A Century from Now John Ewing

Mathematics: A Century Ago—A Century from Now John Ewing

I am not a historian, but I believe in history. His- to sketch mathematical life in America as it was tory lends perspective to the way we view our- near the start of the twentieth century and to use selves and our future. Oliver Wendell Holmes that as a background for a peek at what it might said: be like a century from now. I warn you in advance “When I want to understand what is happen- that I want to draw a moral from this sketch. I ing today or try to decide what will happen to- am therefore heeding the advice of a real histo- morrow, I look back.” rian, Herodotus (450 B.C.): Of course, Holmes also said: “A page of history “Very few things happen at the right time, is worth a volume of logic”…even wise men make and the rest do not happen at all. The conscien- mistakes. If we want to understand our profes- tious historian will correct these defects.” sion today, if we want to guess what will happen In preparing this article, I made all the nec- tomorrow, we ought to study yesterday. It’s almost essary corrections. a new century, and so, in a reflective mood, I decided to look back at the turn of the last century America a Century Ago? for our yesterday. The aim of this article then is What was happening one hundred years ago? In 1895 America was still in the throes of severe depression, which began two years before. The This article is an adaptation of an address given at the Supreme Court ruled 5–4 that the income tax was AMS-MAA Mathfest, Burlington, VT. unconstitutional. A new enterprise, Sears-Roe- buck, was struggling: A. C. Roebuck sold his in- John Ewing is executive director of the American Math- terest for $2,000, getting out while the getting ematical Society. His e-mail address is [email protected]. was good, or so he thought. The Underwood

JUNE 1996 NOTICES OF THE AMS 663 Typewriter Company was founded to market a the democratic community.… [8, p. new machine: a typewriter that actually allowed 118] the typist to see what was being typed. A young A few held out—Princeton was among them. man named Marconi transmitted a message to The elective system had many consequences. his brother on the other side of a hill. People Eliot boasted that it made scholarship possible thought it a hoax because there was nothing not only among undergraduates but among grad- connecting them. An enterprising young farmer uates and faculty as well. It promoted special- ran boiled wheat through his wife’s dough ma- ization; teaching special courses required ex- chine and baked the thin film to produce edible perts in the field, and “fields” might be narrowly flakes. A strange idea, but people bought 10 oz. defined. The old prescribed course of study was bags for 15 cents. The farmer was John Harvey elementary, shallow, superficial. Election per- Kellogg. mitted faculty to become deep, to indulge their All those things took place in 1895. own interests. The American University This in turn promoted departmentalization [7, p. 400]. In 1893 the biology department at the The American university in 1895 was emerging from a quarter century of change that had new University of Chicago split into five: zool- shaped it into a new form. That new form would ogy, botany, anatomy, neurology, and physiol- continue throughout the twentieth century. It ogy. Everywhere, departments of modern lan- was a frantic period of growth, and universities guages became departments of German or madly sought both students and money. French or Spanish. Departments of history be- During that growth, there was an earnest pub- came departments of European history and lic debate about the purpose of a university. In American history and political science. Depart- 1906 the editor of Outlook magazine summa- ments of mathematics sometimes became de- rized the popular view: The English university, partments of pure mathematics and depart- he said, revolved around culture, the production ments of applied. of gentleman aristocrats. The German university In part it was the elective system that led to found its life not so much in culture but in schol- the notion that a university was a place for re- arship, in erudition, in the production of schol- search. The idea was reinforced by the German ars. But in the American university the empha- model, but it was characteristically American, sis often was neither on culture nor scholarship, based on utility. but on service, on the preparation of young Professional requirements changed: by 1893 Americans for active lives of service [7, p. 356]. some amount of graduate work was required to New university presidents made it their mis- hold a permanent position at nearly every promi- sion to sell their universities to the public, and nent institution. By 1900 the Ph.D. degree became to a large extent the sales pitch was utility. In almost universally mandatory. From there, it 1888 David Starr Jordan, then president of In- was a short step to an