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Making MathematicsThe Coffee ConnectionAgnes Arvai Wieschenberg

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Makin g MathematicsThe Coffee Connect ionAgnes Arvai Wieschenberg

e college teachers are end-lessly concerned about moti-vating our students and areconstantly looking for better ways tostimulate students to achieve excellence.The focus of many research efforts todayis the classroom presentation. Should itbe a lecture, discussion, multimedia, orsome combination of them? This essaytakes a look at some unorthodox methodsthat could be used outside the classroomto help students become scholars.Although the focus is on mathematics,most of the ideas can be transferred toother academic subjects. The stories areunconventional, but they are very suc-cessful examples from the twentieth cen-tury literature. I believe that they mayhelp spark new ideas that can be appliedin different college settings.The Premise

In a talk about child prodigies, PaulErdos( 1913-1996), one of the centurysmost prolific mathematicians, once re-marked that in Hungary many mathe-maticians drink strong coffee. Erdosadmitted that he had trouble doing math-ematics without it, claiming that hismathematical notes became blank piecesof paper that he just stared at, unableto work. According to Erdos, AlfredAgnes Arvai Wieschenberg is an associ-ate professor of mathematics at John JayCollege of Criminal Justice CUNY, in NewYork City.

Renyi (1921-1970), another well-recog-nized Hungarian mathematician, wenteven further when he stated a mathe-matician is a machine which turns coffeeinto theorems.It is well documented that Budapest,capital of Hungary, has for centuries hadwonderful coffeehouses where one couldspend many hours in the company offriends drinking coffee, smoking, and dis-cussing current events from politics tolocal gossip. But who would talk mathe-matics in these establishments? The com-ments by Erdos and Renyi intrigued meand started me on an unusual researchproject to find evidence that this mathe-matics-coffee-and-coffeehouse connec-tion is indeed a valid one and to ponderwhether or not a creative atmosphere-with coffee as a main ingredient-hasan effect on formulating mathematicalthought.In the winter of 1983, I had a personalinterview with George P@lya n Palo Alto,California. We discussed mathematicians,and in particular, why so many brilliantmathematicians have emerged from Hun-gary since the turn of the century. Polyamaintained that because Hungary was apoor country and mathematics is thecheapest science-all you need to domathematics is a pencil, some paper, anda waste paper basket-a lot of talentedyoung people turned to mathematics.In fact, that was true of Poland as wellbecause Hungary and Poland were part ofthe Austro-Hungarian Empire until 1918.

The only requirement that may be neededis an appropriate mathematical atmos-phere, which was definitely present in theturn-of-the century Austro-HungarianEmpire, which included Lwow in mathe-matician Banachs time, and Budapestduring Professor Fejers years.Coffeehouses

The exotic drink coffee, followed by acoffeehouse culture, swept Europe with avengeance. The first coffeehouse openedin the middle of the seventeenth centuryin Venice, with the name of La Bottegadel Caffe, under the arches of St. MarksSquare. Other great European cities fol-lowed suit, Oxford in 1650, London in1652, Paris in 1660, Marseilles in 1671,and Vienna in 1683. Actually, the veryfirst European coffeehouse opened itsdoors in Hungary, in Buda (Buda andPest became Budapest in 1898), duringthe century and a half of Turkish occupa-tion. But its patrons were restricted tomen of Turkish origin. Hungarians-toprotest the Turkish occupation-showedno interest in sharing the coffee drinkinghabit with the Turks at that time. Thenumber of coffeehouses, however, keptgrowing all over Europe.Because coffee came from Turkey, itwas first referred to by many as the drinkof the devil. Shakespeare (1564-1616)referred to it as the hot and rebelliousliquor in A s You ike It In the beginningit was vehemently rejected by the Euro-pean society, especially the Hungarians.

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In 1721 Montesquieu ( 1689-1 755) theFrench jurist and political philosopher,protested against the consum ption of cof-fee. He claimed that those who frequent-ed coffeehouses become revolutionariesbecause coffee induces a certain kind ofdrunkenness, which is dangerous for thefuture of the country.At the time of his protest, there werealready about three hundred coffeehousesin Paris. Legend has it that Honor6 deBalzac (1799-1850) carried with him aspecial blend of coffee and coffee-makerwherever he traveled. According to hisown account, he consumed 50,000 cupsof coffee while working on his play LaComkdie Humaine. Rousseau and Voltairewere also avid coffee drinkers. Stendhaleven said that he would rather forgo lovethan give up coffeehouses.

