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Celebratio Mathematica

R H Bing

Topology

R H Bing: A study of his life

by Sukhjit Singh

Introduction

It was dur­ing a vis­it with R H Bing (af­fec­tion­ately called “RH” by his friends and close as­so­ci­ates) while he was hos­pit­al­ized for two weeks in the sum­mer of 1984 that the idea of do­ing a Bing bio­graphy oc­curred to me. As R H greeted me from his bed, he ap­peared weak but in good spir­its, his eyes shin­ing with an­ti­cip­a­tion of some good con­ver­sa­tion about math­em­at­ics. Throughout our talk, R H would tell me of an in­ter­est­ing event in his life ac­com­pan­ied by an an­ec­dote from his child­hood or stu­dent days. After such an en­light­en­ing vis­it, it entered my mind how his­tor­ic­ally cru­cial it was for the math­em­at­ic­al com­munity to have a writ­ten re­cord of R H’s life.

R H re­gained his health and strength and re­turned to his of­fice on the cam­pus of The Uni­versity of Texas at Aus­tin. It was here that I ap­proached him and so­li­cited his help with this pro­ject. My in­tent was to write his bio­graphy un­der his guid­ance, not only for au­then­ti­city but with the hope of cap­tur­ing the “Bing per­sona.” R H re­luct­antly agreed, but be­came more en­thu­si­ast­ic as the pro­ject pro­gressed.

My first goal was to col­lect bio­graph­ic­al data through a series of in­form­al in­ter­views I had with R H sev­er­al times each week, in which he answered ques­tions con­cern­ing the vari­ous stages of his life. However, I soon began to real­ize that my note-tak­ing alone could not serve as a sole re­source for ac­cur­ate bio­graph­ic­al data, much less con­vey the col­or­ful col­lo­qui­al­isms of Bing. It then oc­curred to me that the use of au­di­o­tapes would be an ex­cel­lent me­di­um for gath­er­ing in­form­a­tion.

After a con­sid­er­able amount of per­sua­sion, R H agreed to do a tape, or a se­quence of tapes, fol­low­ing an out­line of top­ics that I provided. However, R H pro­ceeded only un­der the con­di­tion that the tape be used only to fa­cil­it­ate the writ­ing of the bio­graphy, and that it not be made pub­lic. R H was quite con­cerned that his ran­dom com­ments might be mis­un­der­stood, or that he might eas­ily say in­ac­cur­ate things dur­ing the flow of a tap­ing ses­sion.

The bio­graph­ic­al sketch you are about to read is a cul­min­a­tion of years of sep­ar­ate and joint ef­forts by R H and my­self at re­arran­ging, edit­ing, adding, and de­let­ing ma­ter­i­al from the ori­gin­al tran­scrip­tion of this tape.

Ori­gin­ally, I began writ­ing the Bing bio­graphy in the third per­son based on this tape, but soon dis­carded this ap­proach. It oc­curred to me that the first-per­son nar­rat­ive style mani­fest in the tape was more suit­able in ex­em­pli­fy­ing the “Bing per­sona” as well as his unique con­ver­sa­tion­al style.

I star­ted by care­fully re­arran­ging, edit­ing, and pol­ish­ing a tran­scrip­tion of the tape. This draft served as the start­ing point. Sub­sequently, R H and I both jointly and sep­ar­ately kept adding to and de­let­ing from this draft. At one point, we agreed that the pro­ject was fin­ished, and I left a copy of the fi­nal ver­sion with him at his home in Aus­tin. Sure enough, R H re­turned this copy to me with some ad­di­tion­al changes only one month be­fore his death. I have in­cor­por­ated these re­vi­sions.

So, wel­come to a rare in­sight in­to the spe­cial char­ac­ter and spir­it of R H Bing. The strength, love, and dis­cip­line he re­ceived and learned formed the un­der­ly­ing core of his com­mit­ment to math­em­at­ics. Al­though many con­trib­uted to his lifelong com­mit­ment to math­em­at­ics, the most not­able are his moth­er Lula May, his teach­er R. L. Moore, and his wife Mary.

Family background and childhood

My fath­er was reared near Wall­er, Texas. Grand­fath­er Bing died at a re­l­at­ively young age, and I nev­er met him. However, I did vis­it the fam­ily farm when I was very young. He and his broth­er had come from Can­ter­bury, Eng­land. His broth­er, not sat­is­fied with life in this coun­try, re­turned to Eng­land. My fath­er kept in con­tact with the Can­ter­bury Bings un­til World War I. When I vis­ited Eng­land, I tried to find more about the Can­ter­bury Bings and was told that they had been prom­in­ent mem­bers of the com­munity and had owned a bot­tling com­pany. A com­pany there was still called the Bing Bot­tling Com­pany and man­u­fac­tured a product called “Bing Soda”; however, it was not owned by the heirs of the ori­gin­al Bings.

My grand­fath­er Bing was a re­l­at­ively well-edu­cated farm­er for his time. Re­cord keep­ing by farm­ers was a bit un­usu­al at the turn of the cen­tury, but I re­mem­ber see­ing books that he kept show­ing the price that he paid for farm an­im­als and im­ple­ments and the price he got in re­turn for crops that he sold.

My ma­ter­nal grand­fath­er was named Thompson. He had been a bandlead­er and con­duc­ted mu­sic at vari­ous re­viv­als. This caused him to move about quite a bit. By moth­er was born as the fam­ily moved through Em­por­ia, Kan­sas. When I vis­ited Em­por­ia sev­er­al years ago, I looked at news­pa­pers of that time to see if her birth had been re­cor­ded, but it had not. Both of my moth­er’s par­ents died while she was still in high school. This left my moth­er with a young­er sis­ter and broth­er to be reared by kin­folks in Le­onard, Texas.

My moth­er was quite eager to get out and make a liv­ing for her­self. She did tem­por­ary teach­ing while still in high school (at age 15). As soon as she fin­ished high school, she took ex­am­in­a­tions to qual­i­fy for a tem­por­ary teach­er’s cer­ti­fic­ate. My moth­er told me about how she went about look­ing for a job. Tak­ing a pen in hand, she closed her eyes, moved her hand over a map of the state of Texas and put the pen down on the map. She did this three times and wrote to the three schools picked in this fash­ion. My moth­er re­ceived three job of­fers and ac­cep­ted the first, loc­ated in Oak­wood, Texas. This turned out to be the one pay­ing the least, but since she had ac­cep­ted it earli­er, she felt a strong com­mit­ment to take the job. Soon after mov­ing to Oak­wood she sent for her broth­er and sis­ter to come and live with her.

