Rev 1.0 of August 1998; Rev 2.3 of June 1999 Presented at the Homeokinetics 98, a conference held at the University of Connecticut, Storrs, July 24-26, 1998, in honor of the 80th birthday of Arthur Iberall, the creator of homeokinetics. The Origins and Bifurcations of Algebra by Ralph H. Abraham, abraham@vismath.org The Visual Math Institute, www.vismath.org Santa Cruz CA, USA-96051-7920 Abstract.This paper is a fantasy on the origin of algebra around 820 AD at the Bayt al-Hikma (House of Wisdom) in Baghdad, and its later evolution as the abstract symbolic system we know today. Our approach is typical of dynamical historiography, that is, the application of dynamical systems theory (including the mathematical theories of chaos and bifurcations) to history. In this case we view cultural diffusion and bifurcation from the perspective of reaction-diffusion equations, that is, as if layers of culture were diffusing and reacting chemicals. This view is also characteristic of homeokinetics and the style of its creator, Arthur Iberall. Dedicated to Arthur Iberall on his 80th birthday. CONTENTS 1. Introduction 2. Writing systems 3. Number systems 4. Spiritual systems 5. Histomaps 6. The birth of algebra 7. The rise of symbolism 8. Conclusion Bibliography 1. Introduction. Dynamical historiography uses the concepts of bifurcation theory to analyze historical trends and their transformations. Three major bifurcations of world cultural history, · from the paleolithic to the neolithic (that is, agriculture), · the discovery of the wheel (and writing, and patriarchy), and · the chaos revolution (now ongoing), have been so analyzed in recent writings. (Abraham, 1994) The origin of algebra, in comparison, is a minor bifurcation in the history of mathematics. However, due to its key role in the paralyzing epidemic of math anxiety in the United States, together with our conviction that concepts are most easily learned in historical order, we have selected it for a dynamical historiographical analysis. Our analysis, in this case, uses (tacitly) the paradigm of the reaction-diffusion equation, which has been developed to model some of the mysterious natural phenomena of biological morphogenesis, such as: · How does the leopard get its spots? · How does an egg turn into a chicken? This paradigm is the source of the peculiar metaphors we will use, namely, those of: · chemical reactions, as for example, 2H + O -> H2O, and · hydrodynamical diffusion or perfusion, as water through sand. We are going to locate the origin of algebra in a chemical reaction between different layers of cultural reactants diffusing through the geocultural medium: · the writing system (literatic) layer, · the number system (arithmetic) layer, and · the spiritual system (kabbalistic) layer. Again, this approach is typical of homeokinetics. These three layers of signification of Hebrew and Arabic literature, in particular, are well known. (Schimmel, 1984, p. 90) In fact, we wish to model these systems as a point in a geometric space, which is a map for the part of human collective consciousness where writing systems reside. Thus we are thinking of a cultural histomap having three groups of dimensions: one or two spatial dimensions, one temporal dimension, and one or more conceptual dimensions. The conceptual states, in this model, follow the rule of some unknown dynamical system. Cultural/conceptual/mental morphogens diffuse in this histomap, and natural evolution occurs gradually, and occasionally, through catastrophic bifurcations. All this is rather like the reaction-diffusion models used in mathematical biology to account for the formation of a leopard's spots, but we are interested instead in the formation of literacy and numeracy, the very foundations of human consciousness. We will begin with a review of each of these three reactants, one at a time. 2. Writing systems For the sake of background, we may recall that the second of the major bifurcations of world cultural history (the wheel, writing, patriarchy) took place about 6,000 years ago. The morphogenetic sequence of the several aspects of the bifurcation are unknown. One possible sequence, which I have favored in the past, has the pottery wheel first, then its adaptation as the cart wheel, which empowered the urban revolution, the chariot wheel giving advantage to militant patriarchy, and support for the priestly elite, and then writing systems. Another sequence, championed by Leonard Shlain, has writing before patriarchy. (Shlain, 1998) And David Diringer places writing above all in