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Celebratio Mathematica

Marshall Harvey Stone

Marhsall Harvey Stone

by George W. Mackey

Mar­shall Stone made ma­jor con­tri­bu­tions to func­tion­al ana­lys­is that both shaped the field and have guided it for many dec­ades. His first ma­jor con­tri­bu­tion was to the spec­tral the­ory of un­boun­ded self-ad­joint op­er­at­ors on Hil­bert space. The spec­tral the­ory of bounded self-ad­joint op­er­at­ors had been es­tab­lished nearly two dec­ades earli­er by the work of Hil­bert, Hellinger and Hahn, but that work ex­cluded the case when the op­er­at­or was un­boun­ded. Un­boun­ded op­er­at­ors provide some of the most im­port­ant ex­amples and po­ten­tial ap­plic­a­tions of spec­tral the­ory, such as or­din­ary or par­tial dif­fer­en­tial op­er­at­ors, and in par­tic­u­lar the dif­fer­en­tial op­er­at­ors that ap­peared in the then new quantum mech­an­ics. Stone had worked on ei­gen­func­tion ex­pan­sions (i.e., a form of spec­tral the­ory) for or­din­ary dif­fer­en­tial op­er­at­ors in his dis­ser­ta­tion and in his early work up to 1928, so he was primed to at­tack this spec­tral the­ory prob­lem in gen­er­al. He did so start­ing in 1929 and in grand style, first an­noun­cing his res­ults in three notes in the Pro­ceed­ings of the Na­tion­al Academy of Sci­ence [4], [5] and then pub­lish­ing de­tails in a ma­gis­teri­al 600 page book [7] that has re­mained a clas­sic in the sub­ject. The third of these PNAS notes is worthy of spe­cial no­tice as the res­ults here are of par­tic­u­lar im­port­ance for both clas­sic­al and quantum mech­an­ics and were left out of the 1932 book solely for lack of space. This pa­per con­tains a the­or­em on one para­met­er groups of unit­ary op­er­at­ors, known sub­sequently as Stone’s the­or­em, as well as a the­or­em of Stone that was also in­de­pend­ently proved by von Neu­mann [e4], which as­serts the uni­city of a fi­nite set of self-ad­joint un­boun­ded op­er­at­ors sat­is­fy­ing the clas­sic­al com­mut­a­tion re­la­tions of quantum mech­an­ics and act­ing ir­re­du­cibly. This res­ult is uni­ver­sally known now as the Stone–von Neu­mann the­or­em.

After his work on un­boun­ded op­er­at­ors Stone turned his at­ten­tion to Boolean al­geb­ras and an­nounced his new res­ults in two PNAS notes and [9], with the de­tails ap­pear­ing in three lengthy pa­pers [10], [11], [12]. These pa­pers re­vo­lu­tion­ized the study of Boolean al­geb­ras and made the sub­ject part of func­tion­al ana­lys­is; for in­stance Stone showed that any Boolean al­gebra could be real­ized as an al­gebra of sub­sets of its max­im­al ideal space, now called the Stone space. These pa­pers also con­tained im­port­ant ap­plic­a­tions to to­po­lo­gic­al spaces in­clud­ing what is now called the Stone–Wei­er­strass the­orm, which is a vast gen­er­al­iz­a­tion of the 19th cen­tury the­or­em of Wei­er­strass on poly­no­mi­al ap­prox­im­a­tion. The Stone–Wei­er­strass the­or­em is one of the ba­sic tools of func­tion­al ana­lys­is. His res­ults also in­clude ana­lys­is of the dif­fer­ent ways in which a to­po­lo­gic­al space may be be em­bed­ded in a com­pact space (or be com­pac­ti­fied) and to what is now called the Stone–Cech com­pac­ti­fic­a­tion. Two more notes in the PNAS [13], [14] ad­dress the sub­ject of spec­tra in gen­er­al, and which in­clude both his the­ory of self-ad­joint op­er­at­ors and his the­ory of Boolean al­geb­ras. After the war, Stone turned his at­ten­tion to a form of ax­io­mat­ic in­teg­ra­tion and pub­lished four notes in the PNAS [16], [17], [15], [20]. In ad­di­tion Stone pub­lished a num­ber of ex­pos­it­ory art­icles, in­ter­pret­ing and ex­plain­ing many of his pre­vi­ous res­ults. He also turned his in­terests to think­ing and writ­ing about the nature and philo­sophy of math­em­at­ics and to the teach­ing of math­em­at­ics. [21] and [22] are prime ex­amples of his thought­ful and in­sight­ful think­ing and writ­ing on these sub­jects.

