by Della Dumbaugh Fenster
Introduction
In their book, The Emergence of the American Mathematical Research Community (1876–1900): J. J. Sylvester, Felix Klein, and E. H. Moore, Karen Parshall and David Rowe suggest the notion of periodization (as opposed to continuity) as a means of historically investigating mathematics [e53]. They characterize four developmental periods as follows:
1776–1876: Mathematics in the general context of scientific development
1876–1900: Emergence of a research community
1900–1933: Consolidation and growth of research traditions and institutions
1933–1960: Influx of European mathematicians; Government funding ([e53], pp. 427–428).
With their view that the opening three decades of this century witnessed a consolidation and growth of American research traditions and institutions in mathematics, Parshall and Rowe have joined the border pieces of a large interlocking jigsaw puzzle. Their work challenges historians of American mathematics to determine the pieces which make up the remainder of the picture. This interesting and multifaceted project unfolds as we address questions such as: What research traditions came together in this time period? What mathematicians brought cohesion to these traditions and how did they do it? What institutions emerged as research powerhouses in mathematics and why? As these questions have received attention, they have quite naturally brought into sharp focus other unexpected factors which contributed to the development of American mathematics early in this century. The processes of mentoring and role modeling, in particular, represent two such factors and, as a glance at the mathematical life of Leonard Eugene Dickson will reveal, they proved to be among the more successful avenues for the advancement of mathematics in the United States.
By pursuing both his doctoral studies and the majority of his career at the University of Chicago, Dickson helped define an algebraic tradition at that institution. His mathematical contributions — especially his emphasis on the construction of far-reaching theories on firm mathematical foundations1 — brought consolidation and growth to this algebraic endeavor. Moreover, Dickson further advanced the Chicago algebraic school by imparting his standards for solid mathematics to the next generation of aspiring doctoral students.
Dickson’s strong algebraic research program, situated within the far-sighted Chicago mathematics department, fulfilled one of the broader concerns of American mathematics, namely, the effective training of future researchers ([e53], p. 429). From its inception, the University of Chicago provided a conducive environment for future mathematicians to learn not only mathematics but also those extramathematical qualities of import for a career as a productive researcher. A list of the first generation of graduate students at Chicago — Oswald Veblen, Gilbert A. Bliss, Leonard Dickson, and George D. Birkhoff, to name only four — confirms the institution’s success in this regard. This paper highlights the continuation of this success relative to the second generation of Chicago-trained researchers.
Since Dickson acquired many of the ideas and ideals he would pass on to the next generation of aspiring mathematicians from his graduate student days at the then very young, but very visionary, University of Chicago, we initially consider what mathematical and extramathematical attributes Dickson acquired while pursing his doctorate. In our efforts to come to terms with the scope of Dickson’s influence on subsequent generations of American mathematicians, we next consider Dickson as a lecturer and adviser. What was he like in the classroom? What motivated his teaching style? How did he relate to his doctoral students? This information serves to form a characterization of Dickson as an educator. With this in place, we take a closer look at the mathematical pursuits of three of Dickson’s women students — Olive Hazlett, Mayme Logsdon, and Mina Rees — to gain an understanding of both the specifics of Dickson’s algebraic heritage and his broader role in training women in research-level mathematics. Finally, a careful consideration of Dickson’s most successful protege, A. Adrian Albert, brings confirmation to both the consolidation and the growth of the Chicago research tradition.