importance. He wrote that it represented an immense stride forward in the history of mankind, more profound in its own way than the discovery of fire or the wheel. (Diringer, 1962, p. 19) In any case, our story here begins with the creation of writing systems. Our interest now is to extract enough data from the literature on the history of writing so as to be able to picture the diffusion of writing on a histomap. We will proceed by discussing a series of illustrations. The earliest scripts of the Indus valley (still undeciphered), of Mesopotamia, and of Egypt, were originally pictographic. Eventually, all three writing systems evolved in the direction of syllabic, or even phonetic (that is, alphabetic) systems. We wish to distinguish gradual evolution from bifurcation. Figure 2.1. The geographical pattern of the early writing systems, 3,000 BC to 1,000 BC. This map shows the cluster pattern of the early writing systems, adapted from Diringer. Like agriculture, the early systems began in several closely spaced centers of the Old World. The most recent studies (Walker, 1987, Ch. 1) favor a slightly more diffuse pattern, centered on Mesopotamia. In any case, the data suggest a gradual diffusion and evolution, like a spreading stain. Figure 2.2. The tree of writing systems, 3000 BC to early AD. This space-time pattern shows the main bifurcations of the tree of writing systems, from earliest Sumer, before 3,000 B.C.E., to the modern alphabets of today. (Healey, 1990, p. 61) A connection of some importance for us, from Aramaic to the Brahmi of India, 7th C. B.C.E., is not shown. (Diringer, 1962, p. 144) We regard most of these bifurcations (simple branching) as gradual shifts, or subtle bifurcations. The most important catastrophic bifurcations would be: · from pictographic pre-writing to pictographic writing (pictography) · from pictographic writing to phonemic writing (phonograms) · from phonetics to alphabetics Figure 2.3. Development of cuneiform signs from 3,000 B.C.E. Here three stages in the evolution of cuneiform signs are shown in parallel, horizontal rows. These illustrate the first two of the three bifurcations listed above. (Walker,1987, p. 10) Figure 2.4A. Comparison of early alphabets. Here we get to see the shapes of the letters of some of the early alphabets, along the main trunk of the tree, as they evolved from Phoenician around 1,000 B.C.E. (Healey, 1990, p. 29) This figure suggests that between the leap from phonetics to alphabetics up to the present time, primarily gradual shifts have taken place. See also (Naveh, 1982, p. 23). Figure 2.4B. The development of the English alphabet. Here, for comparison, is a tabular view of the morphogenesis from the North-Semitic, 1,000 B.C.E., to the modern English alphabet. (Diringer, 1968, v.2, Fig. 22.1) A lesser, but still discontinuous, leap takes place with the addition of vowels in the Greek alphabet. Figure 2.5. The development of the Greek alphabet. Zooming in on the Greek branch of the tree, here we see the step-by-step changes during the 8th and 7th C. B.C.E. (Healey, 1990, p. 37) Here again, on a microscopic scale, we see the continuity in development of the first complete alphabet. Figure 2.6. The development of the Arabic alphabet. The Arabic alphabet began its development in late Roman times, and evolved into one of the most complete, and widely used, alphabets today. In summary, we may say that long after pictographic pre-writing spread throughout the human-inhabited world, a new idea popped up in Sumer: cuneiform pictography. This was followed rather quickly by a bifurcation to cuneiform phonograms. After this new idea had spread widely, another new idea popped up in Canaan: the alphabet. In between these local catastrophic bifurcations, in addition to geocultural diffusion, there took place a gradual evolution, much like the wearing and polishing of pebbles in a river. 