where both were simultaneously diffused into European culture.

Zero is the symbol for a number that is at once both nothing and something. In his book The Nothing That Is: A Natural History of Zero, Robert Kaplan nicely captures the paradoxical nature of zero: “Names belong to things, but zero belongs to nothing. It counts the totality of what isn’t there.”

Zero as a placeholder—used, for example, in a base ten system to mark the difference between one (1) and ten (10)—was common in the ancient world. But for the ancients a placeholder was not itself a number. Numbers have computational properties: they are used to count things; numbers don’t apply where there’s nothing to count. All of this was turned on its head by Indian mathematicians, who conceived, for the first time, of zero as having computational properties, though admittedly unlike the properties of any other number. First of all, addition and subtraction with zero changes nothing: add zero to any number—including itself—and the sum is that same number; subtract zero from any number and once again the number remains unchanged. But multiplication and division yield even more startling results. Multiply any number by zero and the product is zero; divide by zero and no matter what the dividend the quotient is infinity—which mathematicians still regard not as a number but rather as an exceedingly odd “concept.” As Charles Seife writes in Zero: The Biography of a Dangerous Idea, “Zero is powerful because it is infinity’s twin. They are equal and opposite, yin and yang. They are equally paradoxical and troubling.” In other words, zero is where nothing meets and mingles not just with some particular thing but with everything. As a mathematical concept, zero locates the interface between absence and presence, and in this respect it defies the law of noncontradiction, which states that contradictory propositions cannot both be true in the same sense at the same time. Considered to be one of the “laws of thought” and a cornerstone of reason, the law of noncontradiction finds its classical source in Aristotle’s metaphysics. And so, as Seife has it, “Zero conflicted with the fundamental philosophical beliefs of the West, for contained within zero are two ideas that were poisonous to Western doctrine. Indeed, these concepts would eventually destroy Aristotelian philosophy after its long reign. These dangerous ideas were the void and the infinite.”

No one knows exactly where the idea of “zero” as a placeholder first emerged, but historians agree that regardless of where it originated, India was where zero was transformed from mere placeholder to a legitimate number in its own right. And it was this transformation—a prodigious feat of imagination—that gave zero its mysterious power to absorb and defy all contradictions.

The earliest known reference to mathematical zero appears in the Chandah Shastra, a text on Sanskrit prosody attributed to an otherwise unknown author named Pingala and dated to sometime in the first few centuries BCE. The text unfortunately does not include any example of symbolic notation, but Pingala explicitly uses the Sanskrit word shunya to refer to the result of subtracting a number from itself. The oldest recorded use of symbolic notation for zero as a number is found in a birchbark text known as the Bakhshali manuscript, which has been radiocarbon dated to as early as the 3rd century CE and seems to have been intended for use by merchants as a practical manual on arithmetic. Here zero is indicated by a solid dot.

India was where zero was transformed from mere placeholder to a legitimate number in its own right. And it was this transformation—a prodigious feat of imagination—that gave zero its mysterious power to absorb and defy all contradictions.

By the end of the 5th century the same word shunya appears in another text, the Aryabhatiya, in the context of a fully developed system of decimal place-value notation. In a 7th-century commentary on the Aryabhatiya, the mathematician Bhaskara used a circle to represent shunya, which is the earliest recorded instance of the notation that has now become virtually universal. At that time, however, the circle was perhaps not yet standardized, since a mathematician by the name of Brahmagupta, a contemporary of Bhaskara’s, used the same solid dot that occurs in the Bakhshali manuscript. In his Brahmasphuta Siddhanta, composed around 650 CE, Brahmagupta referred to the dot as shunya or, alternatively, kha—literally a cavity, hollow, or empty space, and by extension, the Sanskrit word for “sky.” Nevertheless, the use of shunya seems at this point to have become more or less fixed. Brahmagupta’s treatise is unprecedented, however, in its meticulous analysis of zero in the context of negative numbers and corresponding algebraic operations. His work leaves no question that by the 7th century, Indian mathematicians had fully conceptualized the role of mathematical zero in the sense familiar to us now.

The Sanskrit word shunya is routinely used in Mahayana Buddhist texts dating back to the first few centuries BCE; which is to say, its appearance both as a revolutionary mathematical term and as the expression of a profound, intuitive understanding of the nature of reality—the mark of “transcendent, liberating wisdom” (prajna-paramita)—seems to have occurred simultaneously in India. The paradoxical characteristic of mathematical zero—as a nothing that is not only something but everything—features in the Buddhist notion of shunya, but its implications are no longer merely abstract or computational. In the scriptures on perfect wisdom, shunya is presented as a fundamental truth of all existence, a truth fully appreciated by spiritual beings known as bodhisattvas, who have achieved this profound insight only as the result of long study and contemplative practice. The famous Heart Sutra opens by telling us that the bodhisattva Avalokiteshvara, “moving in the stream of perfect wisdom,” looked down over the world and saw that “zero-ness” (shunya-ta) is the essential nature of every element of experience—everything that makes up our mental and physical reality.

