Kaplan, Robert; Ellen Kaplan (ill.);
The Nothing that Is: A Natural History of Zero
Oxford University Press, 1999, 240 pages
ISBN 0195128427, 9780195128420
topics: | math | history | zero |
an amazingly euro-centric view of the zero. very grudgingly admits that the zero did appear in india, but not until the 9th c.: You can try pushing back the beginnings of zero in India before 876, if you are willing to strain your eyes to make out dim figures in a bright haze. Why trouble to do this? AryabhaTa may be two characters "with opposite reputations", and there may have been a shadowy third. Brahmagupta is Aryabhata's severest critic (and, by turns, fervent admirer: had you expected less in this play of shadows?) A remarkably negative book. Despite the severely disparaging view of all things Indian, India itself does not appear till chapter 4, and only in relation to alexander's invasion. The rest of the book is focused on India, but in a strictly negative light. For a book written at the cusp of the 21st c., seems to belong to an 18th c. enlightenment mindset.
Supposing we make up signs for the first twenty numbers: 1 2 3 4 5 6 7 8 9 ~| † ‡ ↔ ↑ ↓ • ¶ # ₯ then we can say 6 + 7 is ‡, but ↑ plus • remains hard. difficult - we have to invent twelve new signs. need a more flexible structure... we can mark heaps - e.g. by crossing out a group of four sticks : -||||- is 5. then -||||- -||||- -||||- || can be thought of as ||| of the -||||- plus ||. but then, when we need to do -||||- -||||- -||||- || plus -||||- -||||- we need -||||- of the -||||- plus ||. it gets cumbersome. [and how do we talk about these heaps of heaps?] a definite improvement is: X + XVIII = XXVIII. The Romans used L for 50, so LX was 10 past 50, or 60; and XL was 10 before 50, so 40. C was 100, D 500, M 1,000 and eventually - as debts and dowries mounted - || - a three-quarter frame around an old symbol increased its value by a factor of 100,000. So Livia left 50,000,000 sesterces to Galba, but her son, the Emperor Tiberius — no friend of anyone, certainly not of Galba (and anyway his mother's residual heir) - insisted that |D| be read as D-bar - 500,000 sesterces, quia notata non perscripta erat summa, 'because the sum was in notation, not written in full'. The kind of talk we expect to hear from emperors.
5,000 years ago, a Sumerian father asks his son: F: 'Where did you go?' S: 'Nowhere.' F: 'Then why are you late?' and you realize that 5,000 years are like an evening gone. p.6 Sumerians counted by 1s, 10s, and 60s. The symbol for 60 is just a larger version of the symbol for 1, but different from the symbol for 10. After several centuries, they also started using place notation - going by 60. so 754 would bt 12x60 + 34. But its still difficult to distinguish between 180 and 3 - since there was nothing to tell that the last digit was missing in 180. several centuries later - between 6th and 3d c. BC, a sign that was used to separate words from their definitions, or in bilingual texts, one language to another, was used to hold the empty columns. Let's use the symbol thus 2 5 --> 2x60 + 5 = 125 but 2 : 5 --> 2x60x60 + 5 = 7205 but the endplace zero was never used. also the notation was not really standardized.
Among the Greeks, number syntax was variable. Large numbers were discussed by Archimedes, who had the vision of a"the numbers named by me . . . some exceed the number of the grains of sand ... in a mass equal in magnitude to the universe". This vision as worked out that the earth may hold 1063 grains of sand. p.30
What is surprising for a book on "zero", India is not mentioned till chapter
four, in the context of alexxander's expedition reaching the indus in 326BC.
Kaplan then asks whether Archimedes may have influenced Bhaskara w.r.t. a
celebrated list of large numbers appearing in lalitavistAra:
Can we say that Archimedes' Sand-Reckoner and its zeroless ranks of
numbers influenced a charming story in the lalitavistAra, written at
least three hundred years later?
Guatama (Buddha) has to name all the numerical ranks beyond a koTi (ten
million, i.e., 107), each rank being a hundred times greater than the
last. Gautama answers: ayuta (109); niyuta (1011); kaMkara (1013);
vivara (1015); achobya (1017); utsanga, bahula, nagabAla, titlambha,
vyavaithanaprajnapti (that's 1031), and so through the alluring
samaptalambha (1037) and the tongue-twisting visandjnagati (1047) to
tallakchana (10^{7+46} = 1053) at last.
[But this] is not all. A second level takes Gautama up to 10^{7+2x46} =
1099, and eventually the ninth brings him to 10^{7+9x46} = 10421. p.38
It is clear to anyone who looks that such large terms have been a part of
the Sanskrit lexicon for about two millenia before Bhaskara. The Yajur
veda (c. 1300BC) refers to the parArdha (1 billion, 1012).
As Michel Danino likes to point out, in contrast, the largest named number
word in ancient Greece is a myriad, 10⁴. No word for a billion was to
appear in greek until the 17th c.
Even otherwise, given the general speculation of Archimedes'
grains of sand, versus the linguistic precision entailed in naming each
level, such a remark seems to belong to a rather prejudiced mindset.
