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Comments on Kahane- Krickeberg-Lorch Letter As a reciprocity member of both the American Mathematical Society and the Deutsche Mathematiker-Vere- inigung, I am very disturbed to see an about attack made in the letters to the No- this. As for the tices (vol. 41, number 6) on the DMV. “historically ac- The first question to be asked is how German curate perspective”, the AMS is involved: the letter should mathemati- Bieberbach’s activities surely have been directed to the DMV. cians but with the DMV are fully The authors say that an “historically which brought no described in the DMV cen- accurate perspective” on the past is personal advantage to him. Let it tenary publication Ein being “withheld”. Do they claim to only be said: Bieberbach later rec- Jahrhundert Mathematik, have a superior perspective to those ognized and deeply regretted these er- and his activities as de- who are genuinely trying to under- rors, as is confirmed by well-war- partment head in Berlin are stand what happened under Hitler? ranted assertions.” This is a perfectly described in highly critical terms. No The attacks which they proceed to fair statement, in accordance with the mathematician admires what Bieber- make on their mathematical colleagues usual perception of an obituary notice. bach did in these matters, but to do not suggest that they have. It is surely at least plausible that doubt “current willingness to face the Their first complaint concerns the Bieberbach did regret his activities: it past” in his case is absurd. obituary notice published about would be amazing if he had not re- Then comes Strubecker. Appar- Bieberbach in the Jahresbericht, writ- gretted making such a fool of himself. ently the offence of his obituary ten by the late H. Grunsky. This notice But no, this is not enough for our writ- writer, H. Leichtweiss, is to say that says the following: “This is not the ers, who want evidence of it. But he enjoyed the war years in Stras- place to look closely into these events, Grunsky’s article says “as is confirmed bourg. The obituary notice says that which susceptibly harmed mathemat- by well-warranted assertions”. There one of these enjoyments was his mar- ics in Germany and many individual is no reason to question his integrity riage, but this is not mentioned in

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the letter. Was it a crime, then, to do accusation an example of the anti- matics and descriptive mathematics creative mathematics in Strasbourg German xenophobia which is so com- equal attention from the middle during the Nazi occupation? And our mon in the daily press (in England at school on. The (two-semester) course righteous letter writers are incensed least!) but which has up to now not that I teach to prospective elementary because Leichtweiss speaks of the “oc- reared its ugly head in the mathe- and middle school teachers prepares cupation” of Strasbourg by the Allied matical literature. I sincerely hope them to do deductive mathematics. forces. But that is what they did! To de- that the AMS is not going to change On the final exam (among other mand the use of a word with emotive this. things), I give definitions that they significance instead of “occupation” have not yet seen and new theorems is pure political correctness and has Norman Blackburn that depend on the definitions. The nothing to do with historical under- The University better students are able to provide standing. Manchester, England original proofs for the new theorems. The next attack is on the writers of Some of these better students started a review of Teichmüller’s life and work. (Received October 11, 1994) out with math anxiety. Here our defenders of the truth have The students in my present course a little difficulty, since everybody are much more excited than were the knows that Teichmüller was a genius. Balancing Deductive and students in a traditional section of the They fall back on the old method for Descriptive Mathematics course I first taught. In sections of the getting around this: “Yes, but his re- in the Schools present course, almost half the class sults were anticipated by someone I am writing this to research mathe- rated the course as excellent. In the else.” The writers of the review are maticians to interest as many as I can traditional course, almost none rated then taken to task for not saying this, in a course for prospective elementary it excellent. The course leads up to in particular for “hardly mentioning” and middle school teachers. A recur- certain student investigations. In the Grötzsch. I am in no position to judge ring theme in the letters and forum first investigation, students conjec- the importance of Grötzsch’s work to sections of the Notices is that very ture general set relationships from Teichmüller, but the letter writers ap- few people (educators, politicians, exploring diagrams. The students, on parently think themselves better funding sources) have any idea of their own, discover