Celebratio Mathematica

“Your liv­ing room is your fact­ory.” That’s a slo­gan that Sun-Yung Alice Chang (張聖容) heard of­ten among fam­il­ies like hers when she was grow­ing up in post-World War II Taiwan. “Every­body did some handi­work in the liv­ing room, mak­ing Christ­mas dec­or­a­tions” and oth­er trinkets for ex­port, she re­called. “Every fam­ily did that to have some ex­tra in­come.” Her par­ents came from wealthy fam­il­ies in China, but once they were in Taiwan, said Alice, “they had to start from ground zero.” Her fath­er, who had trained in China as an ar­chi­tect, worked in build­ing con­struc­tion. He did not earn enough to sup­port a fam­ily with a daugh­ter and son, so Alice’s moth­er be­came an ac­count­ant, learn­ing the trade on the job. Most of their com­bined in­come went to­ward food. Sub­sid­ized hous­ing al­lowed the fam­ily to sur­vive. When Alice was a stu­dent at Na­tion­al Taiwan Uni­versity (NTU), she re­ceived a prize con­sist­ing of a set of text­books for the next aca­dem­ic year. “My fam­ily cel­eb­rated that,” Alice re­called. “It was a re­lief.” Text­books were a big strain on the fam­ily budget.

Alice, who today is Eu­gene Hig­gins Pro­fess­or of Math­em­at­ics at Prin­ceton Uni­versity, is one of six wo­men who re­ceived bach­el­or’s de­grees in math­em­at­ics from NTU around 1970, went on to PhDs in the United States, and rose to the top of their pro­fes­sion. The great geo­met­er Shi­ing-Shen Chern (陳省身) once marveled at the ac­com­plish­ments of these six wo­men, call­ing this burst of fe­male math­em­at­ic­al tal­ent in Taiwan a “mir­acle”.

This “mir­acle group,” as I will call them, lived through an in­flec­tion point in Asi­an his­tory. Born around 1950, they grew up in a Taiwan bear­ing little re­semb­lance to the high-tech power­house the is­land has be­come. Taiwan was an im­pov­er­ished land un­der mar­tial law, strug­gling to emerge from the shad­ow of geo­pol­it­ic­al con­flicts that had shaped its fate for half a cen­tury.

Des­pite eco­nom­ic hard­ship and the stric­tures of a con­ser­vat­ive, pat­ri­arch­al so­ci­ety, the mir­acle group pur­sued an ideal­ist­ic dream of in­tel­lec­tu­al achieve­ment in a male-dom­in­ated field. They had few role mod­els; their ca­reer paths were un­cer­tain. But they knew they loved math­em­at­ics. And they had each oth­er, as class­mates at NTU, as loy­al friends, as sources of in­spir­a­tion.

“It was a good thing to have so many girls” study­ing math­em­at­ics at NTU, said an­oth­er mir­acle group mem­ber, Jang-Mei Wu (吳徵眉). “Every­body talked about it be­ing hard, but every­body made an ef­fort… See­ing this group of wo­men class­mates, I kept on go­ing. We all just kept on go­ing.”

Today Jang-Mei is a pro­fess­or emer­ita of math­em­at­ics at the Uni­versity of Illinois at Urb­ana–Cham­paign. Like the oth­er wo­men in the mir­acle group, she grew up at a time when Asia had barely be­gun to heal from the frac­tures of World War II. Her fath­er, who had stud­ied civil en­gin­eer­ing at Fudan Uni­versity, worked in rail­way ad­min­is­tra­tion in Hang­zhou, China. In 1948, the year Jang-Mei was born, he was sent to Taiwan to help re­build rail­ways dam­aged by war­time bomb­ing raids. His wife joined him in Taipei the fol­low­ing year and soon had a second daugh­ter. Eco­nom­ic con­di­tions on the is­land were harsh; even pro­fes­sion­als like Jang-Mei’s fath­er could earn only enough to cov­er ba­sic needs. In those days her moth­er spent all her time on house­work, do­ing everything by hand: sew­ing, knit­ting, wash­ing clothes, bring­ing ice from the mar­ket to keep food cold, cook­ing over a char­coal fire that had to be ten­ded morn­ing to night. “It was really hard work,” said Jang-Mei.

The par­ents of Alice and Jang-Mei, and those of the oth­er four wo­men in the mir­acle group, were among the two mil­lion people who left main­land China for Taiwan around 1949 and entered the flow of his­tory of an is­land that had been mol­ded and ex­ploited by a suc­ces­sion of out­side forces.

For thou­sands of years ab­ori­gin­al people lived on Taiwan and de­veloped their own cul­ture, with little in­flu­ence from China. It was the Dutch col­on­ists, ar­riv­ing in the 17th cen­tury, who en­cour­aged Chinese people to come to Taiwan, as man­power for farm­ing and for ex­ploit­a­tion of the is­land’s re­sources. But Dutch con­trol of the is­land was brief: after a mere four dec­ades, the col­on­ists were forced out and Taiwan was gradu­ally in­teg­rated as a province of China. Two cen­tur­ies later, the Chinese them­selves were com­pelled to re­lin­quish con­trol of the is­land, when Taiwan was ceded to Ja­pan at the end of the first Sino-Ja­pan­ese War in 1895.

Again Taiwan came un­der for­eign rule, as the Ja­pan­ese im­posed their own na­tion­al iden­tity and cul­ture on the is­land. The Ja­pan­ese brought many im­prove­ments, in­clud­ing elec­tri­city, mod­ern medi­cine, and in­fra­struc­ture such as rail­ways. But they also ex­ploited the is­land and treated the nat­ive Taiwanese as second-class cit­izens, com­pel­ling them to ad­opt the Ja­pan­ese lan­guage and lim­it­ing their edu­ca­tion­al and pro­fes­sion­al op­por­tun­it­ies. The Taiwanese who ob­jec­ted to the sys­tem­ic dis­crim­in­a­tion were dealt with harshly or even killed.

In 1945, after its sur­render in World War II, Ja­pan ceded Taiwan back to China, and the is­land was once again com­pelled to take on a new iden­tity. The Kuo­mintang (KMT) Na­tion­al­ist Chinese forces then in power in China un­der­took a pro­cess of “de-Nip­pon­iz­a­tion” of Taiwan, to root out Ja­pan­ese in­flu­ence and to es­tab­lish the Man­dar­in lan­guage and tra­di­tion­al Chinese cul­ture as the found­a­tions of so­ci­ety. Ini­tially these changes were wel­comed on Taiwan. However, the KMT soon be­came sus­pi­cious of the is­landers, see­ing them as col­lab­or­at­ors of the en­emy Ja­pan­ese; the is­landers in turn be­came re­sent­ful of the heavy-handed and some­times bru­tal tac­tics of the new re­gime. To as­sert its au­thor­ity in the in­creas­ingly tense situ­ation, the KMT in 1947 es­tab­lished mar­tial law, which re­mained in place for more than forty years.

In 1949, the KMT Na­tion­al­ists, un­der Chi­ang Kai-Shek, lost the Chinese civil war to Mao Tse-Tung’s Com­mun­ists and re­treated to Taiwan. Along with Chi­ang’s mil­it­ary forces came a wave of ci­vil­ian refugees, in­clud­ing bur­eau­crats and tech­no­crats in the Na­tion­al­ist gov­ern­ment as well as uni­versity stu­dents, schol­ars, and in­tel­lec­tu­als. Taiwan’s ex­ist­ing pop­u­la­tion of six mil­lion ab­sorbed two mil­lion main­land Chinese.

