Celebratio Mathematica

Marshall Harvey Stone

Reminiscences of mathematics at Chicago

by Marshall H. Stone

In 1946 I moved to the Uni­versity of Chica­go. An im­port­ant reas­on for this move was the op­por­tun­ity to par­ti­cip­ate in the re­hab­il­it­a­tion of a math­em­at­ics de­part­ment that had once had a bril­liant role in Amer­ic­an math­em­at­ics but had suffered a de­cline, ac­cel­er­ated by World War II. Dur­ing the war the activ­ity of the de­part­ment fell to a low level and its ranks were de­pleted by re­tire­ments and resig­na­tions. The ad­min­is­tra­tion may have wel­comed some of these changes, be­cause they re­moved per­sons who had op­posed some of its policies. Be that as it may, the uni­versity re­solved at the close of the war to re­build the de­part­ment.

The de­cision may have been in­flu­enced by the plans to cre­ate new in­sti­tutes of phys­ics, me­tal­lurgy, and bio­logy on found­a­tions laid by the uni­versity’s role in the Man­hat­tan Pro­ject. Pres­id­ent Hutchins had seized the op­por­tun­ity of re­tain­ing many of the atom­ic sci­ent­ists brought to Chica­go by this pro­ject, and had suc­ceeded in mak­ing a series of bril­liant ap­point­ments in phys­ics, chem­istry, and re­lated fields. Something sim­il­ar clearly needed to be done when the uni­versity star­ted filling the va­can­cies that had ac­cu­mu­lated in math­em­at­ics. Pro­fess­ors Dick­son, Bliss, and Logs­don had all re­tired fairly re­cently, and Pro­fess­ors W. T. Re­id and Sanger had resigned to take po­s­i­tions else­where. The five va­can­cies that had res­ul­ted offered a splen­did chal­lenge to any­one mind­ful of Chica­go’s great con­tri­bu­tion in the past and de­sirous of en­sur­ing its con­tinu­ation in the fu­ture.

When the Uni­versity of Chica­go was foun­ded un­der the pres­id­ency of Wil­li­am Rainey Harp­er at the end of the nine­teenth cen­tury, math­em­at­ics was en­cour­aged and vig­or­ously sup­por­ted. Un­der the lead­er­ship of Eliakim Hast­ings Moore, Bolza, and Masch­ke it quickly be­came a bril­liant cen­ter of math­em­at­ic­al study and re­search. Among its early stu­dents were such math­em­aticians as Le­onard Dick­son, Os­wald Veblen, George Birk­hoff, and R. L. Moore, destined to fu­ture po­s­i­tions of lead­er­ship in re­search and teach­ing. Some of these stu­dents re­mained at Chica­go as mem­bers of the fac­ulty. Messrs. Dick­son, Bliss, Lane, Re­id, and Mag­nus Hestenes were among them.

Al­gebra, func­tion­al ana­lys­is, cal­cu­lus of vari­ations, and pro­ject­ive, dif­fer­en­tial geo­metry were fields in which Chica­go ob­tained spe­cial dis­tinc­tion. With the pas­sage of time, re­tire­ments and new ap­point­ments had brought a much in­creased em­phas­is on the cal­cu­lus of vari­ations and a cer­tain tend­ency to in-breed­ing. When such out­stand­ing math­em­aticians as E. H. Moore or Wil­czyn­ski, a bril­liant pi­on­eer in pro­ject­ive dif­fer­en­tial geo­metry, re­tired from the de­part­ment, re­place­ments of com­par­able abil­ity were not found. Thus in 1945 the situ­ation was ripe for a re­viv­al.

A second, and per­haps even more im­port­ant, reas­on for the move to Chica­go was my con­vic­tion that the time was also ripe for a fun­da­ment­al re­vi­sion of gradu­ate and un­der­gradu­ate math­em­at­ic­al edu­ca­tion.

The in­vit­a­tion to Chica­go con­fron­ted me with a very dif­fi­cult ques­tion: “Could the elab­or­a­tion of a mod­ern­ized cur­riculum be car­ried out more suc­cess­fully at Har­vard or at Chica­go?”