DeMorgan’s laws, judges of this than the distinguished what mathematics is or what mathe- the distributivity between set inter- mathematicians who wrote the article maticians do. This theme is a symp- sections and union, and some inter- on Teichmüller. Of course, Grötzsch’s tom of a problem I address in my esting relationships that have been work is also discussed in detail in Ein course. For the purpose of explaining new to me. By the time they get to the Jahrhundert Mathematik, the proper the course, I break mathematics into investigation sections, they are able place for it. Again there is no question two aspects (1) discovery/deduction to provide original proofs of their of unwillingness to face the past here; and (2) description computation. Dis- conjectures. A second investigation no one defends Teichmüller. But there covery/deductive mathematics asks has them extend certain order and is mention in the letter of the fact that the questions, What is true about this ring-theoretic properties of the inte- Grötzsch was an “anti-Nazi dismissed thing studied? and How do we know gers to the rational numbers by mak- on that account”; the inclusion of this it’s true? Descriptive mathematics, ing new definitions that are appro- information can only be interpreted as on the other hand, asks us to describe priately analogous to the definitions a suggestion that the authors of the Te- some physical or business situation for the integers. ichmüller article ignored Grötzsch for mathematically and to compute as- Students are taught to write proofs that reason. Such a slur is unaccept- sociated values. To most people who first in terms of a formal system with able. The fact is that the authors of the are otherwise well educated the tech- many inference rules. The rules model letter are unable to come to terms with niques associated with solving equa- what mathematicians do naturally. the fact that it was possible for some- tions, finding antiderivatives, dividing The explicit use of the rules gives one to be both a great mathematician whole numbers, and so on are math- way to implicit use, in somewhat ab- and a fanatical Nazi. ematics. breviated proofs, which look like tra- So the letter ends with an attack The reason for people’s misun- ditional mathematical proofs at a be- on the DMV itself. This is because a derstanding of mathematics is clear. ginning level. It is the explicit rules, “discussion article”—whatever that Their entire experience in twelve years however, which make possible the may be—was rejected by the Jahres- of school and most college courses description of a heuristic system for bericht with no invitation to reply. But will have been with descriptive math- discovering one’s own proof steps. you, sir, rejected a letter of mine about ematics. There are now high school The entire program of learning proof eleven years ago, with no explanation. geometry texts that boast that not is linguistically guided and has noth- Is this not what journals do with math- one proof is to be found in the text. ing to do with students’ quantitative ematical articles all the time? It is im- The solution that I propose to this abilities or familiarity with descriptive possible not to see in such a trivial problem is to give deductive mathe- mathematics.

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Instructors that teach the same I turned my attention toward ele- making high school geometry increas- course here at the University of Maine, mentary teachers because of dis- ingly descriptive. The result of all this and to whom I have described my sys- couragement with math majors. We is that people don’t know what math- tem, are universally put off by the for- are getting fewer of them, and the ematics is. We get mathematics majors mality of the system at its initial ones we do get seem to have the that are discouraged and disappointed stages. I have been told that the trend wrong idea of mathematics. Our be- when they find out. these days is toward increased infor- ginning majors have seen nothing but In the 1960s the “new math” was mality. Students, however, are of ex- descriptive mathematics in their pre- introduced as a curriculum change. It actly the opposite opinion. When we vious education. They have learned was widely misunderstood. Teachers get to the point where we can use rules to solve certain typical “problems” by had never seen it in their own educa- either implicitly or explicitly, students computation and copying “solutions” tion, and parents didn’t understand say that they need the experience with presented to them. Moreover, this has their children’s homework. I take it the explicit use. been what has attracted them to math- that if deductive mathematics is to be Mathematicians here that are in- ematics. made a part of school mathematics, volved with their own research work, The descriptive orientation of thir- the place to start is with courses for and to whom I have described my teen years of mathematics through prospective elementary and middle