Number: The Language of Science by Tobias Dantzig
My rating: 5 of 5 stars

Literate Mathematics

A classic in every sense: a model of style and erudition to rank with Oscar Wilde, as inspiring as Zadie Smith, as concise as a page from George Orwell, and as timeless as any of Dickens’s tales. If you have an interest in mathematics, or if you have been scarred by the imposition of tedious calculating techniques in your school days, or if you simply want to understand an enormous part of intellectual history, this is the single most important book you could have at hand.

The first edition was published almost 90 years ago. Yet it is fresh and witty and simply full of the most remarkable facts and astute observations about the development and use of numbers. Apparently, for example, birds (particularly crows) have a relatively developed sense of number (at least up to five). Dogs, horses and other domestic animals appear to have none. And the English trice has the double meaning of three times as well as simply many, plausibly echoing the Latin ‘tres’ and ‘trans’ - beyond - thus memorializing an ancient method of base 3 counting.

Dantzig‘s factual anecdotes are similarly captivating: “Thus, to this day, the peasant of central France (Auvergne) uses a curious method for multiplying numbers above 5. If he wishes to multiply 9 × 8, he bends down 4 fingers on his left hand (4 being the excess of 9 over 5), and 3 fingers on his right hand (8 – 5 = 3). Then the number of the bent-down fingers gives him the tens of the result (4 + 3 = 7), while the product of the unbent fingers gives him the units (1 × 2 = 2).”

The only misjudgment Dantzig makes is his underestimation of binary arithmetic. “It is the mystic elegance of the binary system,” he says somewhat disapprovingly, “that made Leibnitz exclaim: Omnibus ex nihil ducendis sufficit unum. (One suffices to derive all out of nothing.)” Little could Dantzig (much less Leibniz) have foreseen the rather non-mystical importance of the base-two counting in the age of the digital computer.

Dantzig is acutely sensitive to the cultural matrix of mathematics. That matrix, he points out, is neither commercial nor academic; it is largely religious. “Religion is the mother of the sciences.” The Greeks of course had several mathematically based religious cults. Even the most recent (and difficult) mathematical field, number theory “had its precursor in a sort of numerology” of biblical texts. (See here for more on the religious inspiration in mathematics: https://www.goodreads.com/review/show...).

But he also recognises religion as a major impediment to the development of mathematical knowledge: “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.” Religion, thankfully, shot itself in the foot in interesting ways: “Now, the acquisition of culture was certainly not a part of the Crusader’s program. Yet, this is exactly what the Crusades accomplished. For three centuries the Christian powers tried by sword to impose their “culture” upon Moslem. But the net result was that the superior culture of the Arabs slowly yet surely penetrated into Europe.”

Perhaps most impressive is Dantzig’s intellectual humility. He begs ignorance of the philosophical issue of whether or not numbers exist outside of human thought about them. But he is not without an important philosophical view: “Herein I see the genesis of the conflict between geometrical intuition, from which our physical concepts derive, and the logic of arithmetic. The harmony of the universe knows only one musical form—the legato; while the symphony of number knows only its opposite—the staccato. All attempts to reconcile this discrepancy are based on the hope that an accelerated staccato may appear to our senses as a legato. Yet our intellect will always brand such attempts as deceptions and reject such theories as an insult, as a metaphysics that purports to explain away a concept by resolving it into its opposite.”

To conclude with this sort of poetic image justifies entirely the description of this book as “an ode to the beauties of mathematics.”

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