Mar­shall Har­vey Stone was born in New York City on April 8, 1903 to Har­lan Fiske Stone and Ag­nes Har­vey Stone. Mar­shall had a young­er broth­er, Lauson Har­vey Stone born in 1904, who be­came a law­yer. Har­lan Stone was a prom­in­ent law­yer, leg­al edu­cat­or, and jur­ist. Born in Chester­field, New Hamp­shire, Har­lan grew up on his fam­ily’s farm near Am­h­erst, Mas­sachu­setts, with all the usu­al boy­hood farm chores. The fam­ily had moved to Am­h­erst in or­der to im­prove the edu­ca­tion­al op­por­tun­it­ies for their chil­dren. Har­lan came from a long line of New Eng­land fore­bears, the first be­ing Si­mon Stone, a Pur­it­an who emig­rated from Eng­land and settled in the Mas­sachu­setts Bay Colony in 1635. As Ma­son [e10] put it, Har­lan “nev­er doubted that the rugged qual­it­ies of the Yan­kee hill farm­er were born and bred in him.”

Har­lan at­ten­ded Am­h­erst pub­lic schools and then entered Am­h­erst Col­lege where he ex­celled, gradu­at­ing Phi Beta Kappa in 1894, serving as class pres­id­ent for three years, and as a stal­wart on the foot­ball team. After a year of teach­ing in high school, he de­cided on a ca­reer in law and won ad­mis­sion to the Columbia School of Law, sup­port­ing him­self by part-time teach­ing in his­tory at Ad­elphi Academy in Brook­lyn. He did well in law school and entered law prac­tice in New York City in 1898. Shortly af­ter­ward he was offered a part-time teach­ing po­s­i­tion at Columbia School of Law, and in 1910 he was offered and ac­cep­ted the po­s­i­tion of Dean of the Columbia School of Law. He held this po­s­i­tion un­til 1923 when he resigned and re­turned to the full time prac­tice of law as head of lit­ig­a­tion for the lead­ing Wall Street firm of Sul­li­van and Crom­well. But not for long, as his friend from col­lege days at Am­h­erst, Pres­id­ent Calv­in Coolidge, ap­poin­ted him in 1924 as At­tor­ney Gen­er­al of the United States. Stone was seen as a man of un­im­peach­able in­teg­rity, and was giv­en the job of clean­ing up the scan­dals in the gov­ern­ment from the pre­vi­ous ad­min­is­tra­tion. His nom­in­a­tion was greeted with both re­lief and ap­prov­al, and the New York Times ed­it­or­i­al­ized that “In char­ac­ter, in leg­al at­tain­ments, in ap­prov­al of his brethren of the law, and phys­ic­al vig­or and abil­ity to cope with the severe labor which will be laid upon in the De­part­ment of Justice, Mr. Stone is un­usu­ally well qual­i­fied.” Then in 1925, Coolidge ap­poin­ted him as As­so­ci­ate Justice of the Su­preme Court, where dur­ing the 30s, Stone to­geth­er with Justices Bran­de­is and Holmes, (and later with Holmes’ re­place­ment, Justice Car­doza) made up the lib­er­al wing of the Court. In 1941 Pres­id­ent Roosevelt ap­poin­ted Stone, a life-long re­pub­lic­an, as Chief Justice of the Court, a po­s­i­tion he held un­til his death in 1946 [e10], [e14].

Ag­nes Har­vey Stone was also born in Chester­field, New Hamp­shire, com­ing from a sim­il­ar back­ground as Har­lan. The Stone and Har­vey fam­il­ies were good friends and of­ten vis­ited each oth­er even after the Stones moved to Am­h­erst. Har­lan and Ag­nes were child­hood com­pan­ions and be­came child­hood sweet­hearts, mar­ry­ing in 1899. In ad­di­tion to her re­spons­ib­il­it­ies as wife and moth­er, Ag­nes was an ac­com­plished paint­er, spe­cial­iz­ing in wa­ter­col­or land­scapes. She had a num­ber of shows of her work in­clud­ing at the Corcor­an Gal­lery in Wash­ing­ton D.C. which in 1937, staged a show fea­tur­ing 24 of her works, and staged a second show in 1941 fea­tur­ing 30 of her works. Fi­nally in 1945 the Vir­gin­ia Mu­seum of Fine Art staged a show of sixty-four of her works [e10].

While Har­lan Stone was based in New York, the fam­ily lived in Engle­wood, New Jer­sey, and Mar­shall and his young­er broth­er Lauson at­ten­ded the loc­al pub­lic schools. The fam­ily also built a sum­mer home on the Isle au Haut, an is­land in Pen­ob­scot Bay in Maine. The fam­ily en­joyed their sum­mer va­ca­tions there and found it very re­lax­ing.

As a young boy, Mar­shall of­ten ac­com­pan­ied his fath­er to Columbia Uni­versity sit­ting quietly and ob­serving his fath­er in of­fice and classroom. He and his broth­er had house­hold chores as­signed by their fath­er per­formed for “mod­est” re­mu­ner­a­tion. Mar­shall was a vig­or­ous and healthy lad, play­ing foot­ball, base­ball and ten­nis, and sail­ing small boats at their sum­mer house in Maine [e12]. In 1919 at the age of only 16, Mar­shall entered Har­vard Uni­versity from which he was gradu­ated, summa cum laude, in 1922.