Since this paper focuses on the student — adviser relationship, we provide a sociological framework for our assessment of Dickson as an adviser and perpetuator of a research tradition. In the last 30 or so years, sociologists and others have begun to consider this significant academic relationship in an effort to shed light on the dynamics of this key aspect of both the transmission of knowledge and the development of the profession. As this important avenue for the exchange of ideas attracted increasing attention, terms such as “mentor,” “role model,” and “sponsor” became a part of our language, and, consequently, our thinking ([e36], pp. 693, 708–709). The precise meanings of the terms may vary somewhat from study to study but in general a “role model” describes one who “possesses skills and displays techniques which the student lacks … and from whom, by observation and comparison with his own performance the student can learn” ([e36], p. 693). The term “mentor” has come to refer to those “older people in an organization or profession who take younger colleagues under their wings and encourage and support their career progress until they reach mid-life” ([e36], p. 708). or to indicate someone “higher in the institution or organization who coaches, teaches, advises, provides support and guidance, and helps the mentee (what a dreadful word!) achieve his or her goa[l]s” ([e46], p. 2). Thus, a role model primarily teaches his or her trade by example, while a mentor encourages the up-and-coming professional in a more personal, interactive way. We will adopt this distinction here, while noting that it is, of course, entirely possible for one person to serve simultaneously as a role model and a mentor.
Dickson’s mathematical roots
A seemingly unlikely man, the comparative philologist, William Rainey Harper, found himself in a position of considerable influence on Dickson (and others, for that matter) after he opened the doors of the University of Chicago for instruction in 1892. Harper’s own discovery of academe, however, had come some 15 years earlier, giving him an inside view of the metamorphosis of the American university. Substantial reforms in the American university system had occurred in the decades following the close of the Civil War. The new developments grew out of a discontent which had arisen because of an American desire to match the established European university system, the availability of new wealth, and an increasing concern over the decline of American collegiate influence ([e28], pp. 2–3). Public service and a commitment to abstract research (based on the German model) emerged among the conceptions of the new American university ([e28], p. 12).
Aware of the changing tide in American higher education, Harper expanded the ideas of his trustees and primary financial backer, John D. Rockefeller, by his insistence that the University rise above the level of an undergraduate college by emphasizing the training of future researchers. Despite Harper’s new design of a university, he did not introduce the first research-oriented institution on American soil. Both the Johns Hopkins University, established in 1876, and Clark University, founded in 1889, had already attempted to emphasize teaching as well as graduate studies and research ([e53], pp. 262–263).2 Harper drew from the standards Presidents Daniel Coit Gilman of Johns Hopkins and G. Stanley Hall of Clark had set in establishing their institutions in general and in assembling their faculties in particular.
In Harper’s view, a professor should “be a teacher, but first and foremost a scholar, in love with learning, with a passion for research, an investigator who could produce, and, if what he produced was worthy, would wish to publish” ([e12], pp. 123–124). With his high standards for faculty members, Harper tended to pursue well-established members of the academic world, including university presidents.3 He must have also recognized potential, however, for Harper gave an unproven associate professor from Northwestern University, Eliakim Hastings Moore, the temporary (and, ultimately, permanent) reins of his original research-oriented mathematics department. Moore’s appointment represented a gamble on Harper’s part, but one which would have a “tremendous influence” on the history of American mathematics ([e21], p. 3). Moore secured Oskar Bolza, a professor from the fledgling Clark University, and Heinrich Maschke, the Göttingen Ph.D. then working as an electrician in New Jersey, as his two first colleagues. With Moore’s commitment to the highly successful German university tradition and the first-hand experience of Bolza and Maschke within that system as students of Felix Klein at Göttingen, the Chicago mathematics department opened its doors with its sights set on emulating this tradition.4 Germany served as the model of excellence for virtually all Americans interested in mathematics during the last quarter of the 19th century. Moore himself had observed the German university traditions during the year (1885–1886) he spent at the Universities of Göttingen and Berlin. Thus, Moore saw the German emphasis on pure research and became acquainted with the seminar, the effective German teaching tool. The seminar brought together students and professors for the presentation and discussion of both original and recently published research. The real treasure of the seminar, however, lay in the “close human contact … offered between advanced students and a man of major reputation in the field” ([e28], p. 156). For aspiring mathematicians who traveled to Germany in the latter part of the 20th century, the “men of reputation” included Karl Weierstrass, Leopold Kronecker, Felix Klein, Sophus Lie, and later, David Hilbert. With American universities still largely oriented toward an undergraduate population, the research-minded mathematics students tended to look to Germany for their training.
