Celebratio Mathematica

On April 14 of this year, in the small town of Bryn Mawr (Pennsylvania, U.S.A.), Emmy No­eth­er, former pro­fess­or at Göttin­gen Uni­versity and one of the fore­most math­em­aticians of mod­ern times, died at the age of 53 fol­low­ing an op­er­a­tion.

The death of Emmy No­eth­er is not only a great loss for math­em­at­ic­al sci­ence, it is a tragedy in the full sense of the word. The greatest wo­man math­em­atician who ever lived died at the very height of her cre­at­ive powers; she died, driv­en from her home­land and torn from the sci­entif­ic school which she had cre­ated over the years and which had be­come one of the most bril­liant schools of math­em­at­ics in Europe; she died, torn from her fam­ily, which had been scattered to dif­fer­ent coun­tries be­cause of the same polit­ic­al bar­bar­ism that caused Emmy No­eth­er her­self to emig­rate from Ger­many.

We of the Mo­scow Math­em­at­ic­al So­ci­ety pay our re­spects today to the memory of one of our lead­ing mem­bers, who for more than ten years main­tained con­tinu­al and close ties, con­stant sci­entif­ic in­ter­ac­tion, and sin­cere good will and friend­ship with the So­ci­ety, with the Mo­scow math­em­at­ic­al com­munity, with the math­em­aticians of the So­viet Uni­on… Per­mit me in the name of the So­ci­ety to ex­press our deep­est con­dol­ences to the broth­er of the de­ceased, Pro­fess­or Fritz No­eth­er, formerly of the Tech­nis­che Hoch­schule in Bre­slau and cur­rently of the Tomsk Math­em­at­ic­al In­sti­tute, who is with us today.

The bio­graphy of Emmy No­eth­er is simple. She was born on March 23, 1882 in Er­lan­gen to the fam­ily of the fam­ous math­em­atician Max No­eth­er. Her math­em­at­ic­al tal­ent de­veloped slowly. A stu­dent of Gordan in Er­lan­gen, she de­fen­ded her dis­ser­ta­tion in 1907; her work was on Gordan’s form­al com­pu­ta­tion­al in­vari­ant the­ory. She of­ten al­luded to this dis­ser­ta­tion af­ter­wards, and al­ways re­ferred to it with dis­dain­ful epi­thets, such as “Formelngestriipp” [jungle of form­al­ism] and “Rech­nerei” [routine com­pu­ta­tions]. Des­pite all of this it should be noted that Emmy No­eth­er, the ar­dent foe of com­pu­ta­tion and al­gorithms in math­em­at­ics, her­self was fully cap­able of mas­ter­ing such meth­ods — this is proved not only by her first dis­ser­ta­tion, which in ac­tu­al fact was not a ma­jor work, but also by her sub­sequent pa­pers on dif­fer­en­tial in­vari­ants (1918), which have be­come clas­sics. But in these pa­pers we already see the fun­da­ment­al char­ac­ter­ist­ic of her math­em­at­ic­al tal­ent: the striv­ing for gen­er­al for­mu­la­tions of math­em­at­ic­al prob­lems and the abil­ity to find the for­mu­la­tion which re­veals the es­sen­tial lo­gic­al nature of the ques­tion, stripped of any in­cid­ent­al pe­cu­li­ar­it­ies which com­plic­ate mat­ters and ob­scure the fun­da­ment­al point.

She wrote the pa­pers on dif­fer­en­tial in­vari­ants when she was still in Göttin­gen, where she had moved in 1916. Her work of this peri­od was heav­ily in­flu­enced by Hil­bert. It is of­ten for­got­ten that in this peri­od Emmy No­eth­er ob­tained ex­cel­lent res­ults con­cern­ing the con­crete al­geb­ra­ic prob­lems of Hil­bert. These res­ults and her work on dif­fer­en­tial in­vari­ants would have been enough by them­selves to earn her the repu­ta­tion of a first class math­em­atician and are hardly less of a con­tri­bu­tion to math­em­at­ics than the fam­ous re­search of S. V. Ko­va­levskaya. But when we think of Emmy No­eth­er as a math­em­atician, we have in mind not these early works, im­port­ant though they were in their con­crete res­ults, but rather the main peri­od of her re­search, be­gin­ning in about 1920, when she be­came the cre­at­or of a new dir­ec­tion in al­gebra and the lead­ing, the most con­sist­ent and prom­in­ent rep­res­ent­at­ive of a cer­tain gen­er­al math­em­at­ic­al doc­trine — all that which is char­ac­ter­ized by the words “be­griff­liche Math­em­atik” [ab­stract math­em­at­ics].

Emmy No­eth­er her­self is partly re­spons­ible for the fact that her work of the early peri­od is rarely giv­en the at­ten­tion that it would nat­ur­ally de­serve. With the sin­gle­minded­ness that was part of her nature, she her­self was ready to for­get what she had done in the early years of her sci­entif­ic life, since she con­sidered those res­ults to have been a di­ver­sion from the main path of her re­search, which was the cre­ation of a gen­er­al, ab­stract al­gebra.