Al­though it had been as­sumed that Mar­shall would fol­low his fath­er in­to the law, a grow­ing fas­cin­a­tion with math­em­at­ics led to an ex­traordin­ary ar­range­ment in which he spent the aca­dem­ic year 1922–1923 as a part-time in­struct­or at Har­vard to find out wheth­er he liked teach­ing. It turned out that he did and he pro­ceeded quickly to write a Ph.D. thes­is un­der the dir­ec­tion of G. D. Birk­hoff. The de­gree was awar­ded in 1926 but the work was com­pleted rather earli­er. The very dis­tin­guished math­em­at­ic­al ca­reer of Mar­shall Stone was un­der way. Be­fore set­tling down at Har­vard for the thir­teen-year peri­od 1933–1946 Stone held a vari­ety of po­s­i­tions. He was at Columbia from 1925 to 1927, at Har­vard from 1927 to 1931, at Yale from 1931 to 1933 and at Stan­ford for the sum­mer of 1933. He be­came a full pro­fess­or at Har­vard in 1937. These early years were enorm­ously fruit­ful ones for Stone’s ca­reer as a re­search math­em­atician — so much so that he was elec­ted a mem­ber of the Na­tion­al Academy of Sci­ences in 1938 at the un­usu­ally early age of 35.

His first pa­per was a short note on nor­mal or­tho­gon­al sets of func­tions pub­lished in 1924 and by 1928 had pub­lished a num­ber of ad­di­tion­al pa­pers on vari­ous as­pects of the the­ory of or­tho­gon­al ex­pan­sions — spe­cial em­phas­is be­ing placed on ex­pan­sions in terms of ei­gen­func­tions of lin­ear dif­fer­en­tial op­er­at­ors. See for in­stance [1], [2], [3]. This was one of the prin­cip­al in­terests of G. D. Birk­hoff and Stone’s work was in the same tra­di­tion. Then in 1929 he began to work on the ab­stract the­ory of pos­sibly un­boun­ded self-ad­joint op­er­at­ors in Hil­bert space, an­noun­cing his res­ults with three notes pub­lished in the Pro­ceed­ings of the Na­tion­al Academy of Sci­ences [4], [4], [5]. This work cul­min­ated in a six hun­dred page book which is now one of the great clas­sics of twen­ti­eth cen­tury math­em­at­ics. It was en­titled “Lin­ear trans­form­a­tions in Hil­bert space and their ap­plic­a­tions to ana­lys­is”. This com­pre­hens­ive and beau­ti­fully writ­ten book [7] has been enorm­ously in­flu­en­tial. Mod­ern func­tion­al (or ab­stract) ana­lys­is began with the ideas of Vol­terra on “func­tion­als” in the late nine­teenth cen­tury and was trans­formed and giv­en con­sid­er­able im­petus in the first two dec­ades of the twen­ti­eth cen­tury by the work of Hil­bert and F. Riesz. The very dif­fer­ent book of Banach [e6] to­geth­er with Stone’s book set the stage for the ex­tens­ive de­vel­op­ments since that time. Stone’s fath­er, who was now on the Su­preme Court, was very proud of his son’s achieve­ments in math­em­at­ics, and one of his law clerks re­por­ted that the justice al­ways kept a copy of his son’s book on his desk in the Su­preme Court [e12].

In his in­tro­duc­tion Stone freely ac­know­ledges his sci­entif­ic debt to J. von Neu­mann. Von Neu­mann pub­lished a long pa­per on the same sub­ject [e3], and it is not easy to dis­en­tangle their re­spect­ive con­tri­bu­tions. What is clear is that Stone was ori­gin­ally stim­u­lated by pre­lim­in­ary work of von Neu­mann but had many key ideas quite in­de­pend­ently. Moreover the whole last half of the book, in­clud­ing the chapter on spec­tral mul­ti­pli­city the­ory and the ex­tens­ive ap­plic­a­tions to dif­fer­en­tial and in­teg­ral op­er­at­ors, has no coun­ter­part in von Neu­mann’s writ­ings. The cent­ral point of the work of both men was the ex­ten­sion of Hil­bert’s spec­tral the­or­em from bounded to un­boun­ded op­er­at­ors. This ex­ten­sion was made ne­ces­sary by the prob­lem of mak­ing math­em­at­ic­ally co­her­ent sense of the newly dis­covered re­fine­ment of clas­sic­al mech­an­ics known as quantum mech­an­ics. Here an im­port­ant part of the prob­lem was dis­cov­er­ing the ”cor­rect” defin­i­tion of self-ad­joint­ness for un­boun­ded op­er­at­ors. This cor­rect defin­i­tion is rather del­ic­ate and the ex­ten­sion of the older the­ory of Hil­bert and oth­ers was a ma­jor task.