Who went to coffeehouses? Painters,writers, philosophers, musicians, politi-cians, scientists, students, and men fromevery social status. Women were not partof the scene until the first decades of thetwentieth century. For those who frequent-ed the cafes it was a home away fromhome, a warm, bright, comfortable placewith instant company, conversation, cof-fee, l iqueur, smoke, newspapers, andgames. (Sounds like a mens club?) Al-though in the history of coffeehouses onedoes not find many references to mathe-maticians frequenting these establish-ments, there are a few interesting ones.Coffee,Warmth, and Genius

The best example is that of StefanBanach (1892-1943 , the founder offunctional analysis and one of the greatmathematicians of the twentieth century.His half-sister noted in an interview thather brother drank coffee by the gallon.But even more important, Banach wasknown to have spent most of his wakinghours in cafes surrounded most of thetime by fellow mathematicians and somestudents from nearby universities.

Banach was discovered by Hugo Stein-haus, who was a professor of mathemat-ics at Lwow University from 1920 to1941. These two m athematicians were tobe known later as the founders of whatwe now call the Lwow School of Mathe-matics. And yet Banach had no collegedegree. He was a self-taught mathemati-cian, and he needed a legal exemption

from the minister of education to beallowed to take a masters examination,which was necessary to obtain his Ph.D.

Curiously, Banach never wrote his dis-sertation. He had no interest in writingdown any of his theorems and proofs inpubl ishable form; he was bored withdetails. Although he did plenty of mathe-matics to earn his degree-he had intro-duced by this time what later becameknown as Banach Spaces and the BanachFixed Point Theorem-he never botheredto write his theories in a presentableform. Finally, he had to be literallytricked into completing his thesis, as hisbiographer tells us:

Professor Ruziewicz instructed one of hisassistants to accompany Banach on his fre-quent visits to the coffee houses, query himin a discreet fashion on his work, and after-wards write down Banachs theorems andproofs. When all of this information wastyped out, the notes were presented toBanach, who edited the text. This is howhis Ph.D. dissertation was finally complet-ed (recorded by Turowitz). The title ofBanachs dissertation translated from Pol-ish is On Ope rations on Abstract Sets andTheir Application to Integral Equations.(Kaluza 1996, 32-3)To receive the doctoral degree, Banach

still had to prepare for and pass the doc-toral examination. He dutifully read andmastered the material including a requiredextra topic in astronomy, but he had to betactfully deceived again to take the exam.Nevertheless, at the age of twenty-eight,Banach was awarded the Ph.D. of M athe-matical Scien ces under the advisement ofAntoni Lomnicki. He received his habili-tation in 1922 and thus became a professor extruordinarius at the Jan KazimierzUniversity in Lwow, Poland.

Cafes that were frequented by mathe-maticians were of the Viennese type of-fering large comfortable spaces withnewspapers hanging on bamboo racks.Chess sets, dominoes, and cards wereavailable for the enjoyment of the pa-trons. Comfortable seating with marbletables created a warm and congenial at-mosphere. Some patrons or groups visit-ed the sa me cafe daily, and certain tableswere reserved for them. Most cafesserved coffee, tea, and pastries; somewould offer food and other drinks, too. Atthe turn of the century, most people ineastern Europe had no central heating,

and every room had its own wood burn-ing stove that needed to be nurtured allday long to keep warm. In a cafe thechore of keeping the place warm wastaken care of by some one else, so that onecould sit in comfort with friends, lovers,or colleagues all morning or afternoon f orthe price of a cup of coffee.Banachs Lwow had many such cafes.Cafe Rom a used to be the unofficial gath-ering place of the local chapter of the Pol-ish Mathematical Society. After their offi-cial weekly Saturday meetings, theywould retire to the Cafe Roma on Aka-demicka Street close to the university.Banach later moved the meeting acrossAkademicka Street to the by now leg-endary Scottish Cafe. For the gatheredmathematicians, the little marble top cof-fee tables served as a sketch pad, whichhad to be washed by the proprietor afterthe patrons left. After awhile the ownersbecame annoyed by the scribbles, andfinally in 1935, Banachs wife purchaseda notebook, which Banach then presentedto the group.