insistence upon the con- diana University and soon to be president at tinued publication of scholarly studies. In 1901 Stanford, gave speeches enticing prospective the president of Yale made a statement that students. Among the reasons to go to college: the ought to sound familiar: Promotion at New benefit to society, the virtue of hard work, the Haven, he decreed, depends upon “productive financial worth of a degree [8, p. 342]. work” which gives the professor “a national rep- Utility was bundled with another idea, the utation”. Stanford began publishing an annual elective system. A prescribed curriculum, espe- list of its faculty’s output for each year. Presi- cially a traditional one in the classics and phi- dents kept index cards for every faculty mem- losophy, did not lend itself to the arguments of ber with precise lists of publications and talks. utility and service. So during this period the It was during this period that the term “publish elective system was born—students and faculty or perish” was first used. Research had become were encouraged to choose courses and topics. an essential part of university life [7, p. 404]. By 1900 Harvard had virtually no required There was a continuing debate about the bal- courses; Cornell had switched to an almost en- ance of teaching and research. Abraham Flexner, tirely elective system, as did Wisconsin, Michi- who came down squarely on all sides of the gan, and Yale. The greatest proponent of the issue, wrote an article in the Atlantic Monthly [7, elective system, Charles W. Eliot of Harvard, p. 403]. He told the story of asking a college dean waxed grandly in 1908: who was the best teacher in his institution. The dean named a certain instructor. “What is his rank?” he asked. [Because of the elective system,] teachers and students alike are pro- “Assistant professor.” foundly moved by the desire to serve “When will his appointment expire?”

664 NOTICES OF THE AMS VOLUME 43, NUMBER 6 “Shortly.” Mathematicians were feeling good. And math- “Will he be promoted?” ematicians in America were taking part in the transformation of universities, accommodating “No.” many new roles—utility, service, specialization, “Why not?” research. The elective system had an especially “He hasn’t done anything!” profound effect on mathematics. Before this Service to the community, utility, electives, time, almost all mathematics instruction was specialization, research. It was a logical pro- broad and shallow, often done by faculty as their gression and a major theme in the growth of obligatory extra course (a bit like coaches in American universities. high school who often seem to teach algebra Of course, this is only a partial picture. While and geometry courses). In 1888 Florian Cajori universities were marching forward towards re- conducted a survey. Of 118 college mathemat- search, they were also spending large sums of ics teachers, 73 reported teaching outside mathematics, and 32 of those taught outside the sci- money on boat houses and gymnasiums. They ences altogether—art, music, languages, and needed to attract more students. Was there am- Bible. By the turn of the century, mathemati- bivalence about the role of the university? At the cians taught mathematics. turn of the century, President Eliot at Harvard William Duren wrote a survey of American threw away many of Harvard’s books rather than mathematics for the centennial of the AMS [3, spend money on their storage. Columbia Uni- part II, p. 399]. Looking back, he laments the loss versity built an ornate new library, mainly de- of the traditional importance of mathematics, signed for holding receptions to honor bene- which had been, along with Greek and Latin, factors. At the singularly research-minded part of the required curriculum for many years. University of Chicago, in 1897 in existence only The evidence doesn’t bear this out: mathemat- five years, President Harper complained loudly ics grew, the profession prospered, and there was that his professors were offering entirely too greater concern about teaching—all because of many graduate courses and they had better stop the elective system. [8, p. 178]. The Society American Mathematics Specialization and research had two conse- At the turn of the century, mathematics world- quences: professional societies and professional wide was healthy and vigorous and savoring its journals. They arose naturally and sponta- triumphs—a bit like the present. In the previous neously. There was a new Athletic Research So- quarter century, some major problems had been ciety, for example, holding annual meetings of solved—the transcendence of e and π, the prime coaches and athletic directors to present (and to number theorem, the finiteness theorems in in- publish) research papers on athletics [7, p. 403]. variant theory. The German and French math- In the spring of 1887 an enthusiastic Thomas ematicians, and to a lesser extent the British, were Fiske was a second-year graduate student at Co- in a period of frenetic