Celebratio Mathematica

Saunders Mac Lane

The Mac Lane Lecture

by John G. Thomson

Shortly after the death of Saun­ders Mac Lane in April, Krishna [Al­ladi] asked me if I would be will­ing to speak pub­licly about Saun­ders. I agreed to do so, but asked for time to think about and to pre­pare my re­marks. In the mean­time, Saun­ders’s auto­bi­o­graphy [1] has ap­peared, and it has been help­ful to me.

I ex­pect that every­one here is aware of the book and the movie “A beau­ti­ful mind” which ex­plore the life of John Nash. You will know that for many years, Nash was in­sane with schizo­phrenia. For most of us, and cer­tainly for me, in­san­ity is fright­en­ing and far from beau­ti­ful. I sub­mit that Saun­ders had a genu­inely beau­ti­ful mind. Ex­cept for an elite few of us, Mac Lane’s life and work do not have the drama and punch of Nash’s odys­sey. I see my note today as a re­cord­er, neither a ha­gi­o­graph­er nor a de­bunker.

Mac Lane’s men­tal world had great lu­cid­ity and covered much ter­rit­ory. He took math­em­at­ics ser­i­ously and he was un­swerving in the clar­ity with which he probed the struc­tures and con­cepts that sup­port lo­gic­al thought. I shall try to give some in­dic­a­tions of why I make such an as­ser­tion. His auto­bi­o­graphy makes a good case that my feel­ings about his mind are well foun­ded. There are in­tel­lec­tu­al and psy­cho­lo­gic­al nug­gets in his ac­count which mer­it ex­am­in­a­tion.

I can­not re­call the trans­ition mo­ment when I began to think of him as Saun­ders and ad­dress him by his first name. I would not have thought to ad­dress him as Saun­ders dur­ing the time he was my thes­is ad­visor. We met at my home around my for­ti­eth birth­day, and by that time it came eas­ily to me to call him Saun­ders. In dif­fer­ent peri­ods and places, the pro­gres­sion of an on­go­ing teach­er-stu­dent pair­ing takes dif­fer­ent paths from in­tens­ity to es­trange­ment. With Saun­ders and me, our math­em­at­ic­al paths crossed only briefly, but very in­tensely. Even though we di­verged pro­fes­sion­ally, we saw each oth­er so­cially though the years, and there still re­mained in my mind a sense of be­ing sheltered by him, which I think is one of the gifts that a good fath­er provides to his off­spring. I in­ter­ject here that my own fath­er also provided that shel­ter­ing sense to me.

Sev­er­al ob­it­u­ar­ies of Saun­ders ap­peared shortly after his death, and sev­er­al math­em­aticians were quoted. Here is a sampling:

He was one of the most im­port­ant fig­ures in the Uni­versity of Chica­go Math­em­at­ics De­part­ment, or in­deed in Amer­ic­an math­em­at­ics.

… ex­tremely en­er­get­ic, dy­nam­ic, clear-headed, opin­ion­ated, a ra­con­teur.

With Sammy Ei­len­berg he cre­ated a new way of think­ing about math­em­at­ics

Cat­egory the­ory is still ex­plod­ing in its in­flu­ence after 60 years.

Mac Lane was one of the pi­on­eers of al­geb­ra­ic to­po­logy.

He wrote fam­ous texts.

He has left a unique body of ma­ter­i­al for fu­ture his­tor­i­ans.

These snip­pets have been culled from com­ments of Eis­en­bud, May, Lawvere, and John­stone. Keep­ing in mind that, when per­son A makes a com­ment about per­son B, we may learn something about both A and B, these quotes carry quite a bit of in­form­a­tion.

Saun­ders was born in 1909, the eld­est of his par­ents’ four chil­dren. His pa­ternal grand­fath­er be­came a Pres­by­teri­an min­is­ter in Ohio, who resigned his po­s­i­tion there after be­ing charged with heresy for preach­ing evol­u­tion. He took up a po­s­i­tion as min­is­ter in a Con­greg­a­tion­al church in New Haven, Con­necti­c­ut. Saun­ders’s fath­er also was a min­is­ter. The fath­er fall­ing ill with tuber­cu­los­is and pos­sibly the after-ef­fects of the 1917 in­flu­enza epi­dem­ic, it was de­cided that Saun­ders, his broth­er and his moth­er take up res­id­ence with the grand­fath­er in Leo­m­in­ster, Mas­sachu­setts, while his fath­er tried to re­cu­per­ate in a san­it­ari­um. Con­cern­ing this peri­od in his life, Saun­ders wrote:

Dur­ing my high-school years, the Ku Klux Klan was act­ive and ap­par­ently crit­ic­al of Cath­ol­ic doc­trines. A Con­greg­a­tion­al church down­town re­ceived and praised Klan mem­bers with these views, and the ex­cess­ive at­tacks on Cath­oli­cism dis­pleased my grand­fath­er. He pro­ceeded to open his church on a week­day to give a two-hour lec­ture on the ori­gins of the Prot­est­ant/Cath­ol­ic di­vide, which was well at­ten­ded and im­press­ively done. I ad­mired his de­vo­tion and vig­or in de­fense of tol­er­ance, and marveled at his de­cis­ive activ­ity at nearly 80 years of age.

In this pas­sage, Saun­ders achieves at least two goals. He bows to the in­sight and sens­it­iv­it­ies of his re­mark­able grand­fath­er, and he es­tab­lishes that he was a teen­ager on the qui vive in­tel­lec­tu­ally. He con­tin­ues and qual­i­fies his ad­mir­a­tion of his grand­fath­er’s lec­tures:

This does not mean, however, that I un­der­stood his ideas. I did join his church, and in the pro­cess of do­ing so, re­mem­ber be­ing ques­tioned care­fully by an eld­erly mem­ber about my be­liefs. I answered ser­i­ously, but at the same time, kept some re­ser­va­tion about cer­tain points of doc­trine. I struggled with as­pects of my min­is­teri­al her­it­age, but I did not of­ten ap­proach my grand­fath­er for ad­vice and wis­dom. On one oc­ca­sion, I asked him about the pur­pose of in­di­vidu­al life. He re­spon­ded that we were there to ex­hib­it the glory of God; this con­clu­sion stopped me cold — God’s glory was not vis­ible to me.

This pas­sage is the only place in the auto­bi­o­graphy which men­tions Saun­ders’s re­li­gious be­liefs. His grand­fath­er’s life was roiled oc­ca­sion­ally, and Saun­ders’s reti­cence is an un­der­stand­able re­ac­tion. Giv­en his grand­fath­er’s pu­tat­ive heresy, Saun­ders may have taken the path so poignantly traced out in Mat­thew Arnold’s “Dover Beach.”

Saun­ders’s fath­er died in early 1924, when Saun­ders was 14. His sis­ter died at the age of 4, when Saun­ders was 7. These deaths con­trib­uted, I think, to the ser­i­ous­ness with which Saun­ders faced life.

All five of Saun­ders’s uncles went to Yale, as did both of his grand­fath­ers, so it was nat­ur­al for Saun­ders to go to Yale too. He in­cludes in his auto­bi­o­graphy a pho­to­graph of him­self around the time of his gradu­ation. This pho­to­graph ap­peared in the (Nor­walk) Times on Sunday, March 31, 1929, with the cap­tion: “At­tains highest aca­dem­ic stand­ing in the his­tory of Yale. Saun­ders Mac Lane of Nor­walk, who main­tained an Av­er­age of 384.8 in his stud­ies for the first two and a half years at the Uni­versity, for which he was honored by be­ing elec­ted Pres­id­ent of the New Haven Chapter of Phi Beta Kappa.”

Saun­ders closes his ac­count of the so­cial as­pects of his Yale years:

Em­phas­is on tra­di­tion was still power­ful at Yale, …. The fam­ous Tap Day came at the end of ju­ni­or year. All of the ju­ni­ors con­greg­ated in the old cam­pus hop­ing to be se­lec­ted while the seni­or mem­bers of the secret so­ci­et­ies walked among the ju­ni­ors tap­ping se­lec­ted can­did­ates and say­ing “Go to your room.” I stood there with all my class­mates. As a ju­ni­or Phi Theta Kappa I must have con­sidered I was worth a tap, but it nev­er ar­rived. I wept no tears — I had aimed my un­der­gradu­ate ca­reer in the ap­pro­pri­ate dir­ec­tion. I cared more for les­sons than for stu­dent cus­toms. As a sopho­more, I de­clined to join a fra­tern­ity; as a seni­or, I would not have made a loy­al mem­ber of a secret so­ci­ety. Make no bones about it, any edu­ca­tion in­volves choices, and a col­lege stu­dent must choose.

