Celebratio Mathematica

Mathematics at the University of Chicago: A brief history

by Saunders Mac Lane

The first Moore department

The Uni­versity of Chica­go opened its doors in 1892. Wil­li­am Rainey Harp­er, the first pres­id­ent of the uni­versity, im­me­di­ately star­ted out to em­phas­ize act­ive gradu­ate study, bring­ing in a num­ber of uni­versity pres­id­ents to be his de­part­ment heads. For act­ing head of math­em­at­ics, he found Eliakim H. Moore (1862–1932), a lively young math­em­atician (Ph.D. Yale 1885, stu­dent at Ber­lin, Ger­many, 1885–1886), then an as­so­ci­ate pro­fess­or at North­west­ern. Moore came in 1892 and im­me­di­ately found and ap­poin­ted two ex­cel­lent Ger­man math­em­aticians, then at Clark Uni­versity: Os­kar Bolza (1851–1936), a stu­dent of Wei­er­strass in the cal­cu­lus of vari­ations, and Hein­rich Masch­ke (1853–1908), a geo­met­er — both had been stu­dents at Ber­lin and at Göt­tin­gen. They con­sti­tuted the core of the first de­part­ment at Chica­go. G. A. Bliss [e6] (who stud­ied at Chica­go then) has writ­ten of them: “Moore was bril­liant and ag­gress­ive in his schol­ar­ship, Bolza rap­id and thor­ough, and Masch­ke more bril­liant, saga­cious and without doubt one of the most de­light­ful lec­tur­ers on geo­metry of all times.” This team al­most im­me­di­ately made Chica­go the lead­ing de­part­ment of math­em­at­ics in the United States.

University of Chicago Archives

In the peri­od 1892–1910, Chica­go awar­ded 39 doc­tor­ates in math­em­at­ics (far sur­pass­ing the next in­sti­tu­tions: Cor­nell, Har­vard, and Johns Hop­kins). Even more strik­ing is the qual­ity of the first doc­tor­ates. The first (1896) was Le­onard Eu­gene Dick­son, who sub­sequently did de­cis­ive re­search on al­gebra [e5] and in num­ber the­ory [e2], [e3], [e4]. He and five oth­ers in this group of 39 Ph.D.s sub­sequently held the of­fice of pres­id­ent of the Amer­ic­an Math­em­at­ic­al So­ci­ety (A.M.S.). These five (with their sub­sequent in­sti­tu­tions) were, with year of Ph.D.:

  • 1900: Gil­bert Ames Bliss (Uni­versity of Chica­go).

  • 1903: Os­wald Veblen (Prin­ceton).

  • 1905: Robert Lee Moore (Texas).

  • 1907: George Dav­id Birk­hoff (Har­vard).

  • 1910: Theo­phil Henry Hildebrandt (Uni­versity of Michigan).

All ex­cept Bliss were doc­tor­al stu­dents of E. H. Moore, al­though it ap­pears that Veblen ac­tu­ally dir­ec­ted most of R. L. Moore’s thes­is work! In the next gen­er­a­tion, these math­em­aticians were prob­ably the dom­in­ant fig­ures at their in­sti­tu­tions (Dick­son was also dom­in­ant at Chica­go). Note that this in­cludes the three in­sti­tu­tions gen­er­ally re­garded as the lead­ing ones in math­em­at­ics from 1910–1940: Chica­go, Har­vard, and Prin­ceton. At Texas, R. L. Moore was a great in­di­vidu­al­ist, while Hildebrandt, as a long-time chair­man at Michigan, set the style for a ma­jor state uni­versity. Birk­hoff was in­ter­ested in dif­fer­en­tial equa­tions and dy­nam­ics. In 1912, Henri Poin­caré, the lead­ing French math­em­atician, had for­mu­lated and left un­proven his “last geo­met­ric prob­lem.” Birk­hoff provided the proof in 1913 and was in con­sequence soon re­garded as the lead­ing Amer­ic­an math­em­atician. Veblen also had a ma­jor role in the de­vel­op­ment of to­po­logy and math­em­at­ic­al lo­gic at Prin­ceton Uni­versity, and later (from 1932) ad­min­istered the math­em­at­ic­al group at the In­sti­tute for Ad­vanced Study.

Clearly these res­ults in­dic­ate that re­mark­able and ag­gress­ive ad­vanced math­em­at­ic­al edu­ca­tion took place at Chica­go. G. A. Bliss [e6] writes, “Those of us who were stu­dents in those early years re­mem­ber well the in­tensely alert in­terest of these three men (Bolza, Masch­ke and Moore) in the pa­pers which they them­selves and oth­ers read be­fore the club … Math­em­at­ics … came first in the minds of these lead­ers.” In the files of the math­em­at­ic­al club, which met bi­weekly, I have also found a rap­tur­ous ac­count of a vis­it by Dick­son, who in 1897 re­turned for a brief vis­it after his year of study in Par­is and Leipzig, to re­port on the cur­rent math­em­at­ic­al de­vel­op­ments in Europe. Some echoes of this sense of ex­cite­ment were still present when I was a gradu­ate stu­dent (1930–1931) in Chica­go. I took a sem­in­ar on the “Hellinger in­teg­ral,” con­duc­ted by E. H. Moore, with as­sist­ance from R. W. Barn­ard. Moore, real­iz­ing that I knew little about Hellinger’s in­teg­ral, asked me to present E. Zer­melo’s fam­ous second proof [e1] that the ax­iom of choice im­plies that every set can be well ordered. I gave what I thought was a very clear lec­ture, but after Barn­ard and the two Chinese stu­dents had left, Moore took me aside and spent an hour ex­plain­ing to me what was really in­volved and what I should have said in my lec­ture. It was a thrill­ing ex­per­i­ence — one which re­flects in brief the ex­cite­ment of the early days at Chica­go.

Oskar Bolza
University of Chicago Archives

What sorts of math­em­at­ics were stud­ied then?

The archives at the Uni­versity of Chica­go lib­rary con­tain lec­ture notes taken by Ben­jamin L. Re­mick (1894–1900) on the fol­low­ing sub­jects:

  • By Bolza, with aca­dem­ic quarter in­dic­ated:

    • Func­tions of a Com­plex Vari­able (Au­tumn 1894)

    • Notes on Qua­ternions (Au­tumn 1894)

    • Hy­per­el­lipt­ic Func­tions (Au­tumn 1897)

    • In­vari­ants I and II (Winter, 1897, 1898)

    • The­ory of Ab­stract Groups (Sum­mer 1899)

  • By Masch­ke:

    • High­er Plane Curves (Au­tumn 1894)

    • Ana­lyt­ic­al Mech­an­ics (Spring 1895)

    • Al­geb­ra­ic Sur­faces (Spring 1895)

    • Wei­er­strass on El­lipt­ic Func­tions (Winter 1895)

    • High­er Plane Curves (Winter 1897)

    • Al­geb­ra­ic Sur­faces (Spring 1897)

    • Lin­ear Dif­fer­en­tial Equa­tions (Spring 1897)

  • By E. H. Moore:

    • Pro­ject­ive Geo­metry (Au­tumn 1896)

    • The­ory of Num­bers (Au­tumn 1897)

    • Gen­er­al Arith­met­ic I and II (Winter, Spring, 1898)

Heinrich Maschke
University of Chicago Archives

Also, Har­ris Han­cock (later at the Uni­versity of Cin­cin­nati) lec­tured on the cal­cu­lus of vari­ations (Spring, 1895), while George A. Miller (later at the Uni­versity of Illinois) lec­tured on per­muta­tion groups (Sum­mer, 1898). There was a closely re­lated as­tro­nomy de­part­ment, which em­phas­ized math­em­at­ic­al as­tro­nomy. F. R. Moulton lec­tured there on gen­er­al as­tro­nomy (Au­tumn, 1896), and Kurt Laves on “the three-body prob­lem” (Spring, 1897). In 1902, the full list of gradu­ate courses giv­en reads as fol­lows:

  • Au­tumn:

    • The­ory of Equa­tions (Bolza)

    • Pro­ject­ive Geo­metry (Moore)

    • Mod­ern Geo­metry (Masch­ke)

    • The­ory of Func­tions (Bolza)

    • Fi­nite Groups (Dick­son)

  • Winter:

    • The­ory of Equa­tions II (Bolza)

    • His­tory of Math­em­at­ics (Ep­steen)

    • High­er Plane Curves (Masch­ke)

    • The­ory of Func­tions II (Bolza)

    • Lin­ear Sub­sti­tu­tion Groups (Masch­ke)

  • Spring:

    • His­tory of Math­em­at­ics (Ep­steen)

    • Teach­ing Labor­at­ory (Moore)

    • Vec­tor Ana­lys­is (Lunn)

    • Lin­ear Dif­fer­en­tial Equa­tions (Masch­ke)

These courses cov­er most of the top­ics of math­em­at­ics then of cur­rent in­terest. In the math­em­at­ics club, Moore spoke of fi­nite fields (pre­sum­ably his proof clas­si­fy­ing all such), on Peano’s space-filling curve, and on his el­eg­ant sys­tem of gen­er­at­ors for the sym­metry group \( S_n \). Cur­rent con­cerns in fi­nite-group the­ory star­ted then: J. W. A. Young spoke “On Hold­er’s enu­mer­a­tion of all simple groups of or­ders at most 200.” In 1904, J. H. M. Wed­der­burn, a vis­it­or from Scot­land, proved his fam­ous the­or­em that every fi­nite di­vi­sion ring must be a com­mut­at­ive field. Al­gebra was there with a ven­geance.