6. The birth of algebra Abu Ja`far Muhammed ibn Musa al-Khw×razmä is generally acknowledged the father of algebra. He is called al-Khw×razmä, from which derives our word algorithm. In the early 9th century AD, he was the leader of the Bayt al-Hikhma (House of Wisdom, an official institution of the Caliphate)) of Baghdad, a key figure in the intellectual development of early Islam. (Sayili, 1989) He wrote two books: · On the Hindu Numbers, which survives only in a Latin translation, and · An Abridged Treatise on the Jabr and Muqabala Calculation (Al-Kitab al-Mukhtasar fi Hisab al-Jabr wa al-Muqabala) The latter was one of the first algebra texts in Arabic. Our word algebra comes from al-Jabr in this title. As if by coincidence, these two works reflect the coincident diffusion into Islam of the number system from the Hindus, and outward of the new algebra. Our chemical analysis of the birth of algebra in this intersection of diffusing reactants boils down to this.The Hindu numbers dissolved the attachment of the symbols of the alphabet from numerics, thus liberating them for a new level of significance: the unknown variables and operations of geometric algebra. At the time of the arrival of Euclid's Elements in the home ground of the original Babylonian arithmetic, Euclid's solutions to the algebraic problems of Babylonia by means of geometric constructions, known as geometric algebra, stimulated the mathematicians of the Bayt al-Hikma to make use of this newly liberated symbol system for a fresh expression of the classics. Together with the recent rise of trivalent Islamic calligraphy, two vectors converged on Baghdad in his time, as shown in Figure 5.8. Further, it is certain that al-Khw×razmä knew the recent translations of Euclid's geometric algebra into Arabic, to which he refers in his Abridged Treatise. He may also have seen a translation of Diophantus, who had continued the Babylonian algebra tradition in Greek antiquity. To this we may add the surmise that al-Khw×razmä was aware of a vestigial survival of the original Babylonian geometric algebra, which no doubt was the basis for the geometric algebra of Euclid, and the symbolic algebra of Diophantus. (Neugebauer, 1962, p. 146; Klein, 1968, p. 127) In summary: · Ancient Babylonia had a tradition of geometrical algebra, that is, algebraic problems solved geometrically, by 2000 BC, · Thales and Pythagoras learned this tradition during their travels, stimulating Greek geometry, around 600 BC. · Greek geometry was organized by Euclid as a justification and proof of the constructions of Babylonian geometrical algebra, around 300 BC. · Euclid's Elements reached Babylon and were translated into Arabic by 800 AD. · At about the same time, the Indian number system arrived from India, and sacred calligraphy arrived from the Christians. The numerical layer was replaced by the spiritual in a significant cultural phase transformation. · In this new cultural phase, al-Khw×razmä's efforts to understand Euclid, bolstered by a residuum in situ of the old Babylonian tradition, gave rise to algebra. These vectors are indicated in the histomap of Figure 6.1. 7. The rise of symbolism The algebra of al-Khw×razmä was not yet the algebra we know. It was still rhetorical, that is, the variables and operations were denoted by words, rather than by the signs and symbols which characterize algebra today. It is interesting to note the step-by-step evolution of the modern symbols. The plus and minus signs appeared in Germany around 1500 AD, that is, about three centuries after the arrival in Europe of the Hindu numbers. A few decades later, the equal sign appeared in England. The symbolic notations of Diophantus reemerged in a work of ViÜte in 1591. (Klein, 1968, p. 151; Katz, 1993, p. 339) And in 1637, it all came together in the Discours of Descartes, with modern notations for variables, exponents, operations, and so on. (Cajori, 1928; Katz, 1993, p. 399) This process may be regarded as an enlargement of the number system, and further reduction of the burden placed on the writing system by mathematics. With the diffusion of the several cultural morphogens into the fresh soil of Europe, a burst of chemistry brought forth the Renaissance. The ancient and medieval developments of geometry, arithmetic/algebra, astronomy/astrology, medicine/alchemy, Kabbalah/magic were reborn. And kinematics/dynamics was born. Another spot for the leopard, in the morphogenetic sequence of the human life/consciousness story. 8. Conclusion There is little doubt about the concurrent arrival in the new and spirited cultural milieu of Baghdad, early in the 9th century, of: · Arabic alphabet literacy, · the Hindu numerals, · sacred (esoteric) calligraphy, · Euclid's Elements. The connection between this concurrence and the birth of algebra in the same time and place as a cultural/chemical reaction is plausible perhaps, yet not conclusive. One objection immediately comes to mind: why did we not have this explosion in India, where the liberation of the Devanagari syllabary from its numerical burden came a century or two earlier? Our understanding of this involves the idea of the numerical layer as a sort of electrochemical insulator or barrier between