lecture17 - TheAgeofRigor ,whilebrilliantinits...

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HISTORY OF MATHEMATICS – LECTURE 17 – MONDAY 15 TH NOVEMBER The Age of Rigor We have already looked at examples of how the mathematics of Newton and Euler, while brilliant in its originality, lacked the rigor that characterizes the subject today. The concept of a formal proof was well known for making geometric arguments, but for calculus it was not possible to achieve the same level of precision until the concepts of a limit and convergence were better defined. In the meantime, mathematicians had a hard time resolving the paradoxes that seemed to exist, and the ridicule that they caused. Ex. The Taylor series for ଵି௫ is 1 + x + x 2 + x 3 + … One of the most vocal critics of the methods of calculus was the bishop and philosopher George Berkeley (1685 1753), after whom the famous California university is named. Referring to the work of Newton in his famously critical book The Analyst, Berkeley wrote “And if the first [fluxions] are incomprehensible, what should we say of the second and third? He who can digest a second or third fluxion…need not, methinks, be squeamish about any point in Divinity.” “And what are these fluxions? The velocities of evanescent increments? And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?” Augustin Louis Cauchy The two mathematicians most responsible for providing clarity and rigor to the subject of calculus were Augustin Louis Cauchy (1789 1857) and Karl Weierstrass (1815 1897). Cauchy was originally an engineer, but was persuaded by Lagrange to devote his attention to mathematics. He became a full professor in 1816, and won the grand prize of the Acad é mie des Sciences the same year for his paper on wave propagation. Due to the fragile political situation in France in the early 19th century, Cauchy spent time teaching in Turin and Prague, but he communicated his ideas back to his homeland, and his output was so prodigious that he established his own journal, the Exercises des math é matiques , and inundated the journal of the Acad é mie with so many papers (589 of the 789 that he published in all) that they imposed a rule limiting their length to four pages that is still in effect to this day. Cauchy’s most famous work, the Cours d’analyse , was published in 1821, in which he developed the notions of a limit, continuity, differentiability, and the definite integral which we see today in calculus. For example, Cauchy stated that
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lecture17 - TheAgeofRigor ,whilebrilliantinits...

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