My moth­er con­trac­ted a dis­ease dur­ing her high school years which caused the loss of one eye, but she was still quite pop­u­lar and suc­cess­ful as a primary teach­er in Oak­wood. Dur­ing her first year, however, she thought her­self as nearly be­ing fired when she in­tro­duced phonics. But, her en­thu­si­ast­ic per­sua­sion caused the stu­dents to read bey­ond their par­ents’ ex­pect­a­tions and she be­came very re­spec­ted. After sev­er­al years of teach­ing in Oak­wood, she mar­ried my fath­er who was then su­per­in­tend­ent of the Oak­wood School Dis­trict.

Since my fath­er died when I was five years old, I do not have a good memory of him. However, my moth­er had great re­spect and af­fec­tion for my fath­er and much of what I know about him is prob­ably something I learned form her. My moth­er stopped teach­ing after mar­riage. She and my fath­er had two chil­dren of which I was the older. I also have a sis­ter. Hav­ing a sis­ter has been a valu­able ex­per­i­ence for me, al­though at the time my sis­ter did not ap­pre­ci­ate it — es­pe­cially when I en­cour­aged her to have a make-be­lieve doll fu­ner­al.

I re­call that I loved my fath­er very much and have pic­tures of him put­ting me on a horse and giv­ing me a ride. I do re­mem­ber him telling me stor­ies. Dur­ing his fi­nal ill­ness, I was taken to a neigh­bor’s home and told of his death. I did not com­pre­hend this, and on re­turn­ing home, looed for him in the barn, un­der the house, and oth­er places where I thought he might be hid­ing. Two stor­ies about my Dad come to mind. Some of this I re­call and some I re­mem­ber only be­cause I was told about it later.

Dur­ing those days it was the re­spons­ib­il­ity of the su­per­in­tend­ent of the schools to see that or­der was kept in the classes. The wo­men teach­ers would send stu­dents that had been giv­ing them un­usu­al trouble to the su­per­in­tend­ent for pun­ish­ment. Cor­por­al pun­ish­ment was much more pre­val­ent in those days than it is today and on one oc­ca­sion my fath­er re­sor­ted to whip­ping a boy for some bad of­fense. Sev­er­al days later this boy, along with his fath­er and two of his lar­ger broth­ers, way­laid my fath­er on the way home from school, in­tent on beat­ing him up. My moth­er saw this and came run­ning with sticks and stones. The men re­len­ted and left with a warn­ing that they would get my fath­er later.

My moth­er was very ap­pre­hens­ive for my fath­er’s safety and sug­ges­ted he or­der a gun to pro­tect him­self. My fath­er pre­ten­ded not to be con­cerned and told my moth­er not to worry. However, my moth­er went ahead and ordered a pis­tol from the hard­ware store; not let­ting my moth­er know, my fath­er also ordered a shot­gun. Hence, we soon had both a pis­tol and a shot­gun. As far as I know, these were nev­er used dur­ing my fath­er’s life­time. But I have heard that my moth­er car­ried the pis­tol in her purse and said she would shoot any man that at­tacked my fath­er.

Sev­er­al years after my fath­er’s death, my moth­er had oc­ca­sion to use her pis­tol which led to her re­cog­ni­tion as a marks­man of some mer­it by the loc­al gun club. Hogs be­long­ing to someone down the street fre­quently broke in­to our garden. We re­paired our fence and ran the hogs off, but a day or so later they would get out and come and root up the garden again. My moth­er de­cided that she would try to fright­en the hogs by fir­ing shots near them to keep them away. She got out the pis­tol and asked me to chase the hogs out; her ac­cur­acy, however, was not too good. As I chased out a hog and she shot the pis­tol, the hog fell over on his back and star­ted kick­ing. Soon it was dead. My moth­er called the meat mar­ket and asked if they would get the hog be­cause they did not know its own­er. The daugh­ter of one of the neigh­bors brought over a butcher knife and offered to help butcher the hog but the plan was not pur­sued.

Soon the story was all over town of how my moth­er had killed a hog — what a crack­shot she was with a gun. The own­er of the hog nev­er came for­ward be­cause he was ashamed that his hog had des­troyed my moth­er’s garden. As the hog began to stink, my moth­er had someone put a rope around the car­cass and drag it off to where dead cattle were left. The men of the town had great fun telling this story. They had a gun club that was go­ing away to en­gage in a con­test, and were mak­ing my moth­er cap­tain be­cause she was so ac­cur­ate at shoot­ing. I am not sure of the sig­ni­fic­ance of this story, but thought it quite in­ter­est­ing be­cause it made such a deep im­pres­sion on my moth­er. I do not re­mem­ber my par­ents pur­chas­ing these guns, but I do re­mem­ber the kick­ing hog.

An­oth­er story about my fath­er, per­haps more per­tin­ent, in­volved the loc­al schools for the whites and the schools for the blacks. Al­though these schools were sep­ar­ate in those days, they were in the same dis­trict and hence my fath­er was su­per­in­tend­ent of both. He knew how to or­der books and do oth­er things that were use­ful to the black schools. I re­call that one day, when I was about 10, I met sev­er­al black youths. They asked me if I was the son of Su­per­in­tend­ent Bing. I said that I was. They said, “He was a very good man, he was al­ways fair to us. You can nev­er be as good a man as your fath­er was.” This made an im­pres­sion on me.

My fath­er was suc­cess­ful as a teach­er and the Oak­wood schools turned out many out­stand­ing people dur­ing his ten­ure; one be­came pres­id­ent of Chrysler Mo­tor Com­pany. Sev­er­al years be­fore his death, however, my fath­er de­cided to leave teach­ing and take up farm­ing. He ac­quired sev­er­al farms and a store and was quite suc­cess­ful at man­aging them. My moth­er said that my fath­er was a work­ahol­ic. As a mat­ter of fact, work may have led to his early death. One day he had some plow­ing to do and it began to rain. My moth­er in­sisted that he come in out of the rain but he said that he had just a bit more to do and con­tin­ued plow­ing. As a res­ult of this, he took in­flu­enza or pneu­mo­nia or some oth­er dis­ease of that time and died in 1920 leav­ing my moth­er to take care of two chil­dren — the older be­ing just five.