When Pres­id­ent Hutchins in­vited me to vis­it the uni­versity in the sum­mer of 1945, it was with the pur­pose of in­ter­view­ing me as a pos­sible can­did­ate for the dean­ship of the Di­vi­sion of Phys­ic­al Sci­ences. After two or three days of con­fer­ences with de­part­ment heads, I was called to Mr. Hutchins’s res­id­ence, where he an­nounced that he would of­fer me not the dean­ship but a dis­tin­guished ser­vice pro­fess­or­ship in the De­part­ment of Math­em­at­ics.

The ne­go­ti­ations over this of­fer oc­cu­pied nearly a year, dur­ing which I sought the an­swer to the ques­tion with which it con­fron­ted me. It soon be­came clear that the situ­ation at Har­vard was not ripe for the kind of change to which I hoped to ded­ic­ate my en­er­gies in the dec­ade fol­low­ing the war. However, it was by no means clear that cir­cum­stances would be any more pro­pi­tious at Chica­go than they seemed to be at Har­vard. In con­sult­ing some of my friends and col­leagues, I was ad­vised by the more as­tute among them to come to a clear un­der­stand­ing with the Chica­go ad­min­is­tra­tion con­cern­ing its in­ten­tions.

There are those who be­lieve that I went to Chica­go to ex­ecute plans that the ad­min­is­tra­tion there already had in mind. Noth­ing could be fur­ther from the truth. In fact, my ne­go­ti­ations were dir­ec­ted to­ward de­vel­op­ing de­tailed plans for re­viv­ing the Chica­go De­part­ment of Math­em­at­ics and ob­tain­ing some kind of com­mit­ment from the ad­min­is­tra­tion to im­ple­ment them. Some of the best ad­vice giv­en me con­firmed my own in­stinct that I should not join the Uni­versity of Chica­go un­less I were made chair­man of the de­part­ment and thus giv­en some meas­ure of au­thor­ity over its de­vel­op­ment. Earli­er ex­per­i­ences had taught me that ad­min­is­trat­ive prom­ises of whole­hearted in­terest in aca­dem­ic im­prove­ments were too of­ten un­trust­worthy. I there­fore asked the Uni­versity of Chica­go to com­mit it­self to the de­vel­op­ment pro­gram that was un­der dis­cus­sion, at least to the ex­tent of of­fer­ing me the chair­man­ship.

This cre­ated a prob­lem for the uni­versity, as the de­part­ment had to be con­sul­ted about the mat­ter, and re­spon­ded by vot­ing un­an­im­ously that Pro­fess­or Lane should be re­tained in the of­fice. As I was un­will­ing to move merely on the basis of a prom­ise to ap­point me to the chair­man­ship at some later time, the ad­min­is­tra­tion was brought around to ar­ran­ging the ap­point­ment, and I to ac­cept it. Mr. Lane, a very fine gen­tle­man in every sense of the word, nev­er showed any re­sent­ment. Neither of us ever re­ferred to the mat­ter, and he served as an act­ive and very loy­al mem­ber of the de­part­ment un­til he re­tired sev­er­al years later. I was very grate­ful to him for the grace and self­less­ness he dis­played in cir­cum­stances that might have jus­ti­fied a quite dif­fer­ent at­ti­tude.

Even though the uni­versity made no spe­cif­ic de­tailed com­mit­ments to es­tab­lish the pro­gram I had ready to ac­cept the chair­man­ship as an earn­est of forth­com­ing sup­port. I felt con­fid­ent that with some show of firm­ness on my part the pro­gram could be es­tab­lished. In this op­tim­ist­ic spir­it I de­cided to go to Chica­go, des­pite the very gen­er­ous terms on which Har­vard wished to re­tain me.

Re­gard­less of what, many seem to be­lieve, re­build­ing the Chica­go De­part­ment of Math­em­at­ics was an up­hill fight all the way. The uni­versity was not about to im­ple­ment the plans I had pro­posed in our ne­go­ti­ations without res­ist­ing and rais­ing ob­jec­tions at every step. The de­part­ment’s loy­alty to Mr. Lane had the for­tu­nate con­sequence for me that I felt re­leased from any form­al ob­lig­a­tion to sub­mit my re­com­mend­a­tions to the de­part­ment for ap­prov­al. Al­though I con­sul­ted my col­leagues on oc­ca­sion, I be­came an auto­crat in mak­ing my re­com­mend­a­tions. I like to think that I am not by nature an auto­crat, and that the later years of my chair­man­ship provided evid­ence of this be­lief. At the be­gin­ning, however, I took a strong line in what I was do­ing in or­der to make the de­part­ment a truly great one.

