Mathematical Processes and Perspectives

Question 1: Discuss 19th Century Mathematics.

Answer 1: In Euclidean geometry, given a line and a point not on that line, there is one and only one parallel to the given line through the given point. The Russian mathematician Nikolai Ivanovich Lobachevsky and his rival, the Hungarian mathematician Janos Bolyai, independently discovered hyperbolic geometry, where uniqueness of parallels no longer holds. In this geometry the sum of angles in a triangle add up to less than 180°. Elliptic geometry was developed later in the 19th century by the German mathematician Bernhard Riemann; here no parallel can be found and the angles in a trangle add up to more than 180°. Riemann also developed Riemannian geometry, which unifies and vastly generalizes the three types of geometry, and he defined the concept of a manifold, which generalize the ideas of curves and surfaces. These concepts proved important in Einstein's Theory of Relativity. Also in the nineteenth century William Rowan Hamilton developed noncommutative algebra and in addition to new directions in mathematics, older mathematics were given a stronger logical foundation, especially in the case of calculus by Augustin-Louis Cauchy and Karl Weierstrass. A new form of algebra, called Boolean algebra, was also developed by the British mathematician George Boole; a system in which the only numbers were 0 and 1 which today has important applications in computer science.

Question 2: Discuss and describe Mathematical development in the 20th Century.

Answer 2: In 1900, David Hilbert presented a list of 23 unsolved problems in mathematics at the International Congress of Mathematicians. These problems spanned many areas of mathematics and have formed a central focus for much of 20th century mathematics. In the 1910s, Srinivasa Aiyangar Ramanujan developed over 3,000 theorems, including properties of highly composite numbers, the partition function and its asymptotics, and mock theta functions. He also made major breakthroughs and discoveries in the areas of gamma functions, modular forms, divergent series, hypergeometric series and prime number theory. Wolfgang Haken and Kenneth Appel used a computer to prove the four color theorem in 1976. Andrew Wiles proved Fermat's last theorem in 1995. Entire new areas of mathematics such as mathematical logic, topology, complexity theory, and game theory changed the kinds of questions that could be answered by mathematical methods. The French Bourbaki Group attempted to bring all of mathematics into a coherent rigorous whole, having a controversial influence on mathematical education. Kurt Gödel proved that in any mathematical system that includes the integers, there are true statements that cannot be proved. Paul Cohen proved the independence of the continuum hypothesis from the standard axioms of set theory. By the end of the century, mathematics was even finding its way into art, as fractal geometry produced beautiful shapes never before seen.

Question 3: Describe mathematics.

Answer 3: Mathematics is the study of patterns, including concepts such as quantity, structure, space, uncertainty, and change. It evolved, through the use of abstraction and logical reasoning, to counting, calculation, measurement, and the systematic study of positions, shapes, and motions of physical objects. Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.

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