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Authors: Alex Bellos
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number in terms of doubles facilitated a variety of operations. ‘[It] permits the Assayer to weigh all sorts of masses with few weights and could serve in coinage to give more value with fewer pieces,’ he wrote in 1703. Leibniz did admit that binary had some practical drawbacks. The numbers are much longer when written out: 1000 in decimal, for example, is 1,111,101,000 in binary. But he added: ‘In recompense for its length, [binary] is more fundamental to science and gives new discoveries.’ By looking at the symmetries and patterns in binary notation, he claimed, new mathematical insights are revealed, and number theory is richer and more versatile because of it.
Design for eibniz’s binary medallion, in Johann Bernard Wiedeburg’s Dissertatio mathematica de praestantia arithmeticae binaria prae decimali (1718). As well as the words Imago Creationis , the Latin reads ‘From nothing comes one and everything, but the one is necessary’.
Second, Leibniz marvelled at how the binary system chimed with his religious views. He believed that the cosmos was composed of being, or substance, and non-being, or nothingness. The duality was perfectly symbolized by the numbers 1 and 0. In the same way that God creates all beings from the void, all numbers can be written in terms of 1s and 0s. Leibniz’s conviction that binary exemplified a fundamental metaphysical truth was – to his great delight – strengthened when later in life he was shown the I Ching , the ancient Chinese mystical text. The I Ching is a book of divination. It contains 64 different symbols, each of which comes with an accompanying commentary. The reader randomly selects a symbol (traditionally by casting yarrow sticks) and interprets the related text – a little like one might read an astrological chart. Each symbol in the I Ching is a hexagram, which means it is composed of six horizontal lines. The lines are either broken or unbroken, corresponding to a yin or a yang . The 64 hexagrams in the I Ching are the full set of combinations of yins and yangs when taken in groups of six at a time.
A particularly elegant way of ordering the hexagrams is shown opposite. If each yang is written 0 and each yin is 1, then the sequence matches precisely the binary digits from 0 to 63.
This way of ordering is known as the Fu Hsi sequence. (Strictly speaking, it is the inverse of Fu Hsi, but they are mathematically equivalent.) When Leibniz was made aware of the binary nature of Fu Hsi, it gave him ‘a high opinion of [the I Ching ’s] profundity’. Since he thought that the binary system mirrored Creation, his discovery that it also underlay Taoist wisdom meant that Eastern mysticism could now be accommodated within his own Western beliefs. ‘The substance of the ancient theology of the Chinese is intact and, purged of additional errors, can be harnessed to the great truths of the Christian religion,’ he wrote.
Part of the Fu Hsi sequence of the I Ching and its binary equivalent.
Leibniz’s panegyrics on base two were a rather eccentric preoccupation of the pre-eminent polymath of his day. Yet in ascribing a fundamental importance to the system, he was more prescient than even he could ever have imagined. The digital age runs on binary, as computer technology relies at a most basic level on a language comprised of 0s and 1s. ‘Alas!’ wrote mathematician Tobias Dantzig. ‘What was once hailed as a monument to monotheism ended in the bowels of a robot.’
‘Freedom is the freedom to say two plus two equals four,’ wrote Winston Smith, the protagonist of George Orwell’s Nineteen Eighty-Four . Orwell was making a comment not only about freedom of speech in the Soviet Union, but also about mathematics. Two plus two is always four. No one can tell you it isn’t. Mathematical truths cannot be influenced by culture or ideology.
On the other hand, our approach to mathematics is very much influenced by culture. The selection of
Design for eibniz’s binary medallion, in Johann Bernard Wiedeburg’s Dissertatio mathematica de praestantia arithmeticae binaria prae decimali (1718). As well as the words Imago Creationis , the Latin reads ‘From nothing comes one and everything, but the one is necessary’.
Second, Leibniz marvelled at how the binary system chimed with his religious views. He believed that the cosmos was composed of being, or substance, and non-being, or nothingness. The duality was perfectly symbolized by the numbers 1 and 0. In the same way that God creates all beings from the void, all numbers can be written in terms of 1s and 0s. Leibniz’s conviction that binary exemplified a fundamental metaphysical truth was – to his great delight – strengthened when later in life he was shown the I Ching , the ancient Chinese mystical text. The I Ching is a book of divination. It contains 64 different symbols, each of which comes with an accompanying commentary. The reader randomly selects a symbol (traditionally by casting yarrow sticks) and interprets the related text – a little like one might read an astrological chart. Each symbol in the I Ching is a hexagram, which means it is composed of six horizontal lines. The lines are either broken or unbroken, corresponding to a yin or a yang . The 64 hexagrams in the I Ching are the full set of combinations of yins and yangs when taken in groups of six at a time.
A particularly elegant way of ordering the hexagrams is shown opposite. If each yang is written 0 and each yin is 1, then the sequence matches precisely the binary digits from 0 to 63.
This way of ordering is known as the Fu Hsi sequence. (Strictly speaking, it is the inverse of Fu Hsi, but they are mathematically equivalent.) When Leibniz was made aware of the binary nature of Fu Hsi, it gave him ‘a high opinion of [the I Ching ’s] profundity’. Since he thought that the binary system mirrored Creation, his discovery that it also underlay Taoist wisdom meant that Eastern mysticism could now be accommodated within his own Western beliefs. ‘The substance of the ancient theology of the Chinese is intact and, purged of additional errors, can be harnessed to the great truths of the Christian religion,’ he wrote.
Part of the Fu Hsi sequence of the I Ching and its binary equivalent.
Leibniz’s panegyrics on base two were a rather eccentric preoccupation of the pre-eminent polymath of his day. Yet in ascribing a fundamental importance to the system, he was more prescient than even he could ever have imagined. The digital age runs on binary, as computer technology relies at a most basic level on a language comprised of 0s and 1s. ‘Alas!’ wrote mathematician Tobias Dantzig. ‘What was once hailed as a monument to monotheism ended in the bowels of a robot.’
‘Freedom is the freedom to say two plus two equals four,’ wrote Winston Smith, the protagonist of George Orwell’s Nineteen Eighty-Four . Orwell was making a comment not only about freedom of speech in the Soviet Union, but also about mathematics. Two plus two is always four. No one can tell you it isn’t. Mathematical truths cannot be influenced by culture or ideology.
On the other hand, our approach to mathematics is very much influenced by culture. The selection of
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