That India's mathematics originated in the West is emphasized further: You will find a grudging acknowledgement of the Greek source of Indian astronomy and its accompanying mathematics in the Surya Siddhanta, apparently delivered by the Sun to a gentleman named Maya Asura in 2,163,102 BC. The Sun instructs him to 'go to Romaka-city, your own residence. There, reincarnated as a barbarian (thanks to a curse of Brahma), I will impart the science of astronomy to you.' Romaka: that is, Roman, meaning the Greeks of the Roman or Byzantine Empire; and barbarian: the Greeks again, who 'indeed are foreigners', as the astronomer Varahamihira wrote around 550 AD, 'but with them astronomy is in a flourishing state'. p.41 Zero appears indisputably in India in the 9th c., where a plaque near Gwalior of this gift inscribed on a stone tablet, dated Samvat 933 (876 AD), which shows that the garden measured 187 by 270 hastas. The writing of zero as a circle in the guise of the hollow circle is a reincarnation we know from Greek astronomical papyri. p.41 Documents on copper plates, with the same small o in them, dated as far back as the sixth century AD , abound - but so do forgeries... You won't lack for people, however, who find these copper plates authentic, and wrangle with those who, they say, are just out to have the Greeks triumph over all comers. We can try pushing back the beginnings of zero in India before 876, if you are willing to strain your eyes to make out dim figures in a bright haze. Why trouble to do this? Because every story, like every dream, has a deep point, where all that is said sounds oracular, all that is seen, an omen. Interpretations seethe around these images like froth in a cauldron. This deep point for us is the cleft between the ancient world around the Mediterranean and the ancient world of India.
Aryabhata... the name with two t's would mean 'learned man'; having only one turns him improbably into a mercenary. Whatever the case, AryabhaTa hit on a strange scheme. He made up nonsense words whose syllables stood for digits in places, the digits being given by consonants, the places by the nine vowels in Sanskrit. Since the first three vowels are a, i and u, if you wanted to write 386 in his system (he wrote this as 6, then 8, then 3) you would want the sixth consonant, c, followed by a (showing that c was in the units place), the eighth consonant, j, followed by i, then the third consonant, g, followed by u: CAJIGU. The problem is that this system gives only 9 possible places, and being an astronomer, he had need of many more. His baroque solution was to double his system to 18 places by using the same nine vowels twice each: a, a, i, i, u, u and so on; and breaking the consonants up into two groups, using those from the first for the odd numbered places, those from the second for the even. So he would actually have written 386 this way: CASAGI (c being the sixth consonant of the first group, s in effect the eighth of the second group, g the third of the first group). When next you are tempted to think that there aren't different minds but only Mind, remember Aryabhata. There is clearly no zero in this system — but interestingly enough, in explaining it Aryabh says: 'The nine vowels are to be used in two nines of places' - and his word for 'place' is 'kha'. This kha later becomes one of the commonest Indian words for zero. It is as if we had here a slow-motion picture of an idea evolving: the shift from a 'named' to a purely positional notation, from an empty place where a digit can lodge to 'the empty number': a number in its own right, that nudged other numbers along into their places.
In that bible of our grandparents' generation, The Decline of the West, Oswald Spengler wrote that zero was 'that refined creation of a wonderful abstractive power which, for the Indian soul that conceived it as a base for a positional numeration, was nothing more nor less than the key to the meaning of existence'. The Greek soul, he informs us, is sensual and so could never have come up with this key: it takes a Brahmanic soul to perceive numbers as self-evident. Instead of imagining the Hindus as deriving the hollow circle for zero from the meaning of existence (or is it the other way round?), some scholars, defending its Indian origin, make a fascinating Finneganbeginagain argument based on the old Brahmi numeral for 10 - which was [zero w fish-like tail] perhaps [zero with two tails], (in the almost illegible inscriptions from the second century BC in a cave on the Nanaghat Hill near Poona), and and from the first or second century AD in the sacred caves at Nasik. The historian of mathematics Karl Lang-Kirnberg plays a variation on this that ends up taking the laurel away from the Indians, the Greeks and even the Babylonians and fitting it snugly on Sumerian brows, back in 3000 BC. You remember that before they wrote with a stylus, the Sumerians made their marks in clay with a reed - and their symbol for 10 was this reed's unslanted impress: O. In 270 AD someone named Sphujidhvaja wrote the Yavanajataka, ''The Horoscopy of the Greeks', a translation into verse of Sanskrit prose from 150 AD. The Greek original behind it was almost certainly from Alexandria. In adjacent sentences of the text, restored in 1978, we find the number 60 mentioned twice: first as 'sat binduyutani', then as 'sat khayutani': that is, '6 with 0', i.e. 60. The word for zero, as you see, is 'bindu' the first time and 'kha' the second. 150 AD: the time of Ptolemy's Almagest. The time when a marker of clay - a ball or a bead - was spoken of (as Solon and Polybius once had) as having different values in different positions. What earlier reference could you ask for — but more to the point, what better evidence that the hollow circle of kha and the solid dot of bindu came to India from Greece?