Celebratio Mathematica

Saunders Mac Lane

Category theory  ·  U Chicago

Saunders Mac Lane

by Rob Kirby

This brief bio­graphy is based mostly on Mac Lane’s own A math­em­at­ic­al auto­bi­o­graphy [6], a manuscript “mostly com­pleted” by Saun­ders him­self, but “ex­tens­ively de­veloped” by Janet Beis­sen­ger in con­ver­sa­tions with Saun­ders in his last years, as well as some oth­er math­em­aticians (see [6], ac­know­ledg­ments). It is also writ­ten from the per­spect­ive of a to­po­lo­gist; for a dif­fer­ent view see the ex­cel­lent art­icles by Colin McLarty on Saun­ders’ philo­soph­ic work [e1], [e2], [e3]. I have ad­ded re­l­at­ively little, based on my lim­ited per­son­al know­ledge of Saun­ders and on some in­form­a­tion from oth­er math­em­aticians, e.g., Dav­id Eis­en­bud. The lat­ter wrote a pre­face to the auto­bi­o­graphy which is re­pro­duced in this volume.


The McLean clan came from the High­lands of Scot­land, near Castle Duart over­look­ing the Straits of Mull. The clan was de­feated by the Brit­ish in 1746 in the Battle of Cul­loden (the last pitched battle fought on Brit­ish soil), and even­tu­ally Saun­ders’ an­cest­ors came to west­ern Pennsylvania and Ohio in the early 1800s. Saun­ders’ grand­fath­er, Wil­li­am Ward McLane, born in 1846, be­came a Pres­by­teri­an min­is­ter, and then was charged with heresy due to preach­ing about Charles Dar­win. He es­caped to New Haven, Con­necti­c­ut, and be­came pas­tor of a Pres­by­teri­an church.

Saun­ders’ fath­er, Don­ald McLane, born 1882, stud­ied at Yale and the Uni­on Theo­lo­gic­al Sem­in­ary in New York, and be­came a Con­greg­a­tion­al­ist min­is­ter. He mar­ried Wini­fred Saun­ders in 1908, and Leslie Saun­ders MacLane was born on 4 Au­gust 1909.

His first name, Leslie, was quickly dropped and, from one month of age, Saun­ders had only two names. McLean had morph­ed in­to MacLane a gen­er­a­tion earli­er in or­der to not sound Ir­ish, and MacLane got a space, Mac Lane, when Saun­ders’ wife Dorothy, who typed his PhD thes­is and later pa­pers, found it easi­er to type Mac Lane with a space.

Ex­cept for a few years in Bo­ston around age sev­en, Saun­ders grew up in small towns, which he pre­ferred. He star­ted high school in Utica, New York, where his fath­er fell ill (pos­sibly due to the in­flu­enza epi­dem­ic of 1919), and even­tu­ally died when Saun­ders was 14. The fam­ily then lived with Saun­ders’ wid­owed grand­fath­er where Saun­ders got to know his uncles, one of whom de­cided to send Saun­ders to Yale and fund him at the princely amount of \$1,200 per year. (In­cid­ent­ally, I had a schol­ar­ship for the same \$1,200 per year, nearly 30 years later at the Uni­versity of Chica­go, which still covered most of my ex­penses.)

En­ter­ing Yale in 1926, Saun­ders thought about his fin­an­cial fu­ture and figured he needed to save \$100,000, provid­ing \$4,000 per year for old age — a suf­fi­cient sum, he thought. He wanted an in­tel­lec­tu­al ca­reer, and chose chem­istry for its uses in in­dustry. But then he in­quired about a ca­reer in math and was told that one could be­come an ac­tu­ary, so he switched his ma­jor to math. A ca­reer in math­em­at­ic­al re­search had not yet oc­curred to him.

A young Yale pro­fess­or, Filmer Northrop, who had just stud­ied with Al­fred North White­head at Har­vard, brought Rus­sell and White­head’s three-volume Prin­cipia Math­em­at­ica to Saun­ders’ at­ten­tion. He read and an­not­ated the first volume, and a life-long in­terest in lo­gic, found­a­tions and some philo­sophy was born.

Saun­ders was still of a mind to ac­quire know­ledge, not to cre­ate it. But in his seni­or year, Øys­tein Ore came to Yale bring­ing no­tions from Emmy No­eth­er’s school of ab­stract al­gebra in Göt­tin­gen. Ore’s lec­tures and the read­ing of Otto Haupt’s very ab­stract text in al­gebra, led Saun­ders to shift to the world of dis­cov­er­ing know­ledge, as op­posed to just ac­quir­ing know­ledge that oth­ers had dis­covered.