The first re­com­mend­a­tion sent up to the ad­min­is­tra­tion was to of­fer an ap­point­ment to Hassler Whit­ney. The sug­ges­tion was promptly re­jec­ted by Mr. Hutchins’s second in com­mand. It took some time to per­suade the ad­min­is­tra­tion to re­verse this ac­tion and to make an of­fer to Pro­fess­or Whit­ney. When the of­fer was made, he de­clined it, and re­mained at Har­vard for a short time be­fore mov­ing to the In­sti­tute for Ad­vanced Study.

The next of­fer I had in mind was one to An­dré Weil. He was a some­what con­tro­ver­sial per­son­al­ity, and I found a good deal of hes­it­a­tion, if not re­luct­ance, on the part of the ad­min­is­tra­tion to ac­cept my re­com­mend­a­tion.

In fact, while the re­com­mend­a­tion even­tu­ally re­ceived fa­vor­able treat­ment in prin­ciple, the ad­min­is­tra­tion made its of­fer with a sub­stan­tial re­duc­tion in the salary that had been pro­posed. I was forced to ad­vise Pro­fess­or Weil, who was then in Brazil, that the of­fer was not ac­cept­able. When he de­clined the of­fer, I was in a po­s­i­tion to take the mat­ter up at the highest level. Though I had to go to an 8 A.M. ap­point­ment suf­fer­ing from a fairly high fever, in or­der to dis­cuss the ap­point­ment with Mr. Hutchins, I was re­war­ded by his will­ing­ness to re­new the of­fer on the terms I had ori­gin­ally pro­posed. Pro­fess­or Weil’s ac­cept­ance of the im­proved of­fer was an im­port­ant event in the his­tory of the Uni­versity of Chica­go and in the his­tory of Amer­ic­an math­em­at­ics.

My con­ver­sa­tion with Mr. Hutchins brought me an un­ex­pec­ted bo­nus. At its con­clu­sion he turned to me and asked, “When shall we in­vite Mr. Mac Lane?” I was happy to be able to reply, “Mr. Hutchins, I have been dis­cuss­ing the pos­sib­il­ity with Saun­ders and be­lieve that he would give fa­vor­able con­sid­er­a­tion to a good of­fer whenev­er you are ready to make it.” That of­fer was made soon af­ter­ward and was ac­cep­ted. There were oth­er ap­point­ments, such as that of Pro­fess­or Zyg­mund, that also went smoothly, but what would hap­pen in any par­tic­u­lar case was al­ways un­pre­dict­able.

Hand-to-mouth budgeting

One ex­plan­a­tion doubt­less was to be found in the uni­versity’s hand-to-mouth prac­tices in budget­ing. This would ap­pear to have been the reas­on why one even­ing I was giv­en in­dir­ect as­sur­ances from Mr. Hutchins that S. S. Chern would be offered a pro­fess­or­ship, only to be in­formed by Vice-Pres­id­ent Har­ris­on the next morn­ing that the of­fer would not be made. Such cas­u­al, not to say ar­bit­rary, treat­ment of a cru­cial re­com­mend­a­tion nat­ur­ally evoked a strong protest. In the pres­ence of the dean of the Di­vi­sion of Phys­ic­al Sci­ences I told Mr. Har­ris­on that if the ap­point­ment were not made, I would not be a can­did­ate for re­appoint­ment as chair­man when my three-year term ex­pired. Some of my col­leagues who were in­formed of the situ­ation called on the dean a few hours later to as­so­ci­ate them­selves with this protest. Hap­pily, the protest was suc­cess­ful, the of­fer was made to Pro­fess­or Chern, and he ac­cep­ted it. This was the stormi­est in­cid­ent in a stormy peri­od. For­tu­nately the peri­od was a fairly short one, and at the roughest times Mr. Hutchins al­ways backed me un­re­servedly.