The fol­low­ing story about my moth­er re­veals something of her char­ac­ter. She was a strong dis­cip­lin­ari­an and made it a prac­tice of con­front­ing stu­dents who had done something wrong and hav­ing them con­fess what they had done be­fore re­veal­ing what she knew. This worked well most of the time, but I re­mem­ber one case in which it did not.

I was play­ing in the back­yard and a neigh­bor told my moth­er of something that the neigh­bor thought that I had done. My moth­er called me in­to the house and asked me what I had done wrong. I did not re­call hav­ing done any­thing wrong, so my moth­er gave me a few paddles and told me to go off and re­call what it was. I still was un­able to de­cide what she wanted, so I re­ceived an­oth­er paddle. This happened sev­er­al times and fi­nally my moth­er thought she would give a hint. She said, “What were you say­ing in the back­yard?” Then I re­membered that I had been pound­ing on a stick, driv­ing it in­to the ground with a ham­mer and say­ing, “Go down, go down, go down.” The neigh­bor had heard me and thought I was curs­ing and say­ing, “god­damn, god­damn, god­damn,” and this is what was told to my moth­er. I re­mem­ber see­ing the re­morse on my moth­er’s face when she found that she had pun­ished me need­lessly. She went off and cried and tried to make up to me for the harm that she had done. This im­pressed me in two ways. She was in­ter­ested that I did not use pro­fan­ity, and she was very con­cerned when she found that she had wronged me. I long for­got the pun­ish­ment but not these two things.

My moth­er felt that it was an ex­cel­lent idea for moth­ers to train their chil­dren at an early age. Long be­fore start­ing school, I was able to read and do arith­met­ic and re­garded these things as great fun. I think I owe a great deal to my moth­er’s early train­ing for my in­terest and suc­cess in school.

When I star­ted school, I had already mem­or­ized all of the primary books and found school very, very easy. Dur­ing sev­er­al months of the first year, I was ill, but by the second year I was put in the third grade. Al­though I had to work dur­ing my first few years to stay up with the class, by the time I had got­ten to ju­ni­or high school, I was able to lead my class. One thing that bothered me was my be­ing young­er than my class­mates and hence at a dis­ad­vant­age when com­pet­ing in ath­let­ics with boys who were older.

I cred­it my moth­er with much of my suc­cess in math­em­at­ics. Be­fore start­ing school, I learned that math­em­at­ics was fun. In the sev­enth grade, I entered the county num­ber sense con­test. My moth­er was coach of the team and she taught short­cuts en­abling one to do com­pu­ta­tions quickly. We learned to ap­prox­im­ate an­swers to harder prob­lems. I later learned that my part­ner and I made the highest grade in the state in the num­ber sense con­test that year — more than likely due to my moth­er’s coach­ing.

When I took geo­metry, my moth­er was quite in­ter­ested in how I went about prov­ing the­or­ems. She taught me that the pur­pose of geo­metry was to dis­cov­er proofs rather than to mem­or­ize them. I was nev­er pushed in my pur­suit of math­em­at­ics and wondered wheth­er I would have liked it nearly so well if I had been offered a speeded-up course such as we give some of our bet­ter stu­dents today. Today, as well as then, I am grate­ful that I had the time to think, to con­tem­plate and to work out math­em­at­ics on my own rather than to be pushed ahead to learn proofs that were provided by oth­ers.

College days

Sev­er­al years after my fath­er’s death, our funds had been de­pleted to the ex­tent that my moth­er re­turned to teach­ing to earn a liv­ing. (The pay was only $6.50 per month.) Yet, there was nev­er a ques­tion as to wheth­er or not my sis­ter and I would go to col­lege. The ques­tion was only how it could be af­forded. I re­call that one of our friends wanted my moth­er to bor­row enough money to send me to Prin­ceton. He offered to help in fun­drais­ing; however, my moth­er hes­it­ated about this ob­lig­a­tion and thought it wiser for me to at­tend a school that I could af­ford.

Two of my fa­vor­ite high school teach­ers had fin­ished their col­lege edu­ca­tion at South­w­est Texas State Teach­er’s col­lege, or South­w­est Texas State Uni­versity as it is now called, and ar­ranged for me to get a job at San Mar­cos in the col­lege cafet­er­ia. This is the way that I sup­por­ted my­self dur­ing col­lege. I was quite frugal; my moth­er and I figured it had to cost me less than \$300 to get my bach­el­or’s de­gree, most of which was for tu­ition. I did, however, com­plete col­lege in 2 1/2 years — eager to fin­ish in a hurry so funds could be used to send my sis­ter to school. She also went to South­w­est Texas State Teach­er’s Col­lege.

Sev­er­al years after my sis­ter and I had got­ten our de­grees from South­w­est Texas State Teach­er’s Col­lege, and I had even re­ceived a mas­ters and Ph.D. from The Uni­versity of Texas, my moth­er, by go­ing to school in the sum­mers, also re­ceived a bach­el­or’s de­gree from South­w­est Texas State Teach­er’s col­lege in San Mar­cos.

Student of R. L. Moore: Graduate school

After I fin­ished col­lege, I had teach­ing po­s­i­tions in three Texas high schools. Al­though these 4 1/2 years of teach­ing did not ad to my math­em­at­ic­al pro­gress, it taught me how to get along with people. While teach­ing in high school, I went back in the sum­mers and took courses at the Uni­versity of Texas. It was there that I met Pro­fess­or R. L. Moore. R. L. Moore ap­peared to have a very low opin­ion of courses in ped­agogy or edu­ca­tion, and I think he sus­pec­ted I would not be very good since I had spent so long teach­ing in high school. I was soon able to re­lieve him of this no­tion, be­ing able to prove the­or­ems that his stu­dents, who had not taken time off to teach in high school, could not prove.

R. L. Moore was quite ser­i­ous about his teach­ing, as he was about oth­er things. When any­one talked to him, he looked at the per­son with pen­et­rat­ing eyes. I was care­ful to try to not say any­thing un­less it was ac­cur­ate. He was in­ter­ested in what a per­son could do him­self. He placed books off lim­its for his stu­dents when these books were deal­ing with things on which they could work them­selves. As a mat­ter of fact, in his course on to­po­logy he placed his own Col­loqui­um book off lim­its.