The ori­gin­al fac­ulty at Chica­go in­cluded some ju­ni­or mem­bers: Jac­ob Wil­li­am Al­bert Young (Ph.D. from Clark Uni­versity, pre­sum­ably a stu­dent of Bolza; he re­tired as as­so­ci­ate pro­fess­or in 1926) and Har­ris Han­cock (1892–1900). Then, sev­er­al of Chica­go’s own Ph.D.s were ap­poin­ted to the fac­ulty:

  • Her­bert Ell­s­worth Slaught (1861–1937), Ph.D. 1898, from as­sist­ant pro­fess­or (1894) to pro­fess­or (1913–1931). Slaught was primar­ily con­cerned with math­em­at­ic­al edu­ca­tion and with as­sist­ing stu­dents. He was (with lively sup­port from E. H. Moore) one of the prin­cip­al founders of the Math­em­at­ic­al As­so­ci­ation of Amer­ica in 1916.

  • Le­onard Eu­gene Dick­son (1874–1950), Ph.D. 1896, from as­sist­ant pro­fess­or (1900) to pro­fess­or (1910–1939). His massive and schol­arly His­tory of the The­ory of Num­bers was a land­mark [e2], [e3], [e4], while his mono­graph Al­geb­ras and Their Arith­met­ics was trans­lated in­to Ger­man and had a ma­jor in­flu­ence on the Ger­man school of ab­stract al­gebra [e5]. He was a power­ful and as­sert­ive math­em­atician who dir­ec­ted at least 64 doc­tor­al theses. It is rumored that he con­sciously had two classes of doc­tor­al stu­dents: the reg­u­lar ones and the really prom­ising ones (such as C. C. Mac­Duffee, 1921, who later went to Wis­con­sin; C. G. Latimer 1924, to Ken­tucky; Bur­ton W. Jones 1928, to Cor­nell and Col­or­ado; A. A. Al­bert 1928, to Chica­go; Gor­don Pall 1929, to I.I.T.; Al­ex­an­der Op­pen­heim 1930, to Singa­pore; Arnold E. Ross 1931, to Notre Dame and Ohio State; R. D. James, 1928, to Berke­ley and Brit­ish Columbia; and Ral­ph Hull 1932, to Purdue). One can con­tem­plate with amazement the wide in­flu­ence ex­er­ted by Dick­son. I can also re­call his course (1930) in num­ber the­ory, taught from a book of that title which he had writ­ten with sparse pre­ci­sion: he ex­pec­ted his stu­dents (Hull, James, Mac Lane, et al.) to have un­der­stood every ar­gu­ment and every shift in nota­tion.

  • Ar­thur Con­stant Lunn (1877–1949), Ph.D. 1904, rose from as­so­ci­ate pro­fess­or in ap­plied math­em­at­ics (1902) to pro­fess­or (1923–1942). He had ac­cu­mu­lated a massive know­ledge of all of clas­sic­al math­em­at­ic­al phys­ics, and lec­tured on this in en­thu­si­ast­ic but ram­bling ways that did not end with the form­al end of the class hour. It was rumored among the stu­dents that a 1926 pa­per of his con­tain­ing some of the new ideas of quantum mech­an­ics had been re­jec­ted by some un­com­pre­hend­ing ed­it­or. Wheth­er or not this was true, Pro­fess­or Lunn was dis­cour­aged but very know­ledge­able when I listened to him in 1930–1931.

The second Moore department

Left to right: Dickson, Wilczynski, Lunn, MacMillan, Moore, Slaught and Moulton
Courtesy of Saunders Mac Lane

In the peri­od 1908–1910, the verve and dy­nam­ism of the ori­gin­al Chica­go de­part­ment ap­pears to have been gradu­ally lost. Pro­fess­or Masch­ke died in 1908; in 1910 Pro­fess­or Bolza re­turned to Ger­many (Freiburg in Baden) but kept up an es­sen­tially nom­in­al “non­res­id­ent” pro­fess­or­ship at Chica­go. He was still alert when I vis­ited him in Freiburg in 1933. E. H. Moore de­veloped his in­terest in pos­tu­la­tion­al gen­er­al­ity to a form of “gen­er­al ana­lys­is” which tracked prop­er­ties of in­teg­ral equa­tions in terms of func­tions “on a gen­er­al range.” The first form of his gen­er­al ana­lys­is was presen­ted in his col­loqui­um lec­tures of the A.M.S. at Yale Uni­versity in 1906, and pub­lished (in con­sid­er­ably altered form) in 1910. At that time, it was very much in or­der to find the ideas un­der­ly­ing the ex­ist­ence the­or­ems for solu­tions of in­teg­ral equa­tions — and in­deed this ob­ject­ive led Dav­id Hil­bert to his study of what are now called Hil­bert spaces. Moore’s for­mu­la­tion of these ideas did not suc­ceed, in part be­cause of his delay in pub­lish­ing. He con­tin­ued to work on it for 20 or more years, de­vel­op­ing a second form of gen­er­al ana­lys­is which was writ­ten in a lo­gist­ic nota­tion de­rived from Peano — a nota­tion which was pre­cise but hard to read, and which did not in­clude form­al lo­gic­al rules of in­fer­ence. In 1930–1931, his gen­er­al ana­lys­is was presen­ted in a six-quarter se­quence of courses at Chica­go; there were not many stu­dents. This ver­sion has been writ­ten up by Moore’s stu­dent and as­so­ci­ate R. W. Barn­ard; it is a monu­ment to a timely but failed ini­ti­at­ive [e7], [e8].

This, then, is the sense in which the ini­tial de­part­ment at Chica­go (Moore, Bolza, and Masch­ke) came to its ef­fect­ive end in 1907–1910. A new team ap­peared: Dick­son, as already noted, plus Bliss and Wil­czyn­ski:

  • Gil­bert A. Bliss (1876–1951), Ph.D. 1900, Chica­go, wrote his thes­is with Bolza. After teach­ing at Min­nesota, Chica­go, Mis­souri, and Prin­ceton, he be­came an as­so­ci­ate pro­fess­or at Chica­go in 1908, pro­fess­or in 1913, and chair­man in 1927 un­til his re­tire­ment in 1941. His in­terests covered many fields of ana­lys­is — al­geb­ra­ic func­tions, im­pli­cit func­tion the­or­ems, and the the­ory of ex­ter­i­or bal­list­ics. He was an en­thu­si­ast for the cal­cu­lus of vari­ations.

  • Ernst Ju­li­us Wil­czyn­ski (1876–1932) re­ceived his Ph.D. at Ber­lin in 1897. After teach­ing at the Uni­versity of Cali­for­nia (1898–1907) and at the Uni­versity of Illinois (1907–1910), he be­came an as­so­ci­ate pro­fess­or at Chica­go in 1910, and full pro­fess­or (1914–1926). He pub­lished vo­lu­min­ously and en­thu­si­ast­ic­ally, es­pe­cially in his fa­vor­ite sub­ject of pro­ject­ive dif­fer­en­tial geo­metry, where the loc­al prop­er­ties of curves and sur­faces were ana­lyzed in terms of ca­non­ic­al power series ex­pan­sions. Clearly, Wil­czyn­ski was ap­poin­ted at Chica­go as a suc­cessor to the pre­vi­ous geo­met­er, Masch­ke.

There were also two more ju­ni­or ap­point­ments in this peri­od:

  • May­me Ir­win Logs­don (1881–1967) re­ceived her Ph.D. at Chica­go in 1921, with a thes­is on equi­val­ence of pairs of Her­mitian forms, dir­ec­ted by Dick­son. She was an in­struct­or at Chica­go from 1921, and rose to be an as­so­ci­ate pro­fess­or (1930–1946). It is my own ob­ser­va­tion that one of her du­ties was that of ad­vising and help­ing many wo­men who were gradu­ate stu­dents at Chica­go; moreover, she taught a sur­vey course re­quired of all un­der­gradu­ates. After her re­tire­ment from Chica­go, she taught for many years at the Uni­versity of Miami, in Cor­al Gables, Flor­ida.

  • Ern­est P. Lane (1886–1969) re­ceived his Ph.D. at Chica­go in 1918 with a thes­is in geo­metry, and re­turned to Chica­go as as­sist­ant pro­fess­or in 1923. He was a me­tic­u­lous man, and an en­thu­si­ast for pro­ject­ive dif­fer­en­tial geo­metry.

Thus, in the peri­od 1910–1927, the team at Chica­go, headed by E. H. Moore, con­sisted primar­ily of Bliss, Dick­son, and Wil­czyn­ski. There were many Ph.D.s in this peri­od — 115 of them. Some of the more mem­or­able, clas­si­fied by sub­ject, were the fol­low­ing:

  • E. H. Moore dir­ec­ted theses in gen­er­al ana­lys­is:

    • 1910: Anna Pell (later Anna Pell-Wheel­er), sub­sequently a pro­fess­or at Bryn Mawr Col­lege. In 1927 she de­livered col­loqui­um lec­tures for the A.M.S. on the “The­ory of quad­rat­ic forms in in­fin­itely many vari­ables and ap­plic­a­tions.”