course, have been very encouraging calculus and the deductive orienta- school teachers. I have some material but universally (after questions about tion of advanced undergraduate that has worked well for me. I’d like to the involvement necessary) unwilling courses have led to student difficul- get others interested. to teach the course themselves. The ties in the latter and to the introduc- course doesn’t take any more time tion of courses intended to “bridge” Andrew Wohlgemuth than a section of calculus. Since teach- the gap between the elementary and University of Maine ing the course involves leading stu- the advanced courses. A “bridge” is a (Received November 28, 1994) dents into ground that is totally un- poor metaphor. It implies that de- familiar to them, there is some need scriptive mathematics has at least for emotional involvement. In my been taking students in the right di- Allocation of Resources at U.S. twenty-five years at Maine, I have rection and now they need a little Universities taught graduate and undergraduate extra to proceed to mathematics from I have been reading with great interest courses in my general area and two an “advanced” viewpoint. A better reports and comments which have ap- graduate courses in my own research metaphor would be to consider de- peared in the Notices on the job mar- specialty. Yet, I have never been as ex- scriptive and deductive mathematics ket difficulties for Ph.D. mathemati- cited about a course as I am about this as orthogonal. Mathematics with this cians in the U.S. Many people have course for teachers. I have hopes that metaphor is two-dimensional, and complained that rosy predictions made some day my students will have stu- proceeding ever so far in the de- just a few years ago did not come true. dents that are learning how to do de- scriptive direction gets you nowhere Others have suggested remedies or ductive mathematics. I have hopes that in the deductive direction. This is not have voiced their distress. Current ex- some day students will enter high my armchair philosophy. It is what planations for the unpredicted decrease school geometry courses with an un- students in my introduction to ab- of university jobs leave me uncon- derstanding of what a proof is and stract mathematics course have told vinced. “Rising costs” and “budget cuts” how to go about finding one’s own me of their previous work. are explanations which raise more proof. Perhaps politicians will have The identification of deductive questions than they answer. Of course some idea of what mathematicians do, mathematics with advanced mathe- education is a labor-intensive activity because they themselves have done matics has done damage to mathe- and as such its costs tend to go up the same sort of thing as part of their matics education. Deductive mathe- faster than the costs of industrial pro- education. matics deserves a place in the school duction. But this is true of most service While entertaining all these grand curriculum. It is not harder than de- activities in modern society, most of hopes, I am faced with the reality that scriptive mathematics. It’s just dif- which are expanding. What is the dif- I haven’t interested a single soul here ferent. The habits of minds that go ference, in particular, for education in in teaching the course with my mate- with deductive mathematics—the abil- science and mathematics? Are we sure rial. I have a hunch that if I do inter- ity to articulate things precisely and that available resources are rationally est people in this, it will be people to construct and evaluate logical ar- allocated? And who decides how to al- that don’t normally teach a course for guments—are more needed in a gen- locate these resources? I looked for an- elementary school teachers. The ma- eral education than the ability to use swers in the Digest of Educational Sta- terial is written on a level appropriate mathematics as a language for sci- tistics (1993 edition), a publication of for such students, but it could be help- ence and business. We have rushed off the U.S. Government, which dedicates ful for anyone that doesn’t know what in the descriptive direction. We get to an entire chapter (Chapter III) to “Post- a proof is or how to begin finding one. calculus as soon as possible. We are secondary Education”. My attention

