Numbers: Binary Symbolism
NUMBERS: BINARY SYMBOLISM
Binary numbers are a system of counting and computing using two digits, 1 and 0. This system is known today as the principle of the digital computer that represents numbers through the presence (1) and absence (0) of electrical current. The first electronic digital circuit was created in 1919. As early as 1932 binary numeration was used in electronic counting circuits. However, the first binary calculator was designed in 1679 by the great German mathematician and philosopher Gottfried Wilhelm Leibniz (1646–1716), who invented the binary number system. His plan called for using moving balls to represent binary digits. The first completed statement of the number system and its operations was made eighteen years later in a New Year's greeting Leibniz sent to Duke Rudolph Augustus of Brunswick on January 2, 1697. The letter provided a detailed description of a design that Leibniz hoped the duke would strike in the form of a silver medallion. The image reproduced here (see figure 1) was created in 1734 by Rudolph August Nolte following Leibniz's instructions.
The symbol contains three of the chief number systems: the Roman, the decimal (base 10), and the binary (base 2). Roman numerals provide the date (1697) of the greeting to the duke, no doubt to "copyright" the system for Leibniz. In the table in the center are the binary numbers up to 10001 and their decimal analogs up to 17. At the sides of the table are examples of binary addition and multiplication. The system was an imago creationis ("image of creation"), as Leibniz explained to the duke, because it showed how God, the Almighty One, created the world out of nothing, the zero. Thus the caption over the design reads: "2, 3, 4, 5, etc. /For everything to be drawn out of nothing, the one suffices." Leibniz claimed that the binary system practically proved the Christian doctrine of creation ex nihilo, (the creation of the universe out of nothing, through God's omnipotence) by showing the origin of numbers through the use of one and zero. He decorated the symbol with imagery from the creation myth that appears at the beginning of the Book of Genesis. The rays at the top of the design represent the breath of God, the "almighty one," hovering over the waters, the "nothingness and void," in the moment before creation begins. The system also demonstrated the goodness of creation alluded to several times in Genesis. Binary numbers revealed the innate order of numerical relations hidden by decimal numbers. For example, the relation between 2, 4, 8, 16, (2, 22, 23, 24), is obviously the same as that between 10, 100, 1000, 10000; hence the asterisks.
Leibniz's cosmogonic symbol documents the great themes of science and religion in the seventeenth century and their secularizing and syncretistic aspects. The conviction that the universe was a mathematical artifact was almost unanimous among seventeenth-century scientists and philosophers. But this notion did not have a traditional source; scripture provides scant references to God creating with numbers. Instead, it represents the legacy of the Pythagoreans (sixth century bce), which haunted the Western philosophic tradition for millennia. According to Pythagoras, numbers were the origins of things, proceeding from the relation of the one (limit) and the void (unlimited). The mathematical structure of physical phenomena had been so well realized in the seventeenth century that Leibniz took the equation of numbers and the universe for granted. It was the religious doctrine of creation ex nihilo that needed justification. In the letter Leibniz states that the doctrine was one of the main points of Christianity" that have met with the least acceptance on the part of the worldly wise and are not easily imparted to the heathen," taking a distinction Paul reserved for the doctrine of the son of God crucified (1 Cor. 1:24–25). But in the seventeenth century the Incarnation was no longer the primary manifestation of God. Instead people read of God's activities in the sacred book of nature. Science provided the means to read nature, mathematics, and with it demonstrated the power and wisdom of God in his creation. Leibniz was so convinced of his system's success in making God's creative act transparent that he sent his invention to the Jesuit mathematical tribunal in China. He wrote Duke Rudolph that the emperor of China might now see for himself the mystery of creation and the excellency of the Christian faith. It seems clear that mystery here means a logical conundrum and excellency, rationality. The secularizing of the tradition is confirmed by Leibniz's remark in the letter that he added the imagery from Genesis, the breath of God over the waters, "so that something more pleasing than number be on the design." The mythical imagery only ornamented the now reasonable doctrine of creation out of nothing.
Leibniz's symbol is more than a brilliant reflection of seventeenth-century science and religion. It also documents a historical process made possible by the nature of symbolism itself. By making relations between different realms of meaning and experience specific, transparent, and concrete, symbols can remain of continuing relevance beyond their original cultural manifestations, as the rich histories of the one and the zero demonstrate.
Leibniz's ease in combining the Pythagorean doctrine and the Jewish and Christian teaching on creation was made possible by the sacrality of the one. The paradigmatic command of the religion of Israel demanded: "Hear, O Israel: the Lord our God, the Lord is one" (Dt. 6:4). In the Hebrew and Arabic languages counting began with two, one being reserved for God alone. Likewise the Pythagoreans did not consider one a number because it generated all numbers—a consideration held by Aristotle and repeated up through the Middle Ages. Leibniz was certainly aware of some of these aspects of the symbolism of the one, but there is no evidence he was aware of the religious associations in the history of the zero. However, Leibniz's appropriation of the zero from the decimal system in his binary number system was just one development in a long process of religious and mathematical creativity.