    • 1912: E. W. Chit­tenden, who be­came a lead­er in point-set to­po­logy at the Uni­versity of Iowa.

    • 1916: W. L. Hart, later of Min­nesota, a pro­lif­ic au­thor of text­books.

    • 1924: Mark H. In­gra­ham, later chair­man and dean at Wis­con­sin.

    • 1926: H. L. Smith, later a lead­er at Louisi­ana State Uni­versity; he worked on the Moore–Smith lim­its well known in to­po­logy.

    • 1926: R. W. Barn­ard, later Moore’s amanu­en­sis at Chica­go.

  • G. A. Bliss dir­ec­ted theses in ana­lys­is:

    • 1914: W. V. Lovitt, who taught at Col­or­ado Col­lege, and wrote a book on in­teg­ral equa­tions.

    • 1924: L. M. Graves, sub­sequently pro­fess­or at Chica­go.

  • L. E. Dick­son dir­ec­ted theses in al­gebra; for ex­ample:

    • 1921: M. I. Logs­don.

    • 1921: C. C. Mac­Duffee.

    • 1924: C. G. Latimer.

  • C. J. Wil­czyn­ski dir­ec­ted theses in geo­metry:

    • 1915: Archibald Hende­r­son, who be­came in­flu­en­tial at North Car­o­lina.

    • 1918: E. P. Lane, later pro­fess­or at Chica­go.

    • 1921: Ed­win R. Carus, who later foun­ded the Carus mono­graphs (M.A.A.).

    • 1925: V. G. Grove, sub­sequently chair­man at Michigan State.

  • Ap­plied math­em­at­ics:

    • 1913: E. J. Moulton, to North­west­ern.

Clearly, Moore was still a dom­in­ant in­flu­ence. However, none of the Ph.D.s of this peri­od achieved the pro­fund­ity in math­em­at­ic­al re­search of the best five earli­er Ph.D.s. Many, however, did rise to in­flu­en­tial po­s­i­tions in im­port­ant uni­versit­ies, as in­dic­ated be­low:

  • Bryn Mawr: Anna Pell-Wheller

  • Cor­nell: Bur­ton W. Jones (Ph.D., 1928)

  • Louisi­ana State Uni­versity: H. L. Smith

  • Michigan State: V. G. Grove

  • North­west­ern: E. J. Moulton

  • Uni­versity of Iowa: E. W. Chit­tenden

  • Col­or­ado Col­lege: W. V. Lovitt

  • Wis­con­sin: M. H. In­gra­ham, C. C. Mac­Duffee

In 1924, E. H. Moore re­por­ted proudly that the de­part­ment had by then trained 116 Ph.D.s, plus 15 more in math­em­at­ic­al as­tro­nomy, and that 52 of this total of 131 were already full pro­fess­ors at their re­spect­ive in­sti­tu­tions. In 1928 (ac­cord­ing to the lists in the Bul­let­in of the Amer­ic­an Math­em­at­ic­al So­ci­ety), 45 Ph.D.s were gran­ted in math­em­at­ics in the United States, of whom 12 (ac­cord­ing to the Bul­let­in) or 14 (ac­cord­ing to de­part­ment re­cords) were at Chica­go. The nearest com­pet­ing in­sti­tu­tions were Min­nesota (four Ph.D.s) and Cor­nell and Johns Hop­kins, with three Ph.D.s each.

My con­clu­sion is this: Chica­go had be­come in part a Ph.D. mill in math­em­at­ics.

The Bliss department

In 1927 G. A. Bliss be­came chair­man at Chica­go, while E. H. Moore con­tin­ued as head — by then largely a tit­u­lar form­al­ity. This ushered in a new peri­od which las­ted all dur­ing Bliss’ terms as chair­man (1927–1941). At about this time there were a num­ber of new ap­point­ments to the fac­ulty:

  • E. P. Lane was pro­moted to an as­so­ci­ate pro­fess­or­ship in 1927.

  • R. W. Barn­ard was ap­poin­ted as­sist­ant pro­fess­or in 1926.

  • L. M. Graves was ap­poin­ted as­sist­ant pro­fess­or in 1926.

  • Wal­ter Bartkey (Ph.D. Chica­go 1926) be­came an as­sist­ant pro­fess­or of ap­plied math­em­at­ics and stat­ist­ics in 1927; he was sub­sequently dean of the di­vi­sion of phys­ic­al sci­ences at Chica­go (1945–1955).

  • Ral­ph G. Sanger (1905–1960), Ph.D. Chica­go 1931, be­came in­struct­or in 1930 and as­sist­ant pro­fess­or in 1940–1946; he later moved to Kan­sas State Uni­versity.

I note ex­pli­citly that every one of these ap­pointees had re­ceived his Ph.D. at Chica­go.

In the peri­od 1927–1941, Bliss (who re­tired in 1941) and Dick­son (who re­tired in 1939) were the dom­in­ant fig­ures in the de­part­ment. In the el­ev­en-year peri­od from 1927–1937, there were 117 Ph.D.s awar­ded. Of these theses, Bliss, with the oc­ca­sion­al co­oper­a­tion of Graves, dir­ec­ted 35, of which 34 were in the cal­cu­lus of vari­ations. (In the pri­or 21-year peri­od, 1906–1926, there had been 17 theses de­voted to this cal­cu­lus.) Dick­son dir­ec­ted 32 theses. In this peri­od, Dick­son’s in­terests shif­ted from his earli­er en­thu­si­asms for quad­rat­ic forms and di­vi­sion al­geb­ras to an ex­tens­ive (and some­what nu­mer­ic­al) study of as­pects of War­ing’s prob­lem: for each ex­po­nent \( n \), find a num­ber \( k \) such that every (or every suf­fi­ciently large) in­teger is a sum of at most \( k \) \( n \)-th powers. At that time, the then new­er meth­ods of ana­lyt­ic num­ber the­ory could prove the “suf­fi­ciently large” part; Dick­son was con­cerned with a cor­res­pond­ing ex­pli­cit bound, and with the cal­cu­la­tion of what happened be­low that bound. The top­ics of Dick­son’s 32 theses pro­jects were dis­trib­uted as fol­lows: thir­teen on quad­rat­ic forms, twelve on War­ing’s prob­lem, six on di­vi­sion al­geb­ras, and five on gen­er­al top­ics in num­ber the­ory. There were four ad­di­tion­al al­geb­ra­ic theses dir­ec­ted by A. A. Al­bert. In geo­metry, Lane dir­ec­ted 20 theses and Logs­don, two. Moore and Barn­ard to­geth­er dir­ec­ted six theses. There were sev­en in as­pects of ap­plied math­em­at­ics, and ten oth­ers on as­sor­ted top­ics. All told, this peri­od rep­res­ents an in­tense con­cen­tra­tion on the cal­cu­lus of vari­ations and on num­ber the­ory.

In this peri­od there were some out­stand­ing res­ults. Two Chica­go Ph.D.s went on to be­come pres­id­ent of the A.M.S.: A. Ad­ri­an Al­bert (Ph.D. 1928, with Dick­son) and E. J. Mc­Shane (Ph.D. 1930, with Bliss). In the cal­cu­lus of vari­ations, I note four: W. L. Duren, Jr. (Ph.D. 1929), who soon played an im­port­ant lead­er­ship role at Tu­lane and later at Vir­gin­ia, M. R. Hestenes (Ph.D. 1932) was later in­flu­en­tial at U.C.L.A., Al­ston S. Houshold­er (Ph.D. 1937) who shif­ted his in­terests and be­came a lead­er in nu­mer­ic­al ana­lys­is at Oak Ridge while Her­man Gold­stine (Ph.D. 1936) was as­so­ci­ated with von Neu­mann in the de­vel­op­ment of the stored pro­gram com­puter. As already men­tioned, some half dozen of the Dick­son Ph.D.s did ef­fect­ive work in num­ber the­ory. Mina Rees (a Dick­son Ph.D., 1931) sub­sequently was the first pro­gram of­ficer for math­em­at­ics at the Of­fice of Nav­al Re­search. Her lead­er­ship there set the style for the sub­sequent math­em­at­ics pro­gram at the N.S.F.; sub­sequently, Dr. Rees be­came found­ing pres­id­ent of the Gradu­ate School and Uni­versity Cen­ter of the newly es­tab­lished City Uni­versity of New York. In 1983, she was awar­ded the Pub­lic Wel­fare Medal of the Na­tion­al Academy of Sci­ences.

In func­tion­al ana­lys­is, Le­on Alao­glu (Ph.D. 1938, with L. M. Graves) be­came fam­ous for his the­or­em that the closed unit ball in the dual space of a Banach space is com­pact in the weak-star to­po­logy. After teach­ing at Har­vard and Purdue and do­ing more re­search, he be­came a seni­or sci­ent­ist at the Lock­heed Air­craft Cor­por­a­tion. Mal­colm Smi­ley took his Ph.D. in the cal­cu­lus of vari­ations in 1937, but then switched to act­ive re­search in al­gebra. Ivan Niven, a Ph.D. of Dick­son’s in 1938, stud­ied then with Hans Rademach­er at the Uni­versity of Pennsylvania. After teach­ing at Illinois and Purdue, he went to the Uni­versity of Ore­gon and did de­cis­ive re­search on uni­form dis­tri­bu­tion of se­quences mod­ulo \( m \). Fre­d­er­ick Valentine (Ph.D. 1937, in the cal­cu­lus of vari­ations) was sub­sequently at U.C.L.A. where he pub­lished an im­port­ant book on con­vex sets.