It is not my task to ana­lyze and il­lu­min­ate everything that Emmy No­eth­er did in math­em­at­ics. In the first place, not be­ing an al­geb­ra­ist, I do not con­sider my­self qual­i­fied to per­form such a task. In the second place, to the ex­tent ap­pro­pri­ate for an ob­it­u­ary, this task was per­formed in an ex­cel­lent and com­pletely com­pet­ent man­ner by Her­mann Weyl (in an ad­dress de­livered at a com­mem­or­ative meet­ing in hon­or of Emmy No­eth­er on April 26, 1935 at Bryn Mawr, Pennsylvania, U.S.A., and pub­lished in Scripta Math­em­at­ica, v. III, no. 3, June 1935) and by van der Waer­den (Math­em­at­ische An­nalen, v. III, 1935, p. 469). My pur­pose today is some­what dif­fer­ent: I would like to evoke for you as ac­cur­ate an im­age as pos­sible of the de­ceased, as a math­em­atician, as the head of a large sci­entif­ic school, as a bril­liant, ori­gin­al, and fas­cin­at­ing per­son­al­ity.

Emmy No­eth­er em­barked on her own com­pletely ori­gin­al math­em­at­ic­al path in the years 1919/1920. She her­self dated the be­gin­ning of this fun­da­ment­al peri­od in her work with the fam­ous joint work with W. Schmeidler (Math­em­at­ische Zeits­chrift, v. 8, 1920). In a sense this pa­per serves as a pro­logue to her gen­er­al the­ory of ideals, which she re­vealed in 1921 in the clas­sic mem­oir “Ideal­the­or­ie in Ringbereichen.” I be­lieve that, of everything that Emmy No­eth­er did, it is the found­a­tions of gen­er­al ideal the­ory and everything con­nec­ted with this that has had, is con­tinu­ing to have, and will have in the fu­ture the greatest im­pact on math­em­at­ics as a whole. Not only have these ideas already led to many spe­cif­ic fun­da­ment­al ap­plic­a­tions — for ex­ample, in van der Waer­den’s work on al­geb­ra­ic geo­metry — moreover, they have had an es­sen­tial in­flu­ence on al­geb­ra­ic think­ing it­self, and in cer­tain re­spects on gen­er­al math­em­at­ic­al think­ing, in our times. If the de­vel­op­ment of math­em­at­ics today is mov­ing for­ward un­der the sign of al­geb­ra­iz­a­tion, the pen­et­ra­tion of al­geb­ra­ic ideas and meth­ods in­to the most di­verse math­em­at­ic­al the­or­ies, then this only be­came pos­sible after the work of Emmy No­eth­er. It was she who taught us to think in terms of simple and gen­er­al al­geb­ra­ic con­cepts — homeo­morph­ic map­pings, groups and rings with op­er­at­ors, ideals — and not in terms of cum­ber­some al­geb­ra­ic com­pu­ta­tions; and thereby opened up the path to find­ing al­geb­ra­ic prin­ciples in places where such prin­ciples had been ob­scured by some com­plic­ated spe­cial situ­ation which was not at all suited for the ac­cus­tomed ap­proach of the clas­sic­al al­geb­ra­ists. The­or­ems such as the “homeo­morph­ism and iso­morph­ism the­or­em”, con­cepts such as the as­cend­ing and des­cend­ing chain con­di­tions for sub­groups and ideals, or the no­tion of groups with op­er­at­ors, were first in­tro­duced by Emmy No­eth­er and have entered in­to the daily prac­tice of a wide range of math­em­at­ic­al dis­cip­lines as a power­ful and con­stantly ap­plic­able tool, even though these dis­cip­lines may con­cern sub­jects which have no re­la­tion to the work of Emmy No­eth­er her­self. We need only glance at Pontry­agin’s work on the the­ory of con­tinu­ous groups, the re­cent work of Kolmogorov on the com­bin­at­or­i­al to­po­logy of loc­ally bicom­pact spaces, the work of Hopf on the the­ory of con­tinu­ous map­pings, to say noth­ing of van der Waer­den’s work on al­geb­ra­ic geo­metry, in or­der to sense the in­flu­ence of Emmy No­eth­er’s ideas. This in­flu­ence is also keenly felt in H. Weyl’s book Grup­pen­the­or­ie und Quant­en­mech­anik.

I have pur­posely in­dic­ated areas of math­em­at­ics which are dif­fer­ent from the dir­ect sub­ject of Emmy No­eth­er’s re­search; if one speaks of al­gebra it­self (in­clud­ing group the­ory), then we see that she cre­ated a ma­jor new dir­ec­tion in which a large num­ber of very tal­en­ted math­em­aticians work, con­tinu­ing the re­search of Emmy No­eth­er in many con­crete areas. In par­tic­u­lar, gen­er­al elim­in­a­tion the­ory and the the­ory of al­geb­ra­ic vari­et­ies are among the most sig­ni­fic­ant achieve­ments that have aris­en on the ground­work of Emmy No­eth­er’s gen­er­al ideal the­ory. In con­nec­tion with what I have just said, I would like to re­call that, among the ma­jor works of al­gebra that we have pro­duced here in Mo­scow over the last dec­ade, the fam­ous work of O. Yu. Schmidt on unique dir­ect product de­com­pos­i­tion of groups, and also the work of A. G. Kur­osh, were heav­ily in­flu­enced by Emmy No­eth­er.