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was caught by a statement appearing faculty professionals, which dropped bill for universities (legislators, tax- in the introduction to this chapter. from 52.4 to 29. Another way to look payers, students and their families, “Administrative expenditures (in- at these data is to observe that in alumni) that their money is being di- stitutional support and academic sup- 1976 the proportion of full-time fac- verted to expand activities which have port, less libraries) have been rising ulty to full-time nonfaculty profes- little or nothing to do with the uni- more rapidly than most other types of sionals or administrators was 1.75 versities’ real functions: teaching and college expenditures. At public uni- and that in 1989 the same proportion research. But if we are going to appeal versities, between 1980–1981 and fell to 1.1. to parents and taxpayers and plead 1990–1991, inflation adjusted admin- It is tempting to formulate the fol- for more consideration for instruc- istration expenditures per full-time- lowing hypothesis which could par- tional needs, we should be able to equivalent student rose 26 percent tially explain budget cuts for mathe- offer something. This means that we compared with 12 percent for in- matics. Universities have always had should all pay more attention to our struction expenditures per student. to balance limited resources with un- duties as teachers and give more At private universities during the same limited opportunities for expendi- credit and prestige in our commu- period, the per-student administrative tures. In the late seventies, a downard nity to dedicated teachers. costs rose 45 percent, and the in- fluctuation in resources made bud- struction costs rose by 38 per- get cuts in one form or another im- Allessandro Figà-Talamanca cent…College faculty generally suf- perative. The decision on where to Universita La Sapienza fered losses in the purchasing power cut was taken by a group of people Rome, Italy of their salaries from 1972–1973 to which could be loosely identified with (Received December 7, 1994) 1980–1981, when average salaries fell the “executive/administrative/man- 17 percent after adjustment for infla- agerial staff” of the various universi- tion. During the 1980s, average ties. Ultimately decisions were taken Maurice Auslander salaries were on the rise and have re- by trustees or regents, not by em- Maurice Auslander, a member of the couped most of their losses.” ployees of the universities, and they AMS for over forty years, died in Nor- The increase of “administrative ex- certainly decided on the basis of re- way on Friday, November 18, 1994; penses” seems difficult to explain. A ports and proposals made by the top was buried in Massachusetts on Sun- clue however may be found in another nonfaculty staff. As a group, top ad- day, November 27, 1994; and an obit- statement contained in the introduc- ministrative staff had a collective in- uary was published in the tory note to Chapter III of the Digest: terest to emphasize the importance of Times on Saturday, December 10, “The student-staff ratio at colleges their own work and to provide for 1994. This obituary gave false im- and universities dropped from 5.4 in themselves opportunities for ad- pressions and suffered from serious 1976 to 4.8 in 1989. During the same vancement. This could be easily omissions. Because of Maurice’s close period the student-faculty ratio achieved by expanding activities not and deep relationship to the mathe- dropped from 16.6 to 15.7. The pro- directly related to instruction, which matical community, it seemed to me portion of staff who were administra- required an increase of nonfaculty that the errata should be published tive and other non-teaching profes- professionals and top administrators. in the Notices of the AMS rather than sional staff rose from 15 percent in The expansion was achieved at the in . 1976 to 22 percent in 1989, while the expense of instruction-related activi- The obituary gave the impression proportion of staff identified as non- ties. This meant, of course, that bud- that Maurice’s life’s work was im- professional declined from 42 percent get cuts would affect some areas of in- portant because it could be applied. to 38 percent (Table 215).” struction. There was no particular This gives a completely false im- Unfortunately these data are not reason why mathematics should not pression of his work. He loved the fully comparable with the expenditure be affected. I feel that the hypothesis structures of algebra and could and data referred to previously, because outlined above could be tested by an- did build beautiful algebraic struc- they treat a different time period and alyzing the evolution of resource al- tures before knowing a single con- include all institutions of higher edu- location and personnel growth in a crete example. This philosophy of cation, including 2-year colleges. Nev- representative sample of universities, mathematics is very far from the ex- ertheless it is interesting to observe which should include some of the ample-driven field called “applied that from 1976 to 1989 the ratio of major state schools. If it turned out mathematics”. full-time equivalent students to full- that my conjecture was true, the math- But the obituary’s greatest defect time equivalent faculty dropped from ematics community would have to is its failure to even mention that he 16.6 to 15.7, while the corresponding draw its conclusions. Its first priority had a wife, Bernice Auslander, the ratio for “executive/administra- would no longer be to convince the mother of his children and herself a tive/managerial staff” dropped from public that mathematics needs more professor of mathematics at the Uni- 84 to 69.2. A more dramatic decrease support than other subjects. It would versity of Massachusetts in Boston. It affected the ratio of students to non- rather be to inform whoever foots the failed to mention the great support