To sum­mar­ize: In this peri­od the de­part­ment at Chica­go trained a few out­stand­ing re­search math­em­aticians, and a num­ber of ef­fect­ive mem­bers of this com­munity — plus pro­duced a large num­ber of es­sen­tially routine theses. Was this be­cause there was an un­due con­cen­tra­tion on a few spe­cial fields, or be­cause the pres­ence of so many gradu­ate stu­dents meant that the fac­ulty was forced in­to find­ing routine top­ics? In some cases they may have failed to ap­pre­ci­ate stu­dents’ po­ten­tial; I do not know. I do clearly re­call my own ex­per­i­ence as a gradu­ate stu­dent at Chica­go (1930–1931). Since the cal­cu­lus of vari­ations was evid­ently a ma­jor is­sue there, I signed up for Pro­fess­or Bliss’ two-quarter course in this sub­ject. Some­time well in­to the first quarter I had trouble put­ting the (to my mind ne­ces­sary) \( \varepsilon \)’s and \( \delta \)’s in­to his rather sketchy proof of the prop­er­ties of fields of ex­tremals. So I ven­tured to ask Pro­fess­or Bliss how to do this. At once he pro­duced all the needed ep­si­lons, with great skill — but he also made it very clear to me that I did not need to con­cern my­self with such de­tails; gradu­ate stu­dents were ex­pec­ted to get chiefly an over­all im­pres­sion of the shape of the sub­ject. Some years later, I had oc­ca­sion to study Bliss’ book on al­geb­ra­ic func­tions; I ob­served then that this book cor­rectly re­pro­duced the suit­able Ger­man sources, but did not press on to get a real un­der­stand­ing of why things worked out and what the Riemann–Roch the­or­em really meant.

There were light­er mo­ments. Pro­fess­or Bliss liked to kid his stu­dents. One day in his lec­tures on the cal­cu­lus of vari­ations, he re­coun­ted his own earli­er ex­per­i­ences in Par­is. After he sat down in the large lec­ture am­phi­theatre there, an im­press­ive and form­ally dressed man entered and went to the front. Bliss thought it was the pro­fess­or him­self, but no, it was just his as­sist­ant who cleaned the black­boards and set the lights. When the pro­fess­or fi­nally ar­rived, all the stu­dents stood up. At this point in his story, Bliss ob­served that Amer­ic­an stu­dents do not pay prop­er re­spect to their pro­fess­ors. So the class agreed on suit­able steps; I was the only mem­ber own­ing a tuxedo. The next day, ar­rayed in that tuxedo, I knocked on the door for Pro­fess­or Bliss to re­port that his class awaited him. When he came in they all rose in his hon­or.

Of the six stu­dents of Moore and Barn­ard dur­ing this peri­od, only Y. K. Wong (Ph.D. 1931) con­tin­ued sub­stan­tial activ­ity. With Moore, he had stud­ied matrices and their re­cip­roc­als; in his later re­search (at the Uni­versity of North Car­o­lina) he was con­cerned with the use of Minkowski–Le­on­tief matrices in eco­nom­ics [e13].

There re­mains the fas­cin­at­ing ques­tion: In the early days, Moore had been dy­nam­ic and re­mark­ably ef­fect­ive in train­ing gradu­ate stu­dents. What changed? As I have already noted, he was still an alert crit­ic when I knew him in 1930, and he had con­tin­ued to work di­li­gently on his form of gen­er­al ana­lys­is. But he did not pub­lish. Ac­cord­ing to an ob­it­u­ary by Bliss, he pub­lished only two sub­stan­tial re­search pa­pers after 1915, both in 1922, and one of them with H. L. Smith on the im­port­ant concept of the Moore–Smith lim­it. At the start, Moore had been in lively con­tact with many cur­rent de­vel­op­ments in math­em­at­ics. I con­jec­ture that he had gradu­ally lost that con­tact, in part be­cause of a heavy pre­oc­cu­pa­tion with his own ideas in gen­er­al ana­lys­is, and in part be­cause he may have de­pended on the ex­change of ideas with his con­tem­por­ar­ies Bolza and Masch­ke, while the new­er and young­er ap­point­ments at Chica­go did not provide an ef­fect­ive such ex­change.

Appointments by Bliss

In this peri­od (1927–1941), there were a num­ber of oth­er ap­point­ments to the fac­ulty, as fol­lows:

  • A. Ad­ri­an Al­bert (1905–1972), Ph.D. Chica­go 1928; as­sist­ant pro­fess­or (1931) to pro­fess­or (1941).

  • M. R. Hestenes (1906–), Ph.D. Chica­go 1932; as­sist­ant pro­fess­or (1937), as­so­ci­ate pro­fess­or (1942–1947), later in­flu­en­tial in nu­mer­ic­al ana­lys­is and com­bin­at­or­ics at U.C.L.A..

  • W. T. Re­id (1907–1977), Ph.D. Texas 1927; in­struct­or Chica­go (1931) to as­so­ci­ate pro­fess­or (1942–1944), later a pro­fess­or at North­west­ern, Iowa, and Ok­lahoma.

With these ap­point­ments, note the em­phas­is on the two fields of al­gebra and the cal­cu­lus of vari­ations — and on Ph.D.s from Chica­go. (Re­id came from Texas, but had spent the years 1929–1931 as a postdoc­tor­al fel­low at Chica­go.)

Later on, the de­part­ment made real at­tempts to ap­point math­em­aticians not from Chica­go and in new fields; two of them, as fol­lows, did not last:

  • Saun­ders Mac Lane (Ph.D. Göt­tin­gen 1934); in­struct­or, Chica­go (1937–1938), then to Har­vard. I be­lieve that my ap­point­ment in 1937 at Chica­go was due to the in­ter­ven­tion of Pres­id­ent Hutchins. At any rate, I had met Hutchins in 1929, and he had per­son­ally ar­ranged to get me a gradu­ate fel­low­ship at Chica­go for 1930.

  • Nor­man Earl Steen­rod (1910–1971), Ph.D. Prin­ceton 1936; as­sist­ant pro­fess­or Chica­go (1939–1942), then to the Uni­versity of Michigan as as­sist­ant pro­fess­or (1942); in 1945, to Prin­ceton.

  • Otto F. G. Schilling (1911–1973), Ph.D. Mar­burg 1934; in­struct­or Chica­go (1939) to pro­fess­or (1958); in 1961, to Purdue.

There are, to be sure, ru­mors of ap­point­ments which were not made. Thus, the fam­ous Ger­man ana­lyst and num­ber the­or­ist Carl Lud­wig Siegel left Göt­tin­gen in the spring of 1940 and es­caped via Nor­way to the United States. It then be­came clear that he needed a suit­able po­s­i­tion in this coun­try; ru­mor has it that G. A. Bliss knew this, but did not act on this pos­sib­il­ity; soon Siegel be­came a pro­fess­or at the In­sti­tute for Ad­vanced Study in Prin­ceton.

The ap­point­ment of Steen­rod, who soon be­came a noted to­po­lo­gist, may well have been stim­u­lated by the use of to­po­logy and the re­lated the­ory of crit­ic­al points (Mar­ston Morse) in the cal­cu­lus of vari­ations. Up un­til this point the ap­point­ment policy at Chica­go seems to have been based on what I might call the “in­her­it­ance prin­ciple”: If X has been an out­stand­ing pro­fess­or in field F, ap­point as his suc­cessor the best per­son in F, if pos­sible the best stu­dent of X. Let me re-ex­am­ine the ap­point­ments at Chica­go in this light.

Bolza was out­stand­ing in ana­lys­is and had writ­ten an au­thor­it­at­ive book on the cal­cu­lus of vari­ations. Shortly after he left in 1908, his best stu­dent, G. A. Bliss, was ap­poin­ted. Sub­sequently, three stu­dents of Bliss were ap­poin­ted: Graves, Sanger, and Hestenes, as well as W. T. Re­id from Texas. There res­ul­ted a great con­cen­tra­tion on such top­ics as vari­ants of the prob­lem of Bolza in the cal­cu­lus of vari­ations, but the school at Chica­go missed out on the ma­jor de­vel­op­ment of the sub­ject in the early 1930s, as rep­res­en­ted by the work of Mar­ston Morse on the cal­cu­lus of vari­ations in the large. Chica­go was of course aware of this work, but did noth­ing much about it. Spe­cific­ally, in the spring of 1931, Bliss con­duc­ted a sem­in­ar on this top­ic, and as­signed Mac Lane to re­port on Betti num­bers and their mean­ing. Mac Lane thereupon stud­ied the (then unique) text by Veblen, and re­por­ted on the Betti num­bers, but not on their mean­ing (which he did not really un­der­stand).

