History of Mathematics: The Renaissance PDF

History of Mathematics: The Renaissance Europe Prof. Dr. Deniz Karlı Department of Mathematics IŞIK UNIVERSITY Version #: 28–11–2023 How did the center of science move to Europe? Mathematics entered into Europe through 3 different means: i. By means of Christians who established 4 kingdoms over 200...

History of Mathematics: The Renaissance Europe Prof. Dr. Deniz Karlı Department of Mathematics IŞIK UNIVERSITY Version #: 28–11–2023 How did the center of science move to Europe? Mathematics entered into Europe through 3 different means: i. By means of Christians who established 4 kingdoms over 200 year in the Middle East, ii. By means of students from western civilizations who studied in Arabic schools, iii. Through Andalusia. The biggest factor was Andalusia. Why Andalusia? Education was widespread in Andalusia. Philosophy, Chemistry and Medicine was developed, which created a proper background for positive sciences. For example, Cordoba had a central library with a collection of 400 thousand books, 17 schools and many public libraries in 11th century. Christian and Jewish students were studying here. After Spanish conquered Toledo, Bishop of Toledo established a big translation group of Jewish students who studied in Arabic schools and translated many scientific resources into Latin. Until 12th century, schools were mainly churches in Europe. However, as of middle of 12th century students in Italy come together in communities called “universita” which later became university in modern sense. Scholars, who taught in universita, were Italians who previously studied Arabic schools. Students, educated in these communities laid the foundations of today’s famous universities in Europe: University of Cologne - Germany, Sorbone University - France, Oxford and Cambrigde Universities - UK. Europeans’ main resources between 1200-1500 were the books which were translated from Arabic. This led to the case that Europeans mainly interested in the same questions as the ones of Islamic Mathematics society: some geometry questions, finding roots of 3rd degree polynomials, some questions of Number Theory. After 1450, some Greek materials were brought to Italy. Then translation of ancient Greek books became more dominant as primary resource for research. After 1600, Arabic resources were replaced completely. Original research in Mathematics started after 1500s. The modern numerals, called Hindu–Arabic numeral system, (1,2,...,9,0) were introduced to Europe by Fibonacci (Leonordo de Pisa, 1175 - 1250) through his book called ”Liber Acci” 400 years later than al-Khwarizmi. Fibonacci explains how to do arithmetic with this new representation. These number representations wasn’t used widely by Europeans until French revolution. Indeed they were banned by the church time to time. Hindu–Arabic numeral system revolutionized the development of science in Europe, since it was difficult to write in Roman numerals. They were widely used in public after French Revolution. Between 1500 - 1600 There were 2 main achievement in this century. i. First is finding roots of 3rd degree polynomials algebraically: This was found by Tartaglia (1499 - 1557), however published by Cardano (1501 - 1576). (Tartaglia accused Cardano of stealing his research.) Complex numbers first appeared in the formula that gives the root of 3rd degree polynomials. however, they were not understood at that time. ii. Second is the Algebra book by F. De Viete (1540 - 1603). In this book, letters such as a,b,c,.. were used for constants, and letters such as x,y,.. were used for unknowns for the first time. Algebra problems were described verbally without these letters before. Between 1600 - 1700 There were 3 main achievement in this century. i. 5 centuries after S. al-Tusi, efforts in finding maximum and minimum of a curve led Pierre de Fermat (1601 - 1665) to invention of derivative in 1636. Mathematics society were grown enough to understand the idea at this time. ii. Analytical Geometry and cartesian coordinate system were introduced by René Descartes (1596 - 1650). The word of ”cartesian” comes from Descartes. iii. Most importantly, the relation between ”Derivative” and ”Integral” (The Fundamental Theorem of Calculus) was found by Isaac Newton (1643 - 1727) and Gottfried Leibniz (1646 - 1716) independently from each other. This is the birth of modern Calculus. The Fundemental Theorem of Calculus Integral vs Derivative 6 5 5 4 4 O 3 3 2 2 1 1 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 ”The change in area under the curve” is ”the function value at that point”. 2.5 The golden era of Mathematics began in Europe after the birth of Calculus. Mathematics was rather restricted before derivative. Afterwards, it became a universal scientific major. Differential equations (equations including derivatives) explains nature in terms of mathematical language. hence Physics and Mathematics entered a whole new era after invention derivative. Physical sciences and Engineering sciences were born with Calculus. History of Mathematics: The Classical Age The Golden Age Prof. Dr. Deniz Karlı Department of Mathematics IŞIK UNIVERSITY This era is one of the richest periods of history of Mathematics. IŞIK UNIVERSITY ”History of Mathematics: The Golden Age of Islamic World & The Renaissance and Classical Era in Europe” by Deniz Karlı 13 Classical Age: 1700 - 1800 Amoung the great names in this century, we can bring forward 4 names to attention: Euler, Laplace, Lagrange and D’Alembert. Euler