While Saun­ders was dis­missive of Tap Day at Yale, he was not at all dis­missive of the course work dur­ing his un­der­gradu­ate years, and his high grade point av­er­age re­flects his stu­di­ous­ness. His in­terest in phys­ics nev­er left him, but he found him­self drawn to math­em­at­ics more strongly, not however to the ex­clu­sion of oth­er in­tel­lec­tu­al in­terests.

The Saun­ders Mac Lane whom I came to know fledged at Yale, as the fol­low­ing state­ment shows:

It seemed as if all of our at­ten­tion was dir­ec­ted to­ward know­ledge that was already known; there­fore, dur­ing the first years of my un­der­gradu­ate edu­ca­tion, I put my own em­phas­is on ac­quir­ing uni­ver­sal know­ledge — the as­sim­il­a­tion and or­gan­iz­a­tion of everything known.

Had I come across ref­er­ences to “uni­ver­sal know­ledge” and “everything known” and not had the be­ne­fit of know­ing Saun­ders, I would have writ­ten him off as some sort of in­tel­lec­tu­al Wal­ter Mitty. Since I have some idea of the power of his mind, however, I think that a more real­ist­ic com­par­is­on would be to Leib­n­itz. Leib­n­itz’s mon­ads and Mac Lane’s cat­egor­ies may be viewed as at­tempts to reach some mean­ing­ful bed­rock. For me however, uni­ver­sal know­ledge is only ap­proach­able asymp­tot­ic­ally, and con­ver­gence is very slow.

Cer­tain in­tel­lects strive to ex­plore, unite, and most im­port­antly, to un­der­stand new ideas. Saun­ders was of this stripe. As a spe­cial case, he was ex­cited by the lec­tures of Øys­tein Ore at Yale, who gave an ac­count of Galois the­ory and groups. Ore had re­cently stud­ied at Emmy No­eth­er’s school of ab­stract al­gebra in Göt­tin­gen, and Saun­ders found him­self at­trac­ted to Ger­many.

I in­ter­ject here that Pro­fess­or Ore was sym­path­et­ic to me when I went to him as a fresh­man with a note I had writ­ten about prime num­bers. With his ad­vice, this note was pub­lished in the Amer­ic­an Math Monthly when I was a sopho­more.

Saun­ders de­cided to go to the Uni­versity of Chica­go for gradu­ate work, at least partly be­cause Hutchins urged him to do so. I find it fas­cin­at­ing, even thrill­ing, to note just how much in­tel­lec­tu­al growth Saun­ders main­tained for dec­ades. But he did not com­pletely ex­clude the softer arts dur­ing his Chica­go gradu­ate days:

I en­joyed vari­ous di­ver­sions at Chica­go: I learned to play bridge…, I vis­ited the Lyr­ic Op­era, but found Wag­n­er’s Rhine maid­ens too bux­om; and my friend Man­son Be­ne­dict and I in­ven­ted a game of three-di­men­sion­al chess. In the spring of 1931, I went along with Man­son Be­ne­dict and his date Dorothy Jones, a gradu­ate stu­dent in eco­nom­ics from Arkan­sas, to search for a com­mun­ist meet­ing in down­town Chica­go led by the sur­viv­ors of the Hay­mar­ket Mas­sacre of the late 1880s. We failed to find it, but I later dated Dorothy, as will ap­pear.

His first year of gradu­ate work at Chica­go was a let down:

Over­all, I found this year of gradu­ate work at Chica­go dis­ap­point­ing, es­pe­cially be­cause I could not see any pos­sib­il­ity for a Ph.D. thes­is on lo­gic. I was dis­ap­poin­ted with oth­er as­pects of math­em­at­ics at Chica­go as well…. Per­haps I was con­firm­ing Pro­fess­or Ore’s judg­ment on math­em­at­ics. At any rate, I wrote the In­sti­tute for In­ter­na­tion­al Edu­ca­tion, ap­plied for a fel­low­ship to study in Ger­many and won an award to do so….

My shift from Chica­go to Göt­tin­gen had a strong, pos­it­ive, mo­tiv­a­tion: I wanted to write a thes­is on lo­gic.

Saun­ders did in­deed write an ac­cep­ted thes­is on lo­gic in Göt­tin­gen. He and Dorothy Jones were also mar­ried in Ger­many. Both his mar­riage and his in­terest in lo­gic sur­vived for over 50 years. No­eth­er, Hil­bert, Weyl and oth­ers in­tro­duced Saun­ders to the vi­tal­ity of Ger­man math­em­at­ics, a vi­tal­ity which was de­cap­it­ated ab­ruptly when Hitler took over the Reich.

