The story of the zero

Metaphysics and mathematics in India and beyond

Learning a language is the process of becoming. Babble becomes meaningful as you gain syntactical felicity. Undifferentiated noise becomes words that conjure objects and actions. Where in the beginning there is chaos, study and practice bring forth form.

I am learning a language. I am also reading Mythos by Stephen Fry, an entertaining deep dive into the world of Greek Myths. It begins with explaining the Greek conception of origins.

For the Greeks everything came from chaos. “Think of Chaos as a kind of grand cosmic yawn,” says Fry. “As in a yawning chasm.”

Being and becoming in Hindu philosophy

Which led me to think about being, becoming, and origins in Hindu philosophy, a subject I studied at university. At the core of some schools of Hindu and Buddhist thought there is the doctrine of shunyata, or nothingness, or voidness. It postulates that voidness (shunyata) constitutes Ultimate Reality.

It is the undifferentiated “nothing” out of which all apparent entities, distinctions and forms arise. But this nothing is not nothing in the sense of not existing. Yet, it cannot be described using any positive descriptors. Which is why some ancient Hindu texts deploy, “neti neti,” a Sanskrit expression that means, “not this, not that,” as the formula to talk about this nothingness that is ultimately everythingness.

“Neti neti” negates all descriptions (eg: it is not this body, it is not that idol) about the Ultimate Reality, which is a “nothingness” that cannot be contained by words and therefore not spoken about, except negatively.

Having studied all of this in often befuddling detail I was aware that negation, nothingness and irrationality (in the sense of a belief in the limits of the ability of rational thought to apprehend ultimate truths) were an entrenched part of Hindu/Buddhist metaphysics.

The story of shunya or the zero

All of which made me think about how logical it was that the mathematical zero was an Indian invention. It isn’t such a stretch, after all, to transform a metaphysical concept, shunyata, into a mathematical one, shunya (zero).

The story of the zero takes us to back to India in the 7th century AD, when a mathematician called Brahmagupta became the first to clearly define the arithmetic zero as something that:

 “when added to a number or subtracted from a number, leaves the number unchanged. Moreover, a number multiplied by zero becomes zero. Lastly, it is what remains when a number is subtracted from itself.”

Prior to this, what to us feels like a rather banal epiphany, math was limited in its utility. In the absence of the zero, there was no place-value number system. What does this mean?

Why the zero is a hero

Think of the number 106. The zero in this case stands for, “there’s nothing in the tens column.” It’s a placeholder, helping us understand that this number is one-hundred and six and not 16.

Imagine using Roman numerals, which was what the West was stuck with for centuries, for calculations. With no place-value system, simple addition was not simple at all, particularly when it came to dealing with large numbers. If you wanted to talk about, say, 900 billion, you had to invent a symbol for it.

Greeks and Romans did perform sums, but their method was terribly inefficient involving the   drawing of geometrical figures in the sand and adding or subtracting areas of figures.  (Just for fun try adding CIII to XCIX. You’ll find it isn’t much fun at all).

Why did it take so long for someone to “invent” the zero? Probably because it is what exists without being seen. In the natural world you “see” things starting from one: one car, two birds, three stones and so on. When you see a zero number of these, you are in fact not seeing anything. Yet, although you can’t touch or feel or see zero birds/stones/cars, there is a real something/nothing that is the absence of those birds/stones/cars.

Irrationality and murder

Unlike the Indians, and despite their concept of the undifferentiated chaos from whence the universe arose, the Greeks seemed to have had a philosophical problem with the idea of nothingness. A problem they extended to (mathematical) irrationality as well.

For example, Pythagoras (of right-angled triangles fame) asserted that any number, no matter how large, could always be expressed as a perfect ratio of two natural numbers. In other words, he postulated that all numbers were rational.

When his student, Hippasus, discovered that the square root of 2 could never be exactly expressed as a ratio, it led to a dilemma. Pythagoras’ theorem had already proved that the square root of 2 had a real physical meaning: it was the length of the hypotenuse of an isosceles triangle the other two sides of which were of length 1.

Ergo, if the Pythagorean theorem was valid, all numbers could not be rational. But instead of changing his mind and math, Pythagoras took the easy way out and murdered his student.  

Luckily for Indian mathematicians, irrationality was not existentially threatening. Two hundred years before Pythagoras was assassinating his challengers, in 8th century BC India, a mathematician called Baudhayana had already proved the “Pythagoras’” theorem in his work, the Sulba Sutra. He showed that the square formed by the “diagonal” of a triangle has the combined area of the squares formed by the length and breadth of the triangle — the geometric analogue of the Pythagoras theorem.

Baudhayana also did not have the least trouble accepting that numbers could be irrational. His work provided approximations for the square root of 2, and “pi”, even though neither numbers could be neatly expressed as a ratio of two natural numbers.

Let’s skip forward again, to Brahmagupta.  The “inventor” of the zero also posited negative numbers as a concept in his 628 AD work, the Brahmasphutasiddhanta (The Opening of the Universe).

Enter the Arabs

The world of mathematics would never be the same again. About a century later, al-Mansur, the then Caliph of Baghdad heard about Brahmagupta’s work when a visiting Indian scholar, Kanka, brought along a copy of the Brahmasphutasiddhanta with him. Al- Mansur commissioned an Arabic translation of the book.

Gradually, the Arabs became adept at using the zero, a number they called sifr. However, Europe laboured on zero-less for another 400 years, until the Moors conquered Spain and introduced nothingness.

Suspicious Europe

Accountants and merchants were thrilled. They could finally balance their books by ensuring their assets and liabilities cancelled each other out to equal zero.

The church and some governments, however, were less keen. It was postulated that since God is in everything, a symbol that represented nothing must be satanic

In 1299, Florence banned the use of the zero altogether.  One reason given was that it would encourage fraud, by making it easy to inflate figures simply by adding a zero at the end.

What’s more, the zero was the gateway to negative numbers, and these legitimised the concept of debt and money lending, an anathema to Christianity.

Merchants, however, were not ready to give up on the zero so easily. They continued to use it in secret. Hence, zero or sifr (in Arabic) became associated with secret codes — the origin of the modern term “cipher.”

And finally..

I hope you enjoyed this post. I certainly slotted in a few more pieces of the global jigsaw that we all inhabit while researching it.

Could I ask you to consider becoming a paid subscriber of this newsletter, so that we can continue this journey of making sense out of what looks like disconnected chaos at the outset – just as the ancient philosophers of all cultures attempted to do? Like learning a language, it’s a process of becoming. I hope you will sign up to this project. I think it is an important one.

Until next week,

Pallavi