With their enthusiasm for mathematics and their unique contributions in terms of research and talents, Moore, Bolza, and Maschke offered the steady stream of American mathematics students otherwise headed for Germany a reason to stay home. The trio established their successful department through effective classroom teaching, the organization of the Mathematical Club for the review of books and memoirs and the presentation of original research, and the example of top-quality published work ([e53], pp. 371–372).5 This core of Moore, Bolza, and Maschke not only strove to develop a fine department at Harper’s university, but they also played instrumental roles in the establishment of a more truly American (as opposed to New England) mathematical community. Moore and the midwestern mathematicians helped found a journal for original research (ultimately known as the Transactions of the American Mathematical Society) and promoted the development of various avenues (Colloquia, Sections, and the Chicago Congress, for example) to exchange and stimulate mathematical thought.6 By taking a broad perspective on American mathematics, the Chicago department demonstrated to its students the importance of contributing to the much larger mathematical community ([e53], pp. 401–419). Owing to its successful implementation and continuation of a commitment to high research standards, the University of Chicago held a unique position among American institutions in the closing decade of the 19th century. Moore, Bolza, and Maschke sought to build and succeeded in forming a mathematics department which promoted original research, quality publications, and a broad view of the American mathematical community. The strong institutional and departmental philosophy inherent in these goals and manifest in their achievement made the University of Chicago a viable option for Leonard Eugene Dickson and other aspiring American mathematicians, including Oswald Veblen, Gilbert Bliss, George D. Birkhoff, and R. L. Moore, who might otherwise have traveled to Germany for their training.7 Born to Campbell and Lucy Tracy Dickson in Independence, Iowa in 1874, Leonard Dickson made his home with his family in Texas, where his father worked as a merchant and banker.8 Dickson obtained his early education in the public schools in his hometown of Cleburne and attended the University of Texas for his undergraduate and master’s education. At the University of Texas, Dickson came under the influence of “that extraordinary enthusiast for nonEuclidian geometry,” George Bruce Halsted.9 With his master’s degree in hand and two years of teaching experience under his belt, Dickson chose the strong Moore–Bolza–Maschke triumvirate at Chicago over the up-and-coming Harvard with William Fogg Osgood and Maxime Bôcher as the place to pursue his doctorate.10 One cannot help but wonder how mathematics in America might have been different had Dickson chosen Harvard instead of Chicago.
Dickson headed for Chicago as a native Texan possessing what Garrett Birkhoff described as “much of the dynamic energy and rugged individualism that we associate with that state” [e34], p. 34. These characteristics, along with his mathematical ability, placed him among the talented and persevering students left unintimidated by E. H. Moore’s teaching style.11 In fact, it appears as though Moore’s unusual pedagogical approaches, combined with his research interests in algebra and the foundations of mathematics during the late 19th and early 20th centuries, influenced Dickson profoundly.
Dickson arrived at Chicago in 1894, just in time for Moore’s lectures on group theory. Dickson wrote his dissertation in group theory under the direction of Moore in 1896 [1]. The revised and expanded version of this work appeared in 1901 in the form of his first book, written at the age of 27, Linear Groups with an Exposition of the Galois Field Theory [14]. In the meantime, Dickson had traveled to Leipzig and Paris where he studied under Sophus Lie and Camille Jordan, held a couple of brief appointments at the Universities of California and Texas, and, in 1900, returned to the University of Chicago to begin his 40-year-long professional affiliation with that school. As the 20th century opened, Dickson again followed Moore’s lead — but now a Moore captivated by the ideas inherent in David Hilbert’s Foundations of Geometry [e2] — and considered foundational questions in mathematics. Inspired by his adviser, Dickson pursued algebraic researches which would ultimately define his research for more than forty years. Yet this would mean algebra in its broadest sense, for Dickson would work in group theory, invariant theory, finite field theory, and the theory of algebras. Dickson reflected his adviser’s influence in more ways than research interests, however. The department’s sustained commitment to research, their high standards for publication, and their vision for the American (as opposed to New England) mathematical community came to permeate Dickson’s mathematical persona in these formative years. In his 40-year career, Dickson published 18 books and roughly 300 research articles, and, in so doing, made graduate texts in mathematics readily available to American graduate students. From his post as an editor of the American Journal of Mathematics and the Transactions of the American Mathematical Society, he insisted upon quality publications. He shared the broad vision of the Chicago mathematics department, and, like Moore, served as president of the American Mathematical Society. In the process of forging ahead in mathematics and voicing frustration over what he perceived as roadblocks, he showed his students what he did — and did not — value as a professional mathematician. He thus perpetuated albeit with somewhat of a Dicksonian twist, the mathematical, aesthetic, and professional values modeled by the original triumvirate at Chicago.
