Celebratio Mathematica

Saunders Mac Lane

Preface to “Saunders Mac Lane: A mathematical autobiography”

by David Eisenbud

No man could so stim­u­late oth­ers un­less, along­side an in­cis­ive in­tel­lect, he was pos­sessed of en­thu­si­asm and warmth, a deep in­terest in his fel­low man, and a sym­pathy the more real for be­ing un­sen­ti­ment­al. Those who proudly call them­selves his friends know these things: oth­ers will in­fer them in read­ing [his works].

— Max Kelly1


Saun­ders Mac Lane has been my teach­er, ment­or and mod­el al­most from the be­gin­ning of my math­em­at­ic­al life. It is a re­la­tion­ship I’ve cher­ished. He has been for me a fig­ure of great hon­esty and in­teg­rity, who worked hard to ad­vance re­search and to serve the math­em­at­ic­al com­munity. His be­lief in the good, the right and the ra­tion­al, his care for the es­sence of math­em­at­ic­al ideas, his power­ful en­thu­si­asm and his es­sen­tial op­tim­ism were, and are, deeply at­tract­ive to me.

Nearly everything about Saun­ders-in-ac­tion was col­or­ful, start­ing with the red-and-green plaid sports coat (the Mac Lane tartan, of course) and red pants that he would wear for im­port­ant oc­ca­sions. Per­haps a few an­ec­dotes and re­flec­tions from my ex­per­i­ence of him over 40 years will help the read­er ap­pre­ci­ate this col­or.

First encounter

I first met Mac Lane — in a sense I’ll make pre­cise — in 1963. He was one of the most im­port­ant fig­ures in the Uni­versity of Chica­go Math­em­at­ics De­part­ment, or in­deed in Amer­ic­an math­em­at­ics: His first stu­dent, Irving Ka­plansky, was Chair of the de­part­ment, and two oth­er stu­dents were on the fac­ulty — one, John Thompson, a Fields Medal­ist. Mac Lane was an in­vent­or of group co­homo­logy, a founder of ho­mo­lo­gic­al al­gebra and cat­egory the­ory, known for the Ei­len­berg–Mac Lane spaces in to­po­logy. He was past Pres­id­ent of the Math­em­at­ic­al As­so­ci­ation of Amer­ica, and he would soon be Vice Pres­id­ent of the Na­tion­al Academy of Sci­ences, mem­ber of the Board gov­ern­ing the Na­tion­al Sci­ence Found­a­tion, and Pres­id­ent of the Amer­ic­an Math­em­at­ic­al So­ci­ety as well.

I knew none of this. I was six­teen, an early entrant to the Uni­versity, an un­even stu­dent with a great en­thu­si­asm for math­em­at­ics. It was the be­gin­ning of my second quarter, and I was sched­uled to start a ba­sic lin­ear al­gebra class that morn­ing. I happened to ar­rive a little early, settled down in the first row of the class, and sank peace­fully in­to a day­dream. Be­ing so new, I wasn’t sur­prised not to know the oth­er stu­dents who settled in around me, and I didn’t know the teach­er that I’d have. In due course, Mac Lane walked in and began lec­tur­ing. His style was lively and col­or­ful, and I was im­me­di­ately in­ter­ested — but al­most at once aware that I’d made a big mis­take: this was not an un­der­gradu­ate lin­ear al­gebra course, but an ad­vanced gradu­ate course on cat­egory the­ory. I’d come an hour early.

I un­der­stood noth­ing whatever after a few mo­ments, but was far too em­bar­rassed to get up and leave — in­stead I sank in­to day­dreams, glassy-eyed. Mac Lane, who prided him­self on pay­ing at­ten­tion to his class, later told me he thought he could al­ways see who was fol­low­ing and who was not. In a mo­ment like a thun­der­clap, I looked up from my seat and found him point­ing dir­ectly at me from across the room. “You!” he said per­emp­tor­ily, “you don’t be­lieve this proof, do you?” Be­lief and dis­be­lief were equally bey­ond me; I sat pet­ri­fied. He ad­vanced to­ward me, and I don’t know what I ima­gined — that he would pick me up by the scruff of my neck and throw me from the room? He stopped, turned back to the board, and pro­ceeded to ex­plain the proof to sat­is­fy me. Of course, I still un­der­stood noth­ing — but I sat in rapt at­ten­tion.

