5 December 2011
Earlier, I stopped with the creation of a true zero by Brahmagupta. Now, I have to close the circle and take us back into Europe (last seen dismissing the possibility of something being nothing), a journey that goes through Baghdad, and to an author most known for love poetry, not the poetry of numbers!
It was at this point that Indian mathematics really took off, and reached it's full potential. Between them, Bhaskara I and Brahmagupta started solving every possible classical problem they could lay their hands on, from the Diophantine Equations, to an approximate algebraic formula for the sine of a number, from the volumes of objects to systems of linear equations with multiple unknowns. Where Aryabhatiyam is pretty terse and small in size, Brahmasputasiddhanta is a respectable tome, which expounds on various subjects. This, in about a century and a half.
Most importantly, these two laid the foundation for the next, probably the greatest of India's mathematicians, Bhaskara II. Bhaskara II lived about 500 years after Brahmagupta and Bhaskara I, and worked at Ujjain, so was probably another Maharashtra astronomer like the first of that name.
What Bhaskara II did was to take Brahmagupta's rules, and for the first time, figure out all the modern rules of the zero. It was he who came up with 1/0 = ∞, the axiom that Srinivasa Ramanujam would later anecdotally state to his teacher, and which gave the title of his biography by Robert Kanigel. Bhaskara's reasoning for this statement is quite poetic:
”A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.”
And then, he described other properties, such as 0² = 0, √0 = 0 and so on. With this knowledge in hand, he went on to solve everything that he could get his hands on, and gave general solutions to most of them, something that, as we have seen, all his predecessors failed to do. He could solve equations that had multiple solutions, and equations with multiple unknowns. He even pioneered differential and integral calculus, centuries before Newton or Leibniz, becoming literally the first known person to talk about it. To cut it short, he deserves a whole lot more than a few paragraphs, and would probably drown out everyone else if I continue expanding on him. So, let's leave him alone for now…
During all this time, the five to six hundred years from Aryabhata to Bhaskara, Arab traders had visited the subcontinent quite often. With Ujjain and Maharashtra being on the west coast, I'm quite sure that they would have had the chance to interact with these greats who lived in that area, the mathematically inclined among them exchanging ideas with the local schools. Maybe a few of them even stayed behind to study under the two Bhaskaras or in their schools…
What I can say for sure is that many of their texts were translated by Islamic scholars of the medieval era, into Persian and Arabic, and were to be found in the House of Wisdom in Baghdad during the Abbasid Caliphate.
There was the early 9th century scholar, Yaʻqūb ibn Isḥāq al-Kindi, the first of the Arab scholars of Baghdad, who was a polymath to beat all polymaths - he worked on everything from the existance of the soul to music theory, and from medicine to cryptography. But his important contribution in our context is that he was the author of a book in four volumes, ”On the Use of the Indian Numerals”, which is one of the important links that transmitted Hindu numerals, and the zero, to Arabia, from where it reached the West.
Al-Khwarizmi's book, ”On the Calculation with Hindu Numerals”, written in 825 CE was the first taste that the West had of true Indian numerals, in a proper scientific setting. From his name, we get the word Algorithm, and from the name of his book, ”Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala” (which can be translated as ”The Compendious Book on Calculation by Completion and Balancing”), or more correctly, from the short-form of its name, ”Kitab al gabr wa'l-muqabala”, we get the word ”al-Gabr”, meaing ”the completion” or ”the restoration”, which is the root of Algebra.
Since he wrote before Bhaskara, and probably didn't pay much attention to Brahmagupta, he had no idea of negative numbers, so his solutions for equations completely ignore the possibility of a negative coeffecient. Again, not embracing the zero in its full, held him back, though he's able to produce more general solutions than the Greeks whom he had also studied.
”Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved by propositions five and six of Book two of Elements. –Omar Khayyam”
We know of Omar Khayyam as the author of some extremely erotic poetry (!), but he was also an accomplished mathematician; Building on the work of his predecessors, Khayyam created the new subject of Geometric Algebra, which would later form the basis of Cartesian geometry, and of Newton and Leibniz's calculus, and indeed, quite a large section of modern mathematics. Again, the whole-hearted acceptance of the zero, and of negative numbers, is evident in his work, and it was quite settled in Arab science by his time (about the 11th to 12th century or so).
In 1258, the Mongols invaded Baghdad, and destroyed all the libraries and academies established by the Abbasids, including the House of Wisdom. In many massacres in history, it is said that rivers ran red with blood. Of the invasion of Baghdad by the Mongols, it is said that the Tigris ran black with the ink from the books thrown into it. That was the end of the House of Wisdom, and for a long time, the end of the refined and sophisticated Islamic scholarship that had grown amidst trade links and mutual appreciation between Greece, India and Baghdad.
Fortunately, as this destruction of Arab scholarship, and the simultaneous destruction in India were taking place, knowledge transmission to the West was not stagnant. Books of the East eventually found their way to the west, where they formed the basis for European science of a later era.
Among the Europeans, the first to deal with the place-value numerals and the zero, was Leonardo of Pisa, whom we know better as Fibonacci. He lived anterior to the sack of Baghdad, from about 1170 to 1250. As a young boy, he travelled in the Islamic lands with his father, who was a merchant. There, he learned about Indian numerals, which he introduced to Europe. He called these the ”Modus Indorum”, or ”Method of the Indians”. He writes ,
”After my father's appointment by his homeland as state official in the customs house of Bugia for the Pisan merchants who thronged to it, he took charge; and in view of its future usefulness and convenience, had me in my boyhood come to him and there wanted me to devote myself to and be instructed in the study of calculation for some days. There, following my introduction, as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods; and at these places thereafter, while on business. I pursued my study in depth and learned the give-and-take of disputation. But all this even, and the algorism, as well as the art of Pythagoras, I considered as almost a mistake in respect to the method of the Hindus. (Modus Indorum). The nine Indian figures are: 9 8 7 6 5 4 3 2 1 With these nine figures, and with the sign 0 … any number may be written. ”
The last bit is interesting - he doesn't call it the ”ten figures”, it's ”nine figures and the sign 0”. Again, he hadn't fully realised what zero meant. From Fibonacci introducing the zero (again, finally) as a place-value into European science, to Descartes making the centre of his Cartesian grid a zero in the 17th century, was a long struggle against the forces of status quo.
In 1299, for example, the bankers of Florence were forbidden from using the Hindu-Arabic numerals, and the statutes of the University of Padua required that price lists of books be kept ”non per cifras, sed per literas claros” - ”Not by zeros, but by literal numbers”. Calendars and almanacs of the middle ages are often written entirely in Roman numerals. Europeans seem to have found it very difficult to accept the simple zero, until after Cardano. As I mentioned earlier, it was only after the 16th century that European mathematics really took off, and that coincides with their final acceptance of zero.
And that's the value of nothing. When you accept that something can be nothing, it's easier to come up with general solutions to various problems that would otherwise break down at zero.