Moore taught not only in the classroom but in the halls and else­where. I re­mem­ber my land­lord telling me how in­ter­ested R. L. Moore was in my work and how Moore tried to ex­plain to him some of the things I was do­ing. My land­lord did not un­der­stand any of this. But this made a pro­found im­pres­sion on me that Moore was so in­ter­ested in my work that he would try to ex­plain some of it to my land­lord.

Mrs. Moore was a real as­set to Dr. Moore. I re­mem­ber Mrs. Moore tele­phon­ing my wife, Mary, and telling her that she real­ized what a sac­ri­fice my wife was mak­ing in or­der to make it pos­sible for me to spend so much time on math­em­at­ics. I be­lieve the word “sac­ri­fice” came as a sur­prise to Mary who thought run­ning a house­hold was a 50-50 pro­pos­i­tion. But be­ing told by Mrs. Moore that a wife might en­hance her hus­band’s ca­reer by let­ting him spend full time on it did have an im­pres­sion. Mary has al­ways been a big help in en­ter­tain­ing stu­dents, man­aging the fam­ily, and provid­ing a pleas­ant en­vir­on­ment.

In Moore’s classes, he taught by in­dir­ec­tion. I re­call Moore hav­ing one of his ex-stu­dents vis­it his class. This ex-stu­dent had not pro­duced as a re­search math­em­atician and after he left, Moore said, “How can I stand hav­ing a man like that vis­it my class? He had a real op­por­tun­ity and he did not take ad­vant­age of it!” The stu­dents got the mes­sage.

Moore did not like for people to have sloppy ap­pear­ances. He ex­pec­ted them to sit up straight in their chairs and to be­have like ladies and gen­tle­men. Once he showed a pic­ture of stu­dents at the Uni­versity of Chica­go who were be­ing quite slov­en and his re­marks were, “Do you think the Uni­versity of Chica­go is that bad? Do the stu­dents there think so little of them­selves as to be­have in such an un­dig­ni­fied man­ner?” Again, we got the mes­sage.

One of the teach­ers at The Uni­versity of Texas had a wife who would come to the build­ing after hours to pick up her hus­band and blow on the horn to ad­vise him that she was there. Moore thought it was in­ap­pro­pri­ate to make noise around a build­ing where re­search was be­ing done. He would ask his stu­dents, “What do you think of a man who would per­mit his wife to come to the build­ing and honk the horn out­side?” Again, we got the mes­sage.

Moore was very proud of his stu­dents who had suc­ceeded in re­search. He told us stor­ies of how Ray Wilder had been able to prove a the­or­em in a short peri­od of time. How Gor­don Why­burn had had suc­cesses. How Bur­ton Jones had been able to do such and such. How one of his stu­dents, Miss Miller, had suc­ceeded where oth­ers had failed in prov­ing a cer­tain the­or­em. Again, we got the mes­sage.

Dick An­der­son was a stu­dent at The Uni­versity of Texas who got his de­gree sev­er­al years after I did. He and I fre­quently spent part of the noon hour play­ing chess. Moore did not ap­prove of this and told me that I should not waste An­der­son’s time which could be bet­ter spent on math­em­at­ics. I also got the mes­sage that Moore thought I could make bet­ter use of my time.

On an­oth­er oc­ca­sion in the halls someone had asked me who had won the foot­ball game the pre­ced­ing Sat­urday and wheth­er or not I knew the score. I re­membered who had won the game, but I was un­able to provide the score. Moore’s re­mark was, “At least, thank good­ness for that!”

I will al­ways ap­pre­ci­ate the good in­struc­tion I re­ceived un­der R. L. Moore. It was to my lik­ing be­cause I en­joyed work­ing out math­em­at­ics for my­self and this is what he en­cour­aged. However, I did not like the close su­per­vi­sion he gave my thes­is. When he had me write things, he wanted it in his own way rather than in mine. I felt very glad when my thes­is was fin­ished, for I now felt I had the au­thor’s prerog­at­ive of say­ing things the way I wanted to say them. Moore felt that if it was said cor­rectly, it didn’t really mat­ter wheth­er or not it was eas­ily un­der­stood be­cause it was the read­er’s re­spons­ib­il­ity to dig it out. My feel­ings were that we should say things, of­ten re­peatedly, in an ef­fort to make it more un­der­stand­able. Moore’s con­cise­ness in writ­ten present­a­tion did not al­ways carry over to the or­al dis­cus­sions since he took care to make them clear.

Early research at Texas

After re­ceiv­ing my de­gree from Moore, I vis­ited him many times. We nev­er dis­cussed math­em­at­ics to­geth­er. In one case, shortly after I re­ceived my de­gree from him, I proved one of the big the­or­ems that had been baff­ling quite a few people; namely the Kline sphere char­ac­ter­iz­a­tion. After writ­ing this up, I sub­mit­ted it as an ab­stract to the Amer­ic­an Math­em­at­ic­al So­ci­ety and also to a journ­al without ever hav­ing shown or dis­cussed it with Moore. When it be­came known that I had an­nounced the res­ult, we were flooded with phone calls, tele­grams, and let­ters — mostly ad­dressed to Moore — ask­ing if I had got­ten the res­ult and if it had been checked.

Moore’s reply was to turn these let­ters over to me to be answered be­cause he had not seen proof. Yet, he did not ask me for the proof. I think that he thought that by ask­ing me, he would show lack of con­fid­ence in my abil­ity to check the proof my­self. I also did not show him the proof be­cause I felt that this might show that I did not have self-con­fid­ence and needed someone else to help me check the res­ult.

At Chica­go I an­nounced the res­ult. Oth­ers had an­nounced par­tial solu­tions to the res­ult, and be­fore I got on the pro­gram I was offered ad­di­tion­al time so I could bet­ter ex­plain my proof. I was only able, however, to give the present­a­tion that I had planned. I do re­call Paul Er­dös ask­ing me at that meet­ing wheth­er I had had a first-class math­em­atician check the proof. I rather re­sen­ted this be­cause I thought that it was a first-class math­em­atician who had provided the proof.

