- Archis Anupam Mohapatra
- Apr 15
- 10 min read
Story of Zero Contd.
As zero took root in the Indian subcontinent, its journey was far from over—it now began to travel beyond borders. After being formally recognized and refined by Brahmagupta—and later emphasized by Bhaskaracharya II—we now look toward the Middle East, where a different kind of revolution was brewing.
In the early 7th century, Prophet Muhammad founded a new religion—Islam. He called upon his followers not only to embrace faith but also to spread it, through both preaching and, at times, through conquest as he worked to establish a unified community under a new spiritual and political order. The Arab world happens to be in the middle of many trade routes from the subcontinent to the European kingdoms. Thereby it worked as a great aggregator of wealth and knowledge. At the heart of this intellectual current stood the House of Wisdom (بَيْت الْحِكْمَة Bayt al-Ḥikmah) in Baghdad.[1]
Baghdad became a haven for scholars and scientists from across the Islamic world. Here, began the groundwork of modern scientific method. Observation, experimentation, repetition and mathematical formulation of science began to take shape.
A Gift that Changed History!
Ambassadors and merchants from Sindh province of the subcontinent—would arrive in Baghdad from time to time to not only trade wares but to gain and spread knowledge from their land as well. The year was 773AD, the ambassadors arrived with many gift including a mathematical manuscript, Brāhmasphuṭasiddhānta. Poetic yet cryptic in language the book proved to be difficult challenge for the Arab mathematician to solve. It will lay more or less gathering dust for the next 50 years.[2]
This wasn’t ordinary Sanskrit—it was scientific, dense and yet symbolic. Full of astronomical observation, new system of mensuration and algebraic identities which hadn’t been seen anywhere till now, this magnum opus by Brahmagupta proved to be a difficult challenge to translate and work with. 50 years from its originally date of delivery came a scholar well versed in the language and curious enough to see through the complexity. The scholars was a brilliant mathematician named Muhammad ibn Musa al-Khwarizmi. Or as we call him today, al-Khwarizmi. From his study of the section titled Kuttakādhāya, emerged his foundational text, a discipline he referred to as al-jabr—or as we call it today Algebra.
“Brahmagupta uses the term ganita only for those calculations which are of arithmetical in nature. The science of algebra, the foundations of which was laid by Āryabhata I was named as kuttaka or kuttākāra by Āryabhata, and in the Brāhmapushidhānta also it is separately dealt with under Kuttādhyāya or kuttakādhāya (Chapter XVIII). Later on the term bījaganita was specifically given to the science of algebra.” Chapter VIII, Brahmagupta and Arithmatic, BSS.
Al-Khwarizmi didn’t merely translate Brahmagupta; he refined, reorganized, and expanded it. He synthesized the poetic stanzas to systematic rules—what we now call algorithms—to solve algebraic problems. He renamed the śūnya to the Arabic word ṣifr. Though the roots were Indian, the number system now became known globally as the “Arabic numerals or Indo-Arabic Numeral”.[3]
I will (in time) try to attempt to explain some the various novel astronomical, trignometric and algebric notions that classical Indian text such as Aryabhatiya, Suryasidhānta, Brahmpushidhanta, Mahābhāskariya and text such as Lilavati had uncovered and how the credit for these work is still given to European or Arab mathematician even though many of these treatise were first composed centuries earlier.
Al-Khwarizmi used the word ṣifr (صفر) to describe zero—an Arabic translation of the Sanskrit śūnya, meaning “emptiness” or “void.” This concept clashed directly with Aristotelian doctrine, which refused to acknowledge a true void. For Aristotle, the idea of “nothingness” was incompatible with his philosophy of nature, and his ideas remained highly influential in Abrahamic religions.
Yet by the 10th century, a new school of Islamic theology had began to question Aristotle. This school, influenced by Islamic atomism, was founded by the Asharites. They believed that all matter was composed of indivisible atoms, whose interactions were guided not by deterministic laws, but by the will of God. For the Asharites, God wasn’t just omnipotent—He was intimately involved in every moment of creation.
