From A Random World to a Rational Universe Randomness, Luck, Astragali and Dice In the time before the mathematical idea of randomness was discovered, people thought that everything that happened was part of the will of supernatural beings, the gods, who looked down upon human affairs and decided to ‘tip the balance’ one way or another to influence events. Hence, sacrifices were made and rituals performed to discover the ‘will of the gods’ or to try to influence human affairs. This idea still prevails, and many people all over the world use lucky charms, engage in superstitious practices, use horoscopes, and still have some kind of belief that there are such ways of influencing their lives. The gods may be dead, but ‘Lady Luck still survives. The astragalus is a small bone, about an inch cube, found in the heel of hoofed mammals. Astragali have six sides but are not symmetrical, so there is no way of knowing which way they will eventually come to rest. For many ancient civilizations, astragali were used by priests to discover the opinions of their gods. It was customary in divination rites to roll, or cast, five astragali. Typically, each possible configuration was associated with the name of a god and carried with it the sought-after advice. [Insert Astragali 4] Showing the four positions of rest. The small one in the foreground is made from pottery. Astragali found in excavations typically have their sides numbered or engraved They were also used in board games in the First Dynasty in Egypt, c 3500 BCE; archaeological evidence consists of boards, counters, and astragali for various games, including one similar to Snakes and Ladders, still popular today. [Insert Game of Hounds and Jackals] The astragali have been used from classical times for gambling, and similar stones are still in use today for games like ‘fivestones’ or ‘jacks’. Gradually, over thousands of years, astragali were replace by dice , and pottery dice have been found in Egyptian tombs. The earliest die known was made from pottery and excavated in Northern Iraq dating from about 3,000 BCE. It has dots arranged as in (Die a) [Insert Early Die a] By about 1400 BCE a die from Egypt shows that the markings became the familiar 2 - partitions of 7 (i.e. 1 opposite 6, 2 opposite 5, and 3 opposite 4) (Die b). [Insert Early Die b] Dice with other markings like the names or portraits of gods have been found, probably used for special games or rituals, and others where some numbers are repeated, or ‘loaded’, for special purposes or possibly for cheating. (Die c) [Insert Early Die c] Once the Greeks had worked out the geometry of the polyhedra, dice of other shapes began to be constructed. However, whether cube or polyhedral, the shapes were not entirely regular and were therefore biased. However, over time, gamblers would get used to using the same dice, and have an intuitive idea of how they would fall, but given another set of dice, the odds would be different. Later, as the manufacture of dice became more exact, some ideas of the possible combinations of number began to emerge. The Earth and The Cosmos There were many other forms of rituals hoping to overcome the randomness of nature and man’s condition. A few of these which became of particular mathematical interest are geomancy, the nine square grid or magic square, and temple designs, the ancestors of board games. Geomancy Geomancy means divination of or by the earth, and is a system of 16 mathematically related arrangements of stones, beans or other available small objects used to make decisions, answer questions, or foretell the future. The stones are cast upon the ground and the pattern formed is interpreted. The symbols represent a series of binary ‘opposites’ like good and evil, male or female, sadness and happiness, etc. Combinations of these opposites can be used to represent odd and even numbers. [Insert 16 Figures of Geomancy] As in all methods of divination, each of these figures has a number of interpretations depending on its relation to other figures shown, and many other circumstances like the time of day, the weather, and the kind of person who is asking the question. The Grid of Nine Squares The Nine square grid is said to come from an ancient system for the division of land, probably from feudal India. In China the nine-square configuration was supposed to be an ideal arrangement, with eight farmers’ fields surrounding