Saun­ders had been an ex­cel­lent stu­dent at Yale, ap­par­ently hav­ing the highest GPA ever. He re­ceived an award at an event in 1929 at which Robert Maynard Hutchins, just ap­poin­ted Pres­id­ent of the Uni­versity of Chica­go in 1929 (at the age of 30 after be­ing Dean at the Yale Law School for two years), also re­ceived an award. When Hutchins met Saun­ders, he promptly re­cruited him to start gradu­ate school at Chica­go with a \$1,000 fel­low­ship. Ore was not pleased to hear that Saun­ders chose Chica­go, point­ing out that Har­vard or Prin­ceton (or pre­sum­ably Yale) were bet­ter choices than Chica­go.

Non­ethe­less, Saun­ders went to Chica­go, dis­cov­er­ing upon ar­rival that he had not been ad­mit­ted be­cause Hutchins had not in­formed the math de­part­ment of his de­cision. But ad­mis­sion was quickly ar­ranged.

Saun­ders was dis­ap­poin­ted in his first year at Chica­go, where he per­ceived too much em­phas­is on push­ing stu­dents to com­plete a thes­is as a one-time piece of re­search be­fore go­ing out in the real world, rather than as a pre­par­a­tion for fu­ture re­search.

However there were pos­it­ive as­pects to his year at Chica­go: study with E. H. Moore lead­ing to a sem­in­ar talk on the Zer­melo–Fraen­kel ax­ioms for set the­ory; read­ing the most ac­cess­ible avail­able source for to­po­logy, Veblen’s Ana­lys­is sit­us, where he learned about Betti num­bers, but not that there were con­cepts like ho­mo­logy groups; a Mas­ter’s thes­is in ab­stract al­gebra where he tried to ax­io­mat­ize ex­po­nen­tials; and most im­port­antly, meet­ing Dorothy Jones whom he later mar­ried in 1933. Saun­ders saw little op­por­tun­ity to write a PhD thes­is in lo­gic at Chica­go, so he ap­plied for, and re­ceived, a fel­low­ship to study in Göt­tin­gen.

Saun­ders spent the aca­dem­ic years 1931–32 and 1932–33 at the Uni­versity of Göt­tin­gen. He ex­per­i­enced Göt­tin­gen in its full glory, and then in its dénoue­ment in 1933.

Dav­id Hil­bert, al­though re­tired, lec­tured once a week dur­ing term; Ed­mund Land­au gave care­ful, pre­cise, but un­mo­tiv­ated lec­tures on Di­rich­let series; Emmy No­eth­er in­flu­enced Saun­ders greatly with her ax­io­mat­ic view of al­gebra which fit­ted his in­ter­ested in lo­gic and found­a­tions; Paul Bernays be­came Saun­ders’s thes­is ad­viser; Richard Cour­ant was the ad­min­is­trat­ive head of the Math­em­at­ics In­sti­tute; Her­glotz held the chair in ap­plied math­em­at­ics.

In ad­di­tion to these ten­ured math­em­aticians there were many as­sist­ants and gradu­ate stu­dents such as Hans Lewy, Franz Rel­lich, Ernst Witt, Os­wald Teich­müller and Fritz John.

The Nazis came to power in Feb­ru­ary 1933, and Hitler be­came Chan­cel­lor. On 7 April 1933, a law was passed (Law for the Res­tor­a­tion of the Pro­fes­sion­al Civil Ser­vice) that all Jew­ish fac­ulty mem­bers at uni­versit­ies (all state sup­por­ted) be dis­missed im­me­di­ately, ex­cept for a few with either 25 years of ser­vice or who had fought in the Ger­man army in World War I.

Cour­ant was im­me­di­ately fired. Land­au, hav­ing served in World War I, was spared, but left with­in months after his classes were boy­cot­ted by Nazi stu­dents. No­eth­er went to Bryn Mawr Col­lege. Bernays went to Ei­dgenöss­is­che Tech­nis­che Hoch­schule in Zürich. Weyl left for the In­sti­tute for Ad­vanced Study (he had turned them down a year earli­er but they re­newed the of­fer). Levy and many young­er math­em­aticians left. The only full pro­fess­or re­main­ing was Her­glotz.