Saun­ders and Dorothy re­turned to the United States in 1933, and Saun­ders pur­sued postdoc­tor­al work at Yale for a year, and then went to Har­vard as a Pierce in­struct­or. He taught a course us­ing van der Waer­den’s text “Mod­ern al­gebra.”

In 1936, a res­ult of Saun­ders was pub­lished in a journ­al. This res­ult gives a set of gen­er­at­ors for the fun­da­ment­al group of a planar graph. He notes that he learned some to­po­logy from Whit­ney and some al­geb­ra­ic geo­metry with Walk­er. After stints at Cor­nell and Chica­go, Saun­ders re­turned to Har­vard and with Gar­rett Birk­hoff wrote and pub­lished “Sur­vey of mod­ern al­gebra,” a text with con­sid­er­able in­flu­ence over sev­er­al dec­ades. There is a brief re­mark by Stend­hal which to my mind il­lu­min­ates one of the vir­tues of the Birk­hoff–MacLane text­book.

In Stend­hal’s auto­bi­o­graphy “The life of Henri Brul­ard,” Stend­hal la­ments that he was un­able to un­der­stand why the product of two neg­at­ive num­bers is a pos­it­ive num­ber, and he lam­basts one of his teach­ers for spout­ing some non­sense about this “fact.” Once one has the ax­ioms for a ring, the equal­ity \( (-a) (-b) = ab \) be­comes an ex­er­cise, an ex­er­cise which ap­pears ex­pli­citly in Birk­hoff–MacLane. I men­tion this since the point which troubled Stend­hal also troubled me to the verge of tears in ju­ni­or high school, and was not re­solved un­til I read Birk­hoff–MacLane.

I have men­tioned tears and I re­cord that, dur­ing one of my weekly meet­ings with Mac Lane while writ­ing my thes­is, I broke down in sobs. The pre­cip­it­at­ing cause was no doubt a com­bin­a­tion of my aware­ness that I was not close to where I wanted to be, proof-wise, and Mac Lane’s all too clear re­cog­ni­tion of my vague­ness. My memory is that he was shocked to wit­ness tears eli­cited by his straight­for­ward re­marks.

Dur­ing World War II, Saun­ders and oth­er math­em­aticians were re­cruited to aid in the war ef­fort. Among oth­er prob­lems with a math­em­at­ic­al com­pon­ent, they ex­amined how to shoot down air­craft. Saun­ders men­tions the pur­suit curve, which in two di­men­sions is the curve traced by a point in the plane such that the tan­gent at each point is dir­ec­ted to­ward a second point in the plane which is mov­ing with con­stant ve­lo­city in a straight line. Sim­il­ar prob­lems of fire con­trol are men­tioned by Norbert Wien­er in his book on cy­ber­net­ics.

Saun­ders was some­what dis­missive of the de­gree to which doc­u­ments were clas­si­fied as secret by the gov­ern­ment, and could see no value in clas­si­fy­ing tri­go­no­met­ric tables. He states that the highest clas­si­fic­a­tion was “Burn after read­ing,” and he jokes that there may have been a yet high­er level, “Burn be­fore read­ing.” This an­ec­dote and oth­ers in­dic­ate that Saun­ders was not bowled over by au­thor­ity.

It is in­struct­ive to read Saun­ders’s at­ti­tude to­ward joint work.

Tra­di­tion­ally, most math­em­aticians con­duc­ted re­search by work­ing alone, as in the case of Gauss and Poin­caré and many oth­ers, em­in­ent or not….

In my own case, I came to real­ize that my re­search was hampered by a con­sid­er­able lack of broad know­ledge of math­em­at­ics.

It was some­what of a rev­el­a­tion to me to learn that Saun­ders on sev­er­al oc­ca­sions act­ively sought to col­lab­or­ate with oth­ers. Any­one with even a passing ac­quaint­ance with Saun­ders, however, will know of the fruit­ful and en­dur­ing col­lab­or­a­tion between Ei­len­berg and Mac Lane, which led to Ei­len­berg–MacLane spaces, and even more im­port­antly to cat­egory the­ory. Saun­ders dis­cusses sev­er­al as­pects of cat­egory the­ory, start­ing with the “well known map \( \theta \) send­ing each space \( V \) to its double dual \( V^{**} \).” As the earli­er quote of Lawvere that cat­egory the­ory is still “ex­plod­ing” em­phas­izes, MacLane’s in­flu­ence is widely ap­pre­ci­ated. Saun­ders de­scribes how Sammy Ei­len­berg dumped Lawvere’s thes­is lit­er­ally in his lap dur­ing a flight that they shared, and how Lawvere even­tu­ally con­vinced Saun­ders that it is pos­sible, per­haps ad­vis­able, to elim­in­ate the no­tion \( a\in S \) from the found­a­tions of set the­ory. Set the­ory without ele­ments ex­ists and, if I have it right, leads to topos the­ory.