The last of the three notes men­tioned above was en­titled ”Lin­ear trans­form­a­tions in Hil­bert space III. Op­er­a­tion­al meth­ods and group the­ory”. The ma­ter­i­al it sum­mar­ized was ori­gin­ally meant to be in­cluded as an ex­tra chapter of the book but was omit­ted for reas­ons of space. The two the­or­ems it an­nounces are of suf­fi­cient im­port­ance to be dis­cussed here in some de­tail.

Three years earli­er, in 1927, Her­mann Weyl [e1] and Eu­gene Wign­er [e2] had in­tro­duced group the­or­et­ic­al meth­ods in­to the new quantum mech­an­ics in quite dif­fer­ent ways. Weyl’s idea was to use group the­ory to help cla­ri­fy the found­a­tions. His pa­per, writ­ten in phys­i­cists’ lan­guage, im­pli­citly con­jec­tured two the­or­ems about one-para­met­er groups of unit­ary op­er­at­ors in Hil­bert space. Stone’s note states these con­jec­tures as care­fully for­mu­lated the­or­ems, an­nounces that he is in pos­ses­sion of proofs and gives some in­dic­a­tion of their nature. See [6] for his proof and also [e7]. Both the­or­ems were not only im­port­ant for quantum mech­an­ics, in the man­ner in­dic­ated by Weyl, but were also highly sig­ni­fic­ant early steps in the then nas­cent unit­ary rep­res­ent­a­tion the­ory of non com­pact loc­ally com­pact groups. One of them also played an im­port­ant role in the chain of events lead­ing through a note of B. O. Koop­man [e5] to the er­god­ic the­or­ems of von Neu­mann and Birk­hoff and on to mod­ern er­god­ic the­ory.

At this point it is use­ful to dis­tin­guish between two ver­sions of one of Stone’s the­or­ems. The ver­sion sug­ges­ted by Weyl’s pa­per (and which stim­u­lated Koop­man) as­serts that for every one para­met­er unit­ary group \( t \leq Ut \) there is a unique self ad­joint op­er­at­or \( H \) such that \( Ut = e^{l H t} \). The ver­sion em­phas­ized in Stone’s note is an ana­logue of the spec­tral the­or­em for one para­met­er unit­ary groups. This ver­sion has the great ad­vant­age that it can be gen­er­al­ized al­most ver­batim to ar­bit­rary (sep­ar­able) loc­ally com­pact com­mut­at­ive groups.

The oth­er the­or­em — the cel­eb­rated Stone–von Neu­mann unique­ness the­or­em — states the unique­ness of the ir­re­du­cible solu­tions of the Heis­en­berg com­mut­a­tion re­la­tions in in­teg­rated form see; also [e7]. This res­ult may be in­ter­preted as giv­ing a com­plete de­term­in­a­tion of all ir­re­du­cible rep­res­ent­a­tion of a cer­tain non-com­pact non-com­mut­at­ive loc­ally com­pact group — now well known as the Heis­en­berg group. So in­ter­preted, it is the first ex­ample of such a de­term­in­a­tion by about a dec­ade. Fi­nally a series of nat­ur­al gen­er­al­iz­a­tions of the Stone–von Neu­mann unique­ness the­or­em cul­min­ated in the im­prim­it­iv­ity the­or­em and the ex­ten­sion of the no­tion of in­duced rep­res­ent­a­tion from fi­nite groups to gen­er­al loc­ally com­pact groups.

Shortly there­after Stone pub­lished two more notes in the Pro­ceed­ings of the Na­tion­al Academy [8], [9] which seemed at first to rep­res­ent a com­pletely new de­par­ture. They were en­titled “Boolean al­geb­ras and their ap­plic­a­tions to to­po­logy” and ”Sub­sump­tion of Boolean al­geb­ras un­der the the­ory of rings”. Ac­tu­ally, just as Stone’s work on spec­tral the­ory may be re­garded as a nat­ur­al out­growth of his earli­er work on con­crete ei­gen­func­tion ex­pan­sions, so can his work on Boolean al­geb­ras be re­garded as a nat­ur­al out­growth of his work on spec­tral the­ory. This is be­cause of the role played in spec­tral the­ory by Boolean al­geb­ras of pro­jec­tions. In an en­tirely char­ac­ter­ist­ic at­tempt to get to the bot­tom of things Stone un­der­took a thor­oughgo­ing study of Boolean al­geb­ras and made a num­ber of far reach­ing dis­cov­er­ies re­lat­ing Boolean al­geb­ras to gen­er­al to­po­logy on the one hand and to the the­ory of rings and ideals on the oth­er.