In geo­metry, the death of that not­able geo­met­er Masch­ke in 1908 was soon fol­lowed by the ap­point­ment of an­oth­er geo­met­er, Wil­czyn­ski, in 1910 and then, upon his re­tire­ment in 1926, by the pro­mo­tion in 1927 of his best stu­dent E. P. Lane. The spe­cial em­phas­is on the sub­field of pro­ject­ive dif­fer­en­tial geo­metry (as in Lane’s sub­sequent book) gradu­ally lost its im­port­ance, both in Chica­go and in Shang­hai (where the seni­or pro­fess­or Buchin Su worked in this field). In 1939, George White­head, one of the gradu­ate stu­dents, asked Pro­fess­or Lane for a thes­is top­ic in pro­ject­ive dif­fer­en­tial geo­metry. In­stead of giv­ing him a top­ic, Lane gave White­head the good ad­vice to work in the new­er field of to­po­logy with Steen­rod; White­head later (at M.I.T.) be­came a lead­er in this field.

In gen­er­al ana­lys­is, E. H. Moore had con­sid­er­able in­flu­ence on Lawrence Graves. Then, in 1928, Moore’s stu­dent Barn­ard was ap­poin­ted to the fac­ulty. However, Moore did not work out the pos­sible con­nec­tion between his gen­er­al ana­lys­is and the study (at oth­er cen­ters) of Hil­bert spaces and of func­tion­al ana­lys­is. Moore was a great en­thu­si­ast for in­fin­ite matrices, pos­tu­la­tion­al meth­ods, and Peano. In early work, Peano had the ax­ioms for a (two-di­men­sion­al, real) vec­tor space. I nev­er learned about these ax­ioms from Moore — and had to learn them in 1932 from Her­man Weyl in Göt­tin­gen (who had clearly for­mu­lated them in his 1917 book on re­lativ­ity). Thus is an­oth­er small piece of evid­ence that Moore had lost con­tact.

In math­em­at­ic­al as­tro­nomy and ap­plied math­em­at­ics, Kurt Laves (first ap­poin­ted about 1894) was the first fac­ulty mem­ber. Then vari­ous Chica­go Ph.D.s were ap­poin­ted in as­tro­nomy or in math­em­at­ics. The most out­stand­ing was per­haps F. R. Moulton; oth­ers were W. D. Mac­Mil­lan, A. C. Lunn, and then Wal­ter Bartkey in 1926. Per­haps be­cause of his activ­ity in mil­it­ary re­search dur­ing W.W. II, Bartkey’s in­terests shif­ted to ad­min­is­trat­ive mat­ters; he be­came a dean and fi­nally a vice pres­id­ent for re­search at the uni­versity from 1956 to 1958. In this sense, the line of in­her­it­ance in ap­plied math­em­at­ics died out, not to be re­newed un­til the ap­point­ment in 1963 of two former stu­dents (at Chica­go) of S. Chandrasekhar.

E. H. Moore provided the ini­tial im­petus in al­gebra, group the­ory, and num­ber the­ory; the ap­point­ment of his first Ph.D. stu­dent Dick­son in 1900 was a strong step. Then in 1931. Dick­son’s best stu­dent, Ad­ri­an Al­bert, was ap­poin­ted. In 1945, Al­bert’s re­com­mend­a­tion brought the ap­point­ment of Irving Ka­plansky (Ph.D. from Har­vard; Mac Lane’s first stu­dent). Al­bert kept the in­terest in al­gebra gen­er­ally, and in group the­ory in par­tic­u­lar, alive, and in 1961 or­gan­ized a “spe­cial year” on group the­ory at Chica­go. It was dur­ing this year that Wal­ter Feit (M.S. Chica­go, 1951) and John Thompson (Ph.D. Chica­go 1959) worked out their “odd or­der” pa­per with the re­mark­able proof that every fi­nite simple group is either cyc­lic or of even or­der. This was a vi­tal step to­ward the sub­sequent clas­si­fic­a­tion of all fi­nite simple groups.

Thus, in al­gebra the in­her­it­ance the­ory of ap­point­ments worked splen­didly, while in oth­er fields, as noted, it was not suc­cess­ful in the long run.

The Lane department

When G. A. Bliss re­tired in 1941, E. P. Lane be­came chair­man of the de­part­ment of math­em­at­ics. He made sev­er­al at­tempts to re­vive and strengthen the de­part­ment, but the times were not pro­pi­tious, largely be­cause of the on­set of W.W. II. When Pres­id­ent Robert M. Hutchins (with con­sid­er­able ad­min­is­trat­ive cour­age) brought the Man­hat­tan pro­ject on atom­ic en­ergy to Chica­go, it was soon housed in the de­part­ment’s treas­ured build­ing, Eck­hart Hall, and the math­em­aticians were moved out to one of the towers of Harp­er Lib­rary. There were no new ap­point­ments till the post­war ap­point­ment of Ka­plansky (1945). There were 21 Ph.D.s (1941–1946), in­clud­ing White­head (1941), the al­geb­ra­ists R. D. Schafer (1942) and Daniel Zel­in­sky (1946), and in the cal­cu­lus of vari­ations the very young math­em­atician J. Ern­est Wilkins (1942), who later did not­able re­search in ap­plied math­em­at­ics.

Ph.D.s to women at Chicago

In the 39 years 1908–1946, the de­part­ment awar­ded 51 Ph.D.s to wo­men out of a total of 270 Ph.D.s in math­em­at­ics in that peri­od. It is likely that more Ph.D.s were awar­ded to wo­men at Chica­go than at any oth­er Amer­ic­an uni­versity in this peri­od. Chica­go had been coedu­ca­tion­al from the start, but 1908 was the year when the first Ph.D. was awar­ded to a wo­man — Mary Emily Sin­clair, who sub­sequently be­came pro­fess­or and chair­man at Ober­lin Col­lege. Also, 1946 is the year when Mar­shall Stone, as a new chair­man, came to drastic­ally change the dir­ec­tion of the de­part­ment; this de­term­ines the peri­od 1908–1946 which I chose for this list.

Thanks to Mar­lene Tuttle of the alumni re­la­tions of­fice of the Uni­versity of Chica­go I have been able to col­lect defin­it­ive in­form­a­tion on al­most all of these 51 wo­men math­em­aticians. In par­tic­u­lar, I could loc­ate the col­lege or uni­versity where they sub­sequently taught math­em­at­ics. After clas­si­fy­ing these in­sti­tu­tions as wo­men’s col­leges, coedu­ca­tion­al col­leges (e.g., Ober­lin), uni­versit­ies, or re­search uni­versit­ies, I get the fol­low­ing table based on one chosen in­sti­tu­tion for each Ph.D.; in a few cases there was a change of in­sti­tu­tion:

Sub­sequent Aca­dem­ic Em­ploy­ment of Wo­men Ph.D.s
Ph.D. date:1908–19311932–1946
Wo­men’s col­lege812
Re­search uni­versity61

In one case (in the second peri­od) I was un­able to loc­ate any sub­sequent teach­ing em­ploy­ment; I be­lieve that the in­di­vidu­al was mar­ried and did not take up teach­ing. But note that of the 51 lis­ted, 50 did en­gage in teach­ing, most of them at just one in­sti­tu­tion and for a con­sid­er­able peri­od. The Ph.D.s from Chica­go provided an ef­fect­ive source of fac­ulty — es­pe­cially at wo­men’s col­leges. Note that teach­ing loads at such col­leges were then quite heavy.

I have ven­tured to clas­si­fy sev­en of the in­sti­tu­tions as “re­search uni­versit­ies,” al­though that term was not then in use. The word ap­peared only later as a la­bel for those uni­versit­ies which seek to ac­quire sub­stan­tial re­search funds from the gov­ern­ment. At any rate, the sev­en re­search uni­versit­ies lis­ted above were (in chro­no­lo­gic­al or­der of the de­grees) Wis­con­sin, Berke­ley, Min­nesota, Chica­go, Illinois, North­west­ern, and Illinois again, the last in 1932. It will make my clas­si­fic­a­tion clear if I list the five uni­versit­ies (1932–1946) as Kent State, Uni­versity of Utah, Uni­versity of Ok­lahoma, Uni­versity of Alabama. and Temple Uni­versity. All told, this tab­u­la­tion in­dic­ates clearly that in all this peri­od very few of the wo­men went or were sent to ma­jor re­search uni­versit­ies. (In 1916, the Uni­versity of Cali­for­nia at Berke­ley did not then have its present stand­ing in math­em­at­ics.)

In re­port­ing this situ­ation, I de­lib­er­ately said “were sent,” be­cause in those days po­s­i­tions for new doc­tor­ates in math­em­at­ics were man­aged by what is now called the “old-boy net­work.” At present this is a term of op­pro­bri­um; at that time it re­ferred to a place­ment sys­tem for a small num­ber of gradu­ates that in fact worked much more ef­fi­ciently than the present sys­tem, which in­ev­it­ably is ap­plied to much lar­ger num­bers and in­volves massive em­ploy­ment in­ter­views at the Janu­ary A.M.S. meet­ings, plus pi­ous de­clar­a­tions of equal op­por­tun­ity in ad­vert­ise­ments which (es­pe­cially today) make it clear that op­por­tun­ity beck­ons at X uni­versity only if your re­search lies in a field already X-rep­res­en­ted.