This is Saun­ders’ de­scrip­tion of this dénoue­ment:

Pri­or to 1933, the at­mo­sphere at Göt­tin­gen crackled with en­thu­si­asm for math­em­at­ic­al ideas and dis­cov­er­ies. It was a re­mark­able ex­hib­it of the uni­versity as an am­al­gam of re­search, teach­ing, and in­spir­a­tion. But the glory of math­em­at­ics at Göt­tin­gen came to an ab­rupt end when Hitler came to power. The Nazi gov­ern­ment wrecked what had been the most prom­in­ent and in­flu­en­tial cen­ter of math­em­at­ics in the world.

Saun­ders had an­oth­er year on his Hum­boldt fel­low­ship, but now rushed to fin­ish his thes­is. Bernays, his real ad­viser, had left so it was Weyl who had to ap­prove the thes­is. He did so, but gave it the low­est passing grade. Two days after Saun­ders had passed (very well) his ma­jor and two minor doc­tor­al ex­ams, he and Dorothy were wed in a quiet ce­re­mony at City Hall (21 Ju­ly 1933). (Co­in­cid­ent­ally, at City Hall they met Fritz John and his non-Jew­ish girl­friend who were get­ting mar­ried hast­ily be­fore such mar­riages were out­lawed; Saun­ders and Dorothy were wit­nesses and were sworn to secrecy, and the Johns then left and traveled sep­ar­ately to New York.)

After Göt­tin­gen, Saun­ders bounced around, first at Yale as a postdoc (1933–34), then Har­vard (1934–36), Cor­nell (1936–37), and fi­nally Chica­go (1937–38). Dur­ing those years, Saun­ders’ re­search seems, ret­ro­spect­ively, to be some­what un­focused. There was some lo­gic; some al­gebra, partly un­der the in­flu­ence of Øys­tein Ore at Yale; and some to­po­logy. The lat­ter had be­gun with the study of Haus­dorff’s book Men­gen­lehre as a Yale un­der­gradu­ate, then, with the study of Ana­lys­is sit­us at Chica­go, next with Hassler Whit­ney (who was work­ing on the 4-col­or prob­lem and graphs) at Har­vard, and then with V. W. Adkisson at the Uni­versity of Arkan­sas (where Saun­ders and Dorothy were vis­it­ing her re­l­at­ives). Adkisson was a stu­dent of J. R. Kline, and with Saun­ders stud­ied wheth­er a homeo­morph­ism of a graph \( G \) in \( \mathbb{S}^2 \) could be ex­ten­ded to a homeo­morph­ism of \( \mathbb{S}^2 \). These bits of to­po­logy were a far cry from Saun­ders’ later work in to­po­logy with Ei­len­berg, but they served as a warm-up.

Saun­ders’ daugh­ter Gretchen was born dur­ing his year at Chica­go. Then the lure of an as­sist­ant pro­fess­or­ship brought the fam­ily to Har­vard in 1938. Right away Saun­ders and Gar­rett Birk­hoff began of­fer­ing an un­der­gradu­ate course in al­gebra; Birk­hoff’s course em­phas­ized al­geb­ras, lat­tices and groups, Saun­ders’ course em­phas­ized group the­ory, Galois the­ory and an ax­io­mat­ic treat­ment of vec­tor spaces. Soon they com­bined their notes, and in 1941 Sur­vey of mod­ern al­gebra was pub­lished. It was the first Amer­ic­an text to use the ideas of Emmy No­eth­er, and even­tu­ally spread bey­ond the elite de­part­ments.

Dur­ing his Har­vard years, Saun­ders spent some time study­ing group ex­ten­sions and crossed-product al­geb­ras. In 1941, he gave the Zi­wet lec­tures at the Uni­versity of Michigan, speak­ing about group ex­ten­sions. In the audi­ence was Samuel Ei­len­berg, and in con­ver­sa­tion they real­ized that group ex­ten­sions were re­lated to the Steen­rod ho­mo­logy of the \( p \)-ad­ic solen­oid; thus had their fam­ous col­lab­or­a­tion be­gun, res­ult­ing even­tu­ally in 15 joint pa­pers.

The war years in­ter­vened. Saun­ders’ second daugh­ter, Cyn­thia, ar­rived in 1941. Saun­ders spent time at Columbia both work­ing on war-re­lated ap­plied math prob­lems, and talk­ing with Ei­len­berg who was now at Columbia.