the phonemic and the spiritual layers. With the melting or dissolving of the barrier, the spiritual significance of the characters came into catalytic interaction with the alphabet, and the abstraction of the divine attributes of the spiritual layer flowed into the space vacated by the numerical layer of signification, elevating it into an abstract, or algebraic, significance. And the spiritual layer was much stronger in medieval Islamic, Christian, and Jewish culture than in India. Also, the cultural memory of ancient Babylonian geometrical algebra might have been the critical factor. Bibliography. Abraham, Ralph, Chaos, Gaia, Eros, San Francisco: Harper-Collins, 1994. Develops the ideas of dynamical historiography and applies them to world cultural history. Abraham, R., Bifurcation of the Kalihari !Kung, 1992. A dynamical analysis of the transformation of the Bushmen from stone age to neolithic culture around 1974. Abraham, R., Chaos and the millennium, 1994. Preprint. Bifurcation theory reviewed, and applied to the Chaos Revolution now underway. Backhouse, Janet, The Lindisfarne Gospels, Oxford: Phaidon, 1981. Baker, Arthur, Celtic Hand, Stoke by Stroke: Irish Half-Uncial from ªThe Book of Kellsº, New York: Dover, 1983. How to do it manual for creating your own Book of Kells. Bernal, Martin, Black Athena: the Afroasiatic Roots of Classical Civilization, 2 vols., New Brunswick, N.J.: Rutgers University Press, 1987, 1991. Cajori, Florian, A History of Mathematical Notations, La Salle, Ill.: Open Court Pub. Co., 1928/1951. Diringer, David, Writing, New York: Praeger, 1962. This is the masterful, concise, and complete story of writing, from pictograms to alphabets, with amazing illustrations. Diringer, David, The Alphabet, A Key to the History of Mankind, 3rd edn., New York: Funk & Wagnalls, 1968. Two volume expansion of the alphabet development story. Fowden, Garth, The Egyptian Hermes: a Historical Approach to the Late Pagan Mind, Princeton, N.J.: Princeton University Press, 1986/1993. Healey, John F., The Early Alphabet, Berkeley: University of California, 1990. A short and up-to-date version of the alphabet story. Joseph, George Gheverghese, The Crest of the Peacock, Non-European Roots of Mathematics, London: Penguin, 1992. A unique history of mathematics from a multicultural perspective. Katz, Victor, An Introduction... Well balanced and current text on the history of mathematics. Khatibi, Abdelkebir, and Mohammed Sijelmassi, The Splendour of Islamic Calligraphy, London: Thames and Hudson, 1996. Klein, Jacob, Greek Mathematical Thought and the Origin of Algebra, New York: Dover, 1968/1992. Meehan, Bernard, The Book of Kells: An Illustrated Introduction to the Manuscript in Trinity College Dublin, London: Thames and Hudson, 1994. Menninger, Karl, Number Words and Number Symbols, A Cultural History of Numbers, New York: Dover, 1969, 1992. Naveh, Joseph, Early History of the Alphabet: An Introduction to West Semitic Epigraphy and Paleography, Jerusalem: Magnes Press, 1982. Neugebauer, Otto, The Exact Sciences in Antiquity, 2d ed., New York: Harper, 1962. Rosen, Frederic, tr., Al-Khwarazmi's Algebra, Islamabad: Pakistan Hira Council, 1989. The classic translation of Rosen of 1831 is here reprinted and updated. Sayili, Aydin, Introduction, in (Rosen, 1989). This essay is a recent and very well informed account of Al-Khwarazmi's role in the development of algebra. Schimmel, Annemarie, Calligraphy and Islamic Culture, New York: New York UNiversity Press, 1984. Complete explanation Islamic esoteric calligraphy and the Sufi spiritual values of the letters of the Arabic alphabet by an adept. Scholem, Gershom G., Major Trends in Jewish Mysticism, New York: Schoken, 1941/1946. Walker, C. B. F., Cuneiform, London: The British Museum, 1987. 3. Number systems Contrary to popular belief, it seems that number systems are older than writing systems. TAlly sticks have been found dating from deep prehistory. In fact, it seems to be firmly established that the first writing systems evolved out of number systems, in the context of inventories, accounting, urban administration, bookkeeping, calendar keeping, and the like.