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that she had offered Maurice in his body, then the AMS member should ified women speakers are inadver- youth. Maurice failed his qualifying not accept the assignment. Shepp re- tently overlooked as possible con- examination at Columbia the first time signed his position as chair of the ference speakers. Surely this is a de- he took it and was considered a “late Committee to Select Hour Speakers, so sirable goal for all members of the bloomer” by Sammy Eilenberg. It was the purpose of this letter is not to AMS. only after his student days that his urge Shepp’s resignation but rather to Marie A. Vitulli unique mathematical talents became correct some misconceptions that his University of Oregon apparent. If Bernice had not supported letter may have created. AWM representative to JCW him during this difficult time in his life, AMS policy regarding women Lynne Billard his later flowering might have been speakers at AMS meetings was shaped University of Georgia lost. by a 1972 resolution (an accurate ver- IMS representative to JCW Maurice Auslander at sixty-eight sion of this resolution was reproduced was still full of creative drive and by Shepp at the end of his message). Kathryn M. Chaloner wanted to work on. This was the AMS commitment to this policy was University of Minnesota tragedy of his death. But perhaps his implicitly reaffirmed in August 1993 ASA representative to JCW death can serve a constructive pur- when the Council voted to extend it Don J. Lewis pose if the mathematical community to joint international meetings. Shepp University of Michigan can take the mistakes of his obituary implies that this policy mandates the MAA representative to JCW to heart. These Notices have been full enforcement of an unwritten quota J. Peter May of statements that this is “the right” for women at AMS meetings. In addi- or “important” mathematics. We see tion, he says that if the number of AMS representative to JCW little cliques that claim they are right women on a program does not satisfy and all others are wrong. We see bit- this quota, conference and special ses- Alayne Parson ter infighting and fragmentation of sion organizers are being asked to Ohio State University mathematicians into groups trying to lower standards. The AMS subscribes AWM representative to JCW get support for their important re- to neither of these views. In addition, Lynne Butler search, trying to get jobs for their stu- the AMS-ASA-AWM-IMS-MAA-NCTM- Haverford College dents. It is time to recognize the whole SIAM Joint Committee on Women in SIAM representative to JCW magnificent diverse effort that is math- the Mathematical Sciences (JCW), a Martha Aliaga ematics. We need mathematicians that committee which is openly interested University of Michigan love to study structure or combina- in advancing the status of women in ASA representative to JCW torics, algorithms or formal systems, our profession, has never proposed Tilla Weinstein number theory or analysis, etc. Math- that the AMS adopt these views. Shepp Rutgers University ematicians are people who love math- says that AMS policy will “corrupt the AMS representative to JCW ematics, be they in industry or acade- reputations of those women and mi- mia, be they users, creators, or norities who can succeed without such (Received December 5, 1994) scholars. political advantage.” This is a gross misrepresentation of both AMS policy Louis Auslander and JCW goals. For the benefit of the Brother and Fellow Mathematician readers of the Notices, we include rel- evant excerpts from the texts of recent (Received December 15, 1994) JCW resolutions. “If the preliminary list of special session invitees has few women, the Some Misconceptions in Shepp organizers should consult others, par- Letter ticularly senior women in related AMS member Larry Shepp has recently fields, to ensure that they have not in- served as chair of the AMS Committee advertently overlooked some. to Select Hour Speakers at Eastern Sec- “Because equity for women is based tion Meetings. Shepp wrote a letter to upon fair treatment, rather than dif- the editor which appeared in the Oc- ferential standards, the JCW supports tober 1994 issue of the AMS Notices. the principles espoused in the pro- The letter presents a serious distortion posed AMS ethical guidelines.” of AMS policy regarding women speak- There is neither a hint of a quota ers at AMS meetings. nor a suggestion of lower standards If a prospective member of an AMS in any of JCW’s resolutions or in AMS committee or editorial board cannot policy. The Society is only being asked support AMS policies governing that to diminish the possibility that qual-

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