a central well. The grid of nine squares, or a circle divided into nine sections by straight lines often appears as a central form in Tibetan sacred diagrams. In Scotland, the pattern was used at Beltane (the eve of May) where eight squares were cut out from the turf, and a bonfire lit on the central square. In this way, from practical beginnings in different cultures, the nine-square grid acquired mystic importance and symbolised divine order, and the representation of control by the gods. Magic Squares are directly related to the Sacred Grid, supposedly being the numerical mystery which underlies their physical form. The simplest magic square is the square of nine, ascribed to Saturn, where each row and column adds up to 15; the total of the rows and the columns is 45, and the diagonals 30. The 4x4 square with row and column numbers 34 is assigned to Jupiter, the 5x5 with row or column numbers 65 to Mars, and so on for the Sun, Venus, Mercury and the 9x9 square with row or column numbers 369, to the Moon. As with other devices, these magic squares are all said to have correspondences to different numbers, various deities, days of the week, natural objects, different qualities, and so on. In the Hindu Temple Yantra you can see the nine squares, the ‘sacred space, or source of energy, in the centre. [Insert Hindu Temple Yantra] Board Games are clearly linked with divination, astrology and sacred geometry, and the designs of the boards can show their sacred or occult origins. The popular game of ‘snakes and ladders’ is controlled by the throw of dice, and the ladders and snakes originally referring to good and bad fortune, now refer to good and bad ‘luck’ in the progress of the game. In some cases the designs of the boards are the same as the plans of temples and holy cities with a ‘sacred space’ in the centre. [Insert Korean Game Board] [Insert Nine Men’s Morris] [Insert Robert Burton Horoscope] Mathematics and Magic. In ancient times, few people could understand even the simplest arithmetic and geometry, and the confusion of mathematics with magic has a long history. People who had knowledge of the regular movements of the heavens were able to predict the position of planets, and the particular the times when astronomical events appeared in certain sections of the sky. In ancient civilisations these were highly skilled technicians, called ‘priests’, and their activities were partly scientific, and partly religious. In Europe, after the arrival of Christianity, the religious aspect of these practices was condemned as superstition. Because numbers were used in these processes, anyone who used numbers was regarded with considerable suspicion. In tis way genuine mathematicians were looked upon with suspicion by the ignorant, and the titles of Astrologer, Mathematician and Conjurer were virtually synonymous. An early Bishop of the Church, St. Augustine of Hippo (354-430 CE) once said: "The good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that mathematicians have made a covenant with the devil to darken the spirit and confine man in the bonds of Hell." Augustine was arguing that belief in astrology denies the freedom of the will. Roger Bacon (1214 - 1292), often called England’s first Scientist, had a reputation as a ‘great necromancer’ because of his ingenious experiments and John Dee (1527 - 1609) probably one of the foremost mathematicians in Europe of his time, gained a reputation as a ‘Conjuror’ while he was at Oxford from a simple mechanical device for a play by which an actor appeared to fly, and people claimed he was in league with the devil. [Insert Portraits of Bacon, Dee, Recorde ?] During the sixteenth century in England, mathematicians like Robert Recorde (1510-1558) and Thomas Digges (1546-1595) published many works showing the everyday practical usefulness of mathematical knowledge for ordinary people clearly showing that mathematics was not an occult practice. Following the foundation of the Oxford chairs in mathematics and astronomy in 1619, some parents kept their sons away from the university in fear of them becoming contaminated by the ‘Black Art’! As the predictive power of astronomy and other practical uses of mathematics became apparent, mathematicians were able to dispel the idea that many events were not controlled by the goddess Fortuna, but could be explained in a rational way. [Insert Castle of Knowledge Title page]