Dickson’s mathematical legacy
Dickson left a strong impression on those who occupied the chairs in his classroom or office. Witness, for example, the vivid descriptions students from over half a century ago give today of his mathematical personality. When they describe Dickson succinctly, they tend to use terms like “gruff” ([e53], p. 377) or “hard-bitten” ([e49]) or “energetic and forceful” ([e40], p. 246). For this group, one need only mention the name of Dickson and a flood of personal recollections regarding his teaching and advising style come to the fore.
A discussion of those who employ conventional pedagogical approaches would certainly not include Dickson. He delivered terse and unpolished lectures and spoke sternly to his students. He frequently assigned readings from a textbook (often one of his own), and he either called on students to present and analyze the material, or he lectured the entire hour. This method motivated students to do their best in making classroom presentations [e48]. Given Dickson’s intolerance for student weaknesses in mathematics, however, his comments could be harsh, even though not intended to be personal. He did not aim to make his students feel good about themselves [e50]. In fact, in the words of one of his former students, “[h]e was blunt and straightforward about his expectations from his students. He himself worked very hard and he expected his students to work hard too” [e50]. These general pedagogical comments about Dickson hardly hint at a mentor in the sense of someone higher in the institution who took his students “under his wings” and “coached” or “provided support and guidance” as they pursued their goals. In spite of his “gruff” approach, however, Dickson did successfully serve as a role model as he imparted algebra and number theory to the next generation of American mathematicians. William Duren, a Chicago Ph.D. student in the late twenties and early thirties, underscored the nature of Dickson’s brand of modeling. “In the conventional sense,” Duren reflected, “Dickson was not much of a teacher. I think his students learned from him by emulating him as a research mathematician more than being taught by him. Moreover, he took them to the frontiers of research, for the subject matter of his courses was usually new mathematics in the making. As Antoinette Huston [a graduate student at Chicago] said, ‘Dickson made you want to be with him intellectually. When you are young, reaching for the stars, that is what it is all about’ ” ([e52], p. 178).
Dickson’s teaching, it seems, reflected his lifelong goal to become the most distinguished mathematician possible. He spent his mathematical life at the cutting edge of the field, and he wanted his students to do the same. Since students who could not meet his standards also could not serve his purposes best, Dickson had a sudden death trial for his prospective doctoral students: he assigned a preliminary problem which was shorter than a dissertation problem, and if the student could solve it in three months, Dickson would agree to oversee the graduate student’s work. If not, the student had to look elsewhere for an adviser ([e45], p. 377). Dickson quite clearly designed his three-month test problem as a means of evaluating whether or not a student could make the trip to the “frontiers of research,” and although some students may not have cared for this approach, 67 Chicago Ph.D. students accepted it enough to sign on for his guidance. As a comparison, Gilbert Bliss, the other principal adviser of mathematics Ph.D.’s at Chicago in the early decades of this century, oversaw 46 doctoral students.