I want to bring in Pres­id­ent Clin­ton’s big book “My life,” which I read last sum­mer rather care­fully. The num­ber of people named in this book is prodi­gious. Take me as a babe in the woods, but Pres­id­ent Clin­ton is a stel­lar net­work­er. Per­haps that is in the nature of polit­ics. There is a re­ser­va­tion in me, in­dic­at­ive of the more closed world in which I have moved, that there is a rep­re­hens­ib­il­ity in that ex­cess­ive out­reach. Saun­ders would not have agreed with me, for he was not only an act­ive col­lab­or­at­or, he was also a good net­work­er, to the be­ne­fit of math­em­at­ics. His en­thu­si­asm for good ideas and the people who have them was marked, and he did not hes­it­ate to ex­press him­self in lec­tures and pa­pers.

Saun­ders ad­mit­ted that he may have been too zeal­ous in pro­mot­ing re­form in the teach­ing of math­em­at­ics, and in his auto­bi­o­graphy he mocks him­self by men­tion­ing the kinder­garten child who is taught that a set con­sist­ing of ex­actly the two ele­ments \( a \) and \( b \) might be sym­bol­ized as \( \{a,b\} \). Johnny’s par­ents ask the kinder­garten teach­er how he is do­ing, and she tells them, “Yes, he un­der­stands sets, but he has trouble writ­ing the curly brack­ets.” I do not enter the con­ten­tious arena con­cern­ing the age at which a child can or should be in­tro­duced to set the­ory, but Saun­ders was in­trep­id and not at all a crack­pot. He puts in a good word for the two-column meth­od of teach­ing Eu­c­lidean geo­metry, but he was also con­vinced of the im­port­ance of ab­stract reas­on­ing.

Saun­ders was aware of his Scot­tish her­it­age, and he dressed ac­cord­ingly, trot­ting out the rel­ev­ant tartan from time to time. In ac­cord­ance with the ste­reo­type of the parsi­mo­ni­ous Scot, Saun­ders was tight-fis­ted about money. I can­not re­mem­ber the cir­cum­stances, but on one oc­ca­sion he re­marked that money is a mod­ule over the in­tegers.

On the few so­cial oc­ca­sions that I at­ten­ded at his apart­ment in Chica­go or at the dunes, he did not serve al­co­hol and, as far as I can tell, al­co­hol played no mean­ing­ful role in his life. I sus­pect that he had enough ex­per­i­ence of drink­ing to think that his mind was more in­ter­est­ing when sober than tipsy, a view which many would ques­tion about them­selves. With his well-fur­nished mind, it was surely true of Saun­ders.

Saun­ders en­joyed dogger­el, and he sprinkled it throughout his auto­bi­o­graphy. At math parties, he gave rendi­tions of the ditty about the zeta func­tion, “Where are the zer­oes of zeta of \( s \)?” His en­thu­si­asm was con­ta­gious.

My gradu­ate years at Chica­go brought me in con­tact with Mar­shall Stone who played such an im­port­ant role in build­ing a world-class math­em­at­ics de­part­ment. Saun­ders was a part of this stel­lar con­stel­la­tion and, in his auto­bi­o­graphy, he gives an en­tire chapter to this peri­od. I can do no bet­ter than to read it to you.

By good in­stincts and foresight, Mar­shall Stone suc­ceeded in cre­at­ing an in­nov­at­ive and in­spir­ing new de­part­ment of math­em­at­ics at Chica­go, de­vised by a happy com­bin­a­tion of an in­ter­na­tion­al seni­or fac­ulty, an am­bi­tious ju­ni­or fac­ulty, and an un­usu­ally lively group of gradu­ate stu­dents. For a peri­od, it was ar­gu­ably the best de­part­ment in the world.

The seni­or fac­ulty mem­bers were Ad­ri­an Al­bert, S. S. Chern, Mar­shall Stone, An­dré Weil, Ant­oni Zyg­mund and my­self. Al­bert was a hol­d­over from the old Chica­go de­part­ment, but Stone brought in Chern, me, Weil, and Zyg­mund, a re­mark­able quar­tet of seni­or ap­point­ments in such a short time that would have giv­en a big shot in the arm to any de­part­ment, and it cer­tainly did so to Chica­go.