Des­pite all of the con­crete and con­struct­ive ex­pli­cit res­ults of Emmy No­eth­er that came out of the vari­ous cre­at­ive peri­ods in her life, there can be no doubt that the fun­da­ment­al strength and the fun­da­ment­al dy­nam­ism of her tal­ent went in the dir­ec­tion of gen­er­al math­em­at­ic­al con­cep­tions hav­ing a marked ax­io­mat­ic col­or­a­tion. The present mo­ment is es­pe­cially op­por­tune for sub­ject­ing this side of her cre­ativ­ity to de­tailed ana­lys­is — not only be­cause the ques­tion of the gen­er­al and the spe­cial, the ab­stract and the con­crete, the ax­io­mat­ic and the con­struct­ive is now one of the sharpest ques­tions in our math­em­at­ic­al life. In­terest in this en­tire prob­lem is es­pe­cially per­tin­ent be­cause, on the one hand, the math­em­at­ic­al journ­als have un­doubtedly be­come over-burdened with a great num­ber of art­icles which gen­er­al­ize, ax­io­mat­ize and so on, of­ten without any con­crete math­em­at­ic­al con­tent; on the oth­er hand, from time to time one hears pro­clam­a­tions to the ef­fect that the only true math­em­at­ics is that which is “clas­sic­al.” This slo­gan sweeps aside im­port­ant math­em­at­ic­al prob­lems only be­cause they con­tra­dict one or an­oth­er habit of thought, be­cause they make use of con­cepts which were not in use a few dec­ades ago, such as, for ex­ample, the con­cepts of gen­er­al al­geb­ra­ic rings and fields, of to­po­lo­gic­al and func­tion spaces, and many oth­ers. In the ob­it­u­ary cited above, H. Weyl also poses this ques­tion. His words on this sub­ject so pen­et­rate to the heart of the mat­ter that I can­not re­frain from cit­ing them in their en­tirety:

In a con­fer­ence on to­po­logy and ab­stract al­gebra as two ways of math­em­at­ic­al un­der­stand­ing, in 1931, I said this:

Nev­er­the­less I should not pass over in si­lence the fact that today the feel­ing among math­em­aticians is be­gin­ning to spread that the fer­til­ity of these ab­stract­ing meth­ods is ap­proach­ing ex­haus­tion. The case is this: that all these nice gen­er­al no­tions do not fall in­to our laps by them­selves. But def­in­ite con­crete prob­lems were first conquered in their un­di­vided com­plex­ity, single­han­ded by brute force, so to speak. Only af­ter­wards the ax­io­maticians came along and stated: In­stead of break­ing in the door with all your might and bruis­ing your hands, you should have con­struc­ted such and such a key of skill, and by it you would have been able to open the door quite smoothly. But they can con­struct the key only be­cause they are able, after the break­ing in was suc­cess­ful, to study the lock from with­in and without. Be­fore you can gen­er­al­ize, form­al­ize and ax­io­mat­ize, there must be a math­em­at­ic­al sub­stance. I think that the math­em­at­ic­al sub­stance in the form­al­iz­ing of which we have trained ourselves dur­ing the last dec­ades, be­comes gradu­ally ex­hausted. And so I fore­see that the gen­er­a­tion now rising will have a hard time in math­em­at­ics.

“Emmy No­eth­er,” con­tin­ues Weyl “pro­tested against that: and in­deed she could point to the fact that just dur­ing the last years the ax­io­mat­ic meth­od had dis­closed in her hands new, con­crete, pro­found prob­lems… and had shown the way to their solu­tion.”

Much in this quo­ta­tion de­serves our at­ten­tion. In the first place, one can­not, of course, dis­pute the point of view that every ax­io­mat­ic present­a­tion of a math­em­at­ic­al the­ory must be pre­ceded by a con­crete, one might say a na­ive mas­tery of it; that, moreover, ax­io­mat­ics is only in­ter­est­ing when it relates to tan­gible math­em­at­ic­al know­ledge (what Weyl calls “math­em­at­ic­al sub­stance”), and is not tilt­ing at wind­mills, so to speak. All of this is in­dis­put­able, and, of course, it was not against this that Emmy No­eth­er pro­tested. What she pro­tested against was the pess­im­ism that shows through the last words of the quo­ta­tion from Weyl’s speech of 1931; the sub­stance of hu­man know­ledge, in­clud­ing math­em­at­ic­al know­ledge, is in­ex­haust­ible, at least for the fore­see­able fu­ture — this is what Emmy No­eth­er firmly be­lieved. The “sub­stance of the last dec­ades” may be ex­hausted, but not math­em­at­ic­al sub­stance in gen­er­al, which is con­nec­ted by thou­sands of in­tric­ate threads with the real­ity of the ex­tern­al world and hu­man ex­ist­ence. Emmy No­eth­er deeply felt this con­nec­tion between all great math­em­at­ics, even the most ab­stract, and the real world; even if she did not think this through philo­soph­ic­ally, she in­tu­ited it with all of her be­ing as a great sci­ent­ist and as a lively per­son who was not at all im­prisoned in ab­stract schemes. For Emmy No­eth­er math­em­at­ics was al­ways know­ledge of real­ity, and not a game of sym­bols; she pro­tested fer­vently whenev­er the rep­res­ent­at­ives of those areas of math­em­at­ics which are dir­ectly con­nec­ted with ap­plic­a­tions wanted to ap­pro­pri­ate for them­selves the claim to tan­gible know­ledge. In math­em­at­ics, as in know­ledge of the world, both as­pects are equally valu­able: the ac­cu­mu­la­tion of facts and con­crete con­struc­tions and the es­tab­lish­ment of gen­er­al prin­ciples which over­come the isol­a­tion of each fact and bring the fac­tu­al know­ledge to a new stage of ax­io­mat­ic un­der­stand­ing.