The dis­cov­ery of these con­nec­tions has had sig­ni­fic­ant con­sequences for all three sub­jects. One beau­ti­ful res­ult is the cel­eb­rated Stone–Wei­er­strass the­or­em vastly gen­er­al­iz­ing the the­or­em of Wei­er­strass con­cern­ing the uni­form ap­prox­im­ab­il­ity of ar­bit­rary con­tinu­ous func­tions on a fi­nite in­ter­val by poly­no­mi­als. An­oth­er is the nat­ur­al one to one cor­res­pond­ence between all com­pact Haus­dorff spaces on the one hand and cer­tain rings on the oth­er. Stone’s stud­ies of the re­la­tion­ship between com­pact spaces and rings of con­tinu­ous func­tions an­ti­cip­ated im­port­ant ele­ments in the mod­ern the­ory of com­mut­at­ive Banach al­geb­ras. An­oth­er noted res­ult was what be­came known as the Stone–Cech com­pac­ti­fic­a­tion of a to­po­lo­gic­al space that was dis­covered in­de­pend­ently by Stone and Eduard Cech in 1937 [e8]. The de­tailed de­vel­op­ment of Stone’s ideas on Boolean al­geb­ras, and gen­er­al to­po­logy. were pub­lished in three lengthy pa­pers [10], [11], [12], re­spect­ively. Ap­plic­a­tions of these new ideas to spec­tral the­ory were an­nounced in notes pub­lished in [13] and [14], with some de­tails ap­pear­ing in [18].

Soon after the entry of the United States in­to World War II the char­ac­ter of Stone’s work un­der­went a con­sid­er­able change. For sev­er­al years he was en­gaged in secret work for the U.S. gov­ern­ment, serving as a con­sult­ant for the U.S. Navy, 1942–43 and then in the of­fice of the Chief of Staff of the Army, 1943–45. As part of his du­ties, he was sent on clas­si­fied mis­sions to China, Burma, and In­dia as well as the European theat­er.

The year after the end of the war he resigned his po­s­i­tion at Har­vard to take on the chair­man­ship of the math­em­at­ics de­part­ment at the Uni­versity of Chica­go. This once great de­part­ment had been de­clin­ing in qual­ity and Stone’s mis­sion was to strengthen it and bring it up to its former stature. In this he suc­ceeded ad­mir­ably. Be­fore very long it was re­garded by many as the best math­em­at­ics de­part­ment in the coun­try and, while a po­s­i­tion like that is hard to keep in­def­in­itely, it has re­mained one of the strongest de­part­ments ever since. He brought in An­dré Weil, S. S.  Chern, Saun­ders Mac Lane and, a num­ber of prom­ising young­er men. Moreover in the words of one of the lat­ter “Mar­shall de­voted him­self with both in­tens­ity and breadth — from the largest is­sues to the smal­lest de­tails — to the De­part­ment’s wel­fare and de­vel­op­ment”. In 1952, Stone turned the chair­man­ship of the Chica­go de­part­ment over to Saun­ders Mac Lane but con­tin­ued to be oc­cu­pied with ad­min­is­trat­ive mat­ters. He was a strong force in rees­tab­lish­ing the In­ter­na­tion­al Math­em­at­ic­al Uni­on — was much in­volved in the draft­ing of its con­sti­tu­tion and served as its pres­id­ent from 1952 to 1954. He also in­ter­ested him­self act­ively in the prob­lems of math­em­at­ic­al and sci­entif­ic teach­ing — es­pe­cially at the in­ter­na­tion­al level — and served on vari­ous boards and com­mis­sions, in­clud­ing as pres­id­ent of the In­ter­na­tion­al Com­mit­tee on Math­em­at­ic­al In­struc­tion from 1961 to 1967.

While his vari­ous ad­min­is­trat­ive con­cerns and activ­it­ies pre­ven­ted him from work­ing on math­em­at­ic­al prob­lems with his former in­tens­ity Stone con­tin­ued to work and to pub­lish res­ults rather stead­ily un­til the early 1960s. In par­tic­u­lar one should note his four pa­pers on in­teg­ra­tion the­ory [16], [18], [15], [20] in which he de­vel­ops an ax­io­mat­ic the­ory of in­teg­ra­tion, build­ing on the earli­er work of Dani­ell, and par­al­lel­ing the ax­io­mat­ic the­ory of in­teg­ra­tion that Bourbaki was soon to pub­lish. At the same time his math­em­at­ic­al in­terests ten­ded more and more to­ward the elu­cid­a­tion of ques­tions of great gen­er­al­ity and pro­fund­ity about the true nature of math­em­at­ics and math­em­at­ic­al con­cepts, for in­stance [16], [18], [19]. His pa­pers [21] and [22] ar­tic­u­late his views on the struc­ture and nature math­em­at­ics and his thoughts and re­com­mend­a­tions on math­em­at­ics edu­ca­tion, and also dis­play his splen­did com­mand of Eng­lish prose. His re­view [23] of Con­stance Re­id’s bio­graphy of Cour­ant is a de­light to read. In­deed, there is reas­on to be­lieve that his pub­lic­a­tions of the last thirty years give a very in­com­plete pic­ture of his math­em­at­ic­al activ­ity. At a con­fer­ence in hon­or of his second re­tire­ment (see be­low) Stone gave a re­mark­able two-hour lec­ture out­lining his rather un­usu­al and ori­gin­al views on the nature and struc­ture of math­em­at­ics.