A look at the faces behind the dissertations directed by Dickson uncovers the rich and colorful legacy left hidden by a merely numerical count of his students. In particular, women accounted for 18 (roughly 27%) of these students. Set within the broader context of women in mathematics, these eighteen Ph.D.’s composed slightly more than 8% of all women who earned an American Ph.D. in mathematics between 1900 and 1939 and 40% of those awarded to women at the University of Chicago.12 Olive Hazlett, a 1915 Dickson doctorate, followed her adviser’s lead and studied nilpotent algebras, division algebras, and modular invariants [e11], [e13], [e6], [e8], [e7], [e10], [e9], before she apparently followed Dickson down the road to the arithmetic of algebras. At the 1924 International Congress of Mathematicians in Toronto, she extended his arithmetic of rational algebras to algebras over an arbitrary field [e13]. Hazlett could not claim the exclusive rights for these ideas, however, since Dickson presented the same generalization at the same meeting [10]. (This, perhaps, represented a bittersweet moment for Dickson (and Hazlett, for that matter), bitter in that he was not the only one presenting a more general theory but sweet in that he had trained his student well.) Like Dickson, Hazlett certainly placed a high value on research mathematics. She wrote more papers than any other pre-1940 American woman mathematician ([e44], p. 138) and, reflective of her adviser’s commitment to the broader mathematical community, she served as a cooperating editor of the Transactions of the American Mathematical Society for 12 years, held a two-year term on the AMS Council, and she, Charlotte Scott, and Anna Johnson Pell Wheeler composed the entire group of women mathematicians starred in American Men of Science between 1903 and 1943 ([e38], p. 293). As for her relationship with Dickson, it can be characterized as one of mutual respect. In particular, although she continued to draw from his work, Hazlett apparently neither tended to consult Dickson directly about her employment dilemmas nor to keep him informed of her research.13 In short, for Hazlett, Dickson served as a role model rather than a mentor.
Mayme Logsdon, a 1921 Dickson Ph.D., who subsequently became interested in algebraic geometry, served on the University of Chicago faculty for 25 years. In terms of the broader employment picture for women in mathematics, Logsdon represented what one might call a typical anomaly. Her situation was anomalous in the sense that she enjoyed one of the few positions offered to a woman by a major research institution.14 It was typical in that she generally taught undergraduate courses and never received promotion to full professor during her many years of service to Chicago. Nevertheless, Logsdon continued to pursue research, which included spending an entire academic year (1925–1926) in Italy when it was the most active center of research for algebraic geometry. She maintained a high enough level of research to oversee the work of four Ph.D. students.15 Thus, Logsdon, like Hazlett, pursued an academic career which emphasized many of the qualities modeled by Dickson.
Mina Rees, a 1931 Dickson doctorate, reflected her adviser’s influence in a different way. Like Dickson, she had a vision for the American community of research mathematicians. Although Dickson certainly held important administrative positions — such as the presidency of the AMS and the editorships of the Transactions and the American Journal — he made his most significant contributions to the development of mathematics in this country by way of his research and publications. Rees, on the other hand, largely left research mathematics behind her after the completion of her doctorate [e16] and wielded her greatest influence through her policy-setting leadership roles. She advanced the cause of American mathematics primarily by securing federal funds for the field, recognizing (in the 1950s) the importance of computers in scientific development, and addressing the changing needs of mathematics education. For Rees, then (and perhaps others?), Dickson seems to have served as a role model of leadership.
In addition to Logsdon, Hazlett, and Rees, Dickson advised 15 other women students.16 In so doing, he directed more than one-fifth of all American algebra Ph.D.’s granted to women before 1940 and all but one of the dissertations in number theory ([e39], p. 20). According to Judy Green and Jeanne LaDuke, Dickson served as something of a “cluster point” for aspiring women mathematicians ([e39], p. 20). Alice Schafer has further suggested that a snowball effect may have led to Dickson’s large number of women graduate students. In other words, once a few women completed their degrees under his guidance, favorable word spread about his abilities as an adviser [e51]. The University of Chicago may have also contributed to Dickson’s description as a “cluster point” for women. Dickson’s student, Gweneth Humphreys, noted that the school had a favorable reputation among women seeking higher degrees in mathematics. Following the start of their studies there, they, like so many of the men, found Dickson among the most — if not the most — committed research mathematician on the faculty and selected their adviser based on this criterion.17 Thus, perhaps as a result of his early reputation as an adviser for women, his institution, and his strong research program, Dickson evolved into a notably successful adviser for women pursuing mathematics doctorates in the United States. As the remarks about his pedagogical style and the careers of the women recounted above reveal, however, his success as an adviser lay primarily in his ability to impart his trade by example rather than to encourage his students in a more personal, interactive way.