The old-boy net­work func­tioned as fol­lows: all the act­ive math­em­aticians, such as Veblen at Prin­ceton, Bliss at Chica­go, or Birk­hoff at Har­vard (plus many oth­ers, such as Hildebrandt at Michigan) had pretty shrewd ideas as to the level of math­em­at­ic­al activ­ity at many schools, and they also had quite de­tailed (but per­haps mis­taken) know­ledge of the qual­it­ies of their own cur­rent products. So when they heard that Ober­lin Col­lege, or the wo­men’s col­lege of North Ere­hwon, or the Uni­versity of W had a va­cancy, they knew which of their gradu­ates would be an ap­pro­pri­ate can­did­ate there, and they ac­ted ac­cord­ingly. (Of course, the can­did­ate’s pro­fess­or was also an act­or in this net­work.) The sys­tem did make mis­takes. For ex­ample, in 1957, Michigan sent to Chica­go let­ters of re­com­mend­a­tion for a new Ph.D., one Steven Smale. The let­ters were not es­pe­cially en­thu­si­ast­ic. At that time, the de­part­ment had few va­can­cies for an in­struct­or, so Smale was ap­poin­ted at Chica­go in the col­lege math­em­at­ics staff, then sep­ar­ate from the de­part­ment and in­ten­ded primar­ily for un­der­gradu­ate teach­ing. This goes to show that there can be mis­judge­ments about re­search po­ten­tial.

At any rate, the table above makes it clear that Chica­go did not nor­mally send its wo­men Ph.D.s to uni­versit­ies anxious to ac­quire re­search hot-shots.

I also tab­u­lated pub­lished re­search pa­pers in math­em­at­ics (1931–1960) for the 25 wo­men Ph.D.s in the second peri­od. In ten cases, I found 14 pub­lic­a­tions all told, in most cases the pub­lic­a­tion of the thes­is, but I note that one wo­man had three pub­lic­a­tions. I have not tab­u­lated re­search pub­lic­a­tions for men Ph.D.s from the peri­od (1932–1946); some were pro­lif­ic, while oth­ers hardly pub­lished.

On the evid­ence, I sum­mar­ize thus: In this peri­od, wo­men were en­cour­aged to study for the Ph.D de­gree at Chica­go, and there was a role mod­el on the staff to help and sup­port them (May­me I. Logs­don). But these wo­men stu­dents were not really ex­pec­ted to do any sub­stan­tial re­search after gradu­ation; the doc­tor­ate was it, and in many cases the thes­is top­ic was chosen to suit. This last sen­tence agrees with my own re­col­lec­tion of the situ­ation and at­mo­sphere at Chica­go dur­ing my gradu­ate study there in 1930–1931. I might add that for some of the men-stu­dents there was the same low level of re­search ex­pect­a­tions — but not for all.

For com­plete­ness, I add that in the fol­low­ing peri­od (1946–1960) at Chica­go there were ex­actly four Ph.D.s gran­ted to wo­men; among those, one to Mary Weiss (Ph.D. with Zyg­mund, 1957) who made an im­press­ive re­search ca­reer. I note that the wo­men’s lib­er­a­tion move­ment was yet to come, and that there ap­par­ently were very few wo­men gradu­ate stu­dents present. My own course re­cords (in ba­sic gradu­ate courses in al­gebra and to­po­logy in this peri­od) show 38 wo­men out of 267 stu­dents all told in my courses — about 14 per­cent. It ap­pears that wo­men stu­dents began gradu­ate work, but that few went on to the Ph.D.

The Stone department

Robert Maynard Hutchins, pres­id­ent of the Uni­versity of Chica­go (1929–1951), had brought the Man­hat­tan pro­ject to the uni­versity dur­ing W.W. II, and with it many not­able sci­ent­ists in­clud­ing En­rico Fermi, James Franck, and Har­old Urey. As the war drew to a close, he and his ad­visors de­cided to try to hold these men and their as­so­ci­ates at Chica­go. For this pur­pose, he es­tab­lished two re­search in­sti­tutes, now known as the Fermi and the Franck In­sti­tutes. He and his ad­visors real­ized that there should a much-needed strength­en­ing of the de­part­ment of math­em­at­ics. With the ad­vice of John von Neu­mann (who had been as­so­ci­ated with the Man­hat­tan pro­ject), they ap­proached Mar­shall H. Stone, then a pro­fess­or at Har­vard, sug­gest­ing (after some talk of a dean­ship) that he come to Chica­go as chair­man of math­em­at­ics. Stone had thought deeply about the con­di­tions which would sup­port a great de­part­ment of math­em­at­ics at a level well above that then-present at Har­vard or Prin­ceton. After re­ceiv­ing suit­able as­sur­ances from Pres­id­ent Hutchins, Stone came to Chica­go in 1946. He thereupon brought to­geth­er what was in ef­fect a whole new de­part­ment. In each such new case, I spe­cify the dates of their activ­ity on the Chica­go fac­ulty:

  • As pro­fess­ors:

    • An­dré Weil (1947–1958), a not­able (and con­ten­tious) French math­em­atician, one of the lead­ing mem­bers of the Bourbaki group. He had just pub­lished his fun­da­ment­al book on the found­a­tions of al­geb­ra­ic geo­metry, con­tain­ing his proof of the Riemann hy­po­thes­is for func­tion fields.

    • Ant­oni Zyg­mund (1947–1980), a Pol­ish ana­lyst, in­ter­ested in Four­i­er ana­lys­is and har­mon­ic ana­lys­is. Zyg­mund had been in this coun­try at Mount Holy­oke, and then at the Uni­versity of Pennsylvania.

    • Saun­ders Mac Lane (1947–1982) from Har­vard; he was at that time act­ive in study­ing the co­homo­logy of groups and the re­lated co­homo­logy of Ei­len­berg–Mac Lane spaces in to­po­logy.

    • Shi­ing-Shen Chern (1949–1959), an out­stand­ing Chinese math­em­atician with in­terests in dif­fer­en­tial geo­metry and to­po­logy (for ex­ample, his char­ac­ter­ist­ic classes).

  • As as­sist­ant pro­fess­ors, Stone brought to Chica­go:

    • Paul R. Hal­mos (1946–1961), work­ing in meas­ure the­ory and Hil­bert space, He had just pub­lished his el­eg­ant ex­pos­i­tion, “Fi­nite-Di­men­sion­al Vec­tor Spaces,” with a present­a­tion in­flu­enced by his con­tacts with John von Neu­mann.

    • Irving E. Segal (1948–1960), an en­thu­si­ast for rings of op­er­at­ors in Hil­bert space, and their ap­plic­a­tion to quantum mech­an­ics.

    • Ed­win H. Span­i­er (1948–1959), a young and know­ledge­able al­geb­ra­ic to­po­lo­gist from Prin­ceton; he had re­cently fin­ished his Ph.D. un­der the dir­ec­tion of Nor­man Steen­rod.

Of the pre­vi­ous de­part­ment, Al­bert and Ka­plansky were im­me­di­ately en­thu­si­ast­ic mem­bers of this new team, which then read:

Pro­fess­ors: Al­bert, Chern, Mac Lane, Stone, Weil, and Zyg­mund.

As­sist­ant Pro­fess­ors: Hal­mos, Ka­plansky, Segal, and Span­i­er.

Of the oth­er pre­vi­ous mem­bers, Hestenes soon left for U.C.L.A., and J. L. Kel­ley, who had briefly been an as­sist­ant pro­fess­or, left for Berke­ley. Pro­fess­ors Barn­ard, Graves, Lane, and Schilling stayed on (in most cases un­til re­tire­ment); they co­oper­ated but were not really full mem­bers of the new dis­pens­a­tion,

The new group covered quite a vari­ety of fields. There were ex­cit­ing gradu­ate courses, and some clashes of opin­ion (for ex­ample, between Weil and Segal). Weil, in con­tinu­ing the tra­di­tion of Hadam­ard’s sem­in­ar in Par­is, taught a course called “Math­em­at­ics 400” in which the stu­dents were re­quired to re­port on a pa­per of cur­rent re­search in­terest not in their own field; a few stu­dents were dis­cour­aged by his severe cri­ti­cisms, but many oth­ers were en­cour­aged to broaden their in­terests. Un­der Stone’s en­cour­age­ment, a whole new gradu­ate pro­gram was laid out, with three-quarter se­quences in al­gebra, ana­lys­is, and geo­metry (see [e12]).

This was the im­me­di­ate post­war peri­od, when many ex-sol­diers could take up ad­vanced study un­der the G.I. Bill. Thus, there were many lively stu­dents at Chica­go; in the peri­od 1948–1960 there were 114 Ph.D.s gran­ted by the de­part­ment. Among the re­cip­i­ents were a num­ber of sub­sequently act­ive people, in­clud­ing:

  • 1950: A. P. Calder­ón, R. V. Kadis­on, and I. M. Sing­er.

  • 1951: Mur­ray Ger­sten­hab­er, E. A. Mi­chael, and Alex Rosen­berg.

  • 1952: Ar­len Brown and I. B. Fleis­cher.

  • 1953: Kat­sumi Nom­izu.

  • 1954: Louis Aus­lander and Bert Kostant.

  • 1955: Er­rett Bish­op, Ed­ward Nel­son, Eli Stein, and Har­old Wi­dom.

  • 1956: R. E. Block, W. A. Howard, Anil Ner­ode, and Guido Weiss.

  • 1957: B. Ab­ra­ham­son, Don­ald Orn­stein, Ray Kun­ze, and Mary Weiss.

  • 1958: Paul Co­hen, Moe Hirsch, and E. L. Lima.

  • 1959: Hy­man Bass, John G. Thompson, and Joseph Wolf.