For­tu­nately the class ended soon, and as stu­dents ask­ing ques­tions sur­roun­ded him, it was easy for me to slip out. I didn’t tell Saun­ders this story un­til many years af­ter­wards, when I had the priv­ilege of re-en­act­ing it (from the oth­er side) in a lec­ture at the con­fer­ence in hon­or of his sev­en­ti­eth birth­day. Need­less to say, the event hadn’t left a trace in his memory, though it re­mains sharp for me to this day.

Saunders and tolerance

Saun­ders be­lieves strongly in prin­ciples, in the right­ness of right po­s­i­tions. I nev­er once saw him per­son­ally in­tol­er­ant, but he could some­times be dir­ect and can­did to the point of of­fend­ing. People whose judg­ment I re­spect have felt in­jured by what he said, and some­times by the blunt­ness of his ex­pres­sion. In some way per­haps he didn’t ap­pre­ci­ate the mag­nitude of his po­s­i­tion in math­em­at­ics, or the ser­i­ous­ness with which people took him. In a less­er per­son­age some of his ex­treme po­s­i­tions might have been re­garded as charm­ing ec­cent­ri­cit­ies. But giv­en Saun­ders’ stature, they could in­jure, and he might have been more cau­tious.

An event from late in Saun­ders’ life may give a bit of the fla­vor. It was a spe­cial ses­sion run by him and Richard As­key at the Joint Math­em­at­ics Meet­ing in 1999, a ses­sion boldly en­titled “Math­em­at­ics edu­ca­tion and mis­taken philo­sophies of math­em­at­ics.” The audi­ence was enorm­ous. I found the title charm­ing (and still find it so, even now as I be­come more in­volved with ideas in K-12 edu­ca­tion), and I ima­gine that Saun­ders meant it to be con­tro­ver­sial but play­ful. Pre­dict­ably, it an­noyed and needled some prac­ti­tion­ers. Saun­ders began the ses­sion with in­tro­duct­ory re­marks that I found fas­cin­at­ing: he said that he now con­sidered the ex­tent of his own em­phas­is on cat­egory the­ory as a tool for learn­ing and teach­ing math­em­at­ics to have been too ex­treme. This humble­ness may have helped soften the crit­ic­al tone of the ses­sion.

Saunders and Sammy

One of Saun­ders’ great math­em­at­ic­al friend­ships and col­lab­or­a­tions was with Samuel Ei­len­berg (widely known as Sammy, or even \( S^2P^2 \): “Smart Sammy, the Pol­ish Prodigy”). I got to see them in ac­tion to­geth­er only once, at the AMS Sum­mer Re­search In­sti­tute on Cat­egory The­ory at Bowdoin Col­lege, in 1969. They had spe­cial status at this three-week con­fer­ence, not only as the seni­or mem­bers, but also as the very founders of the sub­ject. So, when they began dis­cuss­ing its ori­gins one even­ing after din­ner, every­one gathered around to listen.

I dearly wish I could re­call the sub­stance of their de­bate, but I don’t; only my sense of the con­trast in the two men’s styles stays with me. Sammy drew Saun­ders out and egged him on, al­ways slightly evas­ive and mock­ing; Saun­ders, whose fath­er and grand­fath­er were Con­greg­a­tion­al min­is­ters, seemed to feel that, since his view was right, his view would pre­vail. Once he had stated it, all he could do was bang his fist. The de­vi­ous and soph­ist­ic­ated European versus the in­no­cent but hon­est Amer­ic­an? That’s how it seemed to me at the time (maybe I was a little in­no­cent my­self). A loy­al stu­dent, I was root­ing from the be­gin­ning for Saun­ders’ point of view, but I came away feel­ing that he was trounced in the con­test.