My thes­is dealt with simple plane webs and I wrote sev­er­al pa­pers on this sub­ject. I learned, however, that the math­em­at­ic­al com­munity was not very in­ter­ested in webs. There was much more in­terest in the Kline sphere char­ac­ter­iz­a­tion; at least in know­ing that it was true. I found that in the case of hard the­or­ems, the math­em­at­ic­al com­munity is quite of­ten in­ter­ested in know­ing wheth­er the res­ult is true. But many math­em­aticians will not go through the work of check­ing to un­der­stand the proof.

I spent two years at The Uni­versity of Texas after get­ting my Ph.D. There were oth­er in­ter­est­ing res­ults that I got while I was there; one of the most in­ter­est­ing dealt with the pseudo-arc. Ed Moise had writ­ten his thes­is about the pseudo-arc and showed that the pseudo-arc had the in­ter­est­ing prop­erty that it was homeo­morph­ic to each of its nonde­gen­er­ate sub­con­tinua. That is the reas­on he called it the pseudo-arc. It turned out later that oth­er math­em­aticians had de­scribed the pseudo-arc earli­er by oth­er meth­ods. I was quite in­ter­ested in Moise’s de­scrip­tion, but wondered if a dif­fer­ent de­scrip­tion might not show that the pseudo-arc was ho­mo­gen­eous and, in­deed, found that this was true.

When Moise was still a stu­dent and I already had my Ph.D., I was re­luct­ant to an­nounce the res­ult that the pseudo-arc was ho­mo­gen­eous and thought that Moise should at least have a chance to make such an an­nounce­ment if her were close to dis­cov­er­ing it. I asked him if he had con­sidered the mat­ter as to wheth­er or not the pseudo-arc was ho­mo­gen­eous (know­ing all the time that it was). He said that he had checked this and found in­deed that it was not. So sev­er­al months later, I an­nounced that a pseudo-arc is ho­mo­gen­eous. I found that while some people were in­ter­ested in the pseudo-arc, many were not be­cause the pseudo-arc did not have the nice geo­met­ric prop­er­ties of dif­fer­en­ti­ab­il­ity or lin­ear­ity of many geo­met­ric ob­jects. I de­cided that math­em­aticians were more in­ter­ested in man­i­folds than they were in exot­ic ob­jects.

Wisconsin years and mathematics of this time

Two years after tak­ing my Ph.D. at Texas, I went to the Uni­versity of Wis­con­sin where I had a very suc­cess­ful ca­reer. After two years at Wis­con­sin, I took a leave of ab­sence and spent a year as a vis­it­or at the Uni­versity of Vir­gin­ia. I had as an of­fice mate, Ed Floyd, and learned quite a bit about man­i­folds from him. I taught a class there in which the mem­bers were Gor­don Why­burn, Ed Floyd, Vic Klee, Tru­man Botts, and Bob Wil­li­ams. This was a very in­ter­est­ing class and we stud­ied cur­rent lit­er­at­ure. I ob­tained sev­er­al very good res­ults dur­ing the year I was at the Uni­versity of Vir­gin­ia.

Floyd told me that people were in­ter­ested in know­ing wheth­er there was an in­vol­u­tion of a 3-sphere onto it­self dif­fer­ent from or­din­ary in­vol­u­tions. There was a sus­pi­cion that prob­ably the uni­on of two sol­id Al­ex­an­der horned spheres sewed to­geth­er along their bound­ary gave \( \mathbb{S}^3 \) and that an in­vol­u­tion that in­ter­changed these two sol­id spheres (or Al­ex­an­der balls) would be an in­vol­u­tion in­equi­val­ent to a stand­ard in­vol­u­tion. By the use of strings, rub­ber bands, and oth­er meth­ods, I was able to show that the uni­on of two sol­id horned spheres sewed to­geth­er along their bound­ar­ies with the iden­tity homeo­morph­ism was in­deed a 3-sphere.

I found that many people were quite in­ter­ested in this res­ult. Since a proof was not very hard, oth­ers also learned the proof. It is only this year (circa 1984) that I have dis­covered a new proof which gives oth­er res­ults along this line. Mike Freed­man and Richard Skora have made use of these new res­ults.

The pseudo-arc is an in­decom­pos­able con­tinuum. It is not the sum of two prop­er sub­con­tinua. In fact, it is hered­it­ar­ily in­decom­pos­able since each of its prop­er sub­con­tinua has this prop­erty. A stu­dent from the Uni­versity of Vir­gin­ia, John Kel­ley, who is now at Berke­ley, had shown that if there is a two-di­men­sion­al hered­it­ar­ily in­decom­pos­able con­tinuum, then there is an in­decom­pos­able con­tinuum of di­men­sion three, one of di­men­sion four, and in fact, for each pos­it­ive in­teger \( n \), a hered­it­ar­ily in­decom­pos­able con­tinuum of di­men­sion \( n \). However, it was not known wheth­er there was a hered­it­ar­ily in­decom­pos­able con­tinuum of di­men­sion two.

I re­mem­ber spend­ing many hours at the Uni­versity of Vir­gin­ia try­ing to solve the prob­lem of wheth­er or not there is a two-di­men­sion­al hered­it­ar­ily in­decom­pos­able con­tinuum. As a mat­ter of fact, late one even­ing, while I was still at my desk think­ing about this prob­lem, Vic Klee came by. I re­mem­ber telling him, “I think I have this prob­lem about solved. If I could put all of the ideas to­geth­er that I have in mind, I be­lieve that I would have a solu­tion. I am go­ing to sit here and work on this un­til my mind is able to put these things to­geth­er and provide a solu­tion.” And, in­deed, I suc­ceeded; in fact, I was able to provide a proof quite dif­fer­ent from that of Kel­ley’s that there are hered­it­ar­ily in­decom­pos­able con­tinua of all di­men­sions. The ex­ample con­struc­ted was an exot­ic con­tinuum and not so many people were in­ter­ested in it as they were be­fore the prob­lem had been solved.