In 1945 They in­tro­duced the no­tion of cat­egory in a pa­per titled “Gen­er­al the­ory of nat­ur­al equi­val­ences” [1]. Saun­ders re­ferred to it later as “off beat” and “far out,” and when I was a stu­dent at Chica­go the sub­ject was some­times re­ferred to as “gen­er­al­ized ab­stract non­sense” by both its afi­cion­ados and de­tract­ors. The ori­gin­al idea of cat­egor­ies with ob­jects and morph­isms, and nat­ur­al trans­form­a­tions between them, has blos­somed al­most bey­ond re­cog­ni­tion. But it was their pa­pers on co­homo­logy of groups and Ei­len­berg–MacLane spaces that made their repu­ta­tions, at least among to­po­lo­gists.

Here is a quote from Alex Heller writ­ing about Ei­len­berg in the No­tices of the AMS [3], [4]:

With Mac Lane he de­veloped the the­ory of co­homo­logy of groups, thus provid­ing a prop­er set­ting for the re­mark­able the­or­em of Hopf on the ho­mo­logy of highly con­nec­ted spaces. This led them to the study of the Ei­len­berg–Mac Lane spaces and thus to a deep­er un­der­stand­ing of the re­la­tions between ho­mo­topy and ho­mo­logy. Their most fate­ful in­ven­tion per­haps was that of cat­egory the­ory, re­spond­ing, no doubt, to the ex­i­gen­cies of al­geb­ra­ic to­po­logy, but destined to ra­di­ate across most of math­em­at­ics.

And here [5] is Saun­ders him­self writ­ing about the gen­es­is of his col­lab­or­a­tion with Ei­len­berg. The set­ting is 1941 at the Uni­versity of Michigan, where Saun­ders is giv­ing the Zi­wet Lec­tures.

At that time, I had been fas­cin­ated with the de­scrip­tion of group ex­ten­sions and the cor­res­pond­ing crossed product al­geb­ras, which had entered in­to my re­search with O. F. G. Schilling on class field the­ory. So group ex­ten­sions be­came the top­ic of my Zi­wet lec­tures. I set out the de­scrip­tion of a group ex­ten­sion by means of factor sets and com­puted the group of such ex­ten­sions for the case of an in­ter­est­ing abeli­an factor group defined for any prime \( p \) and giv­en be gen­er­at­ors \( a_n \) with \( pa_{n+1} = a_n \) for all \( n \). When I presen­ted this res­ult in my lec­ture, Sammy im­me­di­ately poin­ted out that I had found Steen­rod’s cal­cu­la­tion of the ho­mo­logy group of the \( p \)-ad­ic solen­oid. This solen­oid, already stud­ied by Sammy in Po­land, can be de­scribed thus: In­side a tor­us \( T_1 \), wind an­oth­er tor­us \( T_2 \) \( p \)-times, then an­oth­er tor­us \( T_3 \) \( p \)-times in­side \( T_2 \), and so on. What is the ho­mo­logy of the fi­nal in­ter­sec­tion? Sammy ob­served the the \( \operatorname{Ext} \) group I had cal­cu­lated gave ex­actly Steen­rod’s cal­cu­la­tion of the ho­mo­logy of the solen­oid! The co­in­cid­ence was highly mys­ter­i­ous. Why in the world did a group of abeli­an group ex­ten­sions come up in ho­mo­logy? We stayed up all night try­ing to find out “why.” Sammy wanted to get to the bot­tom of this co­in­cid­ence.

It fi­nally turned out that the an­swer in­volved the re­la­tion between the (in­teg­ral) ho­mo­logy groups \( H_n(X) \) of a space \( X \) with the co­homo­logy groups \( H^n(X,G) \) of the same space, with coef­fi­cients in an abeli­an group \( G \). It was then known that there was an iso­morph­ism \( \Theta \), \[ \Theta:H^n(X,G) \to \operatorname{Hom}(H_n(X),G), \] where the right hand group is that of all ho­mo­morph­isms of \( H_n(X) \) in­to \( G \). But we found that this map \( \Theta \) had a ker­nel which was ex­actly my group of abeli­an group ex­ten­sions, \( \operatorname{Ext}(H_{n-1}(X),G) \). In oth­er words, we found and de­scribed a short ex­act se­quence \[ 0 \to \operatorname{Ext}(H_{n-1}(X),G) \to H^n(X,G) \to \operatorname{Hom}(H_n(X),G) \to 0. \] In ef­fect, this “de­term­ines” the co­homo­logy groups in terms of the in­teg­ral ho­mo­logy groups, and this ex­plains why the al­geb­ra­ic­ally in­tro­duced groups \( \operatorname{Ext} \) have a to­po­lo­gic­al use. This ex­act se­quence is now known as the “uni­ver­sal coef­fi­cient the­or­em.”