activity, with Hilbert and lumbia College. For some reason, his advisor in- Poincaré vying for prominence. sisted he leave to spend at least six months There were many analyses of what everyone abroad. He traveled to Cambridge, England, for viewed as a golden era in mathematics: from a semester and immersed himself in the math- 1900 until 1905, the Bulletin published a num- ematical atmosphere. He later recalled: ber of articles on progress during the nineteenth century, including a general summary by James Pierpont, who delivered an address to the In- On my return to New York I was filled ternational Congress of Arts and Sciences in St. with the thought that there should be Louis. Pierpont’s article begins: a stronger feeling of comradeship among those interested in math- The extraordinary development of ematics, and I proposed to my class- mathematics in the last century is mates and friendly rivals, Jacoby and quite unparalleled in the long his- Stabler, that we should try to organize tory of this most ancient of sci- a local mathematical society [3, part ences…entirely new [fields] have I, p. 13]. sprung up in almost bewildering pro- They mailed a general letter. Six people re- fusion, and many of these have sponded to form the New York Mathematical So- promptly assumed proportions of ciety. By the end of 1890 there were twenty- vast extent [Bulletin, 1904, pp. three members. Things were right. In 1891 they 136–159]. launched the Bulletin to record news and math-

JUNE 1996 NOTICES OF THE AMS 665 ematical events as well as articles. The number mathematician—objected to any journal that of members had to grow, and Fiske launched a would compete with the American Journal. campaign increasing membership from twenty- Maxime Bôcher had a stroke of genius: Wasn’t three to two hundred in a matter of months. In the Society supposed to publish its transac- 1893 Felix Klein visited the Mathematics Con- tions? he asked. Certainly, replied McClintock, gress at the Chicago World’s Fair and met with transactions were acceptable. And so a second the members of the New York Mathematical So- research journal was established. The Transac- ciety in a special session [6, p. 295ff]. By 1894 tions published its first issue in January 1900. the Society changed its name to the American The American Mathematical Monthly had ex- Mathematical Society. By the turn of the cen- isted for seven years in 1900. It was founded by tury, the Society claimed hundreds of members a man, not an organization, and it was struggling across the country and continued to grow in to find an identity. The man who created the spite of its high dues—$5 per annum, nearly Monthly, Benjamin Franklin Finkel, was a school ten times those of the equivalent German Soci- teacher with no editorial experience. After mov- ety. ing to a preparatory school in Missouri, he decided to publish a new journal which he called, Journals somewhat grandly, the American Mathematical By 1898 members of the Society realized they Monthly. Finkel had the ambition to publish a needed a research journal in addition to the Bul- journal devoted solely to mathematics, catering letin. The natural course of action (and the eas- to the needs of teachers of mathematics, largely iest) was to assume partial control of the Amer- because of the deplorable condition of high ican Journal of Mathematics, founded by James school teaching. The description of the Monthly Joseph Sylvester and published for a quarter of on the first page began: “Most of our existing a century by the Johns Hop- journals deal almost exclusively with subjects be- kins University. The Council yond the reach of the average student or teacher passed a resolution: of mathematics…The American Mathematical Monthly will endeavor to reach the average mathematician.…” [Monthly, 1894, p. 1] Even then, It is recommended mathematicians complained that articles were that the American too specialized for general interest. Mathematical Soci- Finkel’s plan was at first a failure. Few high ety offer to the school teachers subscribed. Nonetheless, he had Johns Hopkins Uni- the genius to enlist the help of college and university the follow- versity faculty to provide both funding and writ- ing plan of cooper- ten material. The Monthly began to thrive, di- ation, for the rected towards teachers in colleges and purpose of enlarg- universities rather than high schools. ing and improving The Annals of Mathematics was founded by the American Jour- Ormond Stone at the University of Virginia in nal of Mathemat- 1884. At the turn of the century, it was trans- ics…[1, p. 56] ferred from Virginia to Harvard. Its purpose, The “plan of cooperation” the Bulletin of 1899 reported, was not to com- was simple: the AMS would pete with the American Journal or the new Trans- take over editorial control and actions. The Annals was meant to appeal to the Simon Newcomb in return contribute a small general mathematical public rather than to spe- sum of money towards yearly