The Nature Of Mathematics 12th Edition Pdf

[Janeth Counter]

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European science in the Middle Ages comprised the study of nature, mathematics and natural philosophy in medieval Europe. Following the fall of the Western Roman Empire and the decline in knowledge of Greek, Christian Western Europe was cut off from an important source of ancient learning. Although a range of Christian clerics and scholars from Isidore and Bede to Jean Buridan and Nicole Oresme maintained the spirit of rational inquiry, Western Europe would see a period of scientific decline during the Early Middle Ages. However, by the time of the High Middle Ages, the region had rallied and was on its way to once more taking the lead in scientific discovery. Scholarship and scientific discoveries of the Late Middle Ages laid the groundwork for the Scientific Revolution of the Early Modern Period.

According to Pierre Duhem, who founded the academic study of medieval science as a critique of the Enlightenment-positivist theory of a 17th-century anti-Aristotelian and anticlerical scientific revolution, the various conceptual origins of that alleged revolution lay in the 12th to 14th centuries, in the works of churchmen such as Thomas Aquinas and Buridan.[1]

As Roman imperial power effectively ended in the West during the 5th century, Western Europe entered the Middle Ages with great difficulties that affected the continent's intellectual production dramatically. Most classical scientific treatises of classical antiquity written in Greek were unavailable, leaving only simplified summaries and compilations. Nonetheless, Roman and early medieval scientific texts were read and studied, contributing to the understanding of nature as a coherent system functioning under divinely established laws that could be comprehended in the light of reason. This study continued through the Early Middle Ages, and with the Renaissance of the 12th century, interest in this study was revitalized through the translation of Greek and Arabic scientific texts. Scientific study further developed within the emerging medieval universities, where these texts were studied and elaborated, leading to new insights into the phenomena of the universe. These advances are virtually unknown to the lay public of today, partly because most theories advanced in medieval science are today obsolete, and partly because of the caricature of the Middle Ages as a supposedly "Dark Age" which placed "the word of religious authorities over personal experience and rational activity."[2]

In the ancient world, Greek had been the primary language of science. Even under the Roman Empire, Latin texts drew extensively on Greek work, some pre-Roman, some contemporary; while advanced scientific research and teaching continued to be carried on in the Hellenistic side of the empire, in Greek. Late Roman attempts to translate Greek writings into Latin had limited success.[3]

As the knowledge of Greek declined during the transition to the Middle Ages, the Latin West found itself cut off from its Greek philosophical and scientific roots. Most scientific inquiry came to be based on information gleaned from sources which were often incomplete and posed serious problems of interpretation. Latin-speakers who wanted to learn about science only had access to books by such Roman writers as Calcidius, Macrobius, Martianus Capella, Boethius, Cassiodorus, and later Latin encyclopedists. Much had to be gleaned from non-scientific sources: Roman surveying manuals were read for what geometry was included.[4]

De-urbanization reduced the scope of education and by the 6th century teaching and learning moved to monastic and cathedral schools, with the center of education being the study of the Bible.[5] Education of the laity survived modestly in Italy, Spain, and the southern part of Gaul, where Roman influences were most long-lasting. In the 7th century, learning began to emerge in Ireland and the Celtic lands, where Latin was a foreign language and Latin texts were eagerly studied and taught.[6]

The leading scholars of the early centuries were clergymen for whom the study of nature was but a small part of their interest. They lived in an atmosphere which provided little institutional support for the disinterested study of natural phenomena. The study of nature was pursued more for practical reasons than as an abstract inquiry: the need to care for the sick led to the study of medicine and of ancient texts on drugs,[7] the need for monks to determine the proper time to pray led them to study the motion of the stars,[8] the need to compute the date of Easter led them to study and teach rudimentary mathematics and the motions of the Sun and Moon.[9] Modern readers may find it disconcerting that sometimes the same works discuss both the technical details of natural phenomena and their symbolic significance.[10]

Around 800, Charles the Great, assisted by the English monk Alcuin of York, undertook what has become known as the Carolingian Renaissance, a program of cultural revitalization and educational reform. The chief scientific aspect of Charlemagne's educational reform concerned the study and teaching of astronomy, both as a practical art that clerics required to compute the date of Easter and as a theoretical discipline.[11] From the year 787 on, decrees were issued recommending the restoration of old schools and the founding of new ones throughout the empire. Institutionally, these new schools were either under the responsibility of a monastery, a cathedral or a noble court.

The scientific work of the period after Charlemagne was not so much concerned with original investigation as it was with the active study and investigation of ancient Roman scientific texts.[12] This investigation paved the way for the later effort of Western scholars to recover and translate ancient Greek texts in philosophy and the sciences.

Beginning around the year 1050, European scholars built upon their existing knowledge by seeking out ancient learning in Greek and Arabic texts which they translated into Latin. They encountered a wide range of classical Greek texts, some of which had earlier been translated into Arabic, accompanied by commentaries and independent works by Islamic thinkers.[13]

Gerard of Cremona is a good example: an Italian who traveled to Spain to copy a single text, he stayed on to translate some seventy works.[14] His biography describes how he came to Toledo: "He was trained from childhood at centers of philosophical study and had come to a knowledge of all that was known to the Latins; but for love of the Almagest, which he could not find at all among the Latins, he went to Toledo; there, seeing the abundance of books in Arabic on every subject and regretting the poverty of the Latins in these things, he learned the Arabic language, in order to be able to translate."[15]

This period also saw the birth of medieval universities, which benefited materially from the translated texts and provided a new infrastructure for scientific communities. Some of these new universities were registered as an institution of international excellence by the Holy Roman Empire, receiving the title of Studium Generale. Most of the early Studia Generali were found in Italy, France, England, and Spain, and these were considered the most prestigious places of learning in Europe. This list quickly grew as new universities were founded throughout Europe. As early as the 13th century, scholars from a Studium Generale were encouraged to give lecture courses at other institutes across Europe and to share documents, and this led to the current academic culture seen in modern European universities.

Scholastics believed in empiricism and supporting Roman Catholic doctrines through secular study, reason, and logic. The most famous was Thomas Aquinas (later declared a "Doctor of the Church"), who led the move away from the Platonic and Augustinian and towards Aristotelianism (although natural philosophy was not his main concern). Meanwhile, precursors of the modern scientific method can be seen already in Grosseteste's emphasis on mathematics as a way to understand nature and in the empirical approach admired by Roger Bacon.

Grosseteste was the founder of the famous Oxford Franciscan school. He built his work on Aristotle's vision of the dual path of scientific reasoning. Concluding from particular observations into a universal law, and then back again: from universal laws to prediction of particulars. Grosseteste called this "resolution and composition". Further, Grosseteste said that both paths should be verified through experimentation in order to verify the principals. These ideas established a tradition that carried forward to Padua and Galileo Galilei in the 17th century.

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