The defin­i­tion and con­struc­tion of what be­came known as Ei­len­berg–Mac Lane spaces, \( K(G,n) \), oc­curred in the mid 1940s. The ho­mo­topy groups of a to­po­lo­gic­al space \( X \), \( \pi_n(X) \), are defined by the ho­mo­topy classes of con­tinu­ous maps of the \( n \)-sphere in­to \( X \), a much sim­pler defin­i­tion than that of ho­mo­logy groups \( H_n(X;\mathbb Z) \) (which non­ethe­less had been defined much earli­er in the work of Poin­caré and were easi­er to com­pute). The fun­da­ment­al group \( \pi_1(X) \) may be non-Abeli­an, but the high­er ho­mo­topy groups were Abeli­an. Hopf showed that \( \pi_3(\mathbb S^2) = \mathbb Z \) in con­trast to \( H_3(\mathbb S^2,\mathbb Z) =0 \). So ho­mo­topy groups were dif­fer­ent than ho­mo­logy groups and, as it turned out, were much harder to com­pute. To this day not all of the high­er ho­mo­topy groups of even the 2-sphere are known.

Ei­len­berg and Mac Lane stepped in and con­struc­ted, for any fi­nitely presen­ted Abeli­an group \( G \), a space \( K(G,n) \) whose \( n \)-th ho­mo­topy group is \( G \) and all oth­er ho­mo­topy groups are zero. These spaces form es­sen­tial build­ing blocks for more com­plic­ated spaces. There are only a few simple ex­amples, namely, \( K(Z,1) = \mathbb S^1 \), the circle; \( K(\mathbb Z/2,1) = \mathbb RP^{\infty} \), the in­fin­ite-di­men­sion­al real pro­ject­ive space; and \( K(\mathbb Z,2) = \mathbb CP^{\infty} \), the in­fin­ite-di­men­sion­al com­plex pro­ject­ive space. Oth­er ex­amples can be ob­tained from the Dold–Thom the­or­em: if the only non-zero ho­mo­logy of \( X \) is \( G \) in di­men­sion \( n \), then the in­fin­ite sym­met­ric product of \( X \) is a \( K(G,n) \); for ex­ample, take \( X = \mathbb S^n \).

Saun­ders spent 1938–47 at Har­vard (Irving Ka­plansky and Ro­ger Lyn­don were the best known of his Har­vard stu­dents), but then was lured back to Chica­go in 1949 by Hutchins (who re­mained Chan­cel­lor un­til 1952) and the new chair Mar­shall Har­vey Stone. He spent the in­ter­ven­ing year tour­ing Europe, and in par­tic­u­lar talk­ing with J. H. C. White­head in Ox­ford. They sor­ted out the first case of what later be­came Post­nikov sys­tems, de­fin­ing \( k^3 \in H^3(\pi_1, \pi_2) \).

Saun­ders stayed at Chica­go for the rest of his math­em­at­ic­al life. He has of­ten re­coun­ted the Stone Age [2] at Chica­go (1947–59), when Chica­go was ar­gu­ably the best de­part­ment in the world. The seni­or fac­ulty were Ad­ri­an Al­bert, S.-S. Chern, Stone, An­dré Weil, Ant­oni Zyg­mund and of course Mac Lane; the ju­ni­or fac­ulty in­cluded Paul Hal­mos, Ka­plansky, Irving Segal and Ed­win Span­i­er, plus many even­tu­al stars among the gradu­ate stu­dents.

Stone re­tired, and it all came apart in the late 1950s when Weil went to the In­sti­tute for Ad­vanced Study, Chern and Span­i­er op­ted for Berke­ley in 1958, and Hal­mos and Segal left. Still, it had been a great time, and many of the stars re­mained.

I was an un­der­gradu­ate and a gradu­ate stu­dent at Chica­go (1954–1964), but I was pretty much ob­li­vi­ous to the great math­em­aticians of the Stone Age, pre­fer­ring to con­cen­trate on less­er games than math. But I dimly re­call Saun­ders as a lively teach­er. While teach­ing an al­gebra course, prob­ably in the winter quarter of 1958, a new grad stu­dent entered the class in about the fifth week, and promptly star­ted cor­rect­ing Saun­ders and en­ga­ging him in class. It was Robert So­lovay.

