Boethius (480 524) Boethius became an orphan when he was Europe - PDF document

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 competing with His mathematics texts were the best available Pythagoras using an and were used for many centuries at a time abacus. when mathematical achievement in Europe was at a low point. It is from G. Reisch, Margarita Boethius’ Arithmetic taught medieval scholars Philosophica (1508). about Pythagorean number theory. Lewinter & Widulski The Saga of Mathematics 3 Lewinter & Widulski The Saga of Mathematics 4 Boethius (480 – 524) Boethius (480 – 524) One of the first musical works to be printed Boethius was a main source of material for was Boethius's De institutione musica, written the quadrivium, which was introduced into in the early sixth century. monasteries and consisted of arithmetic, It was for medieval authors, from around the geometry, astronomy, and the theory of ninth century on, the authoritative document music. 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 works are anonymous - the names of the probably changed as they were passed down authors were either not preserved or simply orally from generation to generation. 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 With the collapse of the Roman Empire, The Dark Ages, formerly a designation for the Christianity became the standard-bearer of entire period of the Middle Ages, now refers Western civilization. usually to the period c.450–750, also known The papacy gradually gained secular as the Early Middle Ages. 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 Security and prosperity stimulated intellectual political organization of Europe. life, newly centered in burgeoning The new framework gained stability from the universities, which developed under the 11th century, as the invaders became auspices of the church. Christian. 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 the lively intellectual traditions of the Jews contact with Arab culture, which had and Arabs. preserved works of Greek authors whose Translations of ancient Greek texts (and the writings had not survived in Europe. 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) St. Thomas Aquinas Christian values pervaded scholarship 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 The moral argument Thus, the natural law provides a non- The most serious atheological argument is revelatory basis for all human social conduct. the problem of evil. Lewinter & Widulski The Saga of Mathematics 21 Lewinter & Widulski The Saga of Mathematics 22 The Cosmological Argument The Ontological Argument Ontological arguments are arguments, for the An attempt to prove the existence of god by conclusion that God exists, from premises appeal to contingent facts about the world. which are supposed to derive from some The first of Aquinas's five ways (borrowed source other than observation of the world - e.g., from reason alone. from Aristotle's Metaphysics), begins from the Ontological arguments are arguments from fact that something is in motion, since nothing but analytic, a priori and necessary everything that moves must have been put 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