Richard Dedekind Vincent Shetter
Download 38.24 Kb. Pdf ko'rish
|
|
2
Julius Wilhelm Richard Dedekind was a German Mathematician born in Brunswick (Braunschweig) who contributed to several areas of mathematics. These areas include analysis, modern set theory, number theory, and abstract algebra. 1 His most famous contribution, however, was the construction of the real numbers from the rational numbers by method of “Dedekind Cuts.” 2
Born in 1831, Dedekind did not seriously pursue mathematics until 1848 while studying at Caroline College. He would study calculus, algebra, and analytic geometry until 1850 when he began study at the University of Göttingen. It was there that Dedekind would study under Carl Friedrich Gauss. 3 Gauss supervised Dedekind’s dissertation which was completed in 1852. 4
After completing his second dissertation, Dedekind would remain at the University of Göttingen for another four years. Here Dedekind would be strongly influenced by Dirichlet and Riemann. 5
Dedekind then moved to Polytechnic in Switzerland, where he would work as a professor. Although Dedekind was a great mathematician, he turned down offers for professorships at more prestigious universities. 6
1 Reck, Erich, "Dedekind's Contributions to the Foundations of Mathematics", The Stanford Encyclopedia of Philosophy (Summer 2016 Edition), Edward N. Zalta (ed.), URL = 2 Rogers,Robert and Bowman Eugene, “How We Got From There to Here: A Story of Real Analysis” Open Suny Textbooks 2014, http://textbooks.opensuny.org/how-we-got-from-there-to-here-a-story-of-real-analysis/ 3 The Editors of Encyclopædia Britannica, “Richard Dedekind German Mathematician” Encyclopædia Britannica Online, https://www.britannica.com/biography/Richard-Dedekind 4 Ibid. 5 Reck, Erich, "Dedekind's Contributions to the Foundations of Mathematics", The Stanford Encyclopedia of Philosophy (Summer 2016 Edition), Edward N. Zalta (ed.), URL =
6 Gouvêa, Fernando Q., “Was Cantor Surprised?”, The American Mathematical Monthly, Vol. 118, No. 3 (March 2011) The Mathematical Association of America (2011). URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.118.03.198 3
The mathematics that lead to Dedekind Cuts date as far back as the Ancient Greeks. They knew that the real numbers would be necessary for calculations as soon as they saw that the hypotenuse length of a triangle with legs of length one is √2 . In the 1600s, Newton and Leibniz were credited with the invention of calculus, though it had no rigorous foundation. The 1700s would bring new developments in mathematics. Major contributors included Taylor, Lagrange, Cauchey and Fourier. During this time continuity was assumed. If the real world were made up of discrete points “…then the real world is more like the rational number line (Q)… than the real number line (R).” 7 Assumptions of continuity would continue as most believed that a rigorous definition was unnecessary. 8 “However, the complexities associated with Fourier series and the types of functions that they represented caused mathematicians in the early 1800’s to rethink their notions of continuity.” 9 In 1858, while a lecturer of differential calculus at Polytechnic, Dedekind realized that there were flaws in the justification of concepts in calculus. He believed that the geometric interpretation of continuity and limits, however useful, was unsubstantiated. 10 He would discuss his ideas of finding an arithmetic basis of calculus with his colleagues, but he would not publish anything on the topic until later. 11 It was not until 1859 that Karl Weierstrass formally defined continuity. Although defining the real numbers seemingly has no application, According to Rodgers and Bowman, the real numbers were defined due to the
7 Rogers, Robert and Bowman Eugene, “How We Got From There to Here: A Story of Real Analysis” Open Suny Textbooks 2014, http://textbooks.opensuny.org/how-we-got-from-there-to-here-a-story-of-real-analysis/ 8 Ibid. 9 Ibid.
10 Jourdain, Philip E. B. "RICHARD DEDEKIND. (1833-1916.)." The Monist 26, no. 3 (1916): 415-27. http://www.jstor.org/stable/27900599 .
11 Ibid.
4
exploratory nature of mathematics research. 12 Dedekind and others of his time would work in modern analysis and several mathematicians would go on to construct the real numbers from the rational numbers.
While Cantor defined the real numbers from the rational numbers using Cauchey sequences, Dedekind developed “Dedekind Cuts.” 13 “Dedekind Cuts” was first published in “Stetigkeit und Irrationale Zahlen” (Continuity and Irrational Numbers) in 1872. Rodgers and Bowman define “Dedekind Cuts” in How We Got From There to Here: A Story of Real Analysis (Figure 1):
Figure 1: Definition of Dedekind Cut 14
This definition is fairly straightforward. An example to illustrate Dedekind cuts is to define the square root of two in terms of two sets of rational numbers. { ??????|?????? 2
2 > 2} . This would contribute to the reduction of analysis to arithmetic, rather than geometry; 15 thereby relating the discrete to the continuous.
12 Ibid
13 Ibid.
14 Ibid.
15 Reck, Erich, "Dedekind's Contributions to the Foundations of Mathematics", The Stanford Encyclopedia of Philosophy (Summer 2016 Edition), Edward N. Zalta (ed.), URL =
5
Cantor and Dedekind would both construct the real numbers from the rational numbers in, roughly, the same time period. Serendipitously, they would meet while they were vacationing in Switzerland shortly after completing their respective works. They would exchange papers on the irrational numbers and went on to write one another, sharing their results on set theory. 16 As their friendship developed, so did their work in set theory. This friendship would later end, when Dedekind declined Cantor’s offer to work with him in Halle, but it would later resume after Cantor had his first mental break down. 17
Dedekind remained at Polytechnic for the majority, of his career. He would go on to contribute to other branches of mathematics as well. Dedekind was focused on foundation and proof.
18 He worked on rigorous proofs for concepts that seem obvious to most. He had success in shifting mathematical foundations from geometric ideas to arithmetic proofs. Dedekind died in 1916 in Brunswick, the same town in which he was born.
16 Gouvêa, Fernando Q., “Was Cantor Surprised?”, The American Mathematical Monthly, Vol. 118, No. 3 (March 2011) The Mathematical Association of America (2011). URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.118.03.198
17 Grattan-Guinness, I. "Missing Materials concerning the Life and Work of Georg Cantor." Isis 62, no. 4 (1971): 516-17.
http://www.jstor.org/stable/229823 .
18 Jourdain, Philip E. B. "RICHARD DEDEKIND. (1833-1916.)." The Monist 26, no. 3 (1916): 415-27. http://www.jstor.org/stable/27900599 .
6
Bibliography
Gouvêa, Fernando Q., “Was Cantor Surprised?”, The American Mathematical Monthly, Vol. 118, No. 3 (March 2011) The Mathematical Association of America (2011). URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.118.03.198 Grattan-Guinness, I. "Missing Materials concerning the Life and Work of Georg Cantor." Isis 62, no. 4 (1971): 516-17. http://www.jstor.org/stable/229823 .
415-27. http://www.jstor.org/stable/27900599 .
Encyclopedia of Philosophy (Summer 2016 Edition), Edward N. Zalta (ed.), URL =
Rogers, Robert and Bowman Eugene, “How We Got From There to Here: A Story of Real Analysis” Open Suny Textbooks 2014, http://textbooks.opensuny.org/how-we-got- from-there-to-here-a-story-of-real-analysis/ The Editors of Encyclopædia Britannica, “Richard Dedekind German Mathematician” Encyclopædia Britannica Online, https://www.britannica.com/biography/Richard- Dedekind
Download 38.24 Kb. Do'stlaringiz bilan baham: 
|