Al­bert had star­ted as a stu­dent of Le­onard Dick­son in al­gebra; he was aware of the nature and troubles of the old Bliss de­part­ment and was happy to join Stone in the new dir­ec­tion. His re­search in­terests con­tin­ued some of Dick­son’s in group the­ory, lin­ear al­geb­ras, and nonas­so­ci­at­ive al­geb­ras.

Chern, from China, had stud­ied in Europe with Blasch­ke and Élie Cartan. He knew and un­der­stood the strength of Cartan’s use of dif­fer­en­tial forms in geo­metry.

Stone, as already noted, led in the use of Hil­bert-space meth­ods in ana­lys­is and phys­ics. The new de­part­ment was def­in­itely his design — he was, in ef­fect, a dic­tat­or, not just a chair­man.

Weil had grown up in the de­mand­ing and high-reach­ing world of French math­em­at­ics. He found de­cis­ive res­ults in al­geb­ra­ic geo­metry, and he and his young col­leagues in France had real­ized the im­port­ance of the ab­stract meth­ods newly used in Göt­tin­gen. They star­ted the in­flu­en­tial Bourbaki group in the am­bi­tious pro­ject of an ex­pos­i­tion of all of ba­sic math­em­at­ics.

Zyg­mund and his wife had es­caped Po­land at the start of the war. Between the wars, Pol­ish math­em­aticians es­tab­lished a lively school of math­em­at­ics, de­lib­er­ately re­strict­ing their re­search in­terests to chosen top­ics in to­po­logy, lo­gic, and ana­lys­is. Zyg­mund was a de­voted spe­cial­ist in ana­lys­is, par­tic­u­larly in har­mon­ic ana­lys­is. He was equally de­voted to his many stu­dents. Oth­er refugee math­em­aticians from Po­land dis­played a sim­il­ar de­vo­tion to their own spe­cial­ties.

Stone’s ju­ni­or fac­ulty, who all began as as­sist­ant pro­fess­ors, con­sisted of Paul Hal­mos, Irving Ka­plansky, Irving Segal and Ed­win Span­i­er. Hal­mos, a Hun­gari­an, was a protégé of von Neu­mann, and an en­thu­si­ast­ic ex­pos­it­or on vec­tor space the­ory. Ka­plansky, my first Ph.D. stu­dent, had met Al­bert dur­ing the war, and ac­tu­ally had come to Chica­go on Al­bert’s ini­ti­at­ive a year be­fore Stone ar­rived. Segal, an­oth­er von Neu­mann protégé, stud­ied at Prin­ceton and taught briefly at Har­vard. Span­i­er, a re­cent Prin­ceton stu­dent in to­po­logy, had learned the flour­ish­ing new ideas of al­geb­ra­ic to­po­logy from Steen­rod.

At the start of the Stone Age, Otto Schilling was still in Chica­go; his math­em­at­ic­al in­terests in al­gebra had been in­ter­rup­ted by leg­al prob­lems in­volved in a troubled fin­an­cial in­her­it­ance from Ger­many — my col­lab­or­a­tion with him sadly came to an end. Even­tu­ally, he moved to Purdue Uni­versity. A few years later, I. N. Her­stein, an al­geb­ra­ist, joined the de­part­ment, and sev­er­al of the young­er math­em­aticians from the Bliss–Lane de­part­ment left. Pro­fess­or Lawrence Graves re­tired, as did H. S. Ever­ett, who had ef­fect­ively man­aged cor­res­pond­ence courses in math­em­at­ics.

With the start of the Stone de­part­ment, many new, able gradu­ate stu­dents ar­rived in Chica­go. A num­ber of them were sup­por­ted by the G.I. Bill, which provided ad­equate sti­pends, and many stu­dents came be­cause they had used the Chica­go cor­res­pond­ence courses pre­pared by Pro­fess­or Ever­ett dur­ing their war ser­vice. Oth­er stu­dents in Urb­ana or New York City heard of ex­cit­ing things go­ing on in Chica­go, and so came to study there, and there were also sev­er­al fel­lows who came from Europe to vis­it Chica­go. One Brit­ish vis­it­or re­turned to Eng­land to build up a Chica­go-style de­part­ment in War­wick.