Even still, certain aspects of Dickson’s persona as an adviser to women remain unclear. Were there other reasons for which Dickson became a “cluster point” for women graduate students at Chicago? Did he have personal motivations for wanting to direct women? Even with traditional stereotypes taken into consideration, a “hard-bitten” mathematical personality does not seem the most likely magnet for aspiring women (or men?) mathematicians. Did the women doctoral students in the early decades of this century possess a certain amount of fortitude which immunized them against difficult personality traits and kept them focused on their goal?18 Even with these questions outstanding, this study of Dickson as an adviser clearly uncovers his noteworthy contributions to the education of women mathematicians in the opening decades of this century as well as to the 20th-century heritage of women in research-level mathematics in the United States. Moreover, the experiences of Hazlett, Logsdon, and Rees demonstrate that Dickson served as a role model in at least two ways — as a researcher and as a community-minded activist. Thus, the strong research ethic and the vision for the American mathematical community which Moore had modeled for Dickson, became more firmly entrenched in the Chicago algebraic tradition. The direct continuation of this heritage at Chicago, however, lay in the hands of another of Dickson’s doctoral students, A. A. Albert.
Albert, or A-cubed as he was often called, arrived at the University of Chicago in 1922 when the theory of algebras was among Dickson’s main research interests. Dickson’s “considerable” influence ([e40], p. 246) on Albert manifested itself in his 1927 master’s thesis where he determined all 2-, 3-, and 4-dimensional associative algebras over a nonmodular field \( F \) and in his 1928 dissertation entitled “Algebras and Their Radicals, and Division Algebras.” In the latter, Albert proved that every central division algebra of dimension 16 is not necessarily cyclic but is always a crossed product. Albert polished this work and presented it as his first major publication [e14]. Irving Kaplansky commented on the mathematical personality Albert revealed in his early work when he wrote: “Here was a tough problem that had defeated his predecessors; he attacked it with tenacity till it yielded” ([e40], p. 247).
Albert’s thesis research on central division algebras placed him at the center of activity in the field of linear associative algebras. In particular, he, along with the German mathematicians Richard Brauer, Helmut Hasse, and Emmy Noether, strove to determine all central division algebras. In 1931, the German trio reached the central division algebra finish line only seconds before a breathless Albert. They established the principal theorem that every central division algebra over an algebraic number field of finite degree is cyclic [e17]. One year later, Albert and Hasse published a joint work [e15] which gave the history of the theorem and described Albert’s “near miss” ([e30], p. 663). Although Albert would go on to make significant contributions to the theory of Riemann matrices [e18], [e20], [e19] and to introduce singlehandedly the American school of nonassociative algebras [e23], he maintained an interest in associative division algebras throughout his more than 40-year-long career.
The scope of Albert’s talents extended far beyond the production and publication of mathematical results. He, like Dickson and Moore, made significant contributions to both the University of Chicago and the American Mathematical Society. Relative to the former, the members of the mathematics department at Chicago — Dickson, Bliss, and E. H. Moore, among them — recognized Albert’s abilities and secured him as a permanent faculty member soon after he completed his Ph.D. It was, in fact, more than Albert’s ability which landed him the assistant professorship — and obvious position as heir apparent to Dickson — in 1931, however. Indeed, during its first four decades, the procedure for hiring in the Chicago mathematics department seemed to follow what Saunders Mac Lane has called an “inheritance principle” ([e43], p. 141). This appointment procedure worked exceptionally well as Moore, Dickson, and Albert established a strong algebraic tradition at Chicago.