That year Jang-Mei Wu’s moth­er left be­hind her large fam­ily in Hang­zhou, in­clud­ing sev­er­al sis­ters to whom she was very close, and set out by boat for Taiwan, where her hus­band awaited her. She car­ried the baby Jang-Mei with her on the mul­ti­week trip. “When [my moth­er] talked about the time on the boat, she said she was happy…be­cause she had a baby, and she was go­ing to her hus­band’s place,” Jang-Mei said. Her moth­er as­sumed that she, to­geth­er with hus­band and child, would soon re­turn to China and go on liv­ing there. Said Jang-Mei, “She nev­er, nev­er thought that she would be in Taiwan for all her life.” None of the mi­grants did.

Born in Hua-Li­an, a port city on the east coast of Taiwan, in that tu­mul­tu­ous year of 1949 is an­oth­er mir­acle group mem­ber, Chuu-Li­an Terng (滕楚蓮), who today is a pro­fess­or emer­ita of math­em­at­ics at the Uni­versity of Cali­for­nia, Irvine. Chuu-Li­an grew up in Taipei, in what might today be called a refugee camp, al­beit one with houses not tents. Built for fam­il­ies of men in the Na­tion­al­ist army, the houses were tiny mud-and-bam­boo struc­tures without found­a­tion. They were de­signed to last only two years. “After 5 or 6 years, [the walls] star­ted to warp and make a curve,” Chuu-Li­an re­called. “You could hear the neigh­bors. Every time when a hur­ricane came, they moved us to the high school nearby.”

The short shelf-life of Chuu-Li­an’s fam­ily home re­flec­ted the frame of mind of the refugees and in­deed of Chi­ang Kai-Shek him­self, who planned to re­take the Chinese main­land and rees­tab­lish the Na­tion­al­ist gov­ern­ment there. De­vel­op­ing Taiwan be­came part of this plan, as Hsiao-Ting Lin ex­plained in his book Ac­ci­dent­al State: “Chi­ang stressed the ne­ces­sity of cul­tiv­at­ing Taiwan’s cul­tur­al, so­cial, eco­nom­ic, and edu­ca­tion­al as­sets, in what would later be­come known as ‘soft power,’ to be used one day in­stead of mil­it­ary force to over­throw the Com­mun­ist re­gime on the main­land.”1

The mir­acle group be­nefited from this de­vel­op­ment. They were also spared years of up­heav­al and vi­ol­ence on the main­land, as Mao Tse-Tung and his fol­low­ers re­made the eco­nomy and so­ci­ety to con­form to Com­mun­ist prin­ciples. Alice Chang’s par­ents, poor as they were, sent par­cels of sug­ar and oth­er staples back to their fam­il­ies strug­gling to sur­vive on the main­land. All com­mu­nic­a­tion, in­clud­ing mail de­liv­ery, was cut off between Taiwan and main­land China, so the par­cels had to be routed through fam­ily friends in Hong Kong. After US Pres­id­ent Richard Nix­on’s “re­open­ing” of China in 1972, Alice’s par­ents were fi­nally able to vis­it and see how their re­l­at­ives had fared. “We found out they had a really dif­fi­cult time,” she said. Of her fath­er’s three sib­lings who stayed in China, none had been able to send their chil­dren to col­lege. Hav­ing come from rich, land-own­ing fam­il­ies, the kids were for­bid­den high­er edu­ca­tion. Most of them be­came farm­ers.

A harsh­er fate be­fell the re­la­tions of an­oth­er mem­ber of the mir­acle group, Fan-Rong King Chung (金芳 蓉), who is now a pro­fess­or emer­ita of math­em­at­ics and com­puter sci­ence at the Uni­versity of Cali­for­nia, San Diego. Fan’s moth­er too vis­ited China after 1972 and learned that whole branches of her fam­ily — three uncles, to­geth­er with their wives and chil­dren — were gone. Asked if they had been killed, the usu­ally vol­uble Fan replies only with a quiet “mm-hmm.” These re­l­at­ives had be­longed to the well-off class of landown­ers who were sys­tem­at­ic­ally per­se­cuted by the Com­mun­ists. “People really suffered dur­ing those times,” said Fan. “In Taiwan, we were totally shiel­ded from all that,” she went on. “We all were told we were very lucky be­cause in Taiwan, there was peace and prosper­ity.”

Wen-Ching Win­nie Li (李文卿), an­oth­er mem­ber of the mir­acle group, was born in Taiwan in 1948 and is today a pro­fess­or of math­em­at­ics at Pennsylvania State Uni­versity. Her fath­er was a chem­ist who came to Taiwan in 1946 to run a salt com­pany that the Na­tion­al­ist gov­ern­ment had taken over; her moth­er came the fol­low­ing year. In China, salt was his­tor­ic­ally an im­port­ant com­mod­ity, and be­ing a salt mer­chant meant wealth. But this did not ap­ply to Win­nie’s fath­er. Even though her par­ents had only one child, and even though the salt com­pany provided free hous­ing, the fam­ily lived paycheck to paycheck. “The money went to food and cloth­ing, all the ne­ces­sit­ies,” said Win­nie. She nev­er lacked any­thing but nev­er had any little lux­ur­ies either. “I don’t re­call that I went to any movies with my par­ents. Nev­er.”

Win­nie grew up in Tain­an, on the south­w­est coast of Taiwan. Her life was quite re­stric­ted, not only be­cause her par­ents were pro­tect­ive of their only child but also be­cause of the con­ser­vat­ive, au­thor­it­ari­an so­ci­ety. All school­chil­dren wore uni­forms, with name tags for boys and iden­ti­fic­a­tion num­bers for girls. When school ended for the day, guards mon­itored stu­dents in the streets and on pub­lic trans­it to make sure boys and girls did not mingle. “The ID num­bers and names were con­veni­ent for them to take notes about who was do­ing what and where and when,” said Win­nie. The girls had only num­bers, she ex­plained, in or­der to bet­ter pro­tect their pri­vacy.

In most schools in Taiwan at the time, the early grades were coedu­ca­tion­al, and the genders were sep­ar­ated start­ing in the sev­enth grade. Dur­ing the Ja­pan­ese co­lo­ni­al peri­od, school­ing for girls differed from that of boys, in keep­ing with the aim of mak­ing girls in­to “good wives, wise moth­ers.”2 After the KMT took over the schools, the cur­ricula were made the same for both genders, thereby greatly im­prov­ing girls’ edu­ca­tion­al op­por­tun­it­ies.

After 1949, as main­land China began re­shap­ing the basis of its edu­ca­tion­al sys­tem around com­mun­ist and Maoist prin­ciples, Taiwan was build­ing one that would re­tain and strengthen tra­di­tion­al Chinese val­ues. The KMT made Man­dar­in the lan­guage of school in­struc­tion in Taiwan and sup­pressed loc­al lan­guages. Polit­ic­al in­cul­ca­tion in­cluded the teach­ing of the “Three Prin­ciples of the People” of KMT founder Sun Yat-Sen as a re­quired school sub­ject. Stu­dents learned about the dangers of com­mun­ism and the im­port­ance of a Na­tion­al­ist re­vanche of main­land China.

