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DOCEAMUS doceamus . . . let us teach of : Seeking and Inspiring Students Alex M. McAllister and Diana White

Our doctoral programs helped us master the re- as simply creating lists of people, papers, search methodologies and disciplinary norms of accomplishments, and inventions. Yet the history our respective fields in . How- of mathematics is so much more than this! ever, as professors, we both developed a strong In to describing the development of desire to deepen our of the , of mathematics seek to mathematics and its research methodologies. In understand the influence of ambient culture, so- this column, we provide a brief overview, anchored cial conventions and norms, and in concrete examples and based on our experi- as individuals and in relationships with others. ences, of how learning more about the history of They care about the extramathematical work and mathematics as well as the basics of historiography interests of mathematicians, because these often can benefit working mathematicians profession- provide the broader context of and influences on ally. their work. A robust of history is developed through a dynamic process of “zoom- Rigorous Research ing in” on specific details and “zooming out” to As mathematicians, we a great deal of our a broader of contemporary cultural, work on from social, political, and scientific influences. Histori- and . As a , we may adopt a skeptical cal research produces reliable, accurate historical view toward disciplines that we perceive as lack- knowledge via methodologies that, while different, ing the same “rigor.” We might also view historical are every bit as sophisticated and well developed as the practices of mathematicians and scientists. Alex M. McAllister is professor of mathematics at Centre College. His email address is alex.mcallister@centre. Historical Accuracy and Contextual edu. Understanding Diana White is associate professor of mathematics at As students and later as professors, we typically the University of Colorado Denver. Her email address is become familiar with a variety of historical anec- [email protected]. dotes related to mathematics, which we may in The authors first learned about many of these ideas from turn repeat to our students or others. However, the two-day MAA Short Course: Reading, Writing, Doing because of our lack of expertise in the history of the History of Mathematics: Learning the Methods of His- mathematics, we are usually unable to evaluate torical Research, just before the 2014 Joint Mathematics whether these anecdotes are true or mere apoc- Meetings. ryphal tales. Members of the Editorial Board for Doceamus are: David The tales of mathematics involve more than just Bressoud, Roger Howe, Karen King, William McCallum, the mathematics itself. Mathematical anecdotes are and Mark Saul. perhaps akin to family gossip, capitalizing on hu- DOI: http://dx.doi.org/10.1090/noti1207 mans’ innate love of sharing stories. Students often