A pro­found feel­ing for real­ity lay at the found­a­tion of Emmy No­eth­er’s math­em­at­ic­al cre­ativ­ity; her en­tire sci­entif­ic per­son­al­ity op­posed the tend­ency (which is wide­spread in many math­em­at­ic­al circles) to trans­form math­em­at­ics in­to a game, in­to some sort of pe­cu­li­ar men­tal sport. In the nu­mer­ous con­ver­sa­tions I had with her about the nature of math­em­at­ic­al know­ledge and cre­ativ­ity, most of which were na­ive dis­cus­sions in the sense that we did not enter in­to a truly philo­soph­ic­al state­ment of the ques­tion, I of­ten — with her evid­ent sym­pathy — re­called my fa­vor­ite quo­ta­tion from Laplace: “Si l’homme s’était borné à re­cueil­lir des faits, la sci­ence ne serait qu’une no­men­clature stérile et ja­mais il n’eut con­nu les grandes lois de la nature.” [If man had lim­ited him­self to the ac­cu­mu­la­tion of facts, then sci­ence would have been merely a fruit­less no­men­clature, and he would nev­er have learned the great laws of nature.] These words, spoken by one of the most prom­in­ent rep­res­ent­at­ives of phys­ic­al know­ledge, by a sci­ent­ist who had both feet firmly on the ground, con­tain an en­tire pro­gram of in­ter­re­la­tions between the con­crete and the ab­stract in hu­man thought in gen­er­al, and in math­em­at­ics in par­tic­u­lar. And it seems to me that Emmy No­eth­er brought this pro­gram to real­iz­a­tion in her work.

In 1924–1925 Emmy No­eth­er’s school made one of its most bril­liant dis­cov­er­ies: the stu­dent van der Waer­den from Am­s­ter­dam. He was then 22 years old, and was one of the bright­est young math­em­at­ic­al tal­ents of Europe. Van der Waer­den quickly mastered Emmy No­eth­er’s the­or­ies, ad­ded sig­ni­fic­ant new res­ults, and, more than any­one else, helped to make her ideas widely known. The course on gen­er­al ideal the­ory which van der Waer­den gave in Göttin­gen in 1927 was tre­mend­ously suc­cess­ful. In van der Waer­den’s bril­liant present­a­tion, the ideas of Emmy No­eth­er over­whelmed the math­em­at­ic­al pub­lic, first in Göttin­gen and then in the oth­er ma­jor math­em­at­ic­al cen­ters of Europe. It is not sur­pris­ing that Emmy No­eth­er needed a pop­ular­izer for her ideas: her lec­tures were aimed at a small circle of stu­dents work­ing in the area of her re­search and listen­ing to her con­stantly. They were not at all suit­able for a broad math­em­at­ic­al audi­ence. To an out­sider Emmy No­eth­er seemed to lec­ture poorly, in a rap­id and con­fus­ing man­ner; but her lec­tures con­tained a tre­mend­ous force of math­em­at­ic­al thought and an ex­traordin­ary warmth and en­thu­si­asm. The same is true of her re­ports at meet­ings and con­gresses. Her lec­tures con­veyed very much to a math­em­atician who already un­der­stood her ideas and was in­ter­ested in her work; but a math­em­atician who was far from her re­search nor­mally would have great dif­fi­culty un­der­stand­ing her ex­pos­i­tion.

From 1927 on, as the in­flu­ence of Emmy No­eth­er’s ideas on mod­ern math­em­at­ics con­stantly in­creased, the schol­arly renown of the au­thor of those ideas was sim­il­arly in­creas­ing. Mean­while, the dir­ec­tion of her own work was chan­ging, mov­ing more and more to­ward the areas of non­com­mut­at­ive al­gebra, rep­res­ent­a­tion the­ory and the gen­er­al arith­met­ic of hy­per­com­plex sys­tems. The two most fun­da­ment­al works of her later peri­od show this trend: “Hy­per­kom­plexe Grossen und Darstel­lung­sthe­or­ie” (1929) and “Nichtkom­mut­at­ive Al­gebra” (1933), both pub­lished in Math­em­at­ische Zeits­chrift (vol. 30 and 37). These and re­lated pa­pers im­me­di­ately pro­duced nu­mer­ous re­ver­ber­a­tions in the work of al­geb­ra­ic num­ber the­or­ists, es­pe­cially Hasse. Her best known stu­dent of this peri­od was M. Deur­ing, who pub­lished a book “Al­gebren” in Ergeb­n­isse der Math­em­atik, where he gave a sur­vey of Emmy No­eth­er’s work on hy­per­com­plex sys­tems. In ad­di­tion, her school in­cluded many young math­em­aticians who were start­ing their ca­reers, such as Witt, Fit­ting, and oth­ers.