An­oth­er time in 1959, Saun­ders an­nounced with ob­vi­ous pleas­ure that his good friend Henry White­head would give a guest lec­ture, and he said with a grin that J. H. C. stood for “Je­sus, he’s con­fus­ing,” or some­times “Je­sus, he’s crazy.” This was the same time that White­head was vis­it­ing Michigan and heard of Mort Brown’s proof of the to­po­lo­gic­al Schoen­flies con­jec­ture.

Saun­ders was an ebul­li­ent man, and of­ten spoke and lec­tured with a not-quite-sup­pressed smile. As I look at the photo from the cov­er of his book, I see cheeks which have smiled and laughed a lot; at least that’s how I knew him.

I got a teach­ing job at Roosevelt Uni­versity in 1960, mainly on the basis of a let­ter of re­com­mend­a­tion from Saun­ders, and this sup­por­ted me for the next four years. Later in spring 1962 when I fi­nally passed my PhD qual­i­fy­ing ex­am, I asked Saun­ders about work­ing with him. He asked what I was in­ter­ested in, and I men­tioned group the­ory, ho­mo­lo­gic­al al­gebra, and to­po­lo­gic­al man­i­folds. He wisely told me to spend the sum­mer think­ing about these top­ics and then talk with him when I re­turned in the fall. I only thought about to­po­lo­gic­al man­i­folds, and thus nev­er be­came Saun­ders’ stu­dent (per­haps for­tu­nately for me, as Saun­ders ex­pec­ted his stu­dents to come by once a week to talk about what they had ac­com­plished, a re­quire­ment bey­ond me).

At one point he heard I was work­ing on the An­nu­lus Con­jec­ture, and he said that it was a rather hard prob­lem for a PhD thes­is. (It was, and it only paid off sev­er­al years later.)

These small in­ter­ac­tions with Saun­ders were non­ethe­less im­port­ant to me, and I al­ways felt warmly to­wards him.

In 1949 at the re­l­at­ively early age of 39, Saun­ders was elec­ted to the Na­tion­al Academy of Sci­ences. His most in­flu­en­tial work was that with Ei­len­berg on group ex­ten­sions, co­homo­logy of groups and Ei­len­berg–Mac Lane spaces and, maybe, cat­egory the­ory. Yet Ei­len­berg was not elec­ted for an­oth­er 10 years; it may be that it was Saun­ders’ abil­it­ies out­side re­search (plus be­ing highly ac­cep­ted in the “old boys” net­work that still ex­is­ted in those days), that helped ac­count for his early elec­tion.

In 1959 Saun­ders be­came chair of the ed­it­or­i­al board of the Pro­ceed­ings of the Na­tion­al Academy of Sci­ences, and served for eight years. At that time, PNAS had no sys­tem of ref­er­ee­ing, and ac­cep­ted nearly all pa­pers sub­mit­ted by mem­bers or com­mu­nic­ated by mem­bers. As might be ex­pec­ted, the qual­ity was mixed. Un­til that time, math pa­pers formed the largest pro­por­tion of PNAS, again with mixed qual­ity.

Since those days the right of a mem­ber to pub­lish or com­mu­nic­ate be­came an em­bar­rass­ment, and this right has gradu­ally been whittled away. Mem­bers still re­tain a small edge, but since I joined the ed­it­or­i­al board in 2002, that edge has dwindled and will surely dis­ap­pear. The ar­gu­ment on be­half of an ex­tra priv­ilege for an Academy mem­ber is that a mem­ber may au­thor or see a pa­per, re­jec­ted by the main­stream, which the mem­ber be­lieves is fresh and ori­gin­al and there­fore needs the PNAS im­prim­at­ur to gain the at­ten­tion of the world. It’s not a good ar­gu­ment in math, and doubt­ful else­where.

While Saun­ders was chair, the treas­urer of NAS, wor­ried about the fin­ances of PNAS, in­sti­tuted page charges. Without big grants, math­em­aticians stopped pub­lish­ing in PNAS, nearly dis­ap­pear­ing from its pages. These days PNAS will not charge math­em­aticians, and is try­ing hard to in­crease the rep­res­ent­a­tion of math in its pages.