cialists; it would publish short expositions and expenses. It didn’t sound like much of a deal. Be- would avoid highly technical articles. The writer cause the president of the Society, Simon New- complained that there was a scarcity of good ex- comb, was on the faculty at Hopkins, people position. There was great hope that the new An- thought it might work. nals would entice authors of such expositions. The negotiations quickly broke down; New- It seems to me we’re still complaining…and comb accepted the money but said that editor- hoping. ial control had to remain with Johns Hopkins. Be- sides, he said, the appearance of the name the Libraries American Mathematical Society on the title page In 1900 mathematics libraries faced a constant would be distasteful. This didn’t sound like struggle. Departments were often brand new much of a deal for the AMS. Younger members and lacked back issues of basic journals. The of the Society decided to establish a new jour- great depression which began in 1893 had left nal, but J. E. McClintock—a charter member of many institutions short of funds; library budgets the Society, a past president, and an eminent were easier to cut than salaries (although those

666 NOTICES OF THE AMS VOLUME 43, NUMBER 6 were cut as well sometimes). Mathematics chairs the AMS reported that 10 percent of its members complained of inadequate budgets and used ar- had received Ph.D.s from German universities; guments that sound familiar today. 20 percent had done graduate work in Germany. The Indiana University departmental report Why was the German connection so strong? from 1893 gives the mathematician’s laboratory In part, the elective system was the cause. The argument: “In our advanced work we greatly feel German Lernfreiheit system, in which students the need of complete sets of the leading math- could choose lectures and seminars at will, ematical journals. These are as essential in math- helped to make it comfortable for American stu- ematical work as apparatus and specimens are dents. In part, Klein’s magnificent lectures and in scientific work.” the carefully crafted seminars, in which stu- The report from 1895 shifts the argument to dents learned not just about mathematics but its emphasize students: “There are more students presentation as well, attracted students. doing advanced work than ever before. This in- But there were more mundane causes [8, p. creased demand for advance work makes nec- 129]. Economic conditions in Germany had dra- essary better library facilities. …Mathematical pe- matically decreased the number of German stu- riodical literature is to the advanced student in dents; foreign students were needed, especially mathematics what the modern well-equipped at the graduate level. (Sound familiar?) And Ger- laboratory is to the student in science.” many was a bargain. In 1890 it was estimated that And the report from the next year, 1896, a year of study in Germany was cheaper by a sounds pathetic: “Our work is progressing as sat- third than a year at Hopkins, Harvard, or Cor- isfactorily as we could wish with our present nell—and that included travel! equipment. We have need of the standard jour- Mainly, however, a German degree was pres- nals and periodicals for our students; we there- tige. “One goes to Germany still a doubter as to fore beg of you again a liberal appropriation for the possibility of the theoretic life,” read an ar- the purchase of a few standard journals and ticle in Scribner’s magazine current periodicals.…Each year the department (1891). “One returns an ideal- offers a course in solid analytic geometry and ist, devoted for the time to theory of surfaces. Any instructor is handicapped pure learning for learning’s in his presentation of this subject without the sake” [8, p. 129]. use of models. A complete set of Brill’s plaster At the turn of the century, models would be exceedingly valuable to the these were the dominant fea- department.” tures of American mathemat- By 1899 they returned to the laboratory ar- ics: a changing vision of uni- gument: “Seminar work is a new feature versities, the emerging AMS, and…emphasizes the necessity of better library new journals, a close connec- equipment. It would seem that this is not un- tion with German mathematics. reasonable when it is remembered that our li- All these ideas came together brary is also our laboratory.…[The report adds in one man, who shaped math- a new plea.] We believe that nothing would do ematics in America for much of more to encourage and inspire our students the next century. than to give them the opportunity of listening to lectures…by well-known mathematicians of E. H. Moore other institutions. The expense attached to such Eliakim Hastings Moore was a course of lectures would be very small. We born in Marietta, Ohio, in 1862. E. H. Moore therefore ask that $50 be appropriated for this His father was a Methodist purpose.