School days would be­gin with singing the na­tion­al an­them and watch­ing the rais­ing of the flag of the Re­pub­lic of China, sym­bol­iz­ing Taiwan to­geth­er with the soon-to-be-conquered main­land. After that came a hy­giene check. “You had to bring four items to school every day,” ex­plained Jang-Mei Wu: a handker­chief, a drink­ing cup, toi­let pa­per, and a mask. “If you didn’t have these four items, they mark you down, and the next day you must have them.” What was the mask for? At the end of the school day, the kids had to clean the classrooms — sweep floors, clean win­dows, wipe down desks — and were re­quired to wear masks dur­ing these chores.

Schools main­tained strict dis­cip­line, some­times by cor­por­al pun­ish­ment, and kids were ex­pec­ted to work very hard. Still, the mir­acle group en­joyed school. Among the refugees from main­land China were many schol­ars and uni­versity stu­dents, some of whom be­came school­teach­ers in Taiwan and con­trib­uted much to the aca­dem­ic qual­ity of the schools. Chuu-Li­an re­membered get­ting a “won­der­ful clas­sic­al edu­ca­tion.” “I was lucky, all my teach­ers were good teach­ers,” she said. She re­membered them as mostly young and poorly paid, but ded­ic­ated and en­thu­si­ast­ic.

Es­pe­cially mem­or­able was her sixth grade teach­er, Xi­an-Geng Zhou (周賢耕) who after his day job at­ten­ded uni­versity classes at night and later got a PhD in eco­nom­ics in Ja­pan. One time he wrote on the board the num­bers 1, 2, 3, …, 98, 99, 100, and told the kids to add them all up. “I said, ‘How can you add them all up?’,” Chuu-Li­an re­called. “It would take a long time. So he waited for a few minutes, then he just drew a line from 1 to 100, 2 to 99, and so on. A few lines, then you got it. And I thought, That’s fun!”

Teach­ers some­times gave prob­lems from tra­di­tion­al Chinese math­em­at­ics in­struc­tion, such as: Giv­en a cage con­tain­ing rab­bits and chick­ens, and the num­ber of heads and of feet, cal­cu­late the num­ber of each kind of an­im­al. “If you don’t have al­gebra, it’s dif­fi­cult to solve,” Chuu-Li­an said. “We were trained from fifth grade on prob­lems where you really have to think.”

Fan Chung’s par­ents mar­ried in China and went to Taiwan for their hon­ey­moon — and then stayed. Fan was born in 1949 in Ka­ohsiung, Taiwan, where her fath­er, who had trained as an en­gin­eer, worked in man­u­fac­tur­ing for the mil­it­ary. His salary, to­geth­er with that of Fan’s moth­er, who worked as a high school teach­er, meant that Fan’s fam­ily was a bit bet­ter off fin­an­cially than the fam­il­ies of the oth­er mir­acle wo­men. Fan has one broth­er, who got a mas­ter’s de­gree in civil en­gin­eer­ing in the United States and then had a ca­reer in Taiwan.

Fan’s fath­er en­cour­aged her love of math­em­at­ics. “My fath­er said, ‘Math­em­at­ics is the found­a­tion of sci­ence. If you do math, you will be good at any­thing. You can al­ways switch to oth­er areas later if so de­sired.’ And he was right.” Fan learned just how right he had been when years later she began to do re­search at the bor­der between math­em­at­ics and com­puter sci­ence. Her fath­er also poin­ted to teach­ing as a fit­ting ca­reer for a wo­man. Here Fan did not take the hint. After her PhD she spent twenty years at Bell Labor­at­or­ies, which was le­gendary for its thriv­ing re­search-only en­vir­on­ment.

Fan’s par­ents did not fol­low the tra­di­tion­ally pat­ri­arch­al pat­tern of Chinese so­ci­ety and fa­vor their son over their daugh­ter. In fact, that kind of fa­vor­it­ism did not play a strong role in the fam­il­ies of the mir­acle group. It was en­tirely ab­sent for Win­nie Li, as she was an only child, as well as for Jang-Mei Wu, whose par­ents had two daugh­ters. In the fam­il­ies that did have sons, the daugh­ters were not treated very dif­fer­ently. Taiwan his­tor­i­an J. Megan Greene of the Uni­versity of Kan­sas sug­ges­ted that, be­cause the par­ents of the mir­acle wo­men came from edu­cated back­grounds in China, they might already have be­gun to ques­tion re­ceived ideas like pat­ri­archy even be­fore com­ing to Taiwan. In ad­di­tion, the lack of con­tact with their ex­ten­ded fam­il­ies might have at­ten­u­ated the force of such tra­di­tions.3

Still, the par­ents of the mir­acle group of­ten paid great­er at­ten­tion to their sons — some­times to the ad­vant­age of the daugh­ters. Chuu-Li­an Terng’s fath­er was “very old-fash­ioned, very tra­di­tion­al,” she said. He ex­pec­ted all four of his chil­dren to ex­cel in school, but his high ca­reer ex­pect­a­tions were trained only on his three sons and not his daugh­ter. “He had no real ca­reer goal for me,” said Chuu-Li­an. As a res­ult, when it came to col­lege, “I had com­plete free­dom to choose what I wanted to do.”

In Alice Chang’s fam­ily, pat­ri­archy did not reign, for her moth­er was def­in­itely the boss. She was known in the neigh­bor­hood for be­ing ex­tremely strict with her kids. “My broth­er and I united to try to pro­tect each oth­er,” Alice re­called. “We were scared of her.” She re­quired top aca­dem­ic achieve­ment of both chil­dren and of­ten kept them in­doors for ex­tra study or cal­li­graphy prac­tice while oth­er kids went out­side to play. “She would say, ‘My kids have to be the best’,” Alice re­called. “Nowadays in Amer­ica we would say she was a ‘Ti­ger Moth­er’,” said Alice, re­fer­ring to the 2011 best­selling book Battle Hymn of the Ti­ger Moth­er by Amy Chua.

Hav­ing had a young­er broth­er her­self and hav­ing chafed un­der her status as the less­er child, Alice’s moth­er was de­term­ined not to do the same with her own daugh­ter. But there was a twist. “My moth­er tried to treat my broth­er and me the same,” Alice ex­plained. “On the oth­er hand, it was un­der­stood that I was be­ing spe­cially favored to be treated the same. I shouldn’t take it for gran­ted! In oth­er fam­il­ies, boys come first. I was al­ways re­minded of that.”

We have so far met five of the six mir­acle wo­men: Alice Chang, Jang-Mei Wu, Chuu-Li­an Terng, Fan Chung, and Win­nie Li. The sixth is Mei-Chi Shaw (蕭美琪), who is a few years young­er than the oth­ers. Born in 1955 in Taipei, she is today a pro­fess­or of math­em­at­ics at Notre Dame Uni­versity. Mei-Chi’s fath­er stud­ied phys­ics as a young man in China and then be­came a met­eor­o­lo­gist in the Na­tion­al­ist Air Force; later, after he fled with his fam­ily to Taiwan, he be­came ed­it­or of the Force’s magazine. He loved Chinese lit­er­at­ure, the sub­ject that Mei-Chi’s moth­er had stud­ied at Wuhan Uni­versity.

Mei-Chi grew up in a large fam­ily, with four broth­ers and a sis­ter, in a ram­shackle bar­rack much like the one where Chuu-Li­an Terng lived. In an auto­bi­o­graph­ic­al art­icle4 Mei-Chi re­lated happy memor­ies of fly­ing kites on the rice pad­dies in Taipei — as well as the im­port­ance of bring­ing home all the threads used as kite string. “At that time, ma­ter­i­al things were so scarce, everything was pre­cious, even a spool of thread,” she wrote.