The most celebrated Asharite scholar was Abu Hamid Al-Ghazali. Regarded by many as the mujaddid (reviver) of his century, Al-Ghazali directly challenged Greek rationalism. He denounced many teachings of Aristotle and those who followed him, even declaring them heretical. His influence was so profound that Islamic philosophy shifted—away from Greek metaphysics and toward a more mystical theology that could embrace concepts like the void.[2]
And so, zero found its place—not through logic alone, but through theology. It wasn’t just tolerated; it was, in a way, sanctified by the powers that be.
Meanwhile, beginning in the 8th century in the Iberian Peninsula (modern day Spain and Portugal), the Umayyad Caliphate opened a channel through war and persecution for this knowledge to flow into Europe. Though the Christian and Islamic worlds were often at war—from the fall of Al-Andalus to the Crusades( 1st,2nd, 3rd and 4th)—knowledge found its way across enemy lines.[4]
For much of this time, Europe still relied on Roman numerals—complex, clunky, and difficult for serious calculation. Zero remained a heretical notion, challenging Christian ideas of the soul, of God’s existence, and of the nature of being.
Marching into Europe
In the 12th century, a custom official Gugelielmo Bonaccio and his son Leonardo reached the Mediterranean town of Bugia in Algeria. Leonardo’s father wanted him to learn the abacus, but Leonardo fell in love with something far different: Arabic mathematics and the Indian numeral system. We know Leonardo by a different name today, Fibonacci.
Young Fibonacci would travel extensive from Syria to Egypt, from Sicily to Greece—gathering knowledge from every region he visited. Around the year 1202, he returned home to Pisa to settle down and it was there that he wrote his magnum opus: Liber Abaci.[5]
In this book, Fibonacci introduced all that he had learned, including Indian numerals and the mysterious zero, algebra and the system of algorithm to western world. If you look at the picture below you will see how he introduce he number 0.
The number 0 is still put not in its rightful place alongside the rest of the numbers, but separately as if its thought too eccentric.
The Accountants
The Indian numeral system had breached the Christian intellectual defense lines. But resistance remained strong. In Florence, authorities banned the use of these numerals in 1299 in the name of preventing fraud—after all, a zero could be easily altered into a 9 or 6. Yet merchants quietly continued to use them in private, in secret ledgers. Brahmagupta and the many mathematician’s before him not only unleashed the power of zero, but also introduced to algebraic formulas to keep account of loans and interest payments! This allowed the lenders and burrowers to keep track of their accounts with far more ease than before.
If you recall the Greeks too had once banned Sumerian numerals, yet continued using them in private. History, it seems, was repeating itself.
By the end of 13th Century, in the Christian world, the Indian numerals became the hidden script of a new generation of thinkers. As trade expanded and commerce flourished, it became clear that zero and the Indian system were simply too powerful to ignore. But it wasn’t the theologians, mathematicians or the ‘algorist’ who finally sealed zero’s place in the West. It was the accountants.
Double-entry bookkeeping—an elegant system of credits and debits—was introduced to manage the complexities of trade. The earliest surviving record is from Genoa, 1340. In one column you wrote what you owed, and in the other, what was owed to you. The goal was simple: everything must balance to zero.
In 1494, Luca Pacioli—the father of accounting—published Summa de Arithmetica, a practical guide that included everything from numbers to balance sheets. There was no longer room for philosophical doubt. Zero had triumphed—not with fire or force, but through the quiet logic of trade and the balance of books.[7]
And so, the journey of zero—from the poetic stanzas of Brahmagupta to the ledgers of Genoese merchants—was complete.
At the Center of Everything
A need to formalize the nature of these numbers as mathematician struggled with the meaning of it all. Now, many mathematician from the likes of Reimann to Dedekind had worked on this form of mathematics but it wasn’t until Gottlob Frege the ordinary numbers (think natural numbers,0,1,2,3,4….) found a home.
But before all this, what is a Set?
A framework under which all mathematical objects could be constructed from one primitive idea: the set—a collection of distinct entities grouped together as a single object.