…” minister, his grandfather, a Money for libraries, equipment, colloquia. congressman. Moore went to Yale. He graduated This aspect of professional life does not change. valedictorian in his class in 1883 and two years later received his Ph.D. under Hubert Newton. The German Connection The title of his thesis was a bit vague: “Extensions At the turn of the century, American math- of Certain Theorems of Clifford and Cayley in ematics had sturdy ties to the German math- the Geometry of n Dimensions.” ematical community. Many leading mathemati- Moore’s advisor, Newton, urged him to spend cians had received their Ph.D.s in German time in Germany and lent him money to do so. universities; those who had not often traveled He traveled to Göttingen and then on to Berlin, to Germany for a year or two. Felix Klein, first where everyone, including Moore, worshipped at Leipzig and then at Göttingen, was a favorite Weierstrass and Kronecker. Moore returned to of the young American mathematicians: six of America in the next year and soon settled into his students became presidents of the AMS; thir- a permanent position at Northwestern Univer- teen served as vice-president [6, p. 259]. In 1905 sity, where he became associate professor. By

JUNE 1996 NOTICES OF THE AMS 667 1892, seven years after his degree, Moore had public education, he began teaching in a one- five papers and one research announcement— room country school house at the age of 17. He hardly a rocketing career. was well liked, eager, enthusiastic. In the midst Now the same events that drove the expan- of teaching, he managed to attend Indiana Uni- sion of mathematics swept Moore into promi- versity part-time, majoring in mathematics. “Ma- nence. In 1892 William Rainey Harper, the pres- joring” was a new idea, recently introduced at ident-elect of the new University of Chicago, Indiana by its president, David Starr Jordan. asked Moore to be the acting head of the math- Rothrock finished his undergraduate work in ematics department. Moore accepted and re- 1892 and, at the age of 28, immediately became cruited two others, Bolza and Maschke (students an instructor in the department. of Felix Klein). The next year, in 1893, the In that year, the department made a report Columbian Exposition in Chicago held a Congress to the board of trustees, pleading for reunifica- of Mathematicians, to which Felix Klein came as tion of the departments of pure mathematics and the Prussian representative. The new Chicago de- of applied mathematics, and recommending partment and Moore were the natural hosts. some appointments: Shortly after, Moore organized the Chicago section of the American Mathematical Society, serv- ing as its first chairman, and was then elected vice-president of the Society. When negotiations …the teachers of mathematics in the were held to take over the American Journal, university have all been graduates of Moore was involved. Indeed, he was one Indiana University and have under- of the agitators urging a new jour- stood the conditions surrounding the nal, and he became the first edi- university and the needs of the stu- tor-in-chief of the Transactions. dents.…In order that these condi- By the end of that year, in tions may be preserved…we recom- 1900 at the age of 38, E. H. Moore was elected presi- mend that David A. Rothrock be dent of the American made instructor in mathematics at a Mathematical Society. salary of $800. Mr. Rothrock has Whether Moore was given excellent satisfaction as a swept along by these teacher and is one of the best stu- events or caused them dents of mathematics we have had. isn’t certain. What is cer- Other reports of that year were interesting, tain is that E. H. Moore by the way. The English department complained dominated mathematics in their report about class size: “The correction in 1900. and personal criticism by the teacher is to the Moore represents one English student what the personal blackboard face of mathematics—it’s work is to the student in the Department of the face we see when we Mathematics, and thus there should be a section look back at history, because for every twenty students.” we usually view history with And in that same year, the president filed a Carlyle, as the biography of great men. But mathematical life in 1900 report to the board, grumbling that the faculty comprised two cultures, and concen- were leaving due to low salaries, that they needed David Rothrock trating on only one distorts our view—and fewer hours teaching and more time to do schol- misleads us about our own present. arly work, that they needed more laboratory equipment, and that the library budget was dis- David Rothrock graceful. When Klein left Leipzig in 1886, he arranged Rothrock took leave to attend the new Uni- for his friend Sophus Lie to take his place. Lie versity of Chicago in 1894 and again during the had six American students in the next dozen summer of 1896, studying under Moore, Bolza, years, including some well-known mathematical and Maschke. In 1898 he spent a year at Leipzig, figures [6, p. 234ff]. One of the other students receiving his Ph.D. under Lie. The title of his of Lie, however, was a young mathematician thesis: “Invariants of the Finite Continuous from Indiana. In a strange way, his academic ca- Groups of the Plane”. He returned to Indiana in reer explains much about American mathemat- 1899 and took up a heavy load of teaching, tu- ical life of one hundred years ago. torials, and administration. David Rothrock was born in a small Indiana From the start, David Rothrock was a master town in 1864, one state farther west and two teacher. This is from the department’s annual years later than E. H. Moore. After completing his report from 1895:

668 NOTICES OF THE AMS VOLUME 43, NUMBER 6 No better teacher than Assistant Pro- Once the students arrived on campus, acad- fessor Rothrock has ever been in the emic standards varied greatly. In 1903 at Yale a department. He inspires his students survey showed that the average senior spent to their best efforts and is greatly less than an hour a day preparing for classes. At beloved by all of them. …He has had Harvard some students received A’s in courses charge of a number of advanced they had never attended. An observer of Harvard classes this year, and in all of them commented, “Perhaps in no institution is the he has done superior work. value of the A.B. degree so indeterminate.” [8, p. 359] Rothrock was one of four spectacular teach- Letter grades—or any system of course ers in the department, and together they nur- grades—were not uniform. At Indiana it wasn’t tured students into the undergraduate curricu- until 1908 that the faculty recommended the use lum and, at least for some, into mathematics. of letter grades, six of them: A, B, C, D, condi- They taught three to five courses each term, tioned, and failed. A great debate ensued about three terms per year. They were effective. In who should have access to the grades. Surpris- 1930 more members of the American Math- ingly, the students felt grades should not be ematical Society had received their undergrad- shared with anyone, including the students them- uate degrees at Indiana University than at any selves [2, vol. II, p. 47]. other institution except Harvard. During a Almost immediately faculty worried about twenty-year period, from 1888 until 1908, out grade inflation. After one year at Indiana a com- of more than 2,000 students receiving under- parison was made with Harvard, Missouri, and graduate degrees at Indiana, 10 percent had ma- Wisconsin. Indiana didn’t look good, and faculty jored in mathematics. complained about a few of their colleagues who E. H. Moore dominated mathematical life in gave unusually high grades, distorting the sta- 1900. In his own way, so did David Rothrock. tistics for the entire university. The term “snap Rothrock was the other side of mathematics in course” was used for the first time. America. He clearly loved students, but he loved By 1901, amid much impassioned debate, the mathematics too. He belonged to the AMS and first College Entrance Board examinations were contributed regularly to a new and rebellious held, leading eventually to uniform systems of journal, the American Mathematical Monthly. establishing entrance requirements. In 1902 the He taught advanced seminars on material he AMS formed a committee to formulate standard brought back from Leipzig and digested excit- entrance requirements in mathematics; their de- ing new research for the graduate student sem- tailed report was published a year later in the inar. But he also taught analytic geometry—every Bulletin. Several years after that, the American year—and wrote textbooks in algebra and Association of Universities began accrediting trigonometry. Later, he was an active member in graduate schools. the Mathematical Association of America and served as dean of the college for seventeen years Reform from 1920 until his retirement in 1937. One hundred years ago mathematicians complained about the sorry state of teaching. A good Academic Standards sample of those complaints can be found in the What was life like in this other culture? Math- report of a panel discussion, held at the summer ematicians seemed to agree about two things: meeting of the Society in 1900. Here are some students were deplorable, and so was teaching. excerpts from the Bulletin [October 1900, pp. As the number of universities increased, the 14–24]: competition for students grew keen. Even before 1900 there was a great debate about the nature I urge…a thorough revision of college of a college education and which students should courses in differential calculus. The go to college. Articles such as “Boys Who Should subject is too often taught along tra- Not Go to College” appeared in the Atlantic ditional lines; the lecturer should ask Monthly [8, p. 123]. New institutions opened himself whether certain parts of his with an invitation to all comers, no matter how subject possess the same importance ill prepared; this happened at Cornell, at Stan- today that they had in Euler’s time. ford, and (to a lesser extent) at Chicago. Colleges (Harkness) needed money: at one college in Ohio, an enthusiastic donor offered a large sum provided The fundamental principles of the the school enroll a certain minimum number of calculus must be taught in a manner students each year. Not surprisingly, they suc- wholly different from that set forth ceeded [8, p. 357]. in the textbooks if they are to be-