For Mei-Chi, as well as the oth­er mir­acle wo­men, a dreaded rite of pas­sage was the en­trance ex­am­in­a­tion re­quired to pro­gress from sixth to sev­enth grade. Man­dat­ory school­ing ended at sixth grade, and the only way to con­tin­ue to sev­enth grade in a pub­lic school was to score well on the ex­am. Mei-Chi’s fam­ily was un­able to af­ford private school­ing, so fail­ing the ex­am would have meant the end of her edu­ca­tion. She nev­er felt pres­sure over an ex­am like she did for this one. She wrote, “Both teach­ers and par­ents alike warned us that if we did not pass the ex­am to get in­to pub­lic school we would be sent to ‘tend buf­falo’ (fig­ur­at­ively speak­ing).”

Many schools spent the en­tire sixth-grade aca­dem­ic year pre­par­ing stu­dents for the ex­am. Cram­ming ses­sions kept stu­dents at school un­til late in the even­ing, and fam­il­ies of means paid for out­side tu­tor­ing. Chuu-Li­an’s fam­ily could barely af­ford the min­im­al pub­lic school fees, let alone ex­tra tu­tor­ing. She felt the pres­sure of the ex­am start­ing already in the fifth grade, when her fath­er told her that if she did not pass she would have to be­come a maid. Be­fore that time, “I had nev­er worked at home in the even­ing,” she said. That even­ing, “I turned on the light and star­ted to work.”

In Taiwan’s com­pet­it­ive, hier­arch­ic­al edu­ca­tion sys­tem, stu­dents were ranked nu­mer­ic­ally not only on ma­jor ex­am­in­a­tions like the one lead­ing to sev­enth grade but also with­in their classes and with­in their schools. At the high school level, there was also a peck­ing or­der based on which school one was able to get in­to, with the First Girls’ High School be­ing the most pres­ti­gi­ous and the Second and Third Girls’ High School one and two notches lower, re­spect­ively. High-school-level vo­ca­tion­al schools were yet lower on the to­tem pole.

After the ex­am to enter the sev­enth grade came two more ex­ams, for ninth grade and then for col­lege. At each level the weed­ing-out was con­sid­er­able, but for girls, the drop-off was es­pe­cially steep. For ex­ample, in the year 1969, nearly all boys and girls ages 6–11 were en­rolled in school. But for ages 15–17, only 43 per­cent of boys were en­rolled — and just 31 per­cent of girls.5

Dur­ing the 1960s the gov­ern­ment ran an ex­per­i­ment on Taiwan’s off­shore is­lands of Que­moy and Matsu, in which the ex­am for sev­enth grade was elim­in­ated. The res­ult was “no­tice­able gains in weight, height, and gen­er­al health of sixth-grade chil­dren, and also lessened in­cid­ence of eye troubles.”6 Mei-Chi Shaw was in the last co­hort of stu­dents tak­ing that ex­am. In 1968, Taiwan elim­in­ated it en­tirely and ex­pan­ded uni­ver­sal school­ing from six to nine years.

Ma­dame Wu was the nick­name for the phys­i­cist Chi­en-Shi­ung Wu (吳健雄), who car­ried out the ex­per­i­ments that were the basis for the work that earned Yang and Tsung-Dao Lee (李政道) the No­bel Prize in Phys­ics in 1957. In 1964, when Ma­dame Wu re­ceived an award from a cul­tur­al found­a­tion star­ted by a ce­ment com­pany in Taiwan, the monthly magazine Taiwan Re­view car­ried a long art­icle about her — mak­ing sure to men­tion that she was a de­voted wife and moth­er, as well as a good cook. Mei-Chi re­membered see­ing news­pa­per art­icles in 1964 re­port­ing on the first wo­man to earn a PhD from Prin­ceton Uni­versity, Taiwan-born Tsai-Ying Cheng (鄭彩鶯), who stud­ied bio­chem­istry and did can­cer re­search be­fore mov­ing in­to plant bio­logy.

This re­spect and ad­mir­a­tion for wo­men’s achieve­ments co­ex­is­ted with pat­ri­archy. In tra­di­tion­al Chinese cul­ture, the status of wo­men was lower than that of men, Mei-Chi ex­plained. “But his­tor­ic­ally there are a lot of tal­en­ted Chinese wo­men who were po­ets, who were paint­ers, cal­li­graph­ers, and so on,” she noted. “Tal­en­ted wo­men are very much val­ued and re­spec­ted.” As a res­ult, in Taiwan, even though boys were ex­pec­ted to achieve more than girls, those girls who as­pired to aca­dem­ic ex­cel­lence did re­ceive en­cour­age­ment. They were sus­tain­ing the cul­tur­al tra­di­tion of fe­male tal­ent.

The celebrity in­tel­lec­tu­als were an in­spir­a­tion for the mir­acle group but seem not to have func­tioned quite as role mod­els, in the sense the term is com­monly used today. This is il­lus­trated in Alice Chang’s re­col­lec­tion of a lec­ture by Chen-Ning Yang that she at­ten­ded in Taipei when she was in high school in the mid-1960s. In the lec­ture, Yang said that if he were a young per­son en­ter­ing uni­versity at that time, he would opt for math­em­at­ics, be­cause the field was wit­ness­ing rap­id and fas­cin­at­ing de­vel­op­ments in many dir­ec­tions.

“Yang’s com­ment left a deep im­pres­sion on me, as I thought, ‘what a chance, or priv­ilege, I would have to enter an ex­cit­ing, de­vel­op­ing field’,” said Alice. “Wheth­er I could be­come a re­search math­em­atician, or re­motely reach the heights that Pro­fess­or Yang reached, ac­tu­ally did not seem to oc­cur to me.” She did not quite view him as a role mod­el, mean­ing someone who was like her, someone she could identi­fy with and emu­late. What hooked her was less the de­sire to be like him than the ap­peal­ing vis­ion of math­em­at­ics that he presen­ted.

In a sim­il­ar way, Jang-Mei Wu did not see the celebrity in­tel­lec­tu­als as role mod­els. “I heard of all these people when I was young,” she said. “But I chose math just be­cause I like math. That’s it.” In high school she and a friend solved math prob­lems to­geth­er and puzzled over Sher­lock Holmes mys­ter­ies (in Chinese trans­la­tion). These are the kinds of ex­per­i­ences that sparked her in­terest. Her de­sire to do math­em­at­ics “was just from in­side me. It was not be­cause I wanted to be some­body.”

The 1950s and 1960s were a time of huge growth in high­er edu­ca­tion in Taiwan, with the num­ber of col­leges and uni­versit­ies in­creas­ing from 4 to 22. En­roll­ments in­creased by an even lar­ger pro­por­tion. Nev­er­the­less in 1970, only 8 per­cent of the pop­u­la­tion of 18- to 24-year-olds was en­rolled in high­er edu­ca­tion.7 At­tend­ing uni­versity was a great priv­ilege that stu­dents worked very hard to at­tain. This was es­pe­cially true for those who hoped to enter Na­tion­al Taiwan Uni­versity, which was (and re­mains today) Taiwan’s premi­er high­er edu­ca­tion in­sti­tu­tion.