As soon as the de­part­ment had been brought up to strength by this series of new ap­point­ments, we could turn our at­ten­tion to a thor­ough study of the cur­riculum and the re­quire­ments for high­er de­grees in math­em­at­ics. The group that was about to un­der­take the task of re­design­ing the de­part­ment’s work was mag­ni­fi­cently equipped for what it had to do. It in­cluded, in al­pha­bet­ic­al or­der, Ad­ri­an Al­bert, R. W. Barn­ard, Lawrence Graves, Paul Hal­mos, Mag­nus Hestenes, Irving Ka­plansky, J. L. Kel­ley, E. P. Lane, Saun­ders Mac Lane, Otto Schilling, Irving Segal, M. H. Stone, An­dré Weil, and Ant­oni Zyg­mund. Among them were great math­em­aticians, and great teach­ers, and lead­ing spe­cial­ists in al­most every branch of pure math­em­at­ics. Some were new to the uni­versity, oth­ers fa­mil­i­ar with its his­tory and tra­di­tions. We were all re­solved to make Chica­go the lead­ing cen­ter in math­em­at­ic­al re­search and edu­ca­tion it had al­ways as­pired to be. We had to bring great pa­tience and open minds to the time-con­sum­ing dis­cus­sions that ranged from gen­er­al prin­ciples to de­tailed math­em­at­ic­al ques­tions. The pres­ence of a sep­ar­ate and quite in­de­pend­ent col­lege math­em­at­ics staff did not re­lieve us of the ob­lig­a­tion to es­tab­lish a new un­der­gradu­ate cur­riculum be­side the new gradu­ate pro­gram.

Two aims on which we came to early agree­ment were to make course re­quire­ments more flex­ible and to lim­it ex­am­in­a­tions and oth­er re­quired tasks to those hav­ing some edu­ca­tion­al value.

This stream­lined pro­gram of stud­ies, the un­usu­al dis­tinc­tion of the math­em­at­ics fac­ulty, and a rich of­fer­ing of courses and sem­inars have at­trac­ted many very prom­ising young math­em­aticians to the Uni­versity of Chica­go ever since the late 1940s. The suc­cess­ful co­ordin­a­tion of these factors was re­in­forced by the con­cen­tra­tion of all de­part­ment­al activ­it­ies in Eck­hart Hall with its of­fices (for fac­ulty and gradu­ate stu­dents), classrooms, and lib­rary. As most mem­bers of the de­part­ment lived near the uni­versity and gen­er­ally spent their days in Eck­hart, close con­tact between fac­ulty and stu­dents was eas­ily es­tab­lished and main­tained. (This had been fore­seen and planned for by Pro­fess­or G. A. Bliss when he counseled the ar­chi­tect en­gaged to build Eck­hart Hall.) It was one of the reas­ons why the math­em­at­ic­al life at Chica­go be­came so spon­tan­eous and in­tense. By help­ing cre­ate con­di­tions so fa­vor­able for such math­em­at­ic­al activ­ity, Pro­fess­or Bliss earned the etern­al grat­it­ude of his uni­versity and his de­part­ment. Any­one who reads the roster of Chica­go doc­tor­ates since the later 1940s can­not but be im­pressed by the prom­in­ence and in­flu­ence many of them have en­joyed in Amer­ic­an — in­deed in world — math­em­at­ics. It is prob­ably fair to cred­it the Chica­go pro­gram with an im­port­ant role in stim­u­lat­ing and guid­ing the de­vel­op­ment of these math­em­aticians dur­ing a cru­cial phase of their ca­reers.

As I have de­scribed it, the Chica­go pro­gram made one con­spicu­ous omis­sion — it provided no place for ap­plied math­em­at­ics. Dur­ing my cor­res­pond­ence of 1945–46 with the Chica­go ad­min­is­tra­tion I had in­sisted that ap­plied math­em­at­ics should be a con­cern of the de­part­ment, and I had out­lined plans for ex­pand­ing the de­part­ment by adding four po­s­i­tions for pro­fess­ors of ap­plied sub­jects. I had also hoped that it would be pos­sible to bring about closer co­oper­a­tion than had ex­is­ted in the past between the De­part­ments of Math­em­at­ics and Phys­ics.