Stone re­mained at the Uni­versity of Chica­go un­til he re­tired as Pro­fess­or Emer­it­us in 1968. At this time there was a week-long con­fer­ence in his hon­or the pro­ceed­ings of which were pub­lished by Spring­er-Ver­lag [sic] with Fe­lix Browder as the ed­it­or [e11]. He did not wish to stop teach­ing however and forth­with began a new ca­reer as George Dav­id Birk­hoff pro­fess­or of math­em­at­ics at the Uni­versity of Mas­sachu­setts in Am­h­erst. No doubt the fact that Am­h­erst, Mas­sachu­setts had been his fath­er’s child­hood home ad­ded to the at­tract­ive­ness of this move. He taught there for the next twelve years and among oth­er activ­it­ies su­per­vised two Ph.D. theses. Dur­ing his fi­nal year he was honored with a second re­tire­ment con­fer­ence.

One of the many strik­ing ac­com­plish­ments of Mar­shall Stone was a truly ex­traordin­ary com­mand of the Eng­lish lan­guage. This gave a spe­cial fla­vor to his book, his math­em­at­ic­al pa­pers and his many writ­ings on oth­er sub­jects. His skill with the Eng­lish lan­guage also mani­fes­ted it­self in his lec­tures, which were mod­els of clar­ity and or­gan­iz­a­tion.

Stone was only mod­er­ately act­ive in su­per­vising Ph.D. theses. In­deed there are an­ec­dotes about his re­luct­ance to do this sort of teach­ing. On the oth­er hand, he did turn out a re­spect­able num­ber of new Ph.D.s and in­flu­enced many oth­er young math­em­aticians by his writ­ings and through in­form­al per­son­al con­tact. Ac­cord­ing to the Math­em­at­ics Gene­a­logy Pro­ject, Stone had 14 doc­tor­al stu­dents, and cur­rently has 1455 math­em­at­ic­al des­cend­ants.

Stone mar­ried young (in 1927), and he and his first wife Emmy raised three daugh­ters. Dor­is, Cyn­thia, and Phoebe. Re­ports have it that he was ser­i­ous about fath­er­hood in a rather old fash­ioned way. His daugh­ters had reg­u­lar chores to do and in fin­an­cial mat­ters were kept on strict al­low­ances. Like his fath­er, Stone was a New Englander, and one of his daugh­ters wrote that “… his so­cial and polit­ic­al at­ti­tudes, his per­son­al­ity, all re­flect his new Eng­land her­it­age, and one be­comes aware of how deeply his roots are im­bed­ded in this part of the world when we are to­geth­er on the is­land (the Isle au Haut)” [e12]. On the oth­er hand he also be­lieved in fam­ily fun and one of his many side in­terests was gour­met cook­ery and gour­met din­ing. His daugh­ters re­call many del­ic­acies that he con­cocted. This mar­riage dis­solved in di­vorce in 1962 but Stone soon re­mar­ried. His second mar­riage, to Ravi­jojla Kost­ic, las­ted the rest of his life. His step­daugh­ter Svet­lana Kost­ic-Stone also sur­vived him.

Of all Stone’s many in­terests his love of travel was surely dom­in­ant. He began to travel when he was quite young and was on a trip to In­dia when he died. He traveled fre­quently and ex­tens­ively and was in­ter­ested in see­ing all parts of the globe. For ex­ample he vis­ited the Pa­cific is­lands and (while trav­el­ing with Ravi­jojla) was ship­wrecked in Ant­ar­tica. It is very hard to think of a place that he has not come fairly close to at some time or an­oth­er. Stone, of course, was the re­cip­i­ent of many hon­ors. His early elec­tion to the Na­tion­al Academy of Sci­ences in 1938 has already been noted. He was elec­ted, also at an early age to the Amer­ic­an Academy of Arts and Sci­ences in 1933 — the same year in fact that his fath­er was also elec­ted to the Academy [e10], and to the Amer­ic­an Philo­soph­ic­al So­ci­ety. He was the Amer­ic­an Math­em­at­ic­al So­ci­ety Col­loqui­um Lec­turer in 1939, and was honored by be­ing se­lec­ted as the Jo­si­ah Wil­lard B. Gibbs lec­turer in 1956. He served as pres­id­ent of the Amer­ic­an Math­em­at­ic­al So­ci­ety (1943–1944) and re­ceived many hon­or­ary doc­tor­ates, both do­mest­ic and for­eign. In 1982 he was awar­ded the Na­tion­al Medal of Sci­ence. Ac­cord­ing to Ed­win He­witt [e12], two ex­tra hon­ors that gave him spe­cial pleas­ure were his elec­tion to an Hon­or­ary Pro­fess­or­ship at Columbia Teach­ers Col­lege and to mem­ber­ship in the Ex­plorers Club of New York City.

