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Boethius (480 – 524) The Saga of Mathematics A Brief History Boethius (480 – 524) Boethius became an orphan when he was Europe Smells the Coffee seven years old. He was extremely well educated. Boethius was a Chapter 6 philosopher, poet, mathematician, statesman, and (perhaps) martyr. Lewinter & Widulski The Saga of Mathematics 1 Lewinter & Widulski The Saga of Mathematics 2 Boethius (480 – 524) Boethius (480 – 524) He is best known as a translator of and This shows Boethius commentator on Greek writings on logic and calculating with mathematics (Plato, Aristotle, Nichomachus). Arabic numerals His mathematics texts were the best available competing with and were used for many centuries at a time Pythagoras using an when mathematical achievement in Europe abacus. was at a low point. It is from G. Reisch, Boethius’ Arithmetic taught medieval scholars Margarita about Pythagorean number theory. Philosophica (1508). Lewinter & Widulski The Saga of Mathematics 3 Lewinter & Widulski The Saga of Mathematics 4 Boethius (480 – 524) Boethius (480 – 524) Boethius was a main source of material for One of the first musical works to be printed the quadrivium, which was introduced into was Boethius's De institutione musica, written monasteries and consisted of arithmetic, in the early sixth century. geometry, astronomy, and the theory of It was for medieval authors, from around the music. ninth century on, the authoritative document on Greek music-theoretical thought and Boethius wrote about the relation of music systems. and science, suggesting that the pitch of a For example, Franchino Gaffurio in Theorica note one hears is related to the frequency of musica (1492) acknowledged Boethius as the sound. authoritative source on music theory. Lewinter & Widulski The Saga of Mathematics 5 Lewinter & Widulski The Saga of Mathematics 6 Lewinter & Widulski 1 The Saga of Mathematics A Brief History Boethius (480 – 524) Gregorian Chant His writings and translations were the main The term Gregorian chant is named after works on logic in Europe becoming known Pope Gregory I (590–604 AD). collectively as Logica vetus. He is credited with arranging a large number Boethius' best-known work is the of choral works, which arose in the early Consolations of Philosophy which was written centuries of Christianity in Europe. while he was in prison. Gregorian chant is monophonic, that is, music It looked at the “questions of the nature of composed with only one melodic line without good and evil, of fortune, chance, or accompaniment. freedom, and of divine foreknowledge.” Lewinter & Widulski The Saga of Mathematics 7 Lewinter & Widulski The Saga of Mathematics 8 Gregorian Chant Polyphony As with the melodies of folk music, the chants Although the majority of medieval polyphonic probably changed as they were passed down works are anonymous - the names of the orally from generation to generation. authors were either not preserved or simply never known - there are some composers Polyphony is music where two or more whose work was so significant that their melodic lines are heard at the same time in a names were recorded along with their work. harmony. Hildegard von Bingen (1098 - 1179) Polyphony didn't exist (or it wasn't on record) Perotin (1155 - 1377) until the 11th century. Guillame de Machau (1300 - 1377) John Dumstable (1385 - 1453) Lewinter & Widulski The Saga of Mathematics 9 Lewinter & Widulski The Saga of Mathematics 10 The Dark Ages The Middle Ages The Dark Ages, formerly a designation for the With the collapse of the Roman Empire, entire period of the Middle Ages, now refers Christianity became the standard-bearer of usually to the period c.450–750, also known Western civilization. as the Early Middle Ages. The papacy gradually gained secular authority; monastic communities had the Medieval Europe was a large geographical effect of preserving antique learning. region divided into smaller and culturally By the 8th century, culture centered on diverse political units that were never totally Christianity had been established; it dominated by any one authority. incorporated both Latin traditions and German institutions. Lewinter & Widulski The Saga of Mathematics 11 Lewinter & Widulski The Saga of Mathematics 12 Lewinter & Widulski 2 The Saga of Mathematics A Brief History The Middle Ages The High Middle Ages The empire created by Charlemagne As Europe entered the period known as the illustrated this fusion. High Middle Ages, the church became the However, the empire's fragile central unifying institution. authority was shattered by a new wave of Militant religious zeal was expressed in the invasions. Crusades. Feudalism became the typical social and political organization of Europe. Security and prosperity stimulated intellectual life, newly centered in burgeoning The new framework gained stability from the 11th century, as the invaders became universities, which developed under the Christian. auspices of the church. Lewinter & Widulski The Saga of Mathematics 13 Lewinter & Widulski The Saga of Mathematics 14 The High Middle Ages The High Middle Ages From the Crusades and other sources