Number

Title: Number

Author: Tobias Dantzig

Publisher: Plume Printing, 2007

Pages: 371

Tobias Dantzig’s “Number” is an attempt to trace the historical course of the evolution of the concept of numbers, their underlying operations, and mathematics in general. The origins, natural numbers, integers, rational numbers, irrational numbers, transcendental numbers, complex numbers and other types are explained with special emphasis to the development cycle and prominent mathematicians who had played a great part in their evolution. The author has established that a number sense exists not only in man, but in other lesser life forms like birds, chimps etc. The argument that chimpanzee’s number sense is not so developed as that of birds like crow seemed to be incongruent. Considering that the book was first published in 1930 before the advent of modern computers, the range of information provided is impressive. At several places, Dantzig refers to a ‘computer’ by which we should think of a person allocated the task of computing or calculating and not the modern machine which has become such an inseparable part of our daily lives.

Dantzig (1884-1956) was born in Latvia and as a young man, he was caught distributing anti-tsar propaganda and migrated to the US. In France he studied under Henri Poincare, the noted French mathematician of the 20^th century. This work is his magnum opus.

However, the book miserably fails to impress. It lacks the rigour to be attractive to a practising mathematician and not simple enough to appeal even to those casual readers, who have a good basis in college mathematics. The language is terse, which utterly fails to hold reader’s attention. Without regard to the propriety of using symbols, equations and diagrams in reading material, they are distributed according to one of the branches of mathematics – probability or random theory! The historical narratives are poorly researched and even stretches to the point of boredom. If some author can rewrite the book with proper examples and fluent style, it would surely become one of the essentials expected of a decent book shelf. The work does not appeal to the formal practitioners is shown by the fact that several arguments or proofs are cut short by declaring that it is out of scope for the book, as if it’d be wholeheartedly accepted by the general public. The structure of the book is pathetic at least, if not utterly disastrous. The main content spans over 257 pages, whereas the appendices and notes cover 114, which is 44% of the main theme. The structure is self-evident from the page count itself! Moreover, abstract mathematical formulas and proofs litter those appendices which are thrown at random. The notes, given in the last section, are numbered from 1 to 237, but are not referenced in the main section, and we are at a loss to find out which article they are related to. The book was such a boring experience that I had to complete it only to write a review. The pleasurable experience, which is an essential constituent of reading is sorely lacking in this hodgepodge of a compilation. The book cannot be recommended to any class of readers and the time spent on it can be guaranteed to be useless. The only saving grace of the work is the quotations from various authors given at the beginning of each chapter.

There is an interesting quotation from Laplace regarding how the world view of even great thinkers will be affected by the prejudices formed during childhood. Regarding Leibnitz, he says, “Leibnitz saw in his binary arithmetic the image of Creation……He imagined that Unity represented God, and Zero the void; that the Supreme Being drew all beings from the void, just as unity and zero express all numbers in his system of numeration. This conception was so pleasing to Leibnitz that he communicated it to the Jesuit, Grimaldi, president of the Chinese tribunal for mathematics, in the hope that this emblem of creation would convert the Emperor of China, who was very fond of the sciences. I mention this merely to show how the prejudices of childhood may cloud the vision even of the greatest men!”

Laplace praises Indian thinkers for inventing the decimal notation, as “It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to all computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the greandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity”. (The words “ten symbols” was put in bold by me). Laplace has credited ancient Indians with the invention of ten symbols, which doubtless includes zero. Now, consider how Dantzig explains this, as “The Indian term for zero was ‘sunya’, which meant empty or blank, but had no connotation of ‘void’ or ‘nothing’. And so, from all appearances, the discovery of zero was an accident brought about by an attempt to make an unambiguous permanent record of a counting board operation”, also, “How the Indian sunya became the zero of today constitutes one of the most interesting chapters in the history of culture. When the Arabs of the tenth century adopted the Indian numeration, they translated the Indian sunya by their own, ‘sifr’ which meant empty in Arabic. When the Indo-Arabic numeration was first introduced into Italy, sifr was latinized into ‘zephirum’. This happened at the beginning of the 13^th century, and in the course of the next hundred years, the word underwent a series of changes which culminated in the Italian zero.