In 1966, 43 stu­dents entered NTU as math­em­at­ics ma­jors, twelve of them wo­men. Of those twelve, around half had done so well in high school that they were not re­quired to take the uni­versity en­trance ex­am­in­a­tion. There is a spe­cial term for this in Chinese, bao song (保送), which lit­er­ally means “guar­an­teed send.” Bao song stu­dents could choose any uni­versity and any ma­jor. Among the bao song NTU math ma­jors start­ing in 1966 were three from the mir­acle group: Alice Chang, Win­nie Li, and Fan Chung. “Our class was spe­cial,” said Alice. “It had nev­er happened be­fore” that so many of the top wo­men stu­dents chose math­em­at­ics. Chuu-Li­an Terng entered NTU the fol­low­ing year, in 1967, also a bao song. Her class also had an un­usu­ally high pro­por­tion of wo­men math ma­jors, eight out of thirty.

“The gen­er­al thought was that the best stu­dents should go to med­ic­al school” to en­sure a se­cure, well-paid ca­reer, said Win­nie. When she chose math, her par­ents made no protest but her high school teach­ers were dis­ap­poin­ted. “They said, ‘This is such a splen­did op­por­tun­ity for you to go to med­ic­al school, and you’re just not do­ing it!”’ she re­called. At this time the gov­ern­ing KMT was strongly en­cour­aging young people to go in­to sci­ence and en­gin­eer­ing, as a way to build Taiwan’s in­dus­tri­al ca­pa­city. But this en­cour­age­ment seems not to have ex­er­ted a big in­flu­ence on the mir­acle group. Said Jang-Mei, “We didn’t think about prac­tic­al things. We didn’t think about the fu­ture, about wheth­er math has a fu­ture. We liked math, so we chose that.”

In some sense they didn’t know quite what they were choos­ing, be­cause the math­em­at­ics they had learned in high school — the usu­al fare of al­gebra, geo­metry, tri­go­no­metry — was very dif­fer­ent from what they stud­ied in col­lege. And what math­em­at­ics might be like fur­ther on, as a field of re­search, was totally opaque. “I had no idea what it meant to be a re­search math­em­atician,” said Chuu-Li­an. “I didn’t even know what re­search is, when I was an un­der­gradu­ate.”

They got an ink­ling in a fresh­man course called “In­tro­duc­tion to Math­em­at­ics.” In 1966 the course was taught by NTU math pro­fess­or Kung-Sing Shih (施拱星). Shih was born in Taiwan in 1918, dur­ing the Ja­pan­ese co­lo­ni­al peri­od, and grew up in Kyoto. He was one of very few Taiwanese to re­ceive a math­em­at­ics de­gree in Ja­pan be­fore 1945; an­oth­er was Zhen-Rong Xu (許振榮). Both of them re­turned to Taiwan after 1945 and taught math­em­at­ics at NTU. In 1950 Xu went to the United States to work with Shi­ing-Shen Chern, who at the time was at the Uni­versity of Chica­go. Chern then helped Shih to enter the Uni­versity of Illinois, where the lat­ter earned a PhD in math­em­at­ics in 1953 un­der the dir­ec­tion of Ger­hard Hoch­schild. Both Xu and Shih re­turned to Taiwan, Xu to the In­sti­tute of Math­em­at­ics at the Aca­demia Sin­ica, and Shih to NTU, where he served as the dean of the fac­ulty of sci­ence for ten years.

The text­book for Shih’s “In­tro­duc­tion to Math­em­at­ics” course was the ori­gin­al Eng­lish ver­sion of the clas­sic What is Math­em­at­ics?, by Richard Cour­ant and Her­bert Rob­bins.8 Jang-Mei en­joyed Shih’s course greatly. “It opened my eyes to what math­em­at­ics is, not just cal­cu­lus and proofs, but oth­er things,” she re­called. “Num­ber the­ory, to­po­logy, com­plex ana­lys­is — I had nev­er seen them be­fore.” Shih tried to im­part a sense of what math­em­at­ics is really like and how one might think about it, said Win­nie. Among the per­man­ent fac­ulty at NTU at the time, “prob­ably he is the one who had the best over­all view of math­em­at­ics.” A year later, Chuu-Li­an took the same course, this time taught by Dong-Sheng Lai (賴東昇), and also with the Cour­ant and Rob­bins book. Chuu-Li­an re­membered Lai as an in­spir­a­tion to the stu­dents, who ran their own sem­inars to study each top­ic covered in the course.

An­oth­er im­port­ant first-year course was cal­cu­lus, which in 1970 was taught by Ju-Kwei Wang (王九 逵), an NTU gradu­ate who had earned a PhD in math­em­at­ics from Stan­ford in 1960, un­der the dir­ec­tion of Karel DeLeeuw. Wang’s lec­tures were un­usu­al. “Most of the oth­er teach­ers [at NTU] were teach­ing the Chinese way, that you have a book, and you copy from the book to the black­board and you ex­plain a few things,” said Win­nie. “But [Wang] was really try­ing to ex­plain the mean­ing to you, to make it more lively. To me that was very stim­u­lat­ing.”

In that first year, a sub­set of the wo­men math­em­at­ics ma­jors began study­ing to­geth­er. The reg­u­lar study group in­cluded Alice, Win­nie, and Fan, along with two oth­ers: Shou Jen Hu (胡守仁), who went on to earn a PhD at the Uni­versity of Chica­go un­der the dir­ec­tion of Melvin Rothen­berg, re­turned to Taiwan, and spent her ca­reer at Tamkang Uni­versity; and Shao-Yun Liu (劉小詠), who also went to Chica­go for gradu­ate school but died young. When Pro­fess­or Wang told the stu­dents to hand in solu­tions to the odd-numbered prob­lems from the book, the study group would solve all the prob­lems, odd and even. After work­ing in­di­vidu­ally, the group would meet. “Someone would say, I have dif­fi­culty [with one prob­lem], then an­oth­er would say, I have this idea,” said Alice. “And then we would dis­cuss it to­geth­er.” Some­times they even had long Sat­urday ses­sions. “But it was not just work. We had fun…There was some com­pet­i­tion, but on the oth­er hand, we quickly real­ized every­body has their own strengths and thinks dif­fer­ently…We learned from each oth­er.”

One strik­ing as­pect about uni­versity life in Taiwan at the time is that it was un­af­fected by the stu­dent protest move­ments that rocked uni­versity cam­puses around the world in the late 1960s. A 1969 New York Times art­icle (“Stu­dent act­iv­ism is rare in Taiwan,” 15 June 1969) datelined from Taiwan stated: “The Chinese Na­tion­al­ist of­fi­cials here point out with pride that they have no hip­pies, no Red Guards, and no rad­ic­al stu­dent move­ments.” One reas­on was mar­tial law, which was not lif­ted un­til 1988 and which the KMT re­gime used to repress polit­ic­al op­pos­i­tion.

Im­mersed in math­em­at­ics, the mir­acle group had es­sen­tially no con­tact with polit­ic­al is­sues — and little in­cent­ive to seek it out. “It was well un­der­stood, as long as you don’t touch polit­ics, you can do any­thing you please,” said Fan Chung. She knew of people who had been ac­cused of be­ing spies for the Com­mun­ists and wound up in jail. The mir­acle group learned much more about the real­ity of the KMT rule after mov­ing to the United States and read­ing the Amer­ic­an news. That “opened our eyes more,” said Fan, “be­cause the pro­pa­ganda over in Taiwan was very one-sided.”

The bond­ing of the wo­men in the classes and study groups was es­pe­cially im­port­ant at that time at NTU. Most of the math­em­at­ics in­struct­ors did not have PhDs and lacked ex­per­i­ence in the sub­ject. They knew how to choose cur­rent, high qual­ity text­books, but their lec­tures were of­ten dry and for­mu­laic. As a res­ult, stu­dents learned the ma­ter­i­al largely by study­ing on their own or in groups and by work­ing lots of prob­lems.