came Christian Europe finally began to assimilate contact with Arab culture, which had the lively intellectual traditions of the Jews preserved works of Greek authors whose and Arabs. writings had not survived in Europe. Translations of ancient Greek texts (and the fine Arabic commentaries on them) into Latin Philosophy, science, and mathematics from made the full range of Aristotelean the Classical and Hellenistic periods were philosophy available to Western thinkers. assimilated into the tenets of the Christian Aristotle, long associated with heresy, was faith and the prevailing philosophy of adapted by St. Thomas Aquinas to Christian scholasticism. doctrine. Lewinter & Widulski The Saga of Mathematics 15 Lewinter & Widulski The Saga of Mathematics 16 The High Middle Ages Thomas Aquinas (1225 – 1272) Christian values pervaded scholarship St. Thomas Aquinas was an Italian and literature, especially Medieval Latin philosopher and literature, but Provencal literature also theologian, Doctor of the Church, known as reflected Arab influence, and other the Angelic Doctor. flourishing medieval literatures, He is the greatest figure of scholasticism - including German, Old Norse, and philosophical study as Middle English, incorporated the practiced by Christian thinkers in medieval materials of pre-Christian traditions. universities. Lewinter & Widulski The Saga of Mathematics 17 Lewinter & Widulski The Saga of Mathematics 18 Lewinter & Widulski 3 The Saga of Mathematics A Brief History Thomas Aquinas (1225 – 1272) Thomas Aquinas (1225 – 1272) He is one of the principal saints of the Roman Aquinas's unfinished Summa Theologica Catholic Church, and founder of the system (1265-1273) represents the most complete declared by Pope Leo XIII to be the official statement of his philosophical system. Catholic philosophy. The sections of greatest interest include his St. Thomas Aquinas held that reason and views on the nature of god, including the five faith constitute two harmonious realms in ways to prove god's existence, and his which the truths of faith complement those of exposition of natural law. reason; both are gifts of God, but reason has an autonomy of its own. Lewinter & Widulski The Saga of Mathematics 19 Lewinter & Widulski The Saga of Mathematics 20 Natural Law The Existence of God Belief that the principles of human conduct Attempts to prove the existence of god have can be derived from a proper understanding been a notable feature of Western of human nature in the context of the philosophy. universe as a rational whole. The cosmological argument Aquinas held that even the divine will is The ontological argument conditioned by reason. The teleological argument Thus, the natural law provides a non- The moral argument revelatory basis for all human social conduct. The most serious atheological argument is the problem of evil. Lewinter & Widulski The Saga of Mathematics 21 Lewinter & Widulski The Saga of Mathematics 22 The Cosmological Argument The Ontological Argument An attempt to prove the existence of god by Ontological arguments are arguments, for the appeal to contingent facts about the world. conclusion that God exists, from premises which are supposed to derive from some The first of Aquinas's five ways (borrowed source other than observation of the world - from Aristotle's Metaphysics), begins from the e.g., from reason alone. fact that something is in motion, since Ontological arguments are arguments from everything that moves must have been put nothing but analytic, a priori and necessary premises to the conclusion that God exists. into motion by something else but the series St. Anselm of Canterbury claims to derive the of prior movers cannot extend infinitely, there existence of God from the concept of a “being must be a first mover (which is god). than which no greater can be conceived.” Lewinter & Widulski The Saga of Mathematics 23 Lewinter & Widulski The Saga of Mathematics 24 Lewinter & Widulski 4 The Saga of Mathematics A Brief History The Ontological Argument The Ontological Argument St. Anselm reasoned that, if such a being fails In the 17th century, Rene Descartes to exist, then a greater being – namely, a endorsed a different version of this argument. being than which no greater can be conceived, and which exists – can be In the early 18th century, Gottfried Leibniz conceived. attempted to fill what he took to be a shortcoming in Descartes' view. But this would be absurd, nothing can be greater than a being than which no greater Recently, Kurt Gödel, best known for his can be conceived. incompleteness theorems, sketched a revised So a being than which no greater can be version of this argument. conceived, that is, God, exists. Lewinter & Widulski The Saga of Mathematics 25 Lewinter & Widulski The Saga of Mathematics 26 The Teleological Argument The Teleological Argument Based upon an observation of the regularity Let X represent a given species of animal or or beauty of the universe. for a particular organ (e.g. the eye) or a capability of a given species: Employed by Marcus Cicero (106-43 BC), X is very complicated and/or purposeful.
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