Being Saunders’ student

After flirt­ing a while with op­er­at­or the­ory (Paul Hal­mos and Fe­lix Browder were my teach­ers) and group the­ory (learned from Jon Alper­in and Otto Kegel), it was fi­nally time for me, by now a second-year gradu­ate stu­dent, to settle on an area for a PhD thes­is. I ob­sessed about how to make the choice. A close math­em­at­ic­al friend, Joe Neisen­dorfer, ex­plained to me an al­gorithm: for­get the top­ic, look around the fac­ulty for the per­son you like the most. It didn’t take me long to choose Saun­ders.

I wouldn’t say I ever felt per­son­al in­tim­acy with Saun­ders, but he did go out of his way to make me and oth­er stu­dents feel wel­come in more than his of­fice. Saun­ders and his late wife, Dorothy, had a small but com­fort­able cot­tage in the In­di­ana Dunes, a beau­ti­ful area on the shore of Lake Michigan about an hour south of Chica­go, and they oc­ca­sion­ally in­vited stu­dents to spend an af­ter­noon there. Saun­ders was an en­thu­si­ast­ic sail­or, and I can re­port, from a ride in a small sail­boat on rough wa­ter, that he was ready to provide needed in­struc­tion not only in math­em­at­ics, but also on how to handle the ab­sence of a toi­let — or any pri­vacy — in that dif­fi­cult situ­ation.

If you look at the list of Saun­ders’ 39 stu­dents, you’ll see that Irving Ka­plansky, who worked on valu­ation the­ory of fields, came first; I’m near the end, with a thes­is on non­com­mut­at­ive rings. Along the way are such people as John Thompson (fi­nite groups), Anil Ner­ode (lo­gic and com­pu­ta­tion), and Robert Szczar­ba (al­geb­ra­ic to­po­logy). How did this vari­ety come about?

Per­haps the an­swer lies in Saun­ders’ hos­pit­al­ity to these many ideas. He wanted to learn fi­nite groups, and taught a course on them. By the end of the course he’d de­cided that he’d nev­er really un­der­stand the sub­ject, but in Thompson he found a fab­ulously strong stu­dent. Saun­ders might have tried to turn such a stu­dent to­ward in­terests close to his own, but I think he would not, on prin­ciple: he was happy to en­cour­age his stu­dents to do what ex­cited them.

Saun­ders has fol­lowed an in­ter­est­ing, curving tra­ject­ory through math­em­at­ics, from lo­gic and found­a­tions to field the­ory and the be­gin­nings of ho­mo­lo­gic­al al­gebra, through to­po­logy to cat­egory the­ory, with smal­ler di­ver­sions along the way in­to Hamilto­ni­an mech­an­ics, fi­nite groups, and many oth­er sub­jects. Per­haps his stu­dents, or many of them, could be de­scribed as com­ing off on the tan­gents to this path, a kind of de­velop­able sur­face reach­ing broadly across math­em­at­ics. Al­to­geth­er, Saun­ders has more than 1,000 math­em­at­ic­al des­cend­ants lis­ted on the Math­em­at­ics Gene­a­logy Pro­ject.

Some oth­er as­pects of Saun­ders are also re­flec­ted in his stu­dents: Saun­ders was al­ways act­ive on be­half of the com­munity, wheth­er as Chair work­ing to build the de­part­ment at the Uni­versity of Chica­go or, near the end of his ca­reer, as mem­ber of the Na­tion­al Sci­ence Board or as man­ager of the elab­or­ate sys­tem of re­ports for the Na­tion­al Academy of Sci­ences. Many of his stu­dents and grand-stu­dents have fol­lowed him in­to this will­ing­ness for pub­lic ser­vice. When I was wor­ry­ing about wheth­er to move to my cur­rent po­s­i­tion at MSRI, he was one of the first people I called on for ad­vice and bless­ing, and he gave both.

Re­turn­ing to the more fun­da­ment­al mat­ter of be­ing Saun­ders’ math­em­at­ic­al stu­dent: I tried for a while, du­ti­fully, to find a thes­is top­ic in cat­egory the­ory, Saun­ders’ pas­sion in that part of his life. But I failed; some­how, the things I read and learned in that do­main