The notebook over the years became adiary of the mathematical problems andtheorems as they were developed overcoffee and heated d iscussions. It was keptin the cloakroom of the Scottish Cafe andwas available upon request to the mathe-maticians frequenting the cafe. The note-book was later published under the nameof The Scottish Book: A Collection cfProblems; it becam e well known in math-ematical circles as The Scottish Book .

Just who frequented the Scottish cafe?Almost all the local and visiting mathe-maticians, most of whom made a lastingmark in mathematics. They came first atrandom, but pretty soon a certain rhythmand custom were established, as daily se s-sions at the Scottish grew into a perma-nent ritual amo ng many mathematicians(Kaluza 1996, 61). We can recognizemany of the names of the initial partici-pants, which included Banach, Stein-house, Ruziewicz, Kaczmarz, Zylinski,and Mazur, to whom Banach was closest.

Later the circle expanded to includeUlam, Auerbach, Nikliborc, Eilenberg,Orlicz, Eidelheit, Kac, Birnbaum, Schrei-er, Schauder, Kuratowski, and Nikodym.As Kaluza writes,

Usually they began arriving between 5 and7 pm-always occu pying the same

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tables-and for the next several hours they Later, Ulam became a key member ofworked with total concentration, coveringthe marble tabletops with mathematical for-mulas. But saying that they worked withtotal concentration is not completely accu-rate, as there were no meetings withoutjokes, heated discourse, shouting, anddrinking. Banach, for exam ple, drank enor-mous amounts of coffee and cognac andsmoked dozens of cigarettes. 61-2)In the smoke-filled parlors, some

played chess while others drank coffeeand watched. U lam (1 975) recalls:

the well-known Polish School of Mathe-matics. Mathematician Marc Kac re-marked on the unusual circumstance thatLwow offered: a culture based on leisureand discourse-a peculiar kind of Pol-ish existence where you were in cafes allhours of the day or night drawing dia-grams on small pieces of paper. Ulam(1975) recal led how some importantmathematical theorems were developedas a result of these meetings.

D a u I Erdos v is i ted co l leges around the w or ld .He spent m ost o f h is m oney on p r izes fors tudents who fou nd the best so lu t ion to a problemthat he pro po sed. Many students b enef i ted.

Needless to say such mathematical discus-sions were interspersed with a great deal oftalk about science in general (especiallyphysics and astronomy), university gossip,politics, the state of affairs in Poland; or touse one of John von Neumanns favoriteexamples, the rest of the universe. 35)The variety of characters in the energeticgroup generated a mood of high spirits,and the hard work was accomplished inthe presence of the wry h umo r that is usu-ally associated with people of greatminds.In his autobiography, Stanislaw Ulamfondly remembers his youth, spendingcountless hours in cafes with friends inhis home-town of Lwow. Ulam was bornin 1909 and had the good fortune to at-tend the Lwow Polytechnic Institute,which at the time had the most outstand-ing mathematics faculty, including Pro-fessor Kuratowski (Kuratowski studied inWarsaw under Mazurkiewitcz, Janis-zewski, and Spierpinski). Ulam was par-ticularly impressed by the lectures ofKuratowski, and he participated in thediscussions Kuratowski had w ith some ofhis senior students. Ulam attributes hismathematical development to those stim-ulating discussions.

Sometimes we would sit for hours in a cof-fee house. He (Mazur) would write just asymbol or a line like y = f(x) on a piece ofpaper, or the marble table top. We wouldboth stare at it as various thoughts weresuggested and discussed. These symbols infront of us were like a crystal ball to help usfocus o ur concentration. 31)Ulam recalled a meeting with Banachand Mazu r in the Scottish Ca fe that wenton for seventeen hours interrupted only

by necessities such as meals. Ulam wasextremely impressed by the way Banachcould discuss mathematics, reason aboutit, and find proofs-all in the sam e dis-cussion. Ulam claims that most of mymathematical work was really started inconversations with Mazur and Banach(33). Ulam had the good fortune of find-ing himself in the m idst of this cafe soci-ety, which he enjoyed immensely andmissed greatly later in the United States.Years later during the Second WorldWar, working o n the Manhattan Project in