Saun­ders was elec­ted Pres­id­ent of the AMS in 1972, at a time of some tu­mult due to Viet Nam and the emer­ging wo­men’s move­ment in math­em­at­ics. Chica­go had al­ways ad­mit­ted wo­men and had pro­duced quite a few fe­male PhDs over the years, a fact that Saun­ders seemed to be proud of. However I re­call that he showed up at a meet­ing of the Coun­cil of the AMS wear­ing a tie with a bunch of little white pig­lets and a bunch of MCP’s (male chau­vin­ist pig). I’d guess he did it to tease the more ar­dent fem­in­ists, but I’m not sure it went over well. About the same time he nom­in­ated and helped elect Ju­lia Robin­son to the NAS, the first fe­male mem­ber in math­em­at­ics (she shortly went on to be elec­ted Pres­id­ent of the AMS, an hon­or which at that time was tra­di­tion­ally re­served for mem­bers of the NAS). I be­lieve that he also helped get Steph­en Kleene elec­ted at the same time, for Kleene had had a long and dis­tin­guished ca­reer in lo­gic, and was equally de­serving of be­ing elec­ted.

While Pres­id­ent, Saun­ders en­cour­aged the AMS to be more pro­act­ive in gov­ern­ment af­fairs, es­tab­lish­ing with SIAM and the MAA the Joint Pro­jects Board in Math­em­at­ics to in­flu­ence pub­lic policy.

Saun­ders con­tin­ued to be act­ive in math­em­at­ics and math­em­at­ic­al gov­ernance nearly to the end of his life. I re­call him lec­tur­ing with vig­or on something cat­egor­ic­al in the open­ing term of the New­ton In­sti­tute in Cam­bridge in 1992.

Saun­ders wife Dorothy had been in­fec­ted with a vari­ant of en­ceph­al­it­is dur­ing the war, and, as it pro­gressed along with Par­kin­son’s and arth­rit­is, it be­came more and more dif­fi­cult for them to travel to­geth­er. They cel­eb­rated their 50th wed­ding an­niversary in 1983, and on 3 Feb­ru­ary 1985 Dorothy died. Saun­ders writes very warmly of Dorothy and their mar­riage. Saun­ders re­mar­ried in 1986, to Osa Skot­ting (ex-wife of Irving Segal), and they en­joyed his re­main­ing years to­geth­er.

Saun­ders had a fond­ness for clev­er verse, of­ten about math­em­at­ics or math­em­aticians, and the best way to end this es­say is with some ex­amples that he of­ten quoted or sang.

Where are the zeros of zeta of s?

(Sung to the tune of “Sweet Betsy from Pike”; words by Tom Apostol.)

Where are the zer­os of zeta of s?
G. F. B. Riemann has made a good guess,
They’re all on the crit­ic­al line, said he,
And their dens­ity’s one over 2pi log t.

This state­ment of Riemann’s has been like trig­ger
And many good men, with vim and with vig­or,
Have at­temp­ted to find, with math­em­at­ic­al rig­or,
What hap­pens to zeta as mod t gets big­ger.

The ef­forts of Land­au and Bo­hr and Cramer,
And Lit­tle­wood, Hardy and Titch­marsh are there,
In spite of their ef­forts and skill and fin­esse,
(In) loc­at­ing the zer­os there’s been no suc­cess.

In 1914 G. H. Hardy did find,
An in­fin­ite num­ber that lay on the line,
His the­or­em however won’t rule out the case,
There might be a zero at some oth­er place.

Let \( P \) be the func­tion pi minus li,
The or­der of \( P \) is not known for \( x \) high,
If square root of \( x \) times \( \log x \) we could show,
Then Riemann’s con­jec­ture would surely be so.

Re­lated to this is an­oth­er en­igma,
Con­cern­ing the Lindelöf func­tion mu(sigma)
Which meas­ures the growth in the crit­ic­al strip,
On the num­ber of zer­os it gives us a grip.

But nobody knows how this func­tion be­haves,
Con­vex­ity tells us it can have no waves,
Lindelöf said that the shape of its graph,
Is con­stant when sigma is more than one-half.

Oh, where are the zer­os of zeta of \( s \)?
We must know ex­actly, we can­not just guess,
In or­der to strengthen the prime num­ber the­or­em,
The in­teg­ral’s con­tour must not get too near ’em.