During Albert’s tenure as a faculty member at his alma mater, he participated in a variety of committees, organized conferences, chaired the mathematics department, and served as the dean of the Division of Physical Sciences. While chair, he “skillfully” found support to maintain a steady flow of visitors and research instructors ([e40], pp. 251–252) Albert used his influence to persuade the University to donate an apartment building, affectionately known as “the compound,” to house the visitors. Kaplansky claims that “the compound” became the “birthplace of many a fine theorem” ([e40], p. 252). In particular, two visitors calling it home in 1960–1961, Walter Feit and John Thompson, determined that all groups of odd order are solvable ([e40], p. 252). When considering the sum total of Albert’s mathematical career, it should come as no surprise that he strove to attract visitors to Chicago. He realized that a department which relied solely on its permanent faculty had the potential to become stale and narrow in its focus. An infiltration of new ideas frequently encouraged a fresh perspective on mathematics. Albert recognized that mathematical progress often depended on just such an external spark.
Albert’s career also reflected a strong commitment to the mathematical community at large. He served the AMS in a variety of capacities — as a committee member, as an editor of the Bulletin and Transactions, and, like Dickson and Moore, as President in 1965–1966. The concerns of American mathematicians in the middle two quarters of the twentieth century were, however, somewhat different from those in the early years when Moore and Dickson made their contributions, and Albert’s service quite naturally addressed the changing needs of American mathematicians. In particular, Albert helped establish government research grants for mathematics comparable to those existing in other areas of science ([e32], p. 1077). He helped set the National Science Foundation (NSF) budget for mathematics and aided in the creation of the NSF summer research institutes ([e32], p. 1077). He apparently found satisfaction in this nationally oriented work for “[h]e was always pleased to use his influence in Washington to improve the status of mathematicians in general, and he was willing to do the same for individual mathematicians whom he considered worthy” ([e30], p. 665). This latter category surely included his students.
Beyond his service to Chicago, the AMS, and the mathematical community at large, Albert exerted considerable influence in mathematics through his students ([e32], p. 1078). As his colleague, I. N. Herstein, observed, “Adrian was extremely good at working with students. This is attested by the 30 mathematicians who took their Ph.D.’s with him. In their number are many who are well known mathematicians today. His interest in his students — while they were students and forever afterwards — was known and appreciated by them” ([e31], p. 186). Daniel Zelinsky, in particular, described Albert as an adviser who treated his Ph.D. students “almost as members of his family” ([e30], p. 663). Albert’s students and colleagues regarded him warmly, a luxury his own adviser had not often — if ever — known.
From a broader perspective, Albert in many ways fulfilled the role of an “heir” in that his mathematical career looked very much like Dickson’s. Both received their Ph.D.’s from Chicago, had influential postdoctoral years, held a few short appointments at schools other than Chicago, and ultimately spent the bulk of their careers leading the algebra program at their alma mater. They both led seemingly tireless mathematical lives, producing — and exerting a worldwide influence through — exceptionally large numbers of publications and graduate students. (Interestingly, a count of Albert’s publications and graduate students rounds out at roughly half of Dickson’s in all categories. Albert published eight books, and Dickson 18. Albert wrote just over 140 articles and Dickson just under 300. Albert advised 30 Ph.D. students, and Dickson 67.) They both edited the Transactions, served as AMS president, delivered plenary lectures at International Mathematical Congresses, and received the AMS Cole Prize. Their colleagues (independently) described each of them as powerful mathematicians. (On a less formal note, Dickson and Albert both took billiards seriously.) These striking similarities seem to support the more contemporary (1981) research of Blackburn, Chapman, and Cameron regarding the idea of “cloning” in academe. In their study of mentorship,19 they found that “[m]entors overwhelmingly nominated as their most successful proteges those whose careers were essentially identical to their own — i.e., their ‘clones’ ” ([e35], p. 315). Although we have no written evidence that Dickson acknowledged Albert as his “best” student, Albert’s career, the “inheritance principle” for hiring at Chicago, and the descriptions by Birkhoff and Mac Lane substantiate this idea. Best student or not, Albert echoed the professional heartbeat of his adviser — and role model — throughout his long and distinguished career. Yet despite their many similarities in the mathematical world, as personalities Albert and Dickson hardly resembled one another at all.