(Senner, 1989, pp. 8-9; Ch. 2) Within the cuneiform writing system of Mesopotamia, a relatively advanced number symbol system was embedded. Shortly before the Common Era, a rather different system emerged in India, which evolved with impressive rapidity into the modern system. According to Diringer, the Indian numerals evolved from the Aramaic alphabet, which reached India in the 7th C. B.C.E. Again, with the Indian numerals, we see a cone-like pattern of cultural diffusion from India, this time with greater speed of diffusion. Figure 3.1. Numerical notation, preceding the arabic numerals. Here we see, line-by-line, a comparison of the number symbols of various cultures and writing systems, from 3,000 B.C.E. up to medieval times. They show considerable uniformity, perhaps due to their having a common source in finger counting. The Roman numerals, still much used, are rather typical. (Diringer, 1968) Figure 3.2. Numbers after alphabets: letters as number symbols. The cuneiform and hieroglyphic symbols were given an additional burden of number signification. Following the spread of the alphabet after 1,000 B.C.E., the letters of the alphabet inherited this double entendre. Here we see, in parallel columns, a comparison of number values for six writing systems. This double signification phase of cultural evolution is the basis for Hebrew gematria, or Islamic jafr, a kind of error correcting code in sacred texts, in which the number spelled out by a word is consciously combined with the literary meaning. Figure 3.3. The development of the Indian numerals. The Indian numerals developed from the Kharosthi, from 400 B.C.E., which resemble Roman numerals. Kharosthi was a writing system, which included these number signs. This writing system was succeeded by the Brahmi, around 300 B.C.E., which was influenced by the Aramaic alphabet, and is the parent of the syllabary used today in India. (Menninger, 1969, pp. 294-399) These number symbols of the Brahmi writing system evolved into the Gwalior numerals around 850 C.E., similar to the ones universally used today. By 876 C.E., the decimal place value system, with zero, had been established in India. (Joseph, 1991, pp. 239-243) Figure 3.4. The Brahmi numerals in detail. Here is a longer list of the numerals of the Brahmi system. (Menninger, 1969, p. 395) Figure 3.5. The migration of Indian numerals. Here we see a family tree of numeral systems, from the Brahmi of 300 BCE to Medieval Europe. In our view this is a gradualist evolution punctuated by subtle (that is, continuous) branching bifurcations. (Menninger, 1969, p. 418) Figure 3.6. From Brahmi to modern numerals. Here is another comparative sequence from the family tree of Indian numerals. (Menninger, 1969, p. 419 After viewing these illustrations, we are left with an impression of diffusion and reaction of a cultural/intellectual substance through a geocultural space-time matrix. The leopard spots are obtained by the major catastrophic bifurcations, like phase changes, which occur along the way. So far we have noted mainly four of these: · from pictographic pre-writing to pictographic writing (pictography) · from pictographic writing to phonemic writing (phonograms) · from phonetics to alphabetics · from alphabetic numerals to Indian numerals three from the proceeding section on writing systems, and one more for numerals. Note: The diffusions of literacy and of numeracy originate from quite different times and places. Figure 3.1. Numerical notation, preceding the arabic numerals. (Diringer, 1968, v.2, p. 437)(diringerc10.tif) Figure 3.2. Numbers after alphabets: letters as number symbols. (Menninger, Fig. 95, p. 265) (menninger95.tif) Figure 3.3. The development of the Indian numerals. (Joseph, Fig. 8.2, p. 241) (joseph82.tif, 240 dpi) Figure 3.4. The Brahmi numerals in detail. (Menninger, Fig. 223, p. 395) (menninger295.tif sic) Figure 3.5. The migration of Indian numerals. (Menninger, Fig. 239, p. 418) (menninger239.tif) Figure 3.6. From Brahmi to modern numerals. (Menninger Fig. 241, p. 419) (menninger241.tif) Figure 5.1. The cultural diffusion of ancient math, according to the usual interpretation. (Joseph, Fig. 1.1, p. 4) (joseph11.tif, 200 ppi) Figure 5.2. An improved diffusion path, the multicultural interpretation. (Joseph, Fig. 1.2, p. 9) (joseph12.tif, imp at 200 dpi) Figure 5.3. The medieval transmission of mathematics on a two-dimensional geographic chart. (Joseph, Fig. 1.3, p. 10) (joseph13.tif, 200 dpi) Figure 5.4. All the foregoing diffusion paths, combined into a single figure. This is an archetypal histomap. (Joseph, Fig. 1.4, p. 14) (joseph14.tif) Figure 5.8. Histomap (space-time pattern) of writing and number systems. · Black: prealphabetic writing · Blue: alphabetic writing · Red: number systems · Green: divine codes (aljabr4.ai) Figure 2.3. Development of cuneiform signs from 3,000 B.C.E. (Walker, 1987, p. 10) (walker10.tif) Figure 6.1. Three vectors arrive in Baghdad, 820 AD.