  • 1960: Steve Chase, A. L. Li­ulevi­cius, and R. H. Szczar­ba.

The qual­it­ies of this group of gradu­ates, in my view, match the qual­it­ies of the best gradu­ates of the first group of the Moore de­part­ment. For ex­ample, by 1988, eight of those lis­ted just above had been elec­ted to mem­ber­ship in the sec­tion of math­em­at­ics of the Na­tion­al Academy of Sci­ences; by that date in the whole coun­try, about 30 had been elec­ted from those with Ph.D.s from these years 1948–1960; Prin­ceton, with six, con­trib­uted the next largest con­tin­gent.

In this peri­od at Chica­go, there was a fer­ment of ideas, stim­u­lated by the newly as­sembled fac­ulty and re­flec­ted in the de­vel­op­ment of the re­mark­able group of stu­dents who came to Chica­go to study. Re­ports of this ex­cite­ment came to oth­er uni­versit­ies; of­ten stu­dents came after hear­ing such re­ports (I can name sev­er­al such cases). This serves to em­phas­ize the ob­ser­va­tion that a great de­part­ment de­vel­ops in some part be­cause of the pres­ence of out­stand­ing stu­dents there. (This is true also of Göt­tin­gen in 1930–1933 and Har­vard in 1934–1948, in my own ex­per­i­ence.)

By 1952, Mar­shall Stone had grown weary of the con­tin­ued struggle with the ad­min­is­tra­tion for new re­sources; Mac Lane suc­ceeded him as chair­man (1952–1958). The de­part­ment con­tin­ued in sim­il­ar activ­ity un­til about 1959, when it sud­denly came apart. In 1958, Weil left to go to the In­sti­tute for Ad­vanced Study, Chern and Span­i­er left to go to Berke­ley in 1959, Segal left for M.I.T. in 1960, and Hal­mos left for Michigan in 1961. Those de­par­tures es­sen­tially brought to a close the Stone Age. The de­part­ment was soon re­built un­der A. A. Al­bert (chair­man, 1958–1962, and dean, 1962–1974) and Irving Ka­plansky (chair­man, 1962–1967). (This later peri­od will not be de­scribed in this es­say.) But there had been just this one peri­od 1945–1960 when Chica­go, in its new style, was without doubt the lead­ing de­part­ment of math­em­at­ics in the coun­try.

Why the change?

One may won­der why the Stone Age came to such an ab­rupt end. In some part, this may just be the in­ev­it­ab­il­ity of changes in hu­man situ­ations; people grow older and shift their in­terests. Na­tion­ally, Sput­nik in 1957 stim­u­lated much more ex­ten­ded gov­ern­ment sup­port for math­em­at­ics in the peri­od 1958–1960; one res­ult was that there soon were more math­em­at­ics de­part­ments of ma­jor stand­ing — for ex­ample, Berke­ley and M.I.T. There are also ex­plan­a­tions “in­tern­al” to the Uni­versity of Chica­go. After 1950, Mar­shall Stone traveled fre­quently, and clearly the loss of his pres­ence and lead­er­ship made a dif­fer­ence. Mac Lane may have made mis­takes as chair­man; Al­bert (un­pub­lished) and Hal­mos (pub­lished) evid­ently thought so. A ma­jor ob­ser­va­tion is this: In the peri­od 1949–1957, ex­cept for tem­por­ary in­struct­ors, there were no new ap­point­ments to the fac­ulty; there was one ap­point­ment in 1958. This sug­gests that there was not a suf­fi­cient in­flow of ideas.

The top ad­min­is­tra­tion of the uni­versity had changed. Robert Maynard Hutchins resigned as pres­id­ent in 1951; the new pres­id­ent or “chan­cel­lor” (1951–1960) was Lawrence A. Kimp­ton. It seems clear that the trust­ees in­struc­ted Kimp­ton to pay at­ten­tion to the neigh­bor­hood and to achieve a bal­anced budget for the uni­versity. This he did, but there were in­tel­lec­tu­al costs. For ex­ample, about 1954 An­dré Weil noted an im­port­ant pa­per of a young man, Fe­lix E. Browder, on par­tial dif­fer­en­tial equa­tions; Browder came to vis­it and gave a talk. The de­part­ment pro­posed his ap­point­ment as as­sist­ant pro­fess­or, but the ad­min­is­tra­tion de­clined to act: they had ob­served that Browder’s fath­er, Earl Browder, had been head of the com­mun­ist party in the United States; in fact, Fe­lix had been born in Mo­scow. A de­cision on ap­point­ments on such shaky grounds would nev­er have happened while Hutchins was pres­id­ent (and in­deed Browder was sub­sequently ap­poin­ted to the fac­ulty). There are oth­er ex­amples of the in­tel­lec­tu­al in­eptitude of the Kimp­ton ad­min­is­tra­tion. Per­haps uni­versit­ies can­not main­tain great de­part­ments without out­stand­ing aca­dem­ic lead­er­ship at the top — lead­er­ship which was sub­sequently re­stored at Chica­go.

Requirements for good departments

On the basis of this and oth­er ex­amples, it is tempt­ing to spec­u­late: What does it take to make a great de­part­ment of math­em­at­ics?

  1. Out­stand­ing fac­ulty, prefer­ably young­er; in par­tic­u­lar, in­clud­ing some not on ten­ure.

  2. Nu­mer­ous lively stu­dents, help­ing to prod the fac­ulty.

  3. Ex­cit­ing fields of study, prefer­ably some new thrusts, and cer­tainly sev­er­al dif­fer­ent fields — per­haps even a clash of in­terests between fields.

  4. Sev­er­al in­struct­ors (e.g., postdoc­tor­als or tem­por­ary in­struct­ors), again bring­ing in new ideas.

  5. Act­ive con­tacts between people, e.g., col­loqui­ums, math­em­at­ics clubs, sem­inars, and (im­port­ant) meet­ings at tea.

  6. Un­der­stand­ing sup­port by the uni­versity ad­min­is­tra­tion.

  7. An act­ive sense of com­mon pur­pose.

These con­di­tions seem to me to have been met in the ex­amples of great de­part­ments which I per­son­ally know: Göt­tin­gen (1930–1933), Har­vard (1930–1960), Prin­ceton, and Chica­go (1897–1908, Moore; 1947–1960, Stone). When two or more of these con­di­tions fail, a de­part­ment can lose mo­mentum. When they are present, real ad­vance is pos­sible.

The Bliss department reviewed

Was this de­part­ment just a “dip­loma mill,” as as­ser­ted above, or are there oth­er as­pects? This will now be re­con­sidered in the light of Bill Duren’s re­col­lec­tions [e15] and the auto­bi­o­graph­ic­al notes of Bliss [e11] him­self, all as cited be­low. I have also profited from a con­sid­er­able dis­cus­sion with Her­man Gold­stine, who served as a re­search as­sist­ant to Bliss in the mid thirties, when Bliss was pre­par­ing his book on the cal­cu­lus of vari­ations.

In the peri­od 1920–1935 there were many gradu­ate stu­dents at Chica­go, and hence quite large ad­vanced classes; this is very dif­fer­ent from the present case when at Chica­go there may be 15 or 20 ad­vanced (post Mas­ter’s) courses offered in a giv­en quarter, with no re­quire­ment for a min­im­um num­ber of at­tendees. It is re­por­ted that in the twen­ties the de­part­ment of edu­ca­tion at Chica­go ar­ranged for spe­cial trains from Texas to bring the stu­dents for the sum­mer quarter. In some de­part­ments, it be­came the cus­tom for teach­ers else­where to come to Chica­go sum­mer after sum­mer so as to fi­nally ar­rive at a Ph.D., and in­deed this happened then in some cases in math­em­at­ics. It would be un­think­able now, only in part be­cause the sum­mer quarter has shrunk. Math­em­at­ics Ph.D.s from Chica­go were sta­tioned in in­flu­en­tial po­s­i­tions at uni­versit­ies throughout the Mid­w­est and the South, and of course they sent their best un­der­gradu­ate stu­dents to Chica­go for gradu­ate study. The activ­ity of the de­part­ment must be judged in the light of this massive in­put of stu­dents. Ac­cord­ing to Gold­stine, Bliss felt that there was in the United States a great need for well-trained teach­ers of math­em­at­ics, and that Chica­go was ideally placed to fill that need. In his auto­bi­o­graph­ic­al note, Bliss says that the mer­it of a de­part­ment of math­em­at­ics should not be rated by an in­dex such as the av­er­age num­ber of re­search pa­pers per Ph.D.; at Chica­go there were just too many stu­dents to ex­pect them all to do re­search. He im­plies that what really mat­ters is the re­search done by a few out­stand­ing stu­dents, while in the fac­ulty it­self what mat­ters most is the re­search done by a few out­stand­ing pro­fess­ors, such as Dick­son (whom he names). All this took place long be­fore the present wide­spread con­vic­tion that every de­part­ment mem­ber is ex­pec­ted to do re­search to get pro­mo­tions and gov­ern­ment grants. In de­part­ment meet­ings, Bliss of­ten de­pended for ad­vice on H. S. Ever­ett, whose form­al po­s­i­tion was that of ex­ten­sion pro­fess­or, and who was not in­ter­ested in re­search. Ever­ett was in­deed ef­fect­ive in help­fully cor­rect­ing stu­dent’s pa­pers in cor­res­pond­ence courses, and this activ­ity did in­deed bring stu­dents to Chica­go — for ex­ample, I. M. Sing­er, post W.W. II.