I have noted that many oth­er res­ults that are of con­sid­er­able in­terest be­fore they are proved meet sim­il­ar fates. Dick An­der­son and I gave an ele­ment­ary proof that there is a homeo­morph­ism between Hil­bert space and the cartesian product of a count­able num­ber of lines. The ques­tion of wheth­er or not there is such a homeo­morph­ism had been of in­terest not only to to­po­lo­gists but also to many ana­lysts since Hil­bert space was in their arena. However, after the res­ult was proved, in­terest waned, and many who had pre­vi­ously seemed deeply in­ter­ested, now were will­ing to ac­cept the res­ult without caring to know why it is true.

A some­what sim­il­ar re­ac­tion met the dis­cov­ery that all 3-man­i­folds can be tri­an­gu­lated. Many to­po­lo­gists now ac­cept the res­ult without know­ing a proof that it is true. A com­plic­at­ing factor is that there are no known ele­ment­ary proofs. I have great re­spect for those that un­der­stand the basis of their own work.

I have long been in­ter­ested in met­riz­a­tion prob­lems. When I was a gradu­ate stu­dent at The Uni­versity of Texas, I worked on the prob­lem as to wheth­er or not every nor­mal Moore space is met­riz­able. This ques­tion had been called to Moise’s at­ten­tion when he was vis­it­ing a meet­ing of the Amer­ic­an Math­em­at­ic­al So­ci­ety in New York. I noted that an af­firm­at­ive an­swer could be giv­en by us­ing a res­ult of F. B. Jones. Jones, however, did not re­mem­ber how his ori­gin­al proof went and began to sus­pect that maybe it was false. Oth­ers took up the hunt for the solu­tion. On dif­fer­ent days we heard the ru­mor that vari­ous Moore stu­dents had either proved or dis­proved the the­or­em. In fact, Moore him­self let it be known that he had a solu­tion. An ex­am­in­a­tion of his archives in the Hu­man­it­ies Re­search Cen­ter at The Uni­versity of Texas showed that his write up had a gap.

I found many res­ults about nor­mal Moore spaces and fi­nally de­cided to pub­lish a pa­per telling what I knew, even though I had not got­ten a com­plete res­ult. For ex­ample, I pub­lished a pa­per which showed that a Moore space was met­riz­able if it was col­lec­tion-wise nor­mal. For those want­ing to know what this means I re­mark that a Moore space is a reg­u­lar Haus­dorff space which has a count­able fam­ily of open cov­er­ings \( G_1,G_2,\dots \) such as that if \( p \) is a point and \( N \) is a neigh­bor­hood of \( p \), then there is an in­teger \( n \), such that each ele­ment of \( G_n \) that con­tains \( p \) lies in \( N \). A space is col­lec­tion-wise nor­mal if for each dis­crete col­lec­tion of closed sets, there is a col­lec­tion of mu­tu­ally ex­clus­ive open sets such that each of the ele­ments in the dis­crete col­lec­tion lies in one and only one ele­ment of the col­lec­tion of open sets.

My pa­per con­tained the res­ult that a Moore space is met­riz­able if it is col­lec­tion-wise nor­mal. It also con­tained a ne­ces­sary and suf­fi­cient con­di­tion telling when a Haus­dorff space is met­riz­able. It turned out later that both Smirnoff from Rus­sia and Nagata from Ja­pan had got­ten sim­il­ar res­ults at about the same time. This sug­gests that when it is time for a new res­ult to be born, and that if one per­son does not get it, an­oth­er per­son is likely to. No re­search math­em­atician is in­dis­pens­ible for pro­gress in math­em­at­ic­al re­search.

A square has a con­vex met­ric; namely, if \( p \) and \( q \) are two points of a square, there is an­oth­er point of the square halfway between them. People have long been in­ter­ested in the ques­tion as to wheth­er or not each com­pact con­nec­ted, loc­ally con­nec­ted met­ric space has a con­vex met­ric. Karl Menger had been very in­ter­ested in this prob­lem. Gust­av Beer had what some people thought was a solu­tion; however, oth­ers found fault with his proof. He sub­mit­ted his pa­per to a journ­al many times and each time it came back with sug­ges­tions for cor­rec­tions. Menger told me that the de­cision to pub­lish the pa­per was made so that it would have a wider range of crit­ics. I ex­amined this pa­per and wheth­er or not it had er­rors, I did not know. But I did see that Gust­av Beer was con­sid­er­ing only the prob­lem where the space un­der con­sid­er­a­tion was one-di­men­sion­al. After study­ing Beer’s pa­per, I de­cided that in­deed a loc­ally con­nec­ted met­ric con­tinuum of any di­men­sion could be par­ti­tioned and this par­ti­tion could be used to show that the space had a con­vex met­ric. Moise ob­tained a re­lated res­ult at about the same time but his proof had a minor er­ror.

Al­though for many years I was quite in­ter­ested in the nor­mal Moore space prob­lem, I lost some of my in­terest when Co­hen proved that the con­tinuum hy­po­thes­is could not be proved on the basis of the Zer­melo-Fraen­kel ax­ioms. It was later proved that the nor­mal Moore space prob­lem could not be solved on the basis of these. It seems like there were two points of view. One was that something could be con­sidered as true if it could not be shown to be false even if in a lar­ger space it could be proved false. An­oth­er point of view is that a thing can be con­sidered as false un­less it can be proved to be true in the space un­der con­sid­er­a­tion. The no­tion of for­cing is re­lated to these ideas.

While we might find something that at present we do not re­cog­nize as an ax­iom, the above does show that on the basis of the ax­ioms that we do re­cog­nize, we are un­able to prove all the­or­ems in math­em­at­ics. In oth­er words, math­em­at­ics can­not be based on ax­ioms (as we know them) alone.

I will now tell of an idea I got as a boy that helped me in later re­search. I was as­signed the task of draw­ing a map. One way was to take a map, put it on the win­dow, place a piece of pa­per over it and then trace. I had more dif­fi­culty, however, when I was asked to draw a map of a dif­fer­ent size. However, I found that if I di­vided the page con­tain­ing the ori­gin­al map in­to rect­angles and then sim­il­arly di­vided the pa­per on which I was to draw the map, then I could make a reas­on­able copy of the map without re­sort­ing to the use of the win­dow. In fact, even if the pa­per on which the draw­ing was made was of a dif­fer­ent size from the ori­gin­al map and if one put the cor­res­pond­ing parts in­to the cor­res­pond­ing rect­angles, then one got a reas­on­able copy of the ori­gin­al map, es­pe­cially if the rect­angles were small. A mo­sa­ic copy of a pic­ture looks some­what like the ori­gin­al.

