In Progress Essay

History of math Research paper & presentation notes:-

-In EARLY GREEK
Infinity sign called the Laminiscate (ribbon-Latin). Invented in 1655 by John Wallis * Blaise Pascal: “If I consider the short period of time in which my life is intertwined with the eternity preceding and following it, and if I consider the small space in which I dwell, and I consider just that which I see, which is list in infinite space, of which I know nothing, and which knows nothing of me, then I am struck with wonder that I live here and not there.” * However, the rational structure of infinity still needs to be clarified. Many philosophers and mathematicians have struggled to give a rational account of infinity. * The most famous quote derives from German Mathematician David Hilbert (1862-1943) : “The infinite has always stirred the emotions of mankind more deeply than any other question; the infinite has stimulated and fertilized reason as few other ideas have” (Hilbert 1925) (infinity a research) * * Perhaps the most interesting-and most important- feature of Greek mathematics is its treatment of infinity. The Greeks laid the foundation for a rigorous treatment of infinite processes in 19th century calculus. * The most original contributors to the theory of infinity in ancient times were the theory of proportions and the method of exhaustion. Both were devised by Eudoxus and expounded in book V of Euclid’s Elements. * Theory of Proportion develops the idea that a “quantity” lambda (what we would now call a real number) can be known by its position among the rational numbers. I.e. Lambda is known if we know rational number less than and greater than lambda. * The method of exhaustion generalizes this idea from “quantities” to regions of the plane or space. A region becomes “known” (in area or volume) when its position among known areas or volumes is known. For example, we know the area of a circle when we know the areas of the polygons inside and outside of it. * Using this method, Euclid found that the volume of a tetrahedron equals 1/3 of its base area times its height and Archimedes found the area of a parabolic segment. Both of them relied on an infinite process that is fundamental to many calculations of area and volume: the summation of an infinite geometric series. * * Lecture : * To ancient Greeks infinity was problematic and its role in mathematics was unclear * E.g sequence 1,2,3……. Was unending but could not be encompassed ( they avoided saying lets look all the natural numbers at once, they avoided it) * -------------------------------------DARTHMOUTH * The Greek word for infinity was apeiron, which literally means unbounded, but can also mean infinite, indefinite, or undefined. Apeiron was a negative, even pejorative, word. The original chaos out of which the world was formed was apeiron. An arbitrary crooked line was apeiron. A dirty crumpled handkerchief was apeiron. Thus, apeiron need not only mean infinitely large, but can also mean totally disordered, infinitely complex, subject to no finite determination. In Aristotle's words, "... being infinite is a privation, not a perfection but the absence of a limit..."3 * ------------------------------------------------------------------------------------------------------------------------------------------------ Infinity article * We should begin our account of infinity with the "fifth-century Eleatic" Zeno. The early Greeks had come across the problem of infinity at an early stage in their development of mathematics and science. In their study of matter they realized the fundamental question: can one continue to divide matter into smaller and smaller pieces or will one reach a tiny piece which cannot be divided further. Pythagoras had argued that "all is number" and his universe was made up of finite natural numbers. Then there were Atomists who believed that matter was composed of an infinite number of

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