The fe­male solid­ar­ity also helped the wo­men handle the swag­ger of the men stu­dents. In their all-girl high schools, teach­ers had praised the girls’ math­em­at­ic­al prowess and told them they could do any­thing. NTU was dif­fer­ent. Fan Chung re­membered a fresh­man ori­ent­a­tion ses­sion where sev­er­al male stu­dents “jumped to the black­board and star­ted to tell us about all the books that are good for pre­par­a­tion” for the math ma­jor. “It was very in­tim­id­at­ing,” she said. “They wrote down all these names of math­em­aticians and books that we had no idea about.”

But the wo­men worked very hard, and with­in one semester, they were the ones with the best scores. “That greatly helped our con­fid­ence in ourselves,” Fan re­marked. “Be­cause in math­em­at­ics, it doesn’t mat­ter if you are short or you are a wo­man… The math score doesn’t lie, right? That’s the nice part about math­em­at­ics. You can prove your­self.”

There was pos­tur­ing by some of the men stu­dents who, when they did well, saw their suc­cess as due to in­trins­ic smarts. The wo­men stu­dents by con­trast saw suc­cess more as a res­ult of hard work and per­sever­ance. In ad­di­tion, they served as an in­spir­a­tion for each oth­er. Jang-Mei did not study much with the oth­er wo­men but drew con­fid­ence from their col­lect­ive ef­forts. “I saw oth­er people could make it…and I thought I can make it,” she said. “I just kept on go­ing.”

Only one of their courses, in Galois the­ory, was taught by a wo­man. This was Tzee-Nan Kuo (郭子 南), a vis­it­ing pro­fess­or who was an NTU gradu­ate her­self and had earned a PhD in math­em­at­ics in 1966 from the Uni­versity of Chica­go, un­der the dir­ec­tion of Jonath­an Alper­in. Kuo “was a really good teach­er,” Chuu-Li­an said. “She had everything in her head and lec­tured very well… I re­mem­ber I really wanted to per­form well in that course.” Too shy to speak to Kuo, Chuu-Li­an tried to im­press her by work­ing the home­work prob­lems ex­cep­tion­ally well. Kuo “really in­spired me,” Chuu-Li­an re­called. “I thought, ‘yeah, I would like to be a pro­fess­or like her’.” Kuo left aca­demia a few years later and had a ca­reer work­ing for the United States gov­ern­ment.

Mei-Chi Shaw entered NTU in 1973, which was a few years after the rest of the mir­acle group had gradu­ated. Taiwan’s eco­nom­ic situ­ation was im­prov­ing by then, and its in­dus­tri­al sec­tor was grow­ing rap­idly. Stu­dents were also be­com­ing less ideal­ist­ic. When the oth­er mir­acle wo­men had star­ted at NTU, “math and phys­ics were every­body’s dream de­part­ment,” said Mei-Chi. “By the time I went to col­lege, every­body wanted to go in­to elec­tric­al en­gin­eer­ing.” Many of her fel­low math­em­at­ics ma­jors had not lis­ted that sub­ject as their first choice for a ma­jor. In­deed, sev­er­al of the nine wo­men math ma­jors had hoped to be in an en­gin­eer­ing dis­cip­line and settled for math as their second or even third choice. By this time NTU had one wo­man math­em­at­ics pro­fess­or, Sou-Yung Chiu (邱守榕), who had re­ceived her PhD in 1970 at North­west­ern Uni­versity, un­der the dir­ec­tion of Ken­neth Roy Mount. She en­cour­aged Mei-Chi and helped to coun­ter­act the at­ti­tude con­veyed by some of the male stu­dents and pro­fess­ors that math­em­at­ics is not for wo­men.

Mei-Chi did not over­lap with any of the oth­er mem­bers of the mir­acle group at NTU. But she heard about them. People in the math­em­at­ics de­part­ment spoke with awe es­pe­cially about Win­nie Li, be­cause at the time she had a po­s­i­tion at Har­vard Uni­versity, as a Ben­jamin Peirce Fel­low. One of Mei-Chi’s teach­ing as­sist­ants had star­ted as a math ma­jor in 1966 along with Win­nie, Alice, Fan, and Jang-Mei. He re­called that those four, plus six oth­er fe­male math­em­at­ics ma­jors, con­sist­ently placed at the top. This was not long after the re­lease of a movie whose title trans­lates as The Four­teen Amazons (十四女英豪). The movie, made by Hong Kong film pro­du­cer Run Run Shaw, tells the story of the val­or of four­teen Chinese wo­men war­ri­ors. Mei-Chi’s teach­ing as­sist­ant “al­ways joked that there were ‘Ten Amazons’ in his class, and they ranked num­ber 1 to 10,” Mei-Chi said. “All the boys had to fight for num­ber 11.”

When Jang-Mei Wu was grow­ing up, her par­ents worked very hard just to provide the fam­ily with ba­sic ne­ces­sit­ies. Still, Jang-Mei’s moth­er found time to play ping-pong on the floor with the kids; her fath­er taught her to use an aba­cus and mem­or­ize clas­sic Chinese po­etry. Des­pite their straitened cir­cum­stances, Jang-Mei’s par­ents cre­ated an at­mo­sphere in which she felt free to do what she found fun and in­ter­est­ing. They did not pres­sure her to take up a pro­fes­sion — medi­cine be­ing the ca­non­ic­al ex­ample — that would en­sure fin­an­cial se­cur­ity. “They didn’t push me,” Jang-Mei said, de­scrib­ing their at­ti­tude as: “You do well, and if you don’t make much money, it’s okay.”

The six wo­men in the mir­acle group felt this free­dom from the pres­sure to seek wealth and ma­ter­i­al pos­ses­sions. They were aim­ing for something more ex­al­ted. In do­ing so, they also felt a deep ob­lig­a­tion to make the most of the op­por­tun­it­ies avail­able to them. “All of us grow­ing up dur­ing that time had to strive,” said Mei-Chi Shaw. “We had to make it worth­while for our par­ents. I al­ways had the feel­ing that I can­not fail, be­cause of the hard­ship they en­dured.”

It was only in 1976 that Taiwan es­tab­lished its first doc­tor­al pro­gram in math­em­at­ics. So it was with a sense of both free­dom and of ob­lig­a­tion that the mir­acle wo­men, after gradu­at­ing from NTU, left the is­land they knew as home and moved across the ocean to the United States, to pur­sue gradu­ate study in math­em­at­ics.

Sev­er­al of the wo­men fin­ish­ing in 1970 felt that their chances of get­ting ac­cep­ted by a gradu­ate pro­gram in the United States would be high­er if they ap­plied to dif­fer­ent places. So they de­cided to co­ordin­ate their ap­plic­a­tions. As the num­ber one stu­dent, Win­nie Li got first choice of where to ap­ply, the num­ber two stu­dent got to choose next, and so on. Alice Chang spoke of this meth­od as a way of “hon­or­ing” each oth­er. “We co­oper­ated,” she said. “Without a good friend­ship, I think this would not hap­pen. Later on I real­ized this was very spe­cial.”