Upon hear­ing of the zeta func­tion song, Saun­ders com­posed lyr­ics (to the same mu­sic) for

Simple groups

What are the or­ders of all simple groups?
I speak of the hon­est ones, not of the loops
It seems that old Burn­side the or­ders has guessed
Ex­cept for the cyc­lic ones, even the rest.

Groups made up with the per­mutes will pro­duce some more
For \( A_n \) is simple if \( n \) ex­ceeds 4
There is Sir Math­ieu who came in­to view
Ex­hib­it­ing groups of an or­der quite new.

Still oth­ers have come on to study this thing
Of Artin and Che­val­ley now we shall sing
With matrices fi­nite they made quite a list
The ques­tion is: Could there be oth­ers they’ve missed?

Su­zuki and Ree then main­tained it’s the case
That these meth­ods had not reached the end of the chase
They wrote down some matrices just four by four
That made up a simple group; why not make more?

And then came the opus of Thompson and Feit
Which shed on the prob­lem re­mark­able light
A group when the or­der won’t factor by two
Is cyc­lic or solv­able. That’s what is true.

Su­zuki and Ree had caused eye­brows to raise,
But the the­or­eti­cians they just couldn’t faze.
Their groups wer­en’t new; if you ad­ded a twist,
You could get them from old ones with a flick of the wrist.

Still some hardy souls felt a thorn in their side,
For the five groups of Math­ieu all reas­on de­fied;
Not \( A_n \), not twis­ted, and not Che­val­ley,
They called them sporad­ic and filed them away.

Are Math­ieu groups creatures of Heav­en or Hell?
Zvon­imir Janko de­term­ined to tell.
He found out what nobody wanted to know:
The mas­ters had missed 1 7 5 5 6 0.

The floodgates were opened, new groups were the rage,
And twelve or more sprouted to greet the new age;
By Janko, and Con­way, and Fisc­her, and Held,
McLaugh­lin, Su­zuki and Hig­man and Sims.

You prob­ably no­ticed the last lines don’t rhyme.
Well, that is quite simply a sign of the time;
There’s chaos, nor or­der, among simple groups,
And maybe we’d bet­ter go back to the loops.


And two more:

Here’s to Mar­ston, Mickey Morse,
A man ex­per­i­enced in di­vorce.
His opin­ion of him­self, we charge
Like nose and book is in the large.

Here’s to Lef­schetz, So­lomon L.,
Ir­re­press­ible as hell.
When he’s at last be­neath the sod,
He’ll then be­gin to heckle God.


Fi­nally, Steve Awodey [e4] wrote the fol­low­ing ditty to com­mem­or­ate the verses that Saun­ders en­joyed:

To Saun­ders Mac Lane, of the plaid sport coat,
with his thingama­jigs and the Homs he wrote,
in math­em­at­ics he stood alone,
in a cat­egory of his own!


[1]S. Ei­len­berg and S. MacLane: “Gen­er­al the­ory of nat­ur­al equi­val­ences,” Trans. Am. Math. Soc. 58 : 2 (September 1945), pp. 231–​294. MR 0013131 Zbl 0061.​09204 article

[2]S. Mac Lane: “Math­em­at­ics at the Uni­versity of Chica­go: A brief his­tory,” pp. 127–​154 in A cen­tury of math­em­at­ics in Amer­ica, part II. Edi­ted by P. Duren, R. A. As­key, and U. C. Merzbach. His­tory of Math­em­at­ics 2. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1989. MR 1003124 Zbl 0667.​01023 incollection

[3]H. Bass, H. Cartan, P. Freyd, A. Heller, and S. Mac Lane: “Samuel Ei­len­berg (1913–1998),” No­tices Am. Math. Soc. 45 : 10 (1998), pp. 1344–​1352. MR 1646820 Zbl 0908.​01023 article

[4]H. Bass, H. Cartan, P. Freyd, A. Heller, and S. MacLane: “Samuel Ei­len­berg: 1913–1998,” Bio­graph­ic­al Mem­oirs 79 (2000). article

[5]S. Mac Lane: “Samuel Ei­len­berg and cat­egor­ies,” pp. 127–​131 in Cat­egory the­ory 1999 (Coim­bra, Por­tugal, 13–17 Ju­ly 1999), published as J. Pure Ap­pl. Al­gebra 168 : 2–​3. Issue edi­ted by J. Adamek, P. T. John­stone, and M. Sobral. 2002. MR 1887153 Zbl 0992.​18001 incollection

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