Cir­cum­stances were un­fa­vor­able. The uni­versity felt little pres­sure to in­crease its of­fer­ings in ap­plied math­em­at­ics. It had no en­gin­eer­ing school, and rather re­cently had even re­jec­ted a be­quest that would have en­dowed one. Sev­er­al of its sci­entif­ic de­part­ments offered courses in the ap­plic­a­tions of math­em­at­ics to spe­cif­ic fields such as bio­logy, chem­istry, and met­eor­o­logy. The De­part­ment of Phys­ics and the Fermi In­sti­tute had already worked out an en­tirely new pro­gram in phys­ics and were in no mood to modi­fy it in the light of sub­sequent changes that might take place in the Math­em­at­ics De­part­ment.

However, many stu­dents of phys­ics elec­ted ad­vanced math­em­at­ics courses of po­ten­tial in­terest for them — for ex­ample, those deal­ing with Hil­bert space or op­er­at­or the­ory, sub­jects prom­in­ently rep­res­en­ted among the spe­cial­ties cul­tiv­ated in the Math­em­at­ics De­part­ment.

On the oth­er hand, there was pres­sure for the cre­ation of a De­part­ment of Stat­ist­ics, ex­er­ted par­tic­u­larly by the eco­nom­ists of the Cowles Found­a­tion. A com­mit­tee was ap­poin­ted to make re­com­mend­a­tions to the ad­min­is­tra­tion for the fu­ture of stat­ist­ics with Pro­fess­or Al­len Wal­lis, Pro­fess­or Tjalling Koop­mans, and my­self as mem­bers. Its re­port led to the cre­ation of a Com­mit­tee on Stat­ist­ics, Mr. Hutchins be­ing firmly op­posed to the pro­lif­er­a­tion of de­part­ments.

The com­mit­tee en­joyed powers of ap­point­ment and even­tu­ally of re­com­mend­a­tion for high­er de­grees. It was housed in Eck­hart and de­veloped in­form­al ties with the De­part­ment of Math­em­at­ics.

At a some­what later time a sim­il­ar com­mit­tee was set up to bring the in­struc­tion in ap­plied math­em­at­ics in­to fo­cus by co­ordin­at­ing the courses offered in sev­er­al dif­fer­ent de­part­ments and even­tu­ally re­com­mend­ing high­er de­grees.

Long be­fore that, however, the De­part­ment of Math­em­at­ics had soun­ded out the dean of the di­vi­sion, a phys­i­cist, about the pos­sib­il­ity of a joint ap­point­ment for Free­man Dys­on, a young Eng­lish phys­i­cist then vis­it­ing the United States on a re­search grant. We had in­vited him to Chica­go for lec­tures on some bril­liant work in num­ber the­ory that had marked him as a math­em­atician of un­usu­al tal­ent. We were im­pressed by his lec­tures and real­ized that he was well qual­i­fied to es­tab­lish a much needed link between the two de­part­ments.

However, Dean Zachari­asen quickly stifled our ini­ti­at­ive with a simple ques­tion, “Who is Dys­on ?” Dys­on soon be­came a per­man­ent mem­ber of the In­sti­tute of Ad­vanced Study.

By 1952 I real­ized that it was time for the De­part­ment of Math­em­at­ics to be led by someone whose moves the ad­min­is­tra­tion had not learned to pre­dict. It was also time for the de­part­ment to in­crease its ma­ter­i­al sup­port by en­ter­ing in­to re­search con­tracts with the gov­ern­ment.

For­tu­nately there were sev­er­al col­leagues who were more than qual­i­fied to take over. The two most con­spicu­ous were Saun­ders Mac Lane and Ad­ri­an Al­bert. The choice fell first on Pro­fess­or Mac Lane, who served for the next six years.

Un­der the strong lead­er­ship of these two gif­ted math­em­aticians and their young­er suc­cessors the de­part­ment ex­per­i­enced many changes, but flour­ished migh­tily and was able to main­tain its ac­know­ledged po­s­i­tion at the top of Amer­ic­an math­em­at­ics.