Win­nie ended up put­ting off gradu­ate school and stay­ing in Taiwan for one year, so Alice ap­plied to, and was ac­cep­ted by, Win­nie’s first-choice school, the Uni­versity of Cali­for­nia at Berke­ley. They be­came class­mates when Win­nie was ac­cep­ted the fol­low­ing year. Fan Chung went to the Uni­versity of Pennsylvania, and Jang-Mei Wu to the Uni­versity of Illinois at Urb­ana–Cham­paign. The fol­low­ing year, in 1971, Chuu-Li­an Terng was ac­cep­ted by Bran­de­is Uni­versity. Six years later, Mei-Chi Shaw went to Prin­ceton Uni­versity. Mei-Chi’s only-wo­man status in the Prin­ceton math de­part­ment ended when Chuu-Li­an be­came an as­sist­ant pro­fess­or there, in 1978. The two have been close friends ever since.

The mir­acle group were among the very few wo­men go­ing in­to math­em­at­ics in the 1970s. In 1974 — the year Alice, Win­nie, Fan, and Jang-Mei fin­ished their PhDs — wo­men earned just 9 per­cent of the ap­prox­im­ately 1,000 math­em­at­ics doc­tor­ates awar­ded in the United States. The rate was 10 per­cent when Chuu-Li­an fin­ished the fol­low­ing year; by 1981, the year Mei-Chi fin­ished, it had ris­en to 16 per­cent. In the past ten years or so, it has hovered around 30 per­cent.

After fin­ish­ing their de­grees, the mir­acle group faced a tight aca­dem­ic job mar­ket in math­em­at­ics in the United States. Nev­er­the­less with­in a few years they all man­aged to find per­man­ent jobs in good in­sti­tu­tions. They were all in aca­demia, ex­cept for Fan Chung, who joined the re­search staff of Bell Labor­at­or­ies; twenty years later, after the break­up of the labs, she moved in­to aca­demia with ease. All are act­ive and prom­in­ent re­search­ers, each in a dif­fer­ent area of math­em­at­ics: Alice Chang in geo­met­ric ana­lys­is, Win­nie Li in num­ber the­ory, Jang-Mei Wu in com­plex ana­lys­is and po­ten­tial the­ory, Chuu-Li­an Terng in dif­fer­en­tial geo­metry, Fan Chung in dis­crete math­em­at­ics and com­bin­at­or­ics, and Mei-Chi Shaw in sev­er­al com­plex vari­ables.

Launched in 1962 and still pub­lished today, Bio­graph­ic­al Lit­er­at­ure has served as something of a re­pos­it­ory for the his­tory of Taiwan, told through the lives of people con­nec­ted to it. Chern made a few con­tri­bu­tions to the magazine, in­clud­ing a mem­oir about his own life in math­em­at­ics, pub­lished in 1964 when he was in his early fifties. At that time, he’d been a ten­ured pro­fess­or of math­em­at­ics in the US (at the Uni­versity of Chica­go and then Berke­ley) for 15 years. His ties to Taiwan re­mained strong, however, and his form­at­ive in­flu­ence on the mir­acle group was pro­found.

In 1945, Chern helped to found the In­sti­tute of Math­em­at­ics of the Aca­demia Sin­ica in Nank­ing, the then-cap­it­al of the Re­pub­lic of China. In 1948, the in­tens­i­fic­a­tion of the Chinese civil war com­pelled him to leave for the United States. That same year, the in­sti­tute moved to Taiwan, which has been its home ever since. Through nu­mer­ous vis­its to Taiwan, in­clud­ing to the in­sti­tute he helped found, and through many per­son­al in­ter­ac­tions, Chern lent a help­ing and en­cour­aging hand to many math­em­aticians from Taiwan. An­oth­er con­nec­tion to Taiwan came through the mar­riage of his daugh­ter May to su­per­con­duct­iv­ity pi­on­eer Paul Ching-Wu Chu (朱經武), who was born in China and spent his early years in Taiwan.

Al­though Chern was a pro­fess­or at Berke­ley when Alice Chang and Win­nie Li were gradu­ate stu­dents there, he was not the form­al ad­visor of either of them. But he was a be­loved ment­or to both. Win­nie re­membered when she told Chern she might want to go in­to al­gebra. “He said, ‘Well, in al­gebra there are only two areas. One is al­geb­ra­ic geo­metry, the oth­er one is num­ber the­ory’,” she re­called. “At the time, Berke­ley did not have al­geb­ra­ic geo­met­ers, so I should go in­to num­ber the­ory. That’s how I happened to do num­ber the­ory!” Of course, al­gebra has sub­fields oth­er than the two Chern named. But “in a sense he was right, from the view­point of im­port­ance,” said Win­nie. “He was think­ing about the big struc­ture.”

Alice has vivid memor­ies of Thanks­giv­ing din­ners at Chern’s home and of chat­ting with him while walk­ing across the cam­pus. “We learned from him not only aca­dem­ic­ally but also through his open, op­tim­ist­ic at­ti­tude to­ward oth­er people and life,” she said. Her own re­search in geo­met­ric ana­lys­is has been deeply in­flu­enced by Chern. One of the main top­ics of her re­search in con­form­al geo­metry has been the study of the 4-di­men­sion­al Chern–Gauss–Bon­net for­mula. “I ap­pre­ci­ated his work much more, and in­creas­ingly with time, after I gradu­ated from Berke­ley,” she said.

Chuu-Li­an Terng, who got to know Chern while she was a postdoc at Berke­ley, was es­pe­cially close to him. In a me­mori­al art­icle about him, she wrote of his in­stinct for deep and im­port­ant ques­tions, which in­spired her to move her re­search in­to new dir­ec­tions after her PhD thes­is. Chern’s sup­port and en­cour­age­ment were very im­port­ant to her. “He nev­er gave me false hopes,” Chuu-Li­an wrote, “but made it clear to me that hard work counts and that be­ing a math­em­atician was an en­joy­able pro­fes­sion.”

The oth­er three mir­acle wo­men — Fan Chung, Mei-Chi Shaw, and Jang-Mei Wu — knew Chern less well but were still in­flu­enced by him. Mei-Chi re­membered him as friendly and un­pre­ten­tious, gen­er­ous with his praise and sup­port. She also felt his in­flu­ence through the art­icle in Bio­graph­ic­al Lit­er­at­ure, which put his im­prim­at­ur on their achieve­ments. “It was such an en­cour­age­ment to me,” she said. “He didn’t have to write that art­icle, you see? He was a dif­fer­ent kind of math­em­atician and a dif­fer­ent kind of per­son, and that’s why his in­flu­ence is so great,” not only with­in his own field of geo­metry but in oth­er areas as well. “Chern went out of his way to en­cour­age wo­men,” she ad­ded. “We were really lucky to have such a ment­or.”

For Jang-Mei Wu, the art­icle helped her fam­ily un­der­stand what she’d done with her life. “It was very hard to ex­plain to my par­ents what I was do­ing in my of­fice on most days, which was to just sit and think and throw a lot of scratch pa­per in­to the waste bas­ket,” she said. “In some way that art­icle answered their ques­tions.” Her fath­er had passed away by the time of pub­lic­a­tion, but her moth­er and sis­ter en­joyed read­ing it. The oc­ca­sion also re­newed her ap­pre­ci­ation of her par­ents. “In many ways, their quiet sup­port, in an un­as­sum­ing and not pushy way, gave me a lot free­dom to pur­sue my in­terests.”