Emmy No­eth­er lived to see the full re­cog­ni­tion of her ideas. If in 1923–1925 she was striv­ing to prove the im­port­ance of the the­or­ies she was de­vel­op­ing, in 1932, at the In­ter­na­tion­al Con­gress of Math­em­aticians in Zurich, her ac­com­plish­ments were lauded on all sides. The ma­jor sur­vey talk that she gave at the Con­gress was a true tri­umph for the dir­ec­tion of re­search she rep­res­en­ted. At that point she could look back upon the math­em­at­ic­al path she had traveled not only with an in­ner sat­is­fac­tion, but with an aware­ness of her com­plete and un­con­di­tion­al re­cog­ni­tion in the math­em­at­ic­al com­munity. The Con­gress in Zurich marked the high point of her in­ter­na­tion­al sci­entif­ic po­s­i­tion. A few months later, Ger­man cul­ture and, in par­tic­u­lar, Göttin­gen Uni­versity, which had nur­tured that cul­ture for cen­tur­ies, were struck by the cata­strophe of the fas­cist takeover, which in a mat­ter of weeks scattered to the winds everything that had been painstak­ingly cre­ated over the years. What oc­curred was one of the greatest tra­gedies that had be­fallen hu­man cul­ture since the time of the Renais­sance, a tragedy which only a few years ago had seemed im­possible in twen­ti­eth-cen­tury Europe. One of its many vic­tims was Emmy No­eth­er’s Göttin­gen school of al­geb­ra­ists. The lead­er of the school was driv­en from the halls of the uni­versity; hav­ing lost the right to teach, Emmy No­eth­er was forced to emig­rate from Ger­many. She ac­cep­ted an in­vit­a­tion from the wo­men’s col­lege in Bryn Mawr (1933), where she lived for the last year and a half of her life.

Emmy No­eth­er’s ca­reer was full of para­doxes, and will al­ways stand as an ex­ample of shock­ing stag­nancy and in­ab­il­ity to over­come pre­ju­dice on the part of the Prus­si­an aca­dem­ic and civil ser­vice bur­eau­cra­cies. Her ap­point­ment as Privat­dozent in 1919 was only pos­sible be­cause of the per­sist­ence of Hil­bert and Klein, who over­came some ex­treme op­pos­i­tion from re­ac­tion­ary uni­versity circles. The ba­sic form­al ob­jec­tion was the sex of the can­did­ate: “How can we al­low a wo­man to be­come a Privat­dozent: after all, once she is a Privat­dozent, she may be­come a Pro­fess­or and mem­ber of the Uni­versity Sen­ate; is it per­miss­ible for a wo­man to enter the Sen­ate?” This pro­voked Hil­bert’s fam­ous reply: “Meine Her­ren, der Sen­at ist ja keine Bade­nan­stalt, war­um darf eine Frau nicht dor­thin!” [Gen­tle­men, the Sen­ate is not a bath­house, so I do not see why a wo­man can­not enter it!] In ac­tu­al fact, the op­pos­i­tion of in­flu­en­tial rep­res­ent­at­ives of re­ac­tion­ary aca­dem­ic circles was caused not only and not even primar­ily be­cause Emmy No­eth­er was a wo­man, but by her well-known and very rad­ic­al polit­ic­al be­liefs, along with the ag­grav­at­ing cir­cum­stance in their eyes of her Jew­ish na­tion­al­ity. But I shall re­turn to this later.

Even­tu­ally she re­ceived the ap­point­ment as Privat­dozent, and later as hon­or­ary Pro­fess­or; as a res­ult of Cour­ant’s ef­forts she re­ceived a so-called Lehrauftrag, i.e., a small salary (200–400 marks per month) for her lec­tures, which re­quired re­con­firm­a­tion every year by the Min­istry. It was in this po­s­i­tion, without even a guar­an­teed salary, that she lived un­til the mo­ment she was dis­missed from the uni­versity and forced to leave Ger­many. She was not a mem­ber of a single academy, in­clud­ing the academy of the city whose uni­versity was the set­ting for all of her re­search. Here is what H. Weyl writes about this in his ob­it­u­ary:

When I was called per­man­ently to Göttin­gen in 1930,1

I earn­estly tried to ob­tain from the Min­is­teri­um a bet­ter po­s­i­tion for her, be­cause I was ashamed to oc­cupy such a pre­ferred po­s­i­tion be­side her whom I knew to be my su­per­i­or as a math­em­atician in many re­spects. I did not suc­ceed, nor did an at­tempt to push through her elec­tion as a mem­ber of the Göttinger Gesell­schaft der Wis­senschaften.2

Tra­di­tion, pre­ju­dice, ex­tern­al con­sid­er­a­tions, weighted the bal­ance against her sci­entif­ic mer­its and sci­entif­ic great­ness, by that time denied by no one. In my Göttin­gen years, 1930–1933, she was without doubt the strongest cen­ter of math­em­at­ic­al activ­ity there, con­sid­er­ing both the fer­til­ity of her sci­entif­ic re­search pro­gram and her in­flu­ence upon a large circle of pu­pils.