Dickson, apparently, had few close relationships within mathematics, whereas Albert enjoyed many close friends within the mathematical community. Witness, for example, the four warm biographies written upon his death by Herstein, Nathan Jacobson, Kaplansky, and Zelinsky. After all, in addition to his research work, Albert had dedicated himself to such “homey” causes as obtaining a residence for the visiting mathematicians at Chicago, and his students were well aware of his interest in them while under his guidance and later while they pursued their careers. “Everyone who knew him,” Zelinsky wrote of Albert, “will remember his vigorous but round, medium build, curly hair, and often boyish demeanor; but especially one must remember his great, pleased grin that he flashed to welcome news of new successes for any of his extended family anywhere in the world of mathematics” ([e30], p. 665). Dickson’s students did not recall him in such endearing terms, but they did recognize the importance of what he imparted to them. Unlike his adviser, Albert was a role model and a mentor.
Conclusion
When President Harper opened the doors of the University of Chicago in 1892, he had his sights set on building an institution which promoted scholarly research and teaching. His gamble on the unproven E. H. Moore yielded a high return as Moore and his German colleagues, Bolza and Maschke, succeeded in their quest to emulate the German research tradition in the mathematics department. The three quickly shaped their department into an effective training ground for future research mathematicians. They imparted a strong research ethic — along with a commitment to both quality publications and the broader mathematical community — to the first generation of Chicago-trained mathematicians, not the least of whom was Leonard Eugene Dickson.
As the qualities Dickson saw in Moore, Bolza, and Maschke began to shape his own mathematical persona, Dickson specifically pursued algebraic and number-theoretic interests and generally emphasized far-reaching theories built on firm mathematical foundations. Moreover, like the original triumvirate at Chicago, Dickson impressed these and other mathematical and extramathematical qualities upon the second generation of Chicago-trained Ph.D.’s. Thus, in the first four decades of the 20th century, Dickson, as an active role model, helped define, further develop, and continue the strong algebraic research tradition begun by Moore in the last decade of the 19th century.
In the 40 years he served on the Chicago faculty, Dickson advised 67 doctoral students in algebra and number theory. His “gruff” personality and his almost impersonal interactions with his students, together with his high standards for research and publication, clearly suggest this late 20th-century characterization of Dickson as a role model as opposed to a mentor. To be sure, Dickson had qualities worth emulating, but his mathematical persona combined with the general academic climate of his day fostered a more distant adviser-student relationship than one might expect today.
Distant or otherwise, as the experiences of the students recounted in this paper reveal, Dickson showed his students (and colleagues, for that matter) what the life of a professional mathematician could and should be. Whether in their commitment to the pursuit of mathematical research (Logsdon, Hazlett, and, especially, Albert) or to the goals of the broader mathematical community (Rees and Albert), these students reflected their adviser’s attributes. Furthermore, their careers fit securely within the algebraic research tradition which grew and prospered at Chicago under Dickson. All of his students took the Dicksonian gospel with them as they spread across the country in the middle third of the 20th century to no fewer than 45 academic institutions in at least 22 states and three foreign countries. They, like the particular students discussed in detail above, did work in the same spirit as Dickson — whether in research, publication, or leadership positions — and this, after all, is how research traditions form and continue.
What Moore had begun in the last decade of the 19th century with his keen eye for research trends and his vision for the broader mathematical community, Dickson shaped and furthered through the early decades of the 20th century with his tireless commitment to research, graduate students, and quality publications. Dickson’s students — especially Albert — ensured the continuation of this strong algebraic tradition at Chicago and elsewhere as their embodiment of this heritage manifested itself in their varied careers. Thus, an investigation of the whole of Dickson’s mathematical career through the lens of role modeling — from those who modeled the life of a professional mathematician to him, to those for whom he modeled such a life — gives strong evidence that a mathematical community cannot achieve its potential for growth solely from research and publications. These ideas, tools, and values must be fruitfully cultivated in the next generation of mathematicians for the mathematics constituency not only to continue but also to flourish. In short, Dickson’s research and publications of all types, coupled with an entrenchment of his mathematical and extramathematical values by his students as they dispersed across the country, helped to consolidate and strengthen American mathematics in the opening decades of the 20th century.