Chuu-Li­an Terng’s moth­er grew up in Hun­an in a wealthy fam­ily, with ser­vants do­ing all the house­work. In Taiwan, life was ut­terly dif­fer­ent. “I re­mem­ber [my moth­er] do­ing the laun­dry,” Chuu-Li­an re­called. “We didn’t have a wash­ing ma­chine, so she had to do it by hand on a wash­board… My God, it was something.” Daugh­ters would usu­ally help with such chores, but Chuu-Li­an was not al­lowed to. “She didn’t want me to do any house­work,” Chuu-Li­an re­called. “She didn’t even let me wash dishes. She said, ‘You study. I want you to get a good edu­ca­tion, be in­de­pend­ent. I want you to have a good mar­riage, but you have to be able to de­pend on your­self.’ ”

Hav­ing en­dured a wrench­ing dis­lo­ca­tion from their home­land, the moth­ers of the mir­acle group faced a much more dif­fi­cult life in un­cer­tain times in a new land. How did they ad­apt? Win­nie Li said that, while her moth­er would some­times talk about “the good old days,” the pre­vail­ing mind­set was prac­tic­al. “This is the life, they take it, they ac­cept it,” Win­nie said. In this ac­cept­ance, the moth­ers of the mir­acle group em­bod­ied for their daugh­ters an eth­os of hard work, per­sist­ence, and re­si­li­ence.

The mir­acle group de­veloped and drew on these qual­it­ies to suc­ceed. And be­cause they saw hard work as the route to suc­cess, fail­ure looked less scary and was something from which one could re­cov­er — by work­ing yet harder. For the favored son who ex­pects his in­nate tal­ent will carry him to the heights his fam­ily has set for him, a fall might be dev­ast­at­ing. The mir­acle group labored un­der no such ex­pect­a­tions. “You work as hard as you can, and you see what hap­pens,” said Mei-Chi Shaw. “Our egos are not that big… So we are less afraid of fail­ing than oth­ers.”

Their in­defatig­able ef­forts, made with no sense of en­ti­tle­ment, served the ex­al­ted ideals they pur­sued: learn­ing, un­der­stand­ing, schol­ar­ship, mas­tery, wis­dom, free­dom. Their tri­umph is sweet.

1974  Ph.D. Uni­versity of Cali­for­nia Berke­ley, “On the struc­ture of some Douglas sub­al­geb­ras”.

1979  Fel­low­ship, Al­fred P. Sloan Found­a­tion (1979–1981).

1986  In­vited speak­er, In­ter­na­tion­al Con­gress of Math­em­aticians in Berke­ley.

1995  Ruth Lyttle Sat­ter Prize in Math­em­at­ics (Amer­ic­an Math­em­at­ic­al So­ci­ety).

1998  Gug­gen­heim Fel­low­ship.

2002  Plen­ary Speak­er, In­ter­na­tion­al Con­gress of Math­em­aticians, Beijing.

2008  Mem­ber, Amer­ic­an Academy of Arts and Sci­ences.

2009  Fel­low, Na­tion­al Academy of Sci­ences.

2012  Fel­low, Aca­demia Sin­ica.

2015  Fel­low, Amer­ic­an Math­em­at­ic­al So­ci­ety.

2018  Emmy No­eth­er Lec­turer, In­ter­na­tion­al Con­gress of Math­em­aticians, Rio de Janeiro.

2015  Out­stand­ing Alumni Award, Na­tion­al Taiwan Uni­versity.

2019  Fel­low, As­so­ci­ation for Wo­men in Math­em­at­ics.

1974  Ph.D. Uni­versity of Pennsylvania, “Ram­sey num­bers in multi-col­ors and com­bin­at­or­i­al designs”.

1990  Al­lendo­er­fer Award (Math­em­at­ic­al As­so­ci­ation of Amer­ica).

1994  In­vited ad­dress, In­ter­na­tion­al Con­gress of Math­em­aticians, Zurich.

1998  Fel­low, Amer­ic­an Academy of Arts and Sci­ences.

2009  Emmy No­eth­er Lec­ture, As­so­ci­ation for Wo­men in Math­em­at­ics.

2015  Fel­low, So­ci­ety for In­dus­tri­al and Ap­plied Math­em­at­ics.

2016  Aca­dem­i­cian, Aca­demia Sin­ica.

2017  Euler Medal (In­sti­tute of Com­bin­at­or­ics and its Ap­plic­a­tions).

2022  Fel­low, Net­work Sci­ence So­ci­ety.

1974  Ph.D. Uni­versity of Cali­for­nia, Berke­ley, “New­forms and func­tion­al equa­tions”.

1981  Fel­low­ship, Al­fred P. Sloan Found­a­tion (1981–1983).

2010  Chern Prize in Math­em­at­ics (In­ter­na­tion­al Con­gress of Chinese Math­em­aticians).

2012  Dis­tin­guished Pro­fess­or of Math­em­at­ics, Penn State Uni­versity (in re­cog­ni­tion of ex­cep­tion­al teach­ing, re­search and cre­ativ­ity).

2012  Fel­low, Amer­ic­an Math­em­at­ic­al So­ci­ety.

2015  No­eth­er Lec­ture, As­so­ci­ation for Wo­men in Math­em­at­ics.

2018  Spe­cial Con­tri­bu­tion Award (Taiwanese Math­em­at­ic­al So­ci­ety).

1981  Ph.D. Prin­ceton Uni­versity, “Hodge the­ory on do­mains with cone-like or horn-like sin­gu­lar­it­ies”.

2012  Fel­low, Amer­ic­an Math­em­at­ic­al So­ci­ety.

2019  Stefan Berg­man Prize (Amer­ic­an Math­em­at­ic­al So­ci­ety).

1976  Ph.D. Bran­de­is Uni­versity, “Nat­ur­al vec­tor bundles and nat­ur­al dif­fer­en­tial op­er­at­ors”.

1980  Fel­low­ship, Al­fred P. Sloan Found­a­tion (1980–1982).

1997  Hum­boldt Seni­or Sci­ent­ist Award (Al­ex­an­der von Hum­boldt Found­a­tion of Ger­many).

1999  Fal­con­er Lec­turer, As­so­ci­ation of Wo­men in Math­em­at­ics / Math­em­at­ic­al As­so­ci­ation of Amer­ica.

2006  In­vited Ad­dress, In­ter­na­tion­al Con­gress of Math­em­aticians, Mad­rid.

2018  Fel­low, As­so­ci­ation for Wo­men in Math­em­at­ics (in­aug­ur­al class).

1974  Ph.D. Uni­versity of Illinois at Urb­ana-Cham­paign, “An in­teg­ral prob­lem for pos­it­ive har­mon­ic func­tions”.

1990  In­vited Ad­dress, Amer­ic­an Math­em­at­ic­al So­ci­ety.

1992  Er­skine Fel­low, Uni­versity of Can­ter­bury, New Zea­l­and.

2006  Fre­d­er­ick and Lois Gehring Pro­fess­or, Uni­versity of Michigan.

2020  Fel­low, Amer­ic­an Math­em­at­ic­al So­ci­ety.

I would like to thank the six wo­men of the mir­acle group — Sun-Yung Alice Chang, Fan Chung, Wen-Ching Win­nie Li, Mei-Chi Shaw, Chuu-Li­an Terng, and Jang-Mei Wu — for shar­ing their stor­ies. I also grate­fully ac­know­ledge the help of J. Megan Green (Uni­versity of Kan­sas), Hsiao-Ting Lin (Hoover In­sti­tu­tion, Stan­ford Uni­versity), Sheila New­bery (Math­em­at­ic­al Sci­ences Pub­lish­ers), and Hung-Hsi Wu (De­part­ment of Math­em­at­ics, Uni­versity of Cali­for­nia, Berke­ley). Re­spons­ib­il­ity for the ac­cur­acy of the art­icle is my own.