Emmy No­eth­er had close ties to Mo­scow. Her con­nec­tion with Mo­scow began in 1923, when Pavel Samuilovich Uryso­hn who has now also passed away, and I first went to Göttin­gen and im­me­di­ately found ourselves in the math­em­at­ic­al circle led by Emmy No­eth­er. The ba­sic fea­tures of the No­eth­er school struck us right away: the in­tel­lec­tu­al en­thu­si­asm of its lead­er, which was trans­mit­ted to all of her stu­dents, her deep con­vic­tion in the im­port­ance and math­em­at­ic­al fer­til­ity of her ideas (a con­vic­tion which far from every­one shared at that time, even in Göttin­gen), and the ex­traordin­ary sim­pli­city and warmth of the re­la­tions between the head of the school and her pu­pils. At that time the school con­sisted al­most en­tirely of young Göttin­gen stu­dents; the peri­od when it be­came in­ter­na­tion­al in its com­pos­i­tion and was re­cog­nized as the most im­port­ant cen­ter of al­geb­ra­ic thought in terms of its in­ter­na­tion­al im­pact, was still in the fu­ture.

The math­em­at­ic­al in­terests of Emmy No­eth­er (who was then at the height of her work on gen­er­al ideal the­ory) and the math­em­at­ic­al in­terests of Uryson and my­self, which were then centered around prob­lems in so-called ab­stract to­po­logy, had many points of con­tact with one an­oth­er, and soon brought us to­geth­er in con­stant, al­most daily math­em­at­ic­al dis­cus­sions. Emmy No­eth­er was not, however, only in­ter­ested in our to­po­lo­gic­al work; she was in­ter­ested in everything math­em­at­ic­al (and not only math­em­at­ic­al!) that was be­ing done in So­viet Rus­sia. She did not hide her sym­pathy to­ward our coun­try and its so­cial and gov­ern­ment­al struc­ture, des­pite the fact that such ex­pres­sions of sym­pathy were con­sidered shock­ing and im­prop­er by most rep­res­ent­at­ives of West­ern European aca­dem­ic circles. It went so far that Emmy No­eth­er was lit­er­ally ex­pelled from one of the Göttin­gen board­ing-houses (where she lived and dined) at the in­sist­ence of the stu­dent boarders, who did not want to live un­der the same roof as a “pro-Marx­ist Jew” — an ex­cel­lent pro­logue to the drama that came at the end of her life in Ger­many.

And Emmy No­eth­er was sin­cerely glad of the sci­entif­ic and math­em­at­ic­al suc­cesses of the So­viet Uni­on, since she saw in this a de­cis­ive re­fut­a­tion of all of the prattle about how “the Bolshev­iks are des­troy­ing cul­ture,” and she felt the ap­proach­ing blos­som­ing of a great new cul­ture. Though the rep­res­ent­at­ive of one of the most ab­stract areas of math­em­at­ics, she dis­tin­guished her­self with an amaz­ing sens­it­iv­ity in un­der­stand­ing the great his­tor­ic trans­form­a­tions of our times; she al­ways had a lively in­terest in polit­ics, with all of her be­ing she hated war and chau­vin­ism in all of its mani­fest­a­tions, in this area she nev­er wavered. Her sym­path­ies were al­ways stead­fastly with the So­viet Uni­on, where she saw the be­gin­ning of a new era in the his­tory of man­kind and a firm sup­port for everything pro­gress­ive in hu­man thought. This fea­ture was such a shin­ing as­pect of Emmy No­eth­er’s char­ac­ter, it left such a deep im­print on her en­tire per­son­al­ity, that to be si­lent about it would sig­ni­fy a tenden­tious dis­tor­tion of Emmy No­eth­er’s nature as a sci­ent­ist and as a per­son.

The sci­entif­ic and per­son­al friend­ship between Emmy No­eth­er and me which star­ted in 1923, con­tin­ued un­til her death. Re­fer­ring to this friend­ship, Weyl says in his ob­it­u­ary: “She held a rather close friend­ship with Al­ex­an­drov in Mo­scow. I be­lieve that her mode of think­ing has not been without in­flu­ence upon Al­ex­an­drov’s to­po­lo­gic­al in­vest­ig­a­tions.” I am happy to take this op­por­tun­ity to con­firm the ac­cur­acy of Weyl’s sup­pos­i­tion. Emmy No­eth­er’s in­flu­ence on my own and on oth­er to­po­lo­gic­al re­search in Mo­scow was very great and af­fected the very es­sence of our work. In par­tic­u­lar, my the­ory of con­tinu­ous par­ti­tions of to­po­lo­gic­al spaces arose to a large ex­tent un­der the in­flu­ence of con­ver­sa­tions with her in Decem­ber to Janu­ary of 1925–1926, when we were both in Hol­land. On the oth­er hand, this was also the time when Emmy No­eth­er’s first ideas on the set the­or­et­ic found­a­tions of group the­ory arose, serving as the sub­ject for her course of lec­tures in the sum­mer of 1926. In their ori­gin­al form these ideas were not de­veloped fur­ther, but later she re­turned to them sev­er­al times. The reas­on for this delay is prob­ably the dif­fi­culty in­volved in ax­io­mat­iz­ing the no­tion of a group start­ing from its par­ti­tion in­to cosets as the fun­da­ment­al concept. But the idea of a set-the­or­et­ic ana­lys­is of the concept of a group it­self turned out to be fruit­ful, as shown by the re­cent work of Ore, Kur­osh, and oth­ers.

