The seventeenth century witnessed the development of calculus as the study of geometric curves in the hands of Newton, Leibniz and their immediate followers, with algebra (or 'analysis') serving as an aid to that work. This situation changed dramatically in the eighteenth century when the focus of calculus shifted instead to the study of functions, a change due largely to the influence of Euler. In the hands of Euler and his contemporaries, functions became a powerful problem solving and modelling tool in physics, astronomy, and related mathematical fields such as differential equations and the calculus of variations. Soon thereafter, mathematicians began to express concerns about calculus (analysis) and its foundations. The language, techniques and theorems that developed in response to these concerns are precisely those which students encounter in an introductory analysis course – but without the context that motivated nineteenth century analysts.
This mini-Primary Source Project (mini-PSP) employs excerpts from the works of nineteenth century mathematicians as a means to introduce students to that larger context. By offering a glimpse into the problems that motivated mathematicians to shift towards a more formal and abstract study of the concepts underlying various calculus procedures and applications, the project supports students' success in making a similar shift in their own understanding of these concepts. Completing this mini-PSP early in the course can also provide students and instructors with a basis for reflection on and discussion of current standards of proof and rigor throughout the course.
Bernard Bolzano (1781-1848) |
Augustin-Louis Cauchy (1789-1857) |
Richard Dedekind (1831-1916) |
The project begins with excerpts from the writing of Bernard Bolzano [2], Augustin-Louis Cauchy [3,4] and Richard Dedekind [5] in which these mathematicians described their various concerns about the state of analysis. Those concerns included, for example, a growing mistrust of geometric intuition as a valid method of proof for analytic truths. Project questions in this first section of the mini-PSP direct students' attention towards certain specific aspects of the quoted excerpts, and prompt them to compare and contrast the concerns of these three individuals.
In the second section of the project, students examine an extended excerpt from a letter from Niels Abel to his high school teacher, Bernt Michael Holmboe, written in 1826 while Abel was living in Paris. This portion of the mini-PSP again includes questions that prompt students to reflect upon the general nature of the concerns Abel expressed about the state of mathematics at the time, as well as how his concerns relate to those expressed by the authors of the earlier excerpts. Other questions in this section lead students to explore the mathematical details of examples cited by Abel as just cause for his complaints about the lack of firm foundations for power series techniques in particular.
Divergent series are on the whole devilish, and it is a shame that one dares to base any demonstration on them. One can obtain whatever one wants, when one uses them. It is they which have created so much disaster and so many paradoxes. (Excerpted from [1]) |
Niels Abel (1802-1829) |
The complete project Why be so Critical? is ready for student use, and the LaTeX source code is available from the author by request.
A set of instructor notes offering practical advice for the use of the project in the classroom is appended at the end of the student project; these include suggested Summary Discussion Notes that can be used to guide a whole class discussion of the main project themes.
This project is the second in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources that is planned for publication in Convergence, for use in courses ranging from first year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in the series appear below. The full TRIUMPHS collection includes eleven PSPs for use in a real analysis course.
Acknowledgments
The development of the student project Why be so critical? has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation’s Improving Undergraduate STEM Education Program under Grant No. 1523494. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation.
References
[1] Abel, N., "Breve fra og til Abel," in Festskrift ved Hundredeaarsjubilæet for Niels Henrik Abels Fødsel (editors E. Holst, C. Stømer and L. Sylow), Kristiana: Jacob Dybwad, 1902.
[2] Bolzano, B., Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewähren, wenigstens eine reele Wurzel der Gleichung liege, Leipzig: W. Engelmann, 1817.
[3] Cauchy, A., Cours d'Analyse de L'Ecole Royale Polytechnique, Paris: Debure, 1821.
[4] Cauchy, A., Résumé Leçons sur le calcul infinitésimal, Paris: Debure, 1823.
[5] Dedekind, R., Stetigkeit und irrationale Zahlen, Braunschweig: F. Vieweg und Sohn, 1872.