As chair­man, Stone was the lead­er and man­ager; he was force­ful and ju­di­cious — per­haps grow­ing up as a son of a chief justice of the Su­preme Court helped. The story goes that at one point, he was busy in his of­fices while a stu­dent, Bert Kostant, waited pa­tiently at the door. Pro­fess­or Weil came to see Stone on some mat­ter, opened the of­fice door, and walked in, whereupon Stone poin­ted out that the stu­dent had been there first.

Stone wholly re­or­gan­ized the gradu­ate pro­gram in math­em­at­ics; pre­vi­ously, the fi­nal Ph.D. ex­am­in­a­tion form­ally covered all 27 gradu­ate courses the stu­dent had taken. I re­call one ex­ample where a stu­dent had taken all 27 courses, and fi­nally, in ab­sen­tia, fin­ished his thes­is and came back for his fi­nal ex­am. One of his courses had been al­geb­ra­ic to­po­logy, and the ex­am­iner asked the stu­dent for the defin­i­tion of a cov­er­ing space; at first, the stu­dent was stumped, but on fur­ther ques­tion­ing he came up with an ex­ample, the line and the circle. The ex­am­iner probed fur­ther to ask which covered which, and the stu­dent thought the circle covered the line. Dis­aster! (The solu­tion is to wind the line again and again around the circle, which makes the line cov­er the circle.)

In the re­or­gan­ized Ph.D. pro­gram, the fi­nal Ph.D. or­al ex­am no longer probed all the courses the stu­dent had taken. In­stead, it stuck to the thes­is and re­lated things, and was pretty much a form­al­ity, since the stu­dent had already shown his or her breadth of know­ledge in the new Ph.D. qual­i­fy­ing or­als. Un­der this new setup, the situ­ation de­scribed above might not have happened and the stu­dent would have passed.

In the Stone de­part­ment, there was a whole new se­quence of re­quired gradu­ate courses. After the mas­ter’s courses, there were no fur­ther re­quired courses for the Ph.D.; the stu­dent chose ad­vanced post-Mas­ter’s courses, took qual­i­fy­ing ex­ams, and com­pleted his or her re­search. In many cases, fac­ulty pre­pared mi­meo­graphed notes for the new Mas­ter’s courses: Zyg­mund pre­pared notes on ana­lys­is; oth­ers pre­pared notes on set the­ory and met­ric spaces, as well as sev­er­al al­tern­at­ive sets of notes on point-set to­po­logy (one of which was mine). The lat­ter notes re­cog­nized that to­po­logy had a cent­ral po­s­i­tion, com­par­able to that once held by vari­able the­ory. The Stone–Wei­er­strass the­or­em made a vi­tal ap­pear­ance there. Weil com­plained that the clas­sic­al defin­i­tion of de­term­in­ants should be re­placed by a defin­i­tion mak­ing some use of Grass­man al­geb­ras, which I presen­ted in some lec­ture notes. The Mas­ter’s pro­gram rep­res­en­ted a new, sys­tem­at­ic view of math­em­at­ics.

This new mod­el of em­phas­is cropped up all over, and there were new ad­vanced courses. Weil’s book on the found­a­tion of al­geb­ra­ic geo­metry ap­peared in the Col­loqui­um Series of the AMS. He lec­tured on the sub­ject; I read and listened, and in my read­ing, I noted that this ver­sion of al­geb­ra­ic geo­metry could make con­sid­er­able use of cat­egory the­ory, an ob­ser­va­tion that I re­gret­tably did not fol­low up (later, Grothen­dieck de­veloped the same ob­ser­va­tion). Weil also con­duc­ted a sem­in­ar on cur­rent lit­er­at­ure, in­spired by an older Parisi­an sem­in­ar taught by Hadam­ard. The prin­ciple of the sem­in­ar was that each stu­dent should re­port on a cur­rent pa­per not in his field of spe­cialty; the stu­dent re­ports would be open to cri­ti­cism. It is in­ter­est­ing that Weil saved his most dev­ast­at­ing com­ments for those stu­dents he knew to be the ablest.

Zyg­mund taught ana­lys­is and en­cour­aged many not­able stu­dents, such as Al­berto Calder­ón and Eli­as Stein. He im­pressed upon all of them the im­port­ance of har­mon­ic ana­lys­is. Chern taught dif­fer­en­tial geo­metry us­ing dif­fer­en­tial forms, of course. Is­ad­ore Sing­er and oth­er stu­dents learned from this ap­proach, which was not yet read­ily avail­able in texts. It is re­por­ted that at one time a stu­dent told Weil that he (the stu­dent) did not un­der­stand these dif­fer­en­tial forms, to which Weil re­spon­ded by go­ing to the black­board and writ­ing down the Greek let­ter omega (\( \omega \)); I sup­posed he in­ten­ded that this stand­ard sym­bol for a dif­fer­en­tial form might re­call the idea. Span­i­er and I al­tern­ately taught al­geb­ra­ic to­po­logy; in par­tic­u­lar, I struggled with the still-mys­ter­i­ous prop­er­ties of those spec­tral se­quences found by Leray and now used by all to­po­lo­gists.