Sub­sequent years saw a strength­en­ing and deep­en­ing of Emmy No­eth­er’s to­po­lo­gic­al in­terests. In the sum­mers of 1926 and 1927 she went to the courses on to­po­logy which Hopf and I gave at Göttin­gen. She rap­idly be­came ori­ented in a field that was com­pletely new for her, and she con­tinu­ally made ob­ser­va­tions, which were of­ten deep and subtle. When in the course of our lec­tures she first be­came ac­quain­ted with a sys­tem­at­ic con­struc­tion of com­bin­at­or­i­al to­po­logy, she im­me­di­ately ob­served that it would be worth­while to study dir­ectly the groups of al­geb­ra­ic com­plexes and cycles of a giv­en poly­hed­ron and the sub­group of the cycle group con­sist­ing of cycles ho­mo­log­ous to zero; in­stead of the usu­al defin­i­tion of Betti num­bers and tor­sion coef­fi­cients, she sug­ges­ted im­me­di­ately de­fin­ing the Betti group as the com­ple­ment­ary (quo­tient) group of the group of all cycles by the sub­group of cycles ho­mo­log­ous to zero. This ob­ser­va­tion now seems self-evid­ent. But in those years (1925–1928) this was a com­pletely new point of view, which did not im­me­di­ately en­counter a sym­path­et­ic re­sponse on the part of many very au­thor­it­at­ive to­po­lo­gists. Hopf and I im­me­di­ately ad­op­ted Emmy No­eth­er’s point of view in this mat­ter, but for some time we were among a small num­ber of math­em­aticians who shared this view­point. These days it would nev­er oc­cur to any­one to con­struct com­bin­at­or­i­al to­po­logy in any way oth­er than through the the­ory of abeli­an groups; it is thus all the more fit­ting to note that it was Emmy No­eth­er who first had the idea of such a con­struc­tion. At the same time she no­ticed how simple and trans­par­ent the proof of the Euler–Poin­care for­mula be­comes if one makes sys­tem­at­ic use of the concept of a Betti group. Her re­marks in this con­nec­tion in­spired Hopf com­pletely to re­work his ori­gin­al proof of the well-known fixed point for­mula, dis­covered by Lef­schetz in the case of man­i­folds and gen­er­al­ized by Hopf to the case of ar­bit­rary poly­hedra. Hopf’s work “Eine Ver­allge­meiner­ung der Euler–Poin­caréschen Formel,” pub­lished in Gat­tinger Na­chricht­en in 1928, bears the im­print of these re­marks of Emmy No­eth­er.

Emmy No­eth­er spent the winter of 1928–1929 in Mo­scow. She gave a course on ab­stract al­gebra at Mo­scow Uni­versity and led a sem­in­ar on al­geb­ra­ic geo­metry in the Com­mun­ist Academy. She quickly es­tab­lished con­tact with the ma­jor­ity of Mo­scow math­em­aticians, in par­tic­u­lar, with L. S. Pontry­agin and O. Yu. Schmidt. It is not hard to fol­low the in­flu­ence of Emmy No­eth­er on the de­vel­op­ing math­em­at­ic­al tal­ent of Pontry­agin; the strong al­geb­ra­ic fla­vor in Pontry­agin’s work un­doubtedly profited greatly from his as­so­ci­ation with Emmy No­eth­er. In Mo­scow Emmy No­eth­er read­ily fa­mil­i­ar­ized her­self with our life, both sci­entif­ic and day-to-day. She lived in a mod­est room in the KSU dorm­it­ory near the Krym­skii Bridge, and usu­ally walked to the Uni­versity. She was very in­ter­ested in the life of our coun­try, es­pe­cially the life of So­viet youth and the stu­dents.

That winter of 1928–1929 I made fre­quent trips to Smolensk, where I gave lec­tures on al­gebra at the Ped­ago­gic­al In­sti­tute. In­spired by con­tinu­al con­ver­sa­tions with Emmy No­eth­er, that year I gave my lec­tures in her field. Among my listen­ers A. G. Kur­osh im­me­di­ately stood out; the the­or­ies I was present­ing, which were im­bued with the ideas of Emmy No­eth­er, ap­pealed to his spir­it. In this way, through my teach­ing, Emmy No­eth­er ac­quired an­oth­er stu­dent, who has since, as we all know, grown in­to an in­de­pend­ent sci­ent­ist, whose work is still largely con­cerned with the circle of ideas cre­ated by Emmy No­eth­er.