Notes

  1. The Greek word 'axiom' means an agreed starting point. The word 'postulate' is also used to describe 'what is possible', or what basic ideas can be used.
  2. Proposition 5 used to be called the 'Pons Asinorum' (the Bridge of Asses) because it was once regarded as a test of understanding for the rest of the Elements.
  3. This proof has been criticised because Euclid has no axiom which states that there is a point of intersection when two lines cross.
  4. This is often called the contrapositive argument. i.e. If p implies q, then 'not q' implies 'not p'.
  5. Some mathematicians choose to call 'reductio ad absurdum' orproof by contradiction, but others have decided it is a separate case.
  6. A rational number is any one that can be represented by a fraction, like:

    2
    3
    		 , 57
    21
    		 or 3
    3
  7. Pythagoras (c.569-475 BCE) founded a society which continued well into the 5th century.
  8. In-commensurable means that there is no common measure between two given lengths.In this case, a side of a square and its diagonal cannot be measured exactly in the same units, or fractions of the unit.
  9. In this case, a side of a square and its diagonal cannot be measured exactly in the same units, or fractions of the unit. See Pedagogical Notes and Questions 1(c) .
  10. Another name for these numbers is 'surds'.
  11. For details on Euclid see D.E. Joyce's commentary at weblink EUC
  12. See Netz and Noel (2007) for the most startling recent discoveries about Archimedes.
  13. In this kind of projection, the lengths of lines and the shapes of curves change, but the order in which points are connected, remains the same.The term invariant refers to properties that remain the same under some kind of transformation.
  14. The original conjecture by Descartes was investigated by Euler in 1758 when attempting to classify polyhedra.
  15. The publication of Lakatos' original thesis in 1961 was a major factor in the development of Investigations in school mathematics.
  16. Iso-morphic means the 'same shape'. Isomorphic crystals of chemicals with similar composition can grow on each another, as often seen in the school science laboratory.
  17. See Robin Wilson's Four Colours Suffice (2002). This is the history of the map colouring problem first proposed by a student of Augustus De Morgan in 1852. The first proof in 1976 counted 1,936 different maps, but after re-testing the program and discovering mistakes, by 1994 the number had been reduced to 633. There are still some people who doubt the result.
  18. Known also as Bhaskara II or 'Bhaskhara the Teacher'.
  19. In the early 20th century, Hardy, and others at Cambridge found many of the results of the brilliant Indian mathematician Ramanujan (1887-1920) difficult to understand because the proof methods were unlike anything they had seen before. Even today, mathematicians are still discovering important new results from the work of Ramanujan.

References

Berggren, J. L. (2003) Episodes in the Mathematics of Mediaeval Islam New York, Berlin. Springer (original 1986)
D'Amore, B. (2005) Secondary school students' mathematical argumentation and Indian Logic (Nyaya).For the Learning of Mathematics 25 (2) July 2005 (26, 32)
Datta, B. ; Singh, A. N.(1935) History of Hindu mathematics, a source book. Lahore,: Motilal Banarsi Das. (Reprinted by Bharatiya Kala Prakashan, Delhi)
Dauben, J.W. (2007) Chinese Mathematics in Katz, V. J. (ed.) Sourcebook (187-384)
Lakatos, I. (1976) (Eds. Worrall, J. and Zahar, E.) Proofs and Refutations: The Logic of Mathematical Discovery London. Cambridge University Press
Kanigel, R. The Man Who Knew Infinity: A Life of the Genius Ramanujan Johns Hopkins Univerity Press
Katz, V. J. (2007) (ed.) The Mathematics of Egypt, Mesopotamia, China, India and Islam: A Sourcebook. Princeton and Oxford. Princeton University Press
Keller, Agathe (2005) Making diagrams speak, in Bhskara I 's commentary of the Aryabhatiya. Historia Mathematica 32: 275-302.
Keller, Olivier. (2006) Une Archeologie de la Geometrie (An Archeology of Geometry) Paris. Vuibert
Martzoff, J-C. (1987) A History of Chinese Mathematics Berlin, New York. Springer
Newton, Isaac. (1745) Sir Isaac Newton's Two Treatises on the Quadrature of Curves, and Analysis by Equations of an infinite Number of Terms explained: London John Stewart
Netz, R. and Noel, W (2007) The Archimedes Codex. London. Weidenfeld & Nicolson
Plofker, K. (2007) Mathematics in India in Katz, V. J., Sourcebook (385-514)
Li Yan and Du Shiran (1987) Chinese Mathematics : A Concise History. John N. Crossley, J.N. and Lun, A.W.C. Oxford. Oxford Science Publications.
Watson, A. and Mason, J. (1998) Questions and Prompts for Mathematical Thinking Derby. Association of Teachers of Mathematics
Wilson, R. (2002) Four Colours Suffice: How the Map Problem was Solved London. Penguin Books

Web Links
Euclid (EUC) http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html#props
Non-Euclidean Geometry (NEG) http://www-history.mcs.st-andrews.ac.uk/HistTopics/Non-Euclidean_geometry.html
MacTutor has many useful pages on Indian and Chinese mathematicians http://www-history.mcs.st-and.ac.uk/history/
For India and China, there are resources on the French website: http://www.dma.ens.fr/culturemath/index.html . The outstanding scholars here are Agathe Keller for India and Karine Chemla for China. There is very little from these two researchers available in English at the moment.
Snezana Lawrence of Simon Langton School in Kent offers many ideas for teaching mathematics using history on her website: http://www.mathsisgoodforyou.com/
She has also set up a History section on the NCETM website at: http://www.ncetm.org.uk/ and then search for the History of Mathematics Community