may be defined as the most productive scientist in all of human history. He was born in Basel (Switzerland) and spent his life in Petersburg and Berlin. Leonhard Euler (1707 - 1783) Euler turned Mathematical Analysis into a mathematical discipline. IŞIK UNIVERSITY ”History of Mathematics: The Golden Age of Islamic World & The Renaissance and Classical Era in Europe” by Deniz Karlı 14 Euler He applied power of Calculus to numerous areas from Number Theory to differential equations and applications in Engineering. Mathematics became universal after Euler. Euler prepared more than 30,000 pages of scientific research. Some of these kept being published after his death. Even after 50 year of his death, some of his papers were still being published. Even today, some of Euler’s work is being cited by scientists. Euler’s Formula: An equation with three important numbers e, π and i together IŞIK UNIVERSITY ”History of Mathematics: The Golden Age of Islamic World & The Renaissance and Classical Era in Europe” by Deniz Karlı 15 Pierre-Simon Laplace (1749 - 1827) He was Napoleon’s examiner when Napoleon attended the École Militaire in Paris in 1784. Laplace was a French scholar whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. His five-volume Mécanique céleste (Celestial Mechanics) was one of most comprehensive works in Mechanics. Laplace formulated Laplace’s equation, and pioneered the Laplace transform which appears in many branches of mathematical physics. “Theorie Analytique des Probabilites” was the first important book of Probability Theory. IŞIK UNIVERSITY ”History of Mathematics: The Golden Age of Islamic World & The Renaissance and Classical Era in Europe” by Deniz Karlı 16 Classical Age: 1800 - 1900 Gauss Dirichlet Cauchy Fourier Riemann Abel Weierstrass Galois This century hosted for many brilliant minds of Mathematics, such as Carl Friedrich Gauss, Augustin Louis Cauchy, Bernhard Riemann, Karl Weierstrass, Peter Gustav Lejeune Dirichlet, Jean Baptiste Joseph Fourier, Niels Henrik Abel, Évariste Galois and many other. They solved centuries old questions and raised revolutionary new questions. They matured Mathematics so that it is ready for the Modern Era. IŞIK UNIVERSITY ”History of Mathematics: The Golden Age of Islamic World & The Renaissance and Classical Era in Europe” by Deniz Karlı 17 Classical Age: 1800 - 1900 - Big Crisis At the beginning of 1800s, Mathematics was in a deep crisis. The reasons were that i. there appeared a term called ”infinitesimally small” after Fermat’s introduction of derivative. This term was used by mathematicians inconsistently. ii. At this time, the notion of ”limit” was not defined yet. iii. Altough it had been used for 100 year then, the term ”function” was not properly defined and hence it was used in different contexts by various mathematicians. IŞIK UNIVERSITY ”History of Mathematics: The Golden Age of Islamic World & The Renaissance and Classical Era in Europe” by Deniz Karlı 18 Classical Age: 1800 - 1900 - Big Crisis iv. Continuity of functions were not understood. v. Integration was considered as just the inverse of derivative. Hence it couldn’t be studied independent from derivative. vi. Theory of complex functions was not born yet. IŞIK UNIVERSITY ”History of Mathematics: The Golden Age of Islamic World & The Renaissance and Classical Era in Europe” by Deniz Karlı 19 Crisis Solved Cauchy Cauchy (1789 - 1855) defined the notion of limit as we know it today. Using the definition of limit notion, he also properly defined derivative, continuous function and integral of a continuous function, Cauchy’s work saved the society of Mathematics from the crisis due to ”infinitesimally small”. Together with Riemann and Weierstrass, Cauchy created Theory of Complex Functions, which became one of the fundamental disciplines of Mathematics. IŞIK UNIVERSITY ”History of Mathematics: The Golden Age of Islamic World & The Renaissance and Classical Era in Europe” by Deniz Karlı Crisis Solved Dirichlet (1805 - 1859) defined the notion functions, which ended another factor in te crisis. One of the greatest scientists of all time, Gauss, proved The Fundamental Theorem of Algebra, which was a great advancement in the century. Gauss contributed Number Theory, Differential Geometry, Mathematical Physics and Astronomy greatly. Gauss 20 IŞIK UNIVERSITY ”History of Mathematics: The Golden Age of Islamic World & The Renaissance and Classical Era in Europe” by Deniz Karlı 21 Crisis Solved Riemann Another one of the greatest mathematicians is Riemann. He created and contributed many areas of Mathematics. Only some of his contributions: Riemann Integral, Riemann Surfaces, Riemann Geometry, Differential Geometry, Riemann Hypothesis in Number Theory, Cauchy-Riemann Equations, Algebraic Geometry, Mathematical Physics, and Topology. There were many important contributions in this age. Due to the technicality of them we skip these. IŞIK UNIVERSITY ”History of Mathematics: The Golden Age of Islamic World & The Renaissance and Classical Era in Europe” by Deniz Karlı Characteristics of the Golden Age i. A great leap forward in mathematical sciences were observed in this age. ii. Many new fundamental theorems were born. iii. Certainty and clarity was important in the statements and proofs. iv. Conceptual approach replaced computational approach. 22 IŞIK UNIVERSITY ”History of Mathematics: The Golden Age of Islamic World & The Renaissance and Classical Era in Europe” by Deniz Karlı The Golden Age ended up with another big crisis. What is infinity? 23