Place value notation is often hailed as one of humanity's great inventions. In numerical place value the position a number symbol occupies determines its value. As a result a minimum of symbols can convey a maximum of numbers. The success of this mode of numbering depends upon the zero, the symbol of the empty place in a number that preserves the value of the position. As the uncounted counter it makes rapid calculation possible. Though possibly invented independently in several civilizations, the Babylonian and Indian inventions of place value notation were the ones that influenced Leibniz's system.
By 1600 bce the Babylonian sexagesimal number system (base 60) employed a marker for the empty place. A functional place value symbol was employed in astronomical observations recorded in sexagesimal numbers by 300 bce. These observations became available to the Greeks, who then used an empty circle for the place value. The sexagesimal number system is still employed in astronomy and in calculations involving circles (degrees, minutes, seconds). The earliest Indian translations of Greek astronomical texts (c. 150 ce) use the Sanskrit words kha ("sky") and bindu ("dot") for the sexagesimal place value. At the same time, the Indian decimal system was so well developed and widely known that a Buddhist text used place value, the marker of the empty position, to explain how dharma s ("elements") exist in time. This was at the time when Nāgārjuna, the founder of Mādhyamika Buddhism, described the reality of dharma s by śūnyatā ("emptiness"). Śūnya, from the Sanskrit root śvi (to "swell" and hence "hollow out"), had been used since Vedic times (c. 1000 bce) as a synonym for words describing the sky or celestial vault, for example, kha and ākāśa ("ether"). But these words along with bindu, were used to name the place value symbol. The subsequent evidence suggests a gradual process of syncretic symbolization. By the third century ce the bindu had been used as the decimal place value notation in an Indian astronomical text. In the sixth century śūnyabindu was used to name the zero in a metaphor about the stars being ciphers scattered in the sky. Śūnya is thereafter found with increasing frequency as the name for the zero. The bindu (the dot) was incorporated into the typical Buddhist shrine, the stūpa. As the summit it symbolized the point where śūnyatā and dharmadhātu (the realm of element), were unified as ākāśa, the all-pervading ether. Emptiness and plenum were one. It was the realization of the idea of enlightenment.
It remains difficult to specify the exact relation between the religious symbolism of emptiness and the mathematical zero. The mathematical symbol of an emptiness that bears a value seemed an obvious representation of the Buddhist insight into phenomenal and conceptual reality. Interestingly enough, Leibniz's use of the zero in his binary number design gives to it a meaning not altogether different from the Buddhist value and thus helps to clarify what is centrally important. The place value suggested how conditioned or created being was absolutely distinguished from what is ultimately real, yet inseparable from it.
Leibniz took the zero from the decimal system brought to the West from India by the Muslims in the twelfth century. Zero and cipher both come from Latin transliterations of the Arabic ṣifr ("empty"), a straightforward translation of śūnya. Its symbols were the dot and the empty circle. Dots are still used today in the ellipsis, to indicate omission.
Knowingly or not, Leibniz drew upon ancient religious and mathematical expressions, the achievements of the cultures of Babylon, Greece, Israel, Arabia, and India, to fashion a number system of unforeseen usefulness. The history of the system manifests the processes of secularization, syncretism, and symbolization, as well as the processes of mathematical invention and discovery. It is a useful reminder of the global nature of the relations of the religions and the sciences. Few today may see the image of creation in their video display terminals, but the changes wrought by the technology employing the binary number system testify to the cosmogonic effectiveness of Leibniz's system.
The complete text of Leibniz's letter describing his invention is found in his Deutsche Schriften, edited by G. E. Guhrauer, vol. 1 (Berlin, 1838), pp. 394–407. An English translation of part of the letter is provided in Florian Cajori's "Leibniz's 'Image of Creation,'" The Monist 26 (October 1916): 557–565. It is accompanied by a patronizing discussion of its religious significance. He mentions how Leibniz's system caused the Jesuits in China to interpret the figures of the Yi jing as a binary number system and thus the invention of the zero and binary numbers was attributed to the Chinese. A great part of Leibniz's letter is translated in Anton Glaser's History of Binary and Other Nondecimal Numeration rev. ed. (Los Angeles, 1981), pp. 31–35, but he refrains from including two paragraphs where the references to Genesis are quite explicit. He also discusses the history of the Yi jing as a binary system. The book includes a chapter on seventeenth-century experimentation with number systems and an account of the application of binary numbers to electronic computation. The best introduction to the problems inherent in discussing the origin of the zero is Carl B. Boyer's "Zero: The Symbol, the Concept, the Number," National Mathematics Magazine 18 (May 1944): 323–330. For a summation of the controversy over the Indian origin of the zero with bibliographic references, see Walter Eugene Clark's "Hindu-Arabic Numerals," in Indian Studies in Honor of Charles Rockwell Lanman (Cambridge, Mass., 1929), pp. 217–236. David S. Reugg's "Mathematical and Linguistic Models in Indian Thought: The Case of the Zero and Śūnyatā," Wiener Zeitschrift für die Kunde Südasiens und Archiv für Indische Philosophie 22 (1978): 171–181, examines new information concerning the history of place value in India and its connection to Buddhism, though he declines to specify any relationship between the mathematical zero and Buddhist doctrines of "emptiness." The symbolism of bindu in Buddhist architecture is discussed in Lama Anagarika Govinda's Psycho-Cosmic Symbolism of the Buddhist Stūpa (Emeryville, Calif., 1976), esp. pp. 92–98.
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