JUNE 1996 NOTICES OF THE AMS 669 come flesh and Here’s the opening paragraph on series from a blood to the stu- well-known and heavily used book on college al- dent. (Osgood) gebra: By the term “series” we mean a se- Our textbooks quence of terms connected by + abound…with signs. If the number of terms in the proofs that are no sequence considered be finite, the proofs and with ex- corresponding series is finite; if the tremely misleading number be infinite, the correspond- statements. (Hark- ness) ing series is called an infinite series. A “sequence connected by plus signs?” Here All this time applied was the progenitor of all modern college alge- mathematics have bra texts…the bloodlines are evident. been out of fashion, New teaching journals (including the Math- but there are many ematics Teacher) were formed in the first years signs that a change of the new century. is coming, and the Committees toiled tirelessly, producing re- intelligent college ports in the Bulletin and the Monthly. The In- William Fogg Osgood instructor will do ternational Commission on Teaching of Math- well to anticipate ematics was formed in 1908. It formed this change.… (Harkness) subcommittees to study primary, secondary, The lecture method was hotly debated. Moore, and college teaching in various countries. The of Chicago, spoke disparagingly about lecturing: subcommittees produced great mounds of re- he was throughout his life an outspoken pro- ports, and those in this country were published ponent of the laboratory method, leading stu- by the United States Bureau of Education. They dents to discover ideas for themselves. Osgood, are interesting reading for historical purposes. of Harvard, was the focus of opposition to such Six years after its beginning, we are told that the new-fangled ideas. He fulminated: Commission had produced 10,293 pages of reports [5, p. 26]. G. A. Miller, himself a prolific au- thor, reported on these ten thousand pages with As to how early the lecture method a clear sense of admiration. may be used with profit I may say that at Harvard the lecture method Research versus Teaching has been employed with success for What was it like to be a mathematician in Amer- over twenty years in practically all ica at the turn of the century? If you were with of the courses in mathematics.… (Os- E. H. Moore at Chicago in a new department that good) had quickly come to dominate mathematics in H. E. Slaught looked back at the first decade the Midwest, professional life was research with of the twentieth century in the Monthly: a heady view of a new future. It was exciting. If you were with David Rothrock or at dozens of other universities and colleges across the coun- This decade of remarkable activity try, professional life was teaching, but mixed on the part of teachers of math- with admiration for the growing mathematical ematics in this country can be ac- community in America. That was exciting too. counted for only on the hypothesis But make no mistake. There were two cultures of a widespread and deep-seated feel- in the American mathematical community, and ing of unrest and dissatisfaction with the tensions between research and teaching were the present methods and results, a already felt in 1900. Fiske wrote in a summary conviction that in some way the math- of mathematical progress in America, which ap- ematics teaching in the schools fails peared in the Bulletin in 1905: to connect vitally with the preparation for living in the twentieth century needed by the vast majority of the Notwithstanding the great progress pupils in the schools. [Monthly 20 recently made in America by our sci- (1913), p. 171] ence, we are far from being in a position that we can regard as entirely Were there problems? You bet. In the nine- satisfactory.… the most pressing de- teenth century, calculus books were sloppy and mand seems to be that those engaged fat. There was a five-volume text for calculus. in lecturing…at American universities

