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In ancient history, the concepts of chance and randomness were intertwined with that of fate. Many ancient peoples threw dice to determine fate, and this later evolved into games of chance. At the same time, most ancient cultures used various methods of divination to attempt to circumvent randomness and fate.〔''Handbook to Life in Ancient Rome'', Lesley Adkins, 1998 ISBN 0-19-512332-8 p. 279〕〔''Religions of the Ancient World'', Sarah Iles Johnston, 2004 ISBN 0-674-01517-7 p. 370〕 The Chinese were perhaps the earliest people to formalize odds and chance 3,000 years ago. The Greek philosophers discussed randomness at length, but only in non-quantitative forms. It was only in the sixteenth century that Italian mathematicians began to formalize the odds associated with various games of chance. The invention of modern calculus had a positive impact on the formal study of randomness. In the 19th century the concept of entropy was introduced in physics. The early part of the twentieth century saw a rapid growth in the formal analysis of randomness, and mathematical foundations for probability were introduced, leading to its axiomatization in 1933. At the same time, the advent of quantum mechanics changed the scientific perspective on determinacy. In the mid to late 20th-century, ideas of algorithmic information theory introduced new dimensions to the field via the concept of algorithmic randomness. Although randomness had often been viewed as an obstacle and a nuisance for many centuries, in the twentieth century computer scientists began to realize that the ''deliberate'' introduction of randomness into computations can be an effective tool for designing better algorithms. In some cases, such randomized algorithms are able to outperform the best deterministic methods. ==Antiquity to the Middle Ages== In ancient history, the concepts of chance and randomness were intertwined with that of fate. Pre-Christian people along the Mediterranean threw dice to determine fate, and this later evolved into games of chance.〔''What is Random?: Chance and Order in Mathematics and Life'', Edward J. Beltrami, 1999, Springer ISBN 0-387-98737-1 pp. 2-4〕 There is also evidence of games of chance played by ancient Egyptians, Hindus and Chinese, dating back to 2100 BC.〔''Encyclopedia of Leisure and Outdoor Recreation'', John Michael Jenkins, 2004 ISBN 0-415-25226-1 p. 194〕 The Chinese used dice before the Europeans, and have a long history of playing games of chance.〔''Audacious Angles of China'', Elise Mccormick, 2007 ISBN 1-4067-5332-7 p. 158〕 Over 3,000 years ago, the problems concerned with the tossing of several coins were considered in the I Ching, one of the oldest Chinese mathematical texts, that probably dates to 1150 BC. The two principal elements yin and yang were combined in the I Ching in various forms to produce ''Heads and Tails'' permutations of the type HH, TH, HT, etc. and the Chinese seem to have been aware of Pascal's triangle long before the Europeans formalized it in the 17th century.〔The Nature and Growth of Modern Mathematics'', Edna Ernestine Kramer, 1983 ISBN p. 313〕 However, Western philosophy focused on the non-mathematical aspects of chance and randomness until the 16th century. The development of the concept of chance throughout history has been very gradual. Historians have wondered why progress in the field of randomness was so slow, given that humans have encountered chance since antiquity. Deborah Bennett suggests that ordinary people face an inherent difficulty in understanding randomness, although the concept is often taken as being obvious and self-evident. She cites studies by Kahneman and Tversky; these concluded that statistical principles are not learned from everyday experience because people do not attend to the detail necessary to gain such knowledge.〔''Randomness'', Deborah J. Bennett, Harvard University Press, 1998. ISBN 0-674-10745-4 pp. 8-9 and 24〕 The Greek philosophers were the earliest Western thinkers to address chance and randomness. Around 400 BC, Democritus presented a view of the world as governed by the unambiguous laws of order and considered randomness as a subjective concept that only originated from the inability of humans to understand the nature of events. He used the example of two men who would send their servants to bring water at the same time to cause them to meet. The servants, unaware of the plan, would view the meeting as random.〔''Design and Analysis of Randomized Algorithms'', Juraj Hromkovič, 2005 ISBN 3-540-23949-9 p. 1〕 Aristotle saw chance and necessity as opposite forces. He argued that nature had rich and constant patterns that could not be the result of chance alone, but that these patterns never displayed the machine-like uniformity of necessary determinism. He viewed randomness as a genuine and widespread part of the world, but as subordinate to necessity and order.