Now I want you think of the number 3 and everything that counts to the number 3. What all did you think of? Lets go with 3 chairs, 3 dogs, 3 tridents or 3 Jonas brothers. Now, we can
connect the sets of 3 dogs with the 3 Jonas brothers, one to one. This is called one to one mapping. You can create any number one to one maps as long as both the sets contain same number of objects. We can now use any of the these sets, to define the number 3. But which comes first, the set of brothers, or the chairs or the dogs? Frege showed that all stood equivalent in the eyes of mathematics (sorry Jonas brothers fans). He put all the sets of 3 quantities, and declared the number 3 to be it all together! A set of all 3 numbered sets! Everything is settled, or it felt like that for a while.
Enter Bertrand Russell. The famous philosopher and polymath, Bertrand Russell saw this paper by Frege and felt compelled to write a letter to show his displeasure!
He proposed a puzzle.
Imagine a small town with a peculiar rule: there’s only one barber, and he has a very specific job. He must shave beards of all the men in town who do not shave themselves—and only those men.
At first, it seems straightforward. If Mr. Sharma doesn’t shave his beard himself, the barber shaves it for him. If Mr. Das shaves his own beard, the barber leaves him alone. All good.
But then someone asks a strange question:“What about the barber? Does he shave his own beard himself?”
Let’s think it through:
If the barber shaves himself, then according to the rule, he can’t—because he only shaves those who don’t shave themselves.
But if the barber doesn’t shave himself, then he must—because the rule says he shaves all those who don’t shave themselves.
So either way, we end up with a contradiction. If he does, he shouldn't. If he doesn't, he should.
The problem isn’t just tricky—it’s impossible. Such a barber can’t exist. The whole setup is self-contradictory, like saying “this sentence is false.” It sounds okay at first, but it collapses when you try to apply logic to it.
This is known as Russell’s Paradox, and it's not just a clever riddle—it revealed a deep problem in the way people were thinking about sets and logic in mathematics.
Russell’s Paradox was more than just a quirky puzzle—it shook the very foundations of mathematics in the early 20th century. Before this, mathematicians believed that any collection of objects could be treated as a set. But Russell showed that if you allow sets to contain themselves—or not—you could end up with impossible contradictions. Like the barber, such sets just couldn't exist.[7]
To fix this, mathematicians needed a new, more careful foundation for set theory—one that wouldn't allow such self-referential loops.
One of the major responses came from the mathematician Ernst Zermelo, and later Abraham Fraenkel. Together, their ideas became what's now called Zermelo-Fraenkel set theory (ZF), often paired with the Axiom of Choice to form ZFC (not to be confused with KFC).
Instead of allowing any imaginable collection to be a set, ZFC imposed strict rules—called axioms—to say what kinds of sets are allowed. One of the most important was the Axiom of Foundation ( also known as the axiom of regularity), which basically said:
“No set can contain itself, or be built in a circular way.”
Another breakthrough came from the von Neumann construction of the natural numbers using sets. It starts with the empty set (∅)—the set with nothing in it—and defines the following:
Imagine an empty box, this represents 0. We then fill this empty box with another empty box. This new box is now called one, cause of the number boxes in it. We now have two boxes, lets put this and one more empty box in a new box, and the number of objects inside is 2. So, this new box becomes 2. And so on and so forth, we can now create the entire system of natural number.
This clever trick showed how the entire number system could be built from nothing, layer by layer, with no circular logic.
Even more importantly, this approach avoided Russell’s paradox because it carefully constructed sets only from previously existing sets, always keeping the foundation solid and avoiding the idea of sets containing themselves.
In set theory, zero’s emptiness becomes generative: the empty set contains nothing, yet it gives rise to every element of ℕ through the successor operation. This formalism unifies zero’s dual nature—both void and origin—within a rigorous mathematical framework.
From śūnya and ṣifr to 0
In our modern notation, zero has transcended cultures and epochs. It is the paradoxical hero of mathematics: the very symbol of nothingness that underpins everything, from the axioms of arithmetic to the algorithms of our digital age. Zero is symmetry itself. Symmetry a concept so deeply entwined with the fabric of the universe, something that posed as a challenge to Albert Einstein and later solved by Emmy Noether. All this and more will be a topic of exploration for another day.
So the next time you see a zero—on a page, a screen, or a ledger—remember: you’re looking at the quiet architect of modern thought. What began as a void became the very foundation upon which we built infinity.
🕺🕺🕺