Here… form is zero-ness and zero-ness itself is form; zero-ness does not differ from form, and form does not differ from zero-ness. Whatever is form, that is zero-ness; whatever is zero-ness, that is form. The same is true of feelings, perceptions, impulses, and consciousness.

Nor is the ancient notion of zero as a placeholder marginalized in this literature. In the scriptures on perfect wisdom, this characteristic of zero is an integral component of its status as both absence and presence simultaneously. To say that every element of experience—every dharma—is zero is to say that, like zero, the appearance of individual, self-sufficient things is nothing more than appearance; there is no actual “thing,” no individual physical or mental object that truly exists as it appears. The mental or physical object that seems to exist separately from other such things in fact exists only as a placeholder. Which is to say, the individual exists only in relation to what it is not, and what it is not is literally everything else—an infinitude of other apparently individual things. This is the sense in which dharmas are said to be “devoid of essential nature,” which is the same as saying that their essential nature is zero-ness.

And so in the Perfection of Wisdom in Eight Thousand Lines—perhaps the oldest surviving text of this genre—we are asked, rhetorically: “To what dharma could I point and say that ‘it exists’ or ‘it doesn’t exist?’”

It is precisely through their essential nature that dharmas are not a thing. Their essential nature is no-nature, and their no-nature is their essential nature. All dharmas have only one characteristic, which is no characteristic at all.

“‘All things are no-things,’ taught the Tathagata [the Buddha], ‘therefore they are things.’” Perfect wisdom, then, is a deep understanding that breaks free of our normal habits of thinking and speaking, habits that compel us to both conceive and perceive individual things literally as either existing or not existing, as either this or that. Rather, as seen through the eye of perfect wisdom, things are not things, and not things are things, which means that they only seem to arise and pass away. This is true, according to the Diamond Sutra, for living beings as well, who merely appear to be self-contained individuals subject to birth and death: “‘Beings, beings’… the Tathagata has taught that they are all no-beings. In this way has he spoken of ‘all living beings.’” Nothing whatsoever is exempted: “This entire universe the Tathagata has taught as no-universe. In this sense it is called a ‘universe.’”

Therefore, Shariputra, in zero-ness there is no form, nor feeling, nor perception, nor impulse, nor consciousness; no eye, ear, nose, tongue, body, mind; no forms, sounds, smells, tastes, tactile objects, or objects of mind.… There is no ignorance, no extinction of ignorance … no decay and death and no end to decay and death. There is no suffering, no origin, no cessation, no spiritual path. There is nothing to realize, nothing to attain.

“It is on account of this,” explains the Perfection of Wisdom in Eight Thousand Lines, “that the Tathagata does not fully know the character of any dharma.” What is literal or concrete can be fully known or grasped, its character can—at least in principle—be understood empirically, rationally analyzed and explained; what is metaphorical is “as if,” and “as if” can only be intuited.

Consider, in this context, what Robert Kaplan has to say about zero as the interface of nothing and everything:

It is as if there were a layer behind appearances that had no qualities, but took on the character of its surroundings, accommodating itself to our interpretations, as ambergris acquires and retains fugitive fragrances, giving us perfume. Shunya isn’t so much vacancy, then, as receptivity, a womb-like hollow ready to swell—and indeed it comes from the root shvi, meaning swelling. Its companion kha derives from the verb “to dig,” and so carries the sense of “hole”: something to be filled. . . . This is the zero of the counting board: a column already there, but with no counters yet in it. This is the zero of the place-holder notation, having no value itself but giving value by its presence to other numerals. These same qualities belong to the variable, the unknown: a potential which the different circumstances of the equations it lies in will differently realize. The background shift is from counters taking their value from being in different places, to a single, receptive place whose circumstances will reveal its hidden value.

The concept of shunya evokes the ambiguous, ungraspable nature of what only appears to be literal, concrete truth or reality. The Indian mathematician Bhaskara acknowledges as much when, in a discussion of mathematical zero, he writes: “The arithmetic of known quantity . . . is founded on that of unknown quantity; and . . . questions to be solved can hardly be understood by any, and not at all by such as have dull apprehensions, without the application of unknown quantity.” Perhaps the most eloquent classical passage on this aspect of the zero-ness of things comes from the Diamond Sutra:

A phantom’s mask, a shooting star, a guttering flame.
A sorcerer’s trick, a bubble swept
On a swiftly moving stream.
A flash of lightning among dark clouds.
A drop of dew,
A dream.
So should one view all conditioned things.

— © Tricycle magazine, Winter 2022, by C. W. Huntington Jr., professor of religious studies.