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