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The calculus was the first achievement of modern mathematics, and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.
John von Neumann
Georg Cantor at the Dawn of Point-Set Topology
Conclusion / Links to Related Resources
Cantor's 1872 paper made an important contribution towards the development of point-set topology. His detailed and meticulous construction of the real numbers made it possible to build a rigorous foundation for what we might now refer to as "nearness without distance." These ideas can be used to place point-set topology within the larger mathematical world and motivate the study of topology via analysis. It is our hope that this beautiful branch of mathematics will not be lost on students because it appears to them to be totally removed from what they believe to be "real" mathematics.
The author would like to thank anonymous referees for their helpful suggestions and comments as well as Loci: Convergence editor Janet Beery for greatly improving the exposition of this paper.
About the Author
Nicholas Scoville earned his bachelor's degree in mathematics from Western Michigan University in 2003 and his Ph.D. in algebraic topology from Dartmouth College in 2010. He is assistant professor of mathematics at Ursinus College in Collegeville, Pennsylvania, where he advises summer research projects in discrete Morse theory as part of the NSF-funded Ursinus College Research Experience for Undergraduates (REU).
Links to Related Resources
Topology Papers Project: Collection of historic papers important to the development of point-set topology, compiled by the author of the present article, Nicholas Scoville, Ursinus College
ODE original sources bibliography containing many of the original papers in differential equations, compiled by Adam Parker, Wittenberg University
Related article: "Peano on Wronskians: A Translation," by Susannah Engdahl and Adam Parker, both of Wittenberg University, here in Loci: Convergence, is the result of a student-faculty collaboration. It offers a model for student translation projects and ideas for how to use the translation resulting from this particular collaboration in class.
Teaching Discrete Mathematics via Primary Historical Sources describes class projects from Phase I of this project of Jerry Lodder, David Pengelley, and colleagues, New Mexico State University. Phase II includes projects based on original sources for a variety of mathematics and computer science courses.
Mathematics of Leonhard Euler
The Euler Archive contains hundreds of Leonhard Euler's papers, many in English translation.
"Teaching and Research with Original Sources from the Euler Archive," by Dominic Klyve, Lee Stemkoski, and Erik Tou, here in Loci: Convergence, offers ideas for how to use Euler's papers with students.
"Investigating Euler’s Polyhedral Formula Using Original Sources," by Lee Stemkoski of Adelphi University, here in Loci: Convergence, offers more specific ideas on how to use Euler's papers with students.
Scoville, Nicholas, "Georg Cantor at the Dawn of Point-Set Topology," Loci (March 2012), DOI: 10.4169/loci003861