Mar­shall Stone was a man with a very broad out­look and wide range of in­terests who seems to have thought rather deeply about a num­ber of is­sues. One had only to talk to him at length or read his non-math­em­at­ic­al writ­ings to come away with the im­pres­sion that here was an un­usu­ally thought­ful man with a high de­gree of pen­et­ra­tion and in­sight. More than most he seemed well en­dowed with a qual­ity, which one can only de­scribe as wis­dom. While on a vis­it to Madras, In­dia he died quickly of a sud­den ill­ness on Janu­ary 9, 1989.

Bibliographic Note

Ac­cord­ing to sev­er­al of his friends and col­leagues, George Mackey had be­gun work on an NAS bio­graph­ic­al mem­oir for his ment­or, dis­ser­ta­tion su­per­visor, and friend Mar­shall Stone. However health is­sues in­ter­vened and he was not able to com­plete the mem­oir be­fore his death. The Academy asked me, Calv­in Moore, who had just com­pleted an NAS bio­graph­ic­al mem­oir for George Mackey to take on the task of com­plet­ing the Stone mem­oir. However, a di­li­gent search of George’s Nachlass failed to turn up any manuscript or draft of the mem­oir he was work­ing on. But George had writ­ten a very com­plete sci­entif­ic ob­it­u­ary of Stone that was pub­lished in the No­tices of the Amer­ic­an Math­em­at­ic­al So­ci­ety in 1989 [e13]. In ad­di­tion George had writ­ten an art­icle about Stone and about his re­la­tion­ship with Stone [e15] that amp­li­fies some­what on what is in [e13]. Fi­nally Ed­win He­witt gave a speech at Stone’s second re­tire­ment sym­posi­um at the Uni­versity of Mas­sachu­setts in 1979 en­titled Mar­shall Stone in Ret­ro­spect and Pro­spect that has much ad­di­tion­al ma­ter­i­al es­pe­cially about Stone’s life out­side of math­em­at­ics. A tran­script of this speech [e12] was found in George’s nachlass, and pre­sum­ably was something that he was go­ing to make use of in writ­ing his mem­oir. George re­tained the copy­right to his 1989 ob­it­u­ary of Stone, which in turn passed to his wid­ow Alice. What I have done is to use, with Alice’s per­mis­sion, the text of [e13] as the base of this mem­oir, with some minor ed­its and with ad­di­tions to his text, in­clud­ing an in­tro­duct­ory para­graph and ad­di­tion­al ma­ter­i­al about Stone’s par­ents and his fam­ily life largely from the bio­graphy of his fath­er [e10], of which Mackey was aware as it is cited in [e15], and from [e12]. I hope the res­ult above bears some reas­on­able re­semb­lance to what George would have writ­ten had he been able to com­plete the task, and it is pub­lished here un­der his name as it is over­whelm­ingly his work. Fi­nally, I thank the Amer­ic­an Math­em­at­ic­al So­ci­ety for per­mis­sion to use the por­trait of Stone that ap­pears at the be­gin­ning of this mem­oir, and I thank Ann Mackey, George’s daugh­ter for per­mis­sion to use her pho­to­graphs in the body of the text.

Works

[1] M. H. Stone: “An un­usu­al type of ex­pan­sion prob­lem,” Trans. Am. Math. Soc. 26 : 3 (1924), pp. 335–​355. An ini­tial re­search re­port was pub­lished in Bull. Am. Math. Soc. 30:7 (1924). MR 1501281 JFM 50.​0313.​01 article

[2] M. H. Stone: “Ir­reg­u­lar dif­fer­en­tial sys­tems of or­der two and the re­lated ex­pan­sion prob­lems,” Trans. Am. Math. Soc. 29 : 1 (1927), pp. 23–​53. A re­search an­nounce­ment for this art­icle was pub­lished in Bull. Am. Math. Soc. 31:5–6 (1925). MR 1501375 JFM 53.​0429.​01 article

[3] M. H. Stone: “The ex­pan­sion prob­lems as­so­ci­ated with reg­u­lar dif­fer­en­tial sys­tems of the second or­der,” Trans. Am. Math. Soc. 29 : 4 (1927), pp. 826–​844. MR 1501416 JFM 53.​0420.​01 article

[4] M. H. Stone: “Lin­ear trans­form­a­tion in Hil­bert space, I: Geo­met­ric­al as­pects,” Proc. Natl. Acad. Sci. U.S.A. 15 : 3 (March 1929), pp. 198–​200. JFM 55.​0824.​01 article

[5] M. H. Stone: “Lin­ear trans­form­a­tions in Hil­bert space, III: Op­er­a­tion­al meth­ods and group the­ory,” Proc. Natl. Acad. Sci. U.S.A. 16 : 2 (February 1930), pp. 172–​175. JFM 56.​0357.​01 article

[6] M. H. Stone: “On one-para­met­er unit­ary groups in Hil­bert space,” Ann. Math. (2) 33 : 3 (July 1932), pp. 643–​648. MR 1503079 JFM 58.​0424.​01 Zbl 0005.​16403 article