Math­em­at­ic­al dis­cus­sions con­tin­ued in the cor­ridor and at the daily tea. For ex­ample, I re­call try­ing to per­suade some stu­dents at tea of the lo­gic­al ex­ist­ence of a max­im­al at­las for any dif­fer­en­tial man­i­fold — lo­gic was not, as be­fore and af­ter­wards, isol­ated from math­em­at­ics. In this hot­house at­mo­sphere, ideas and proofs were prom­in­ent and many gradu­ate stu­dents flour­ished, in­clud­ing Is­ad­ore Sing­er, Bert Kostant, Richard Kadis­on, John Thompson, and many oth­ers.

As an il­lus­tra­tion, I will de­scribe Thompson’s work: He had been an out­stand­ing un­der­gradu­ate in math­em­at­ics at Yale, and the repu­ta­tion of the new de­part­ment prob­ably at­trac­ted him to Chica­go for gradu­ate study. Shortly after his ar­rival, I wanted a chance to do something dif­fer­ent, and de­cided that group the­ory was due for a re­viv­al; hence, I taught a two-quarter course on group the­ory. By the end of the second quarter, in which I had ten­ded to em­phas­ize in­fin­ite groups, I had es­sen­tially come to the end of my know­ledge on group the­ory. Thompson, who was in the course, came to me to say that he wished to write a thes­is on group the­ory. I en­cour­aged him, but did not trouble to say that my own know­ledge of the sub­ject was some­what lim­ited. But not to worry — I ar­ranged for em­in­ent the­or­ists, such as Richard Brauer, Re­in­hold Baer and Mar­shall Hall, to vis­it Chica­go. Each Sat­urday morn­ing, I listened to Thompson tell me what he had been up to with groups; the sub­ject fit­ted his in­terest, and he chose his own prob­lems.

At the time, An­dré Weil was lec­tur­ing in group rep­res­ent­a­tions with par­tic­u­lar at­ten­tion to the so-called Che­val­ley groups. One of Weil’s stu­dents heard what Thompson was up to, and took him aside to tell him that he should not be study­ing fi­nite groups alone, but should look at rep­res­ent­a­tions as well. However, Thompson per­sisted, and turned out a Ph.D. thes­is that settled an out­stand­ing prob­lem of fi­nite groups, a con­struc­tion of a nor­mal \( p \)-com­ple­ment for cer­tain fi­nite groups.

Ad­ri­an Al­bert was also in­ter­ested in fi­nite group the­ory, a sub­ject act­ive at Chica­go from the very be­gin­ning in 1892, and found spe­cial fund­ing to or­gan­ize a spe­cial year on group the­ory at Chica­go. Thompson and Wal­ter Feit, a re­cent Ph.D. of Richard Brauer at Michigan, were two of the par­ti­cipants; dur­ing the year, they solved a fam­ous prob­lem by prov­ing the Odd Or­der The­or­em: A fi­nite simple group that is not cyc­lic can­not be of odd or­der, which was a fam­ous con­jec­ture of Burn­side. This res­ult was the start­ing point of a ma­jor ef­fort to clas­si­fy all fi­nite simple groups.

Group the­ory, as here ex­em­pli­fied, was just one of the top­ics that flour­ished dur­ing the Stone Age at Chica­go. The people in­volved — the stu­dents and pro­fess­ors — were ex­cited by the pur­suit of new ideas in math­em­at­ics, which was, in all, a happy com­bin­a­tion of tal­ent, cir­cum­stance, and lead­er­ship.

Saun­ders has giv­en us an ac­count of his life and work in this auto­bi­o­graphy, and through his pub­lic­a­tions, his stu­dents, and his col­lab­or­at­ors, he has left a splen­did, in­flu­en­tial re­cord. Thank you.


[1]S. Mac Lane: Saun­ders Mac Lane: A math­em­at­ic­al auto­bi­o­graphy. A K Peters (Welles­ley, MA), 2005. With a pre­face by Dav­id Eis­en­bud. MR 2141000 Zbl 1089.​01010 book