I noted that a per­son could take a map whose bound­ary lines were curved (or even squig­gly) and ap­prox­im­ate it with a map whose bound­ary lines were poly­gon­al; then if one did not ex­am­ine the changed map too care­fully, it re­sembled the ori­gin­al. This is the idea be­hind the fact that if one looks at a mo­sa­ic from a dis­tance, it looks like an ob­ject which is not poly­gon­al. Also, if one looks at a TV set, one does not see the dots, but a pic­ture.

Let me dis­cuss how I used this no­tion many years later. In the 1950’s, I learned that Moise had proved that if \( h \) is a homeo­morph­ism of a 3-cell in­to Eu­c­lidean 3-space, then this homeo­morph­ism could be ap­prox­im­ated by a piece­wise lin­ear homeo­morph­ism. This res­ult has many ap­plic­a­tions, and I felt that it was very im­port­ant. Read­ing Moise’s pa­pers proved very dif­fi­cult; in fact, I have heard that oth­ers had dif­fi­culty in read­ing these pa­pers, even the ref­er­ee whose name was Hu. Hu is said to have met with Moise fre­quently in or­der to have Moise ex­plain the proofs to him. There is some­what of a re­lated joke. Some people make the re­mark that Hu (who) ref­er­eed Moise’s pa­pers. It sounds as though they were ask­ing a ques­tion, if “Hu” was mis­taken to mean “who”. “Hu ref­er­eed Moise’s pa­per. Who? Hu! Who? Hu!”

The first step in Moise’s proof seems to show that if \( h \) is an em­bed­ding of the cartesian product of a 2-sphere \( \mathbb{S}^2 \) and an in­ter­val \( [0,1] \) in­to \( \mathbb{R}^3 \), then there is a poly­hed­ral 2-sphere in this im­age which sep­ar­ates the two bound­ary com­pon­ents of \( h(\mathbb{S}^2\times[0,1]) \). His proof goes something as fol­lows. Let \( X \) be the clos­ure of the bounded com­pon­ent of \( \mathbb{R}^3-h (\mathbb{S}^2\times\{\tfrac{1}{2}\} \). The first Vi­et­or­is ho­mo­logy of \( \mathrm{Bd}(X) \) is trivi­al. In a se­quence of steps Moise changed \( X \) to a poly­hed­ral 3-man­i­fold \( M \) with bound­ary so that \( \mathrm{Bd}(M) \) has trivi­al first Vi­et­or­is ho­mo­logy and sep­ar­ates \( h(\mathbb{S}^2\times\{0\}) \) from \( h(\mathbb{S}^2\times\{1\}) \) in \( \mathbb{R}^3 \). Then \( \mathrm{Bd}(M) \) is the re­quired 2-sphere. Us­ing this as a start, he showed that any homeo­morph­ism of a 3-cell in­to \( \mathbb{R}^3 \) could be ap­prox­im­ated by a piece­wise lin­ear homeo­morph­ism.

As I wrestled with Moise’s pa­per, I de­cided that there might be a dif­fer­ent way of prov­ing the same res­ult and re­called the tech­niques I had learned for copy­ing a map when I was a boy. Hence, I de­cided to tri­an­gu­late \( \mathbb{R}^3 \) and use the 3-sim­plexes as I had used the rect­angles. I re­garded \( h(\mathbb{S}^2\times\{\tfrac{1}{2}\}) \) as the bound­ar­ies of the states and pro­duced a poly­hed­ral 2-sphere ap­prox­im­at­ing \( h(\mathbb{S}^2\times\{\tfrac{1}{2}\}) \) by aping the meth­ods I used as a boy to copy a map. With this step I got the same res­ults that Moise did and was able to make the same ap­plic­a­tions that he did; namely, that a homeo­morph­ism of a 3-cell in­to \( \mathbb{R}^3 \) can be ap­prox­im­ated by a piece­wise lin­ear homeo­morph­ism and that any 3-man­i­fold can be tri­an­gu­lated. Later I was able to prove the side ap­prox­im­a­tion the­or­em and got even more res­ults. Many pa­pers res­ul­ted from the side ap­prox­im­a­tion the­or­em. I sus­pect that my res­ults on the ap­prox­im­a­tion the­or­em and the side ap­prox­im­a­tion the­or­em even have a lar­ger im­pact on 3-di­men­sion­al to­po­logy than any oth­er of the res­ults I have ob­tained in this area.

Collaboration with others

Let me make some re­marks about col­lab­or­a­tion with oth­er math­em­aticians. First, I would like to talk about col­lab­or­at­ing with one of my own stu­dents, Jim Kister. I men­tioned to him as I have to my oth­er stu­dents that it was the re­spons­ib­il­ity of my stu­dents to keep me in­formed of what was go­ing on as I be­came older. Kister and I were vis­it­ors at the In­sti­tute for Ad­vanced Study at the same time and we dis­cussed the em­bed­ding of ob­jects in Eu­c­lidean 3-space. I was pleased to learn that Kister had learned many things since I had him as a stu­dent, and we put some of these no­tions in­to a joint pa­per.

An­oth­er per­son with whom I wrote a joint pa­per was Ed Floyd. When I was vis­it­ing the Uni­versity of Vir­gin­ia and had Floyd as an of­fice mate, I dis­cussed with him par­ti­tion­ing of sets and how the no­tion of par­ti­tion­ing might be used to du­plic­ate some of the things we knew about man­i­folds. Now, the tri­an­gu­la­tion of a man­i­fold is a spe­cial ex­ample of a par­ti­tion­ing of a 3-man­i­fold. Floyd knew much more about man­i­folds than I did and as a res­ult of some ob­ser­va­tions of his, we were able to use the no­tion of par­ti­tion­ings and ex­pand them.

Bur­ton Jones and I also wrote a joint pa­per. Ac­tu­ally, this pa­per did not res­ult from joint work, but res­ul­ted from the fact that both Jones and I had got­ten some res­ults about the pseudo-arc of the same nature. It seemed in­ap­pro­pri­ate for Jones and me to write sep­ar­ate pa­pers cov­er­ing es­sen­tially the same top­ic. We then de­cided to write a joint pa­per with each of us writ­ing one part of it based on work we had done in­de­pend­ently.