670 NOTICES OF THE AMS VOLUME 43 NUMBER 6 should be given greater opportunities Moore, Fiske, Rothrock, and for private study and research. At Finkel found a way to accom- present, the time of almost every uni- modate the two cultures of versity professor is taken up to a very mathematics, and the math- large extent with…the care of com- ematical community gained. It paratively young students. [Bulletin, was an important lesson. February 1905, p. 245] We’ve forgotten that lesson from time to time during the Less teaching, more time for research. past one hundred years. Our The Genius of American Mathematics profession has always suffered from a lapse of memory. One hundred years ago there were two mathematical cultures in America. Call them “research” and “teaching”, but the differences are The Future? much more complicated. The genius of Ameri- What will mathematical life be can mathematics in 1900 was that so many math- like one hundred years from ematicians succeeded in combining those cul- now? I began by telling you I tures. In that same Bulletin article in which he wasn’t a historian. Well, I’m lamented the burdens of teaching, Fiske went on: not a futurist either. Futurists make profound predictions such as this in a recent news- Thomas Fiske paper: THE FUTURE HOLDS …we must [also] have improved BOTH PROGRESS AND PROBLEMS. I think they methods of teaching, better text- are right. books, and more good treatises on Nonetheless, I will make some predictions. I advanced subjects. [Bulletin, February can’t tell you what mathematics will be fash- 1905, p. 246] ionable in 2095, but I can tell you that our pre- E. H. Moore was the master, and perhaps the sent lists of important mathematics from the leader, in combining mathematical cultures. He twentieth century will look quaint then. was clearly the central figure in research math- I can’t tell you how mathematics will be com- ematics in America, and yet he spent much ef- municated in the year 2095, but I can tell you fort and time immersed in teaching. From the that journals as we know them will not exist. start, Moore was a special friend to the Monthly, Libraries will not exist either, but the need for and it was largely Moore’s influence that shaped storing and disseminating mathematics will. the Monthly in years to come: not a research Someone will have to pay for this, and math- journal, but connected to research nonetheless. ematicians will likely be complaining about the Moore gave his retiring presidential address at lack of money. I hope they will have found a vari- the end of 1902. The topic? Mathematics edu- ation on the argument that the library is the cation. mathematician’s laboratory. Will students learn in classrooms, sitting in lectures? Maybe not. But I’ll bet that students still learn by interacting with humans—with teach- The American Mathematical Society ers and with other students. And I’ll bet there has, naturally, interested itself chiefly are complaints in 2095 that students are ill pre- in promoting the interests of research pared and that teaching is badly in need of re- in mathematics. It has, however, rec- form. Committees will set to work, but being elec- ognized that those interests are tronic, reports will likely be far longer—a closely bound up with the interests frightening thought. of education in mathematics. …Do What about the two cultures? Mathematics is you not feel with me that the AMS, as vertical, with the frontiers of research at the the organic representative of the top. Most mathematicians are helping students highest interests of mathematics in climb from farther down. That will not change, this country, should be directly re- and it’s the fundamental cause of the two cul- lated with the movement of re- tures. The two cultures will still be around in form?…[Bulletin, May 1903, p. 412] 2095. But I hope we have learned some lessons And in the same year, as Moore spoke these from the twentieth century. Looking back, we can words, David Rothrock was lecturing to gradu- see the genius of American mathematics in 1900: ate students on new ideas from invariant theory, it embraced both cultures. We ought to recog- and Benjamin Finkel was building an expository nize the vitality this brings to the profession and journal for research mathematicians. the richness it lends to the mathematics itself.

JUNE 1996 NOTICES OF THE AMS 671 There will be two cultures in 2095, but I pre- dict most mathematicians will find it natural to participate in both. I hope I’m right. Both E. H. Moore and David Rothrock would approve.

References [1] Raymond Archibald, A semicentennial history of the American Mathematical Society, 1888-1938, Amer. Math. Soc., New York, 1938. [2] Thomas D. Clark, Indiana University: Midwest pioneer, Vols. I, II, Indiana Univ. Press, Blooming- ton, IN, and London, 1973. [3] Peter Duren (ed.), A century of mathematics in America, Vols. I, II, III, Amer. Math. Soc., Provi- dence, RI, 1988. [4] John Ewing (ed.), A century of mathematics: Through the eyes of the Monthly, Math. Assoc. America, Washington, DC, 1994. [5] G. A. Miller, Historical introduction to mathematical literature, Macmillan, New York, 1916. [6] Karen Parshall and David Rowe, The emergence of the American mathematical research community, 1876–1900, History of Mathematics, vol. 8, Amer. Math. Soc., Providence, RI, 1994. [7] Frederick Rudolph, The American college and university, Alfred A. Knopf, New York, 1962. [8] Laurence R. Veysey, The emergence of the Amer- ican university, Univ. of Chicago Press, Chicago, IL, and London, 1965.

672 NOTICES OF THE AMS VOLUME 43, NUMBER 6