172 Notices of the AMS 62, 2 benefit from this humanizing of mathematics and ,” Rickey helps them understand the error in can come to perceive mathematics as a dynamic Cauchy’s proof and then develops the of evolving of study and inquiry rather than just . some static, staid collection of facts. At the same Studying the history of mathematics can also , though, each story we tell should be true help students understand and embrace mathemati- rather than hagiography or a sort of false fishing cal traditions. For example, the notions of proof tale about “the one that got away.” As mathemati- and rigor have had different meanings in various cians we aspire to share mathematical with and cultures. In the fourth century BCE, Eu- our students, and this goal should also motivate clid provided a deductive proof of the Pythagorean our fidelity to accurately representing the history (in Elements, see also ’s annotated of mathematics. version of Jiu zhang suan shu Nine Chapters on the Some scholars, especially novices, have a ten- Mathematical Art) from 263 CE provided a “proof dency to read modern and perspectives into by picture” of this result that was held in similarly the writings and results of preceding generations high regard by his peers. Approaching the modern of mathematicians. For example, the geometric re- era, in 1899 CE, introduced a “more sults found in ’s Elements can be interpreted rigorous” development of Euclidean in and explained in terms of . A modern math- his Grundlagen der Geometrie. A comparison of ematician might take this algebraic these various proofs provides a wonderful, focused based on techniques as evidence that the opportunity for students to discuss the evolving knew algebra. This observation might lead of proof itself. them to mistakenly posit that Elements is a geomet- We regularly create and teach proofs by induc- ric garland covering the underlying algebra rather tion. Yet many prior generations of mathemati- than acknowledge its true nature as a synthesis of cians were content with using some variation of the -of-the-art in geometry. the phrase “and so on” when it appeared clear that Historians strive to adopt the unique perspec- the arguments would continue indefinitely in the tives of the mathematicians from each time, same fashion. For example, Euclid’s original proof place, and social context that is being studied. that there exist infinitely many primes considers Babylonian, Greek, Chinese, Indian, Islamic, Eu- only the case of there being exactly three primes. ropean, and modern mathematicians all explored We now explain “his” proof with the supposition different questions, developed distinct approaches that there exists some arbitrary, finite number of for grappling with these questions, and varied in primes, but in Euclid’s time and for centuries after- their perceived need for and rigor of mathematical wards, his original proof was regarded as sufficient arguments. In short, historians must be careful to and rigorous. Our students can better understand investigate mathematical developments in their ap- the contemporary notion of rigorous proof when propriate context to avoid creating a “whig” history we contrast modern proofs with the arguments of of mathematics which presents the past as an in- our mathematical ancestors. evitable progression toward the more enlightened present while ignoring mathematical pathways Resources for Deepening Knowledge that are not part of our current methodologies. Just as in mathematical research, the quest for new historical knowledge can require great persistence. Effective Teaching Using the History of Some good starting places include the Dictionary Mathematics of Scientific Biography (2) and the MacTutor History The history of mathematics can inform our ap- of Mathematics archive (3), both of which contain proach to teaching entire fields of mathematics references to original sources for further study. or specific topics within a particular field. For the A must go deeper, though, and examine former, a example is ’s Genetic extant, original manuscripts housed in archives. Approach to , which presents the ideas Fortunately, libraries continue scanning their ar- of calculus from a historical perspective of ideas chival materials and providing access through the evolving over centuries by examining the particular Web. In , translators are making an increas- results of , Kepler, Galileo, Fermat, and ing number of manuscripts available in diverse culminating with Newton and Leibniz. For the lat- languages, although translations can present their ter, we discuss an example shared by Fred Rickey own challenges in their accuracy and reliability. at one of last year’s Joint Mathematics Meetings’ The Mathematical Association of America’s minicourses (1). He presents his Real History of Mathematics Special Interest with Cauchy’s “proof” that if a of continuous (HOMSIGMAA) provides many resources on their functions converges, then the series converges to website (4). These links include references for a continuous . For homework, his stu- mathematicians interested in the basics of read- dents explore to this supposed ing, writing, and doing the history of mathematics. “theorem.” During the next class period, after his Another starting place might be attending a talk on students express their frustration with the “false the history of mathematics at the next Joint Math

February 2015 Notices of the AMS 173 Meetings or at MathFest. Exploring the history of mathematics through these various resources enables us to grow professionally and to come to think differently about our profession and our mathematical ancestors.

Final Thoughts Our doctoral programs provide solid training for mathematical research, and increasingly our profession augments this learning with prepara- tion for teaching. We propose an additional step in our ongoing professional development both as teachers and scholars: learning how our discipline evolved into our contemporary study of mathemat- ics as well as how historical research is conducted. The history of mathematics can inform our understanding of basic topics, strengthen our teaching, and help us better understand our place in the overall story of mathematics. Not only will we grow as mathematicians, we will also build our capacity to help our students understand mathematics as a human endeavor, developing in response to and driving various aspects of so- ciety. Some of us may even decide to go beyond being consumers of the history of mathematics and to learn the in-depth research procedures of historiography, making original contributions to this field.

References [1] Fred Rickey at fredrickey.info/ shared this exam- ple at the MAA Short Course: Reading, Writing, Doing the History of Mathematics: Learning the Methods of Historical Research, just before the 2014 Joint Math- ematics Meetings. [2] Dictionary of Scientific Biography at historyofmath- ematics.org. [3] MacTutor History of Mathematics archive at www- history.mcs.st-andrews.ac.uk/. [4] HOMSIGMAA at historyofmathematics.org/.

174 Notices of the AMS Volume 62, Number 2