The author explains how the ancient Indians carried forward the intellectual questions posed by Greek thinkers, who themselves was reluctant to pursue it. See Dantzig’s style when describing this free flow of intellectual curiosity, as “The Hindus may have inherited some of the bare facts of Greek science, but not the Greek critical acumen. Fools rush in where angels fear to tread. The Hindus were not hampered by the compunctions of rigour, they had no sophists to paralyze the flight of their creative imagination. Such racist remarks should have been omitted from a literary work by a noted writer. Perhaps the notions of culture and sophistication may not apply for an author who had to flee one of the political backyards of Europe where people were almost slaves of one neighbouring power for nearly most of their history!

The description of ‘amicable numbers’ is curious to learn. Pythagoras, when asked what a friend was, replied: “One who is the other I, such are 220 and 284”. This means that the factors of 284 are 1, 2, 4, 71 and 142 and these add up to 220, while the factors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, and these in turn add up to 284. Such numbers the Pythagoreans called ‘amicable numbers’.

With the Renaissance, mathematics flourished, as Dantzig says, “When, after a thousand year stupor, European thought shook off the effect of the sleeping powders so skillfully administered by the Christian Fathers, the problem of infinity was one of the first to be revived”. He describes why such a lethargy existed even in the 17^th century, as “Kepler reluctantly engaged in astronomy after his hopes of becoming an ecclesiastic were frustrated; Pascal gave up mathematics to become a religious recluse; Descartes’ sympathy for Galileo was tempered by his faith in the authority of the church; Newton in the intervals between his masterpieces wrote tracts on theology; Leibnitz was dreaming of number schemes which would make the world safe for Christianity. To minds whose logic was fed on such speculations as Sacrament and Atonement, Trinity and Trans-substantiation, the validity of infinite processes was a small matter indeed.

A commendable piece of insight in the book is the role of mathematicians described. “The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this was long ago; to this day a shape will occasionally appear which will fit into the garment as if the garment had been made for it. Then there is no end of surpirse and of delight! There have been quite a few such delightful surprises. The conic sections, invented in an attempt to solve the problem of doubling the altar of an oracle, ended by becoming the orbits followed by planets in their courses about the sun. The imaginary magnitudes invented by Cardan and Bombelli describe in some strange way the characteristic features of alternating currents. The absolute differential calculus, which originated as a fantasy of Riemann, became the mathematical vehicle for the theory of Relativity. And the matrices which were a complete abstraction in the days of Cayley and Sylvester appear admirably adapted to the exotic situation exhibited b the quantum theory of the atom”.

Even though the publishers have claimed that Einstein has remarked the book as “This is beyond doubt the most interesting book on the evolution of mathematics which has ever fallen into my hands”, we may safely conclude that it requires the genious of Einstein to appreciate this book as it will be simply tossed away by lesser intellects like you and me!

Overall rating: 1 Star

Civilization in Ancient India

The Civilization in Ancient India

Author: Louis Renou
Publisher: Munshiram Manoharlal Publishers
Pages: 182

The Civilization in Ancient India, by Louis Renou in French and translated by Philip Spratt is a work on the polity, life and thought in ancient India. Based on several references, the book is a solid piece of historical scholarship. A brief overview of the history, religion and philosophy is given in the first four chapters as an introduction and the probe into various activities of ancient Indian life begins thereafter in full earnest. The author has based his references mainly on Kautilya’s Arthashasthra. Several passages and conclusions are founded on the anecdotes in Mahabharata, Ramayana, Kathasaritsagara, Mricchakatika and other Buddhist texts. The authenticity of a work which is based on such fictional literature is naturally to be doubted. However, Renou has indicated at several places, that the veracity is to be doubted and suggests possible alternatives. The work makes it an essential object of desire for anybody intending to dig to the roots of the stories and morals depicted in the most prominent epics of Hinduism. Various topics such as Caste, family, civil law, penal law, state, politics and war, economics and public as well as private life are discussed in detail.