Two joint pa­pers were writ­ten with Ka­rol Bor­suk. One of these res­ul­ted from a dis­cus­sion we had in Warsaw, Po­land, and the oth­er from dis­cus­sions we had when Bor­suk was a vis­it­or at the Uni­versity of Wis­con­sin. I was amazed by the amount of know­ledge that Borsu had about 3-space and the ded­ic­a­tion with which he worked at math­em­at­ics. I learned a great deal from Bor­suk — not only about math­em­at­ics but what de­vo­tion to math­em­at­ics really means.

Still an­oth­er per­son with whom I wrote a joint pa­per was Joe Mar­tin. We were both work­ing on the Poin­caré con­jec­ture and had fre­quent dis­cus­sions about it. Most of the work in our joint pa­pers was done in­de­pend­ently.

I have re­ceived much sat­is­fac­tion in writ­ing joint pa­pers with Dick An­der­son, Steve Ar­men­trout, Woody Bled­soe, Ka­rol Bor­suk, Mort Curtis, Ed Floyd, Bur­ton Jones, Jim Kister, An­drzej Kirkor, Vic Klee, Joe Mar­tin, Dan Mauld­in, Mike Star­bird, and oth­ers. Math­em­at­ics has been pro­duced and friend­ships strengthened. But, much of the joint work was cre­ated when part­ners were work­ing sep­ar­ately. In gen­er­al, I think bet­ter res­ults are ob­tained by those who stand on their own feet and work alone.

Teaching methods and related philosophy

Let me make a few re­marks about train­ing stu­dents and how my meth­ods of train­ing are sim­il­ar as well as dif­fer­ent from R. L. Moore’s. We both try to train stu­dents by hav­ing them do math­em­at­ics. Moore, however, would have his stu­dents work on things in which he was in­ter­ested — wheth­er or not these were of great in­terest in the math­em­at­ic­al com­munity. I have tried to lead my stu­dents to work on things that were of gen­er­al in­terest to oth­er math­em­aticians so that if they were to go to any one of the lead­ing math­em­at­ic­al cen­ters in the coun­try, they would find people with kindred in­terests.

My meth­od agreed with Moore’s in that I tried to get stu­dents to dis­cov­er their own thes­is rather than as­sign these top­ics to them. As we study pa­pers I fre­quently ask them, “Why did not the per­son writ­ing this pa­per prove more? What would hap­pen if one changed the hy­po­thes­is? Are there ex­amples that would show that the the­or­em is false if something is dropped?” If a per­son stud­ies a pa­per in this frame of mind, most good pa­pers will lead to oth­er re­search.

Also, like Moore, I tried to make all of the stu­dents in my classes par­ti­cip­ate in prov­ing the­or­ems. Those in­ept at prov­ing the­or­ems can at least have the ex­per­i­ence of try­ing and en­coun­ter­ing dif­fi­culties be­fore see­ing the proofs of oth­ers. They can learn (and per­haps ap­pre­ci­ate) defin­i­tions. However, I give great­er praise to those who can make ori­gin­al con­tri­bu­tions them­selves, who can dis­cov­er ex­amples on their own and un­earth new at­tacks or new the­or­ems. I re­call one un­usu­ally suc­cess­ful class of 40 stu­dents I taught at the Uni­versity of Wis­con­sin. The class was sup­posed to meet 3 times a week. I lec­tured to the stu­dents for one of the peri­ods and di­vided the class in­to the oth­er 2 peri­ods, let­ting half of them come one day to give their res­ults and the oth­er half come an­oth­er day for their res­ults. Sev­er­al mem­bers of this class went on to get Ph.D.’s and to pro­duce even more sig­ni­fic­ant res­ults. I can’t help but think that this tech­nique I used en­cour­aged them.

An­oth­er in­nov­a­tion I have used in teach­ing sem­inars, par­tic­u­larly when study­ing new res­ults, was to have the sem­in­ar meet for about 2 weeks and on the last day only those people who could bring a con­jec­ture re­lated to what had come be­fore were per­mit­ted to at­tend. The idea be­ing that those who were not able to come with a con­jec­ture had not been think­ing enough about what was go­ing on to mer­it be­ing per­mit­ted to at­tend the fi­nal con­jec­ture ses­sion.

I have a ba­sic feel­ing that math­em­at­ics is fun and should be fun for the par­ti­cipant. It does few people any good to force feed them in­to learn­ing things that they do not en­joy. They are un­likely to make much use of the ma­ter­i­al they learned — only hav­ing to do it in or­der to make a grade or meet a re­quire­ment.

At lower levels stu­dents will not be able to prove many the­or­ems, but they should be able to show what they have learned. We should be in­ter­ested in what they are do­ing and praise them for their pro­gress. I do not have much sym­pathy for pro­grams where we try to give a strong dose of math­em­at­ics to stu­dents who find men­tal activ­ity dis­taste­ful. What I do have great sym­pathy for are those who do not find math­em­at­ics easy. If I were to lim­it my friends to those who are good in math­em­at­ics, I would lose many of those that I have. I be­lieve we should strive to have stu­dents en­joy math­em­at­ics wheth­er they are gif­ted in it or not.

Let me make a few re­marks about teach­ing. I think that as we lec­ture to people, we should try to put ourselves in the po­s­i­tion of the audi­ence and try to see how its mem­bers view what we are say­ing. I like to be able to look in­to the eyes of my audi­ence so I can gauge listen­ers’ in­terest. There is no need to lec­ture to ourselves and go off and leave stu­dents. It seems that too many teach­ers are lec­tur­ing to them­selves.

Not just in lec­tur­ing but in talk­ing to people about math­em­at­ics and oth­er things, one should try to look at things from their point of view. I’ve heard that a per­son does not really un­der­stand an­oth­er’s prob­lems un­til they have stood in that per­son’s shoes. I re­call a word of wis­dom giv­en to me by a clean­ing wo­man work­ing at my home. She was a mem­ber of a minor­ity. I had heard an­oth­er mem­ber of that minor­ity say something that soun­ded very ri­dicu­lous — something that I thought even reas­on­able mem­bers of that minor­ity would not ac­cept, and dis­cussed this with her. She said this, “You should listen to what people feel rather than to what they say.” This is good ad­vice.