Los Alamos, Ulam once again found theatmosphere that resembled that of Lwow.Working with s ome of the greatest scien-tific minds of the twentieth century andliving in a small secured area reserved fo rthe scientists and their families reminded

him of his student years and the heateddiscussions in the cafes of Lwow. Ulamsmost valuable contributions a t Los Alam-0s came after the w ar in the developmentof the hydrogen bomb.In Budapest, the cafes flourished. Atthe turn of the century there were six hun-dred coffeehouses in Budapest; some ofthem were open twenty-four hours a dayand some for 365 days a year. Many ofthem were the centers of intellectual life.Students spent countless hours in cafes,sometimes with professors who stimulat-ed the conversation. Professor Lip ot Fejer(1880-1959) spent many afternoons atthe neighborhood cafes with some of hisstudents. Fejer has a remark able legacy ofbrilliant students, among them GeorgePglya, Eugene Wigner, Gabor Szego,Marcel Riesz, and later Paul Erdos andPal Turan. John von Neumann preparedhis doctoral dissertation under Fejer.While still in the Lutheran Gymnasium(secondary school), Neumann frequentedthe gathering of the top mathematiciansof the time. Their meetings w ere held at acoffeehouse in Budapest (Radnai andKunfalvi 1988, 48).Eugene Wigner, a future Nobel laure-ate, after arriving in the United States,like Ulam, missed the con genial meetingsand discus sions at the cafes. As a studentin Berlin in 1920 Wigner had attendedMax Wolmers colloquium; he remem-bered that afterwards many of us wentout to a coffeehouse and sat around alarge table, talking further. Conversationwas not always rigorous, but it touchedon all of the things we loved, not onlyphysics but nature, family and culture(Szanton 1992, 75).Fejer, who worked with distinction onFourier series, was a well-known mathe-matician with a magnanimous nature.P@ya recalled in ou r 198 3 interview,Almost everybody of my age grou p wasattracted to mathematics by Fejer; hewould sit in a cafe in Budapest with hisstudents and solve interesting p roblem s inmathematics or tell stories about mathe-maticians he had known. Although hislectures were considered the experienceof a lifetime, his influence outside theclassroom was even more significant be-cause it was around him that the firstmathematical school developed in Hun-gary (Wieschenberg 1984, 8).

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Princeton:Tea Will DoAt Princeton, the first wave of research

mathematicians began to arrive in 1930,thanks to the generosity of the Rocke-feller Foundation. Among the first toarrive from Hungary were John von Neu-mann, a brilliant mathematician, andEugene Wigner, the physicist who later,in 1963, won the Nobel Prize in physics.Wigner often expressed longings for thecongenial discussions in the cafes ofEurope as well as the informal meetingsand chats with the professors about thelatest research. He complained, Thetown had no coffeehouses in the Euro-pean sense, where scholars and their s tu-dents went for a lively, extended conver-sation (Szanton 1992 120).

When John von Neumann joined thePrinceton crowd, his house became agathering place for the mathematiciansand physicists. His parties were known tobe fun, as well as intellectual events.The Bam bergers philanthropy allowedPrinceton to create an independent Insti-tute for Advanced Study at Princeton. Th einstitute would provide the researchers anenvironment without any pressure or offi-cial duties. There were no students toteach, no responsibilities other than tothink. But some scholars did not believethat this sterile atmosphere was the idealcondition for research. Richard Feynman(Ma crae 1992), a Nobel laureate who hada great sense of humor, made the follow-ing pithy remark upon turning down aprofessorship at Princetons Institute forAdvanced Study:

When I was in Princeton in the 1940s Icould see what happened to those greatminds at the Institute for Advanced Study,who had been specially selected for theirtremendous brains and were now given theopportunity to sit in this lovely house bythe woods there, with no classes to teach,with no obliga tions whatsoever. These poorbastards could now sit and think clearly allby themselves. Okay? So they dont get anidea for a while. They have every opportu-nity to do something and they are not get-ting any ideas. I believe that in a situationlike this a k ind of guilt or depressionworms inside you, and you begin to worryabout not getting any ideas. And nothinghappens. Still no ideas come. (176 )In the United States, Princeton became

the center of mathematics. When profes-sor Lefschetz arrived in the 1920s, hisVol 47/No. 3

unusual philosophy changed the focus ofteaching and learning. Grades and classattendan ce lost their importa nce; his mosturgent requirement of his students wasthat they come to tea. The students wererequired to attend tea every afternoon,where they would meet the best mathe-maticians in the world . Tea was the highpoint of the day. On Wednesdays it washeld at Fine Hallin, the professors room;on other days in the east common room,otherwise known as the students room.It was very much a family gathering,small and intimate (Nasar 1998, 63).