In the spring of 1929 she left Mo­scow for Göttin­gen with the firm in­ten­tion of pay­ing us a re­turn vis­it with­in the next few years. Sev­er­al times she came close to real­iz­ing this in­ten­tion, es­pe­cially in the last year of her life. After her ex­ile from Ger­many, she ser­i­ously con­sidered fi­nally set­tling in Mo­scow, and I had a cor­res­pond­ence with her on this ques­tion. She clearly un­der­stood that nowhere else were there such pos­sib­il­it­ies of cre­at­ing a bril­liant new math­em­at­ic­al school to re­place the one that was taken from her in Göttin­gen. And I had already been ne­go­ti­at­ing with Narkom­pros about ap­point­ing her to a chair in al­gebra at Mo­scow Uni­versity. But, as it hap­pens, Narkom­pros delayed in mak­ing the de­cision and did not give me a fi­nal an­swer. Mean­while time was passing, and Emmy No­eth­er, de­prived even of the mod­est salary which she had had in Göttin­gen, could not wait, and had to ac­cept the in­vit­a­tion from the wo­men’s col­lege in the Amer­ic­an town of Bryn Mawr.

With the death of Emmy No­eth­er I lost the ac­quaint­ance of one of the most cap­tiv­at­ing hu­man be­ings I have ever known. Her ex­traordin­ary kind­ness of heart, ali­en to any af­fect­a­tion or in­sin­cer­ity; her cheer­ful­ness and sim­pli­city; her abil­ity to ig­nore everything that was un­im­port­ant in life — cre­ated around her an at­mo­sphere of warmth, peace and good will which could nev­er be for­got­ten by those who as­so­ci­ated with her. But her kind­ness and gen­tle­ness nev­er made her weak or un­able to res­ist evil. She had her opin­ions and was able to ad­vance them with great force and per­sist­ence. Though mild and for­giv­ing, her nature was also pas­sion­ate, tem­pera­ment­al, and strong-willed; she al­ways stated her opin­ions forth­rightly, and did not fear ob­jec­tions. It was mov­ing to see her love for her stu­dents, who com­prised the ba­sic mi­lieu in which she lived and re­placed the fam­ily she did not have. Her con­cern for her stu­dents’ needs, both sci­entif­ic and worldly, her sens­it­iv­ity and re­spons­ive­ness, were rare qual­it­ies. Her great sense of hu­mor, which made both her pub­lic ap­pear­ances and in­form­al as­so­ci­ation with her es­pe­cially pleas­ant, en­abled her to deal lightly and without ill will with all of the in­justices and ab­surdit­ies which be­fell her in her aca­dem­ic ca­reer. In­stead of tak­ing of­fense in these situ­ations, she laughed. But she took ex­treme of­fense and sharply pro­tested whenev­er the least in­justice was done to one of her stu­dents. The en­tire reser­voir of her ma­ter­nal feel­ings went to them!

So­ci­able, good-willed and simple in re­la­tions with oth­ers, she was able to com­bine ex­pans­ive­ness with a cer­tain calmness and the ab­sence of any van­ity. Glory-seek­ing and the pur­suit of worldly suc­cess were ali­en to her. But she knew her worth, and fought for sci­entif­ic in­flu­ence.

In her house — more pre­cisely, in the mansard-roofed apart­ment she oc­cu­pied in Göttin­gen (Fried­lander­weg 57) — a large group would get to­geth­er eagerly and of­ten. People of di­verse schol­arly repu­ta­tions and po­s­i­tions — from Hil­bert, Land­au, Brauer and Weyl to the young­est stu­dents — would gath­er at her home and feel re­laxed and un­con­strained, as in few oth­er sci­entif­ic salons in Europe. These “fest­ive even­ings” in her apart­ment were ar­ranged on any pos­sible oc­ca­sion; for ex­ample, in the sum­mer of 1927 it was the fre­quent vis­its of her stu­dent van der Waer­den from Hol­land. The even­ings at Emmy No­eth­er’s, and the walks with her out­side town, were a shin­ing and un­for­get­table fea­ture of the math­em­at­ic­al life of Göttin­gen for an en­tire dec­ade (1923–1932). Many lively math­em­at­ic­al con­ver­sa­tions were held dur­ing these even­ings, but there was also much gaiety and laughter, good Rhine wine would some­times be on the table and many sweets would be con­sumed…

Such was Emmy No­eth­er, the greatest of wo­men math­em­aticians, a lead­ing sci­ent­ist, won­der­ful teach­er and un­for­get­table per­son. She did not have the char­ac­ter­ist­ics of the so-called “wo­man schol­ar” or “blue stock­ing.” To be sure, Weyl said in his ob­it­u­ary, “No one could con­tend that the Graces had stood by her cradle,” and he is right, if we have in mind her well-known heavy build. But at this point Weyl is speak­ing of her not only as a ma­jor sci­ent­ist, but as a ma­jor wo­man! And this she was — her fem­in­ine psyche came through in the gentle and del­ic­ate lyr­i­cism that lay at the found­a­tion of the wide-ran­ging but nev­er su­per­fi­cial re­la­tion­ships con­nect­ing her with people, with her avoca­tion, with the in­terests of all man­kind. She loved people, sci­ence, life with all the warmth, all the joy, all the self­less­ness and all the ten­der­ness of which a deeply feel­ing heart — and a wo­man’s heart — was cap­able.