There are several aspects which the book makes amply clear. For example, while even the most adroit philospher may be at a disadvantage if asked to define Hinduism, this work gives a nice one as “a complete organisation of society and of thought”. The family is defined as “the community of residence and meals (those who live at the same fire – Brihaspati Smriti, XXV-6)”. The practice of Stridhana in ancient times is described as, “Property of certain kinds forms Stridhana, “the property of the woman”. Kautilya defines it briefly as cost of maintenance and things attached to the body like clothes and jewels. The interesting fact about this special kind of property is that the woman can dispose it of freely; and passes by right to the daughters according to the compex rules which take into account the class f the marriage etc”. Obviously, the practice has changed a lot from those times!

A weird aspect of the judicial system is explained. The plantiff was penalised if he lost the appeal, and if he wins, the judge who ruled against him were punished. Our judicial officers definitely won’t like to relive the past!. As Renou says, “The cost of justice was high. The plaintiff who loses an appeal pays twice the fine to which he was sentenced in the lower court; if he wins, his opponent pays double, and the judge whose decision is thus reversed are themselves liable to penalty. To the legal costs must be added the sums, at stake on the on side or the other, or on both: the profit from it all goes to the king, or sometimes to the magistrate”.

Islam, which became prominent in the 7^th century CE prohibits depiction of life forms in portraits and sculpture. Curiously, Buddha also seems to have issued such an edict as, “The existence of frescoes representing the human figure is proved by the Vinaya, such representation was forbidden by the Buddha, who allowed only floral designs to be painted (Chullavagga, VI-3)” (page 163). This may be the reason why early Buddhist paintings show the enlightened one by symbols only. It is also true that the practice was grossly abused in the later centuries.

However, on several points, the assertions of the author are liable to objections. Especially in the case of war and military preparations, the author seems to have accepted the puranic descriptions without any qualifications. The constitution of the Akshouhini (a legion of ancient warfare) is given as 21,870 chariots and elephants, 65,610 horses and 1,09,350 infantrymen. Eighteen such akshouhinis participated in the Mahabharata war. This is highly exaggerated and the ancient population was not so numerous as to account for the infantry men, let alone the logistic requirements of managing such huge military force! Similar is the case when Renou asserts that a strong central government existed, as “The provisions summarized here (regarding commercial law) allow us to infer the existence of a highly centralized state, in which the private activitywas subjected to control, and in a certain degree ‘directed’” page 81. We know that India was a decentralized state, with the village councils growing into prominence when the central authority waned. This was the established custom through the ages. So, where is the evidence for a centralized state?

The claims of the proponents of vedic science has grown so extravagant that aviation science was also known in ancient India. But Renou has conclusively puts the idea to rest, as “Certain Indian authors have believed or claimed to believe in aviation in ancient India, on the ground of the mention of vimana or aerial chariots (which are sometimes flying houses, such as the chariot Pushpaka possessed by the demon Ravana in the epics. Simple poetical fantasies are given a trifle more solidity by the pseudotechnical description of ‘flying machines’ found in the Samaranganasutra, a treatise on architecture attributed to king Bhoja (11^th century). In any case, there is nowhere any clear mention of the use of such machines in war”. (Page 125)

Slavery was prominent, and they were valued at very low rates. ”An ox cost 12 pana, a horse 24 and a female slave 50”. No Arya was to be made a slave, but the evidence, which contradicts or qualifies such statements, implies at most that the Arya does not lose his quality of Arya, that the notions of Arya and dasa are incompatible”. The taxation was rigid, but not cruel, as the author says, “The normal taxation, the ‘bhaga’ of one-sixth, the reality of which Hiuen-Tsang establishes for his time, cannot have been generally exceeded, and is tolerable. Many kings prided themselves on their moderation: even Rudradaman, whose palace shone with jewels, boasted that he had acquired them by regular taxes, he was able to build the Sudrasana dam without having recourse to extaordinary contributions. The picture drawn by Fa-hien, and especially by Hiuen-Tsang, is definitely favourable. It would be arbitrary to give a gloomy account of the situation of the peasant in antiquity on the basis of the misery which followed the Muslim invasions” (Page 116). Those magnificent building of Mughal era came at price – the life and blood of the millions of agricultural workers!

Overall rating: 3 Star