Although friendly, the atmos phere washighly competitive . They were disc ussingmath, and math problems; they sharedtheir reading of current math papers andgossiped, as well.

The students w h o gathered at teatime wereas remarkable, in a way, as the faculty. Poorjews, new immigrants, wealthy foreigners,sons of working classes, veterans in theirtwenties, and teenagers, the students were adiverse as well as a brilliant group. (Nasar 1998. 64)Unfortunately not surprisingly, therewere no women among the students. In

1948, John Nash, Nobel laureate, whowas among the fortunate s tudents atPrinceton, remarked, I was awkwardsocially, shy and isolated. Everything waswonderful. This was a whole new world.Here was a whole community in whichfelt very much at home (64).Coffees Lasting Effects

A stimulating atmosphere can do won-ders for mathematics, as well as for otheracademic disciplines. Coffee may have anextra ingredient that can help stimulate themind. It seems that interaction betweenstudents and professors and visiting math-ematicians not only develops the interestin the subject but also stimulates newthinking. If one is totally immersed inmathematical topics, it can become a wayof life. In fact, Paul Erdos visited manycolleges all over the world, saw many stu-dents, and recognized the talented. Hespent most of the money that he earned onprizes for students who came up with thebest solution to a mathematics problemthat he proposed. Many students benefited.When he visited Brooklyn College, I heardhim lecture on one occasion and saw theway the students looked up to him.

Professors PQlya and Szego at Stan fordorganized compet i t ions for Cal i forniastudents, modeled on the Hungarian E ot-vos Com petition, to discover the very tal-ented ones. Unfortunately, the competi-tion had to be discontinued because oflack of money. But students of PQlya willnever forget him because of his deepcommitment to them and to mathematics.One of them, John Kemeny, was afavorite amon g Dartmouth students. Theyadmired him as a professor, as well as thecollege president. Even during his presi-dential years, he kept his door open sothat students could reach him all the time .He also insisted on teaching an under-graduate mathematics course to keep intouch with the students.

A dedicate d professor who is also will-ing to be a mentor to students outside theclassroom provides an invaluable experi-ence. The most important roles of a pro-fessor in a college setting are the one s heor she does not get paid for. Each of uscan probably remember that one profes-sor who profoundly changed our livesand choice of careers. Those of us whoare lucky had a good mentor-some. un-fortunately, had none.REFERENCESErki, E. 1998. Pest Budatol Burlupe.s t ig.From Pest-Buda to Budapest .) Budapest:

Officiana.96 Kiado.Erki, E. ed. 1998. Kriveha:-Siriito. tor:.she-lyek irok muhelyek. ( A memorial for cof-feehou ses.) Budapest: Officiana96 Kiado.Kaluza, R. 1996. The l i fr of Stefan Banuch.Boston: Birkhauser.Lukacs, J. 1988. Budapest 19 : A historicidportrait i f a city its culture. New York:Weidenfeld and Nicolson.Macrae, N. 1992. John I W I Neumann. NewYork: Pantheon Books.Nasar, S . 1998. A beautiful mind. New York:Simon Schuster.Radnai, G., and R. Kunfalvi. 1988. Physics inBudapest: A survey. Amsterdam: North-Holland Physics Publishing.Szanton,A . 1992. The recollec~tions ?fEugeneF W i p e : New York: Plenum.Szentes, E.. and E. Hargitay. 1998. IrodiiltniKavehazak Pesten es Budan. (Literary cof-feehouses in Pest and Buda). Budapest:Universitas Kiado.Ulam, S. M . 1975. Adventures o 61 muthematician Berkeley: University of Califor-nia Press.Wieschenberg, A. A . 1984. Identification an ddevelopment i f the mathematically talent-ed: The Hungarian experience. Ph.D. Diss.,Columbia University.

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