[7] M. H. Stone: Lin­ear trans­form­a­tions in Hil­bert space and their ap­plic­a­tions to ana­lys­is. AMS Col­loqui­um Pub­lic­a­tions 15. Amer­ic­an Math­em­at­ic­al So­ci­ety (New York), 1932. Re­pub­lished (with ab­bre­vi­ated title) in 1990. JFM 58.​0420.​02 Zbl 0005.​40003 book

[8] M. H. Stone: “Boolean al­geb­ras and their ap­plic­a­tion to to­po­logy,” Proc. Natl. Acad. Sci. U.S.A. 20 : 3 (March 1934), pp. 197–​202. JFM 60.​0108.​02 Zbl 0010.​08104 article

[9] M. H. Stone: “Sub­sump­tion of the the­ory of Boolean al­geb­ras un­der the the­ory of rings,” Proc. Natl. Acad. Sci. U.S.A. 21 : 2 (February 1935), pp. 103–​105. JFM 61.​0054.​03 Zbl 0011.​05104 article

[10] M. H. Stone: “Pos­tu­lates for Boolean al­geb­ras and gen­er­al­ized Boolean al­geb­ras,” Am. J. Math. 57 (October 1935), pp. 703–​732. MR 1507106 JFM 61.​0975.​05 Zbl 0012.​29002 article

[11] M. H. Stone: “The the­ory of rep­res­ent­a­tions for Boolean al­geb­ras,” Trans. Am. Math. Soc. 40 : 1 (July 1936), pp. 37–​111. MR 1501865 JFM 62.​0033.​04 Zbl 0014.​34002 article

[12] M. H. Stone: “Ap­plic­a­tions of the the­ory of Boolean rings to gen­er­al to­po­logy,” Trans. Am. Math. Soc. 41 : 3 (1937), pp. 375–​481. MR 1501905 JFM 63.​1173.​01 Zbl 0017.​13502 article

[13] M. H. Stone: “A gen­er­al the­ory of spec­tra, I,” Proc. Natl. Acad. Sci. U. S. A. 26 : 4 (April 1940), pp. 280–​283. MR 0002023 JFM 66.​1284.​03 Zbl 0063.​07208 article

[14] M. H. Stone: “A gen­er­al the­ory of spec­tra, II,” Proc. Natl. Acad. Sci. U. S. A. 27 : 1 (January 1941), pp. 83–​87. MR 0004092 JFM 67.​0413.​03 Zbl 0063.​07209 article

[15] M. H. Stone: “Notes on in­teg­ra­tion, I,” Proc. Natl. Acad. Sci. U. S. A. 34 : 7 (July 1948), pp. 336–​342. MR 0025552 Zbl 0031.​01402 article

[16] M. H. Stone: “The gen­er­al­ized Wei­er­strass ap­prox­im­a­tion the­or­em,” Math. Mag. 21 : 4 (March–April 1948), pp. 167–​184. First of two parts, con­tin­ued in fol­low­ing is­sue. MR 0027121 article

[17] M. H. Stone: “On the com­pac­ti­fic­a­tion of to­po­lo­gic­al spaces,” Ann. Soc. Po­lon. Math. 21 (1948), pp. 153–​160. MR 0026316 Zbl 0031.​41701 article

[18] M. H. Stone: “Bounded­ness prop­er­ties in func­tion-lat­tices,” Ca­na­dian J. Math. 1 (1949), pp. 176–​186. MR 0029091 Zbl 0032.​16901 article

[19] M. H. Stone: “The al­geb­ra­iz­a­tion of har­mon­ic ana­lys­is,” Math. Stu­dent 17 (1949), pp. 81–​92. MR 0037928 Zbl 0037.​35301 article

[20] M. H. Stone: “Notes on in­teg­ra­tion, IV,” Proc. Natl. Acad. Sci. U.S.A. 35 : 1 (January 1949), pp. 50–​58. MR 0027830 Zbl 0034.​03101 article

[21] M. H. Stone: “Math­em­at­ics and the fu­ture of sci­ence,” Bull. Am. Math. Soc. 63 : 2 (1957), pp. 61–​76. 1956 Gibbs Lec­ture. A Pol­ish trans­la­tion was pub­lished in Wia­dom. Mat. 10 (1961). An Itali­an trans­la­tion was pub­lished in Archimede 10 (1958). A Rus­si­an trans­la­tion was pub­lished in Mat. Pros­veshchenie 4 (1959). MR 0086013 Zbl 0077.​00202 article

[22] M. Stone: “The re­volu­tion in math­em­at­ics,” Am. Math. Mon. 68 : 8 (October 1961), pp. 715–​734. MR 0130792 Zbl 0100.​00105 article

[23] M. H. Stone: “Book Re­view: ‘Cour­ant in Göttin­gen and New York: The story of an im­prob­able math­em­atician’,” Bull. Amer. Math. Soc. 84 : 2 (1978), pp. 234–​241. MR 1567039 article