〔''Aristotle's Physics: a Guided Study'', Joe Sachs, 1995 ISBN 0-8135-2192-0 p. 70〕 Aristotle classified events into three types: ''certain'' events that happen necessarily; ''probable'' events that happen in most cases; and ''unknowable'' events that happen by pure chance. He considered the outcome of games of chance as unknowable.〔''A History of Probability and Statistics and Their Applications before 1750'', Anders Hald, 2003 ISBN 0-471-47129-1 p. 30〕 Around 300 BC Epicurus proposed the concept that randomness exists by itself, independent of human knowledge. He believed that in the atomic world, atoms would ''swerve'' at random along their paths, bringing about randomness at higher levels.〔''Epicurus: an Introduction'', John M. Rist, 1972 ISBN 0-521-08426-1 p. 52〕 For several centuries thereafter, the idea of chance continued to be intertwined with fate. Divination was practiced in many cultures, using diverse methods. The Chinese analyzed the cracks in turtle shells, while the Germans, who according to Tacitus had the highest regards for lots and omens, utilized strips of bark.〔''The Age of Chance'', Gerda Reith, 2000 ISBN 0-415-17997-1 p. 15; Tac. Germ. 10〕 In the Roman Empire, chance was personified by the Goddess Fortuna. The Romans would partake in games of chance to simulate what Fortuna would have decided. In 49 BC, Julius Caesar allegedly decided on his fateful decision to cross the Rubicon after throwing dice.〔''What is Random?: Chance and Order in Mathematics and Life'', Edward J. Beltrami, 1999, Springer ISBN 0-387-98737-1 pp. 3-4〕 Aristotle's classification of events into the three classes: ''certain'', ''probable'' and ''unknowable'' was adopted by Roman philosophers, but they had to reconcile it with deterministic Christian teachings in which even events unknowable to man were considered to be predetermined by God. About 960 Bishop Wibold of Cambrai correctly enumerated the 56 different outcomes (without permutations) of playing with three dice. No reference to playing cards has been found in Europe before 1350. The Church preached against card playing, and card games spread much more slowly than games based on dice.〔''A History of Probability and Statistics and Their Applications before 1750'', Anders Hald, 2003 ISBN 0-471-47129-1 pp. 29-36〕 The Christian Church specifically forbade divination; and wherever Christianity went, divination lost most of its old-time power.〔A general history of the Christian church'' Volume 2 by Joseph Priestley 1804 ASIN B002KW4M6O page 11〕〔Catholic encyclopedia ()〕 Over the centuries, many Christian scholars wrestled with the conflict between the belief in free will and its implied randomness, and the idea that God knows everything that happens. Saints Augustine and Aquinas tried to reach an accommodation between foreknowledge and free will, but Martin Luther argued against randomness and took the position that God's omniscience renders human actions unavoidable and determined.〔''The Case for Humanism'', Lewis Vaughn, Austin Dacey, 2003 ISBN 0-7425-1393-9 p. 81〕 In the 13th century, Thomas Aquinas viewed randomness not as the result of a single cause, but of several causes coming together by chance. While he believed in the existence of randomness, he rejected it as an explanation of the end-directedness of nature, for he saw too many patterns in nature to have been obtained by chance.〔''The treatise on the divine nature: Summa theologiae I'', 1-13, by Saint Thomas Aquinas, Brian J. Shanley, 2006 ISBN 0-87220-805-2 p. 198〕 The Greeks and Romans had not noticed the magnitudes of the relative frequencies of the games of chance. For centuries, chance was discussed in Europe with no mathematical foundation and it was only in the 16th century that Italian mathematicians began to discuss the outcomes of games of chance as ratios.〔''A History of Probability and Statistics and Their Applications before 1750'', Anders Hald, 2003 ISBN 0-471-47129-1 pp. 30-4〕〔''World of Scientific Discovery'', Kimberley A. McGrath and Bridget Traverspage, 1999 ISBN 0-7876-2760-7 p. 893〕〔''Randomness'', Deborah J. Bennett, Harvard University Press, 1998. ISBN 0-674-10745-4 p. 8〕 In his 1565 ''Liber de Lude Aleae'' (a gambler's manual published after his death) Gerolamo Cardano wrote one of the first formal tracts to analyze the odds of winning at various games.〔''A Dictionary of Scientists'', John Daintith, Derek Gjertsen, 1999 ISBN 0-19-280086-8 p. 88〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「History of randomness」の詳細全文を読む スポンサード リンク
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