Bliss said: “The real pur­pose of gradu­ate work in math­em­at­ics, or in any oth­er sub­ject, is to train the stu­dent to re­cog­nize what men call the truth, and to give him what is usu­ally his first ex­per­i­ence in work­ing out the truth in some spe­cif­ic field.”

If gradu­ate work at Chica­go in this peri­od is judged on this basis, it must be ac­coun­ted a rous­ing suc­cess — as for ex­ample with the Ph.D.s to wo­men noted above.

The auto­bi­o­graph­ic­al note [e11] by Bliss also ex­hib­its the de­vel­op­ment of his in­terest in the cal­cu­lus of vari­ations. After study­ing math­em­at­ic­al as­tro­nomy with F. R. Moulton, he switched to math­em­at­ics and Bolza, and soon came across a copy of the 1879 ten­ure notes by Wei­er­strass on the cal­cu­lus of vari­ations. They were fas­cin­at­ing, as might well be, be­cause it was there that rig­or­ous proof was fi­nally brought to fruition in this cen­tur­ies-old sub­ject; the dis­sem­in­a­tion of such Wei­er­strass notes had a wide ef­fect. It may be that this ini­tial en­thu­si­asm was the lead­ing prin­ciple of all of his ca­reer — there he found ad­di­tion­al prob­lems in a more gen­er­al set­ting in which the truth could be teased out, and which stu­dents could handle. All these truths were brought to­geth­er in his treat­ise [e9], pub­lished at the end of his life, which can be viewed as a sys­tem­at­ic ex­ten­sion of the Wei­er­strass meth­od to all the vari­ants of the “prob­lem of Bolza.” Moreover, the ideas there were then ready to hand, so that when Pon­trja­gin and oth­ers much later saw that the cal­cu­lus of vari­ations was ad­ap­ted to the study of op­tim­al con­trol, Bliss’s stu­dent Hestenes brought it all to­geth­er in his 1966 book [e14].

In a re­cent is­sue of the Math­em­at­ic­al In­tel­li­gen­cer, I have ar­gued that many math­em­aticians today may spe­cial­ize so nar­rowly on their first re­search field that they miss im­port­ant con­nec­tions. This may not be new.

As noted above, Bliss be­came chair­man in 1927; I have ar­gued that there might then have been more widely spread ap­point­ments to the fac­ulty, with less em­phas­is on in­her­it­ance (and in­deed, Sanger may have been re­garded as the suc­cessor to Slaught). But in 1927 there may have been a dif­fer­ent ob­ject­ive: a new math­em­at­ics build­ing. Up un­til that time, the de­part­ment had been housed on the up­per floors of Ry­er­son, the phys­ics lab. Bliss laid the plans for con­struct­ing Eck­hart Hall next door as a build­ing for math­em­at­ics, with a fine com­mon room, cent­ral of­fices for math­em­at­ics and math­em­at­ic­al as­tro­nomy, ample fac­ulty of­fices, and even space for gradu­ate stu­dents. (In 1930, as a be­gin­ning gradu­ate stu­dent, I oc­cu­pied a fourth-floor of­fice which 40 years later served as the of­fice for a full pro­fess­or). Eck­hart Hall may well have set a pat­tern for math­em­at­ics build­ings; at any rate, it is re­por­ted that Os­wald Veblen in Prin­ceton kept track of Eck­hart as he planned for the con­struc­tion of Fine Hall for the Prin­ceton Math­em­at­ics de­part­ment.

These im­port­ant things said, I re­turn to my harsh judg­ment that by 1930 the de­part­ment at Chica­go had ceased to be really first class. This con­clu­sion is not so much based on the vari­ous items of evid­ence as­sembled above, but on my own dir­ect ex­per­i­ence.

In the fall of 1929, as a seni­or at Yale, I chanced to meet Robert Maynard Hutchins, re­cently law dean at Yale and newly pres­id­ent at Chica­go. He knew of my aca­dem­ic in­terests; find­ing that I in­ten­ded to study math­em­at­ics, he told me that Chica­go had an out­stand­ing de­part­ment, and that I should come there. Some weeks later, he wrote me to of­fer me a fel­low­ship in the (then hand­some) amount of \$1,000. I ac­cep­ted.

When I came in the fall of 1930, I at­ten­ded Moore’s sem­in­ar, as above, and signed up for courses with the lead­ing mem­bers of the de­part­ment:

Dick­son’s course on num­ber the­ory presen­ted a good treat­ment of the rep­res­ent­a­tion of in­tegers by quad­rat­ic forms, but there was no in­dic­a­tion of the con­nec­tions of this with al­geb­ra­ic num­ber fields, a sub­ject with which I had a passing ac­quaint­ance. Dick­son’s own cur­rent in­terests were in the com­pu­ta­tions for the War­ing prob­lem, but when Land­au came to give a vis­it­ing lec­ture, I could see that the cen­ter of in­terest was with the new ideas of ana­lyt­ic num­ber the­ory (Hardy–Lit­tle­wood, Vino­gradoff, and Land­au). I learned something about ap­prox­im­a­tions on ma­jor and minor arcs of the unit circle, but that was not a Chica­go sub­ject.

Lane en­dured on pro­ject­ive dif­fer­en­tial geo­metry. I had nev­er stud­ied dif­fer­en­tial geo­metry, nor did Lane teach it; this left a ser­i­ous gap in my back­ground, not even ad­equately filled when in 1933 Her­man Weyl’s warn­ing be­fore my or­al ex­am led me to bone up on the first and second quad­rat­ic forms of a sur­face. Des­pite this lack of back­ground, I took Lane’s course. I soon no­ticed an older stu­dent up in the front row with an open note­book in which he made only oc­ca­sion­al care­ful entries; even­tu­ally, I learned that he had taken the course once be­fore, and was now bring­ing his notes up to date with the latest re­fine­ments. At the time I was deeply of­fen­ded by this dis­play of ped­antry.

The cal­cu­lus of vari­ations with Bliss (two quar­ters) taught me all about the bra­chis­tro­chrone (I did not care) and about fields of ex­tremals (I did), but I did not really learn any­thing about the con­nec­tions with geo­met­ric­al op­tics (I found this out in Göt­tin­gen) or about the con­nec­tions with Hamilto­ni­an mech­an­ics, which I had to tease out later on my own. Bliss knew that there was Morse the­ory, but it was not taught at Chica­go.

When I signed up for a course in the philo­sophy de­part­ment with Mor­timer Adler, Bliss dis­ap­proved.

Barn­ard su­per­vised my M.A. thes­is, which was an un­suc­cess­ful at­tempt to dis­cov­er uni­ver­sal al­gebra. Barn­ard was then much taken up with Moore’s use of func­tions on a gen­er­al range, mean­ing func­tions \( X \to F \), where \( X \) is an ar­bit­rary set and \( F \) is a field — reals, com­plexes, or qua­ternions. Gold­stine and I both think that Moore’s em­phas­is on this “gen­er­al” idea may have blinded him to oth­er ax­io­mat­ic ap­proaches to func­tion­al ana­lys­is; I did not learn about Banach spaces un­til 1934 at Har­vard.

Moore him­self was in poor health.

At that time, Pres­id­ent Hutchins was be­gin­ning to press for his new col­lege de­voted to gen­er­al edu­ca­tion; Bliss and oth­er seni­or fac­ulty mem­bers strongly op­posed his ideas. This did not help the de­part­ment.

My con­clu­sions were not clearly for­mu­lated at that time, but they really came to this: The de­part­ment of math­em­at­ics at Chica­go in 1930–1931 was no longer out­stand­ing in at­ten­tion to cur­rent re­search. With Moore ill, there was no one on the fac­ulty un­der whose dir­ec­tion I would have liked to write a Ph.D. thes­is. I did not say this but simply put it that I had the the wan­der­lust and that I wanted to study lo­gic — so I took off for Göt­tin­gen. There I did in­deed find a great math­em­at­ics de­part­ment. I can still re­call the ex­cite­ment at the start of each new semester with many new courses at hand: Lie groups (Her­glotz), or group rep­res­ent­a­tions (Weyl), or Di­rich­let series (Land­au), or PDE (Lewy), or rep­res­ent­a­tion of al­geb­ras (No­eth­er), or lo­gic (Bernays). And there were many lively fel­low stu­dents (many more, and on a level hardly present at Chica­go): Ger­hard Gentzen, Fritz John, Hans Schwer­dtfeger, Kurt Schütte, Peter Sherk, Os­wald Teich­müller, and Ernst Witt, for ex­amples [e16].

The con­clu­sion seems to be that there are times when cer­tain de­vel­op­ments achieve a vi­brancy and ex­cite­ment with ample con­tacts with cur­rent de­part­ments which serve to stim­u­late fac­ulty and stu­dents alike. May this ana­lys­is per­haps help to en­cour­age more such cases.


Though the opin­ions voiced in this art­icle are my own, I have been much helped by in­cis­ive com­ments from Bill Duren, Her­man Gold­stine, E. J. Mc­Shane, Ivan Niven, Mina Rees, Arnold Ross, Alice Turn­er Schafer and George White­head. All of them had dir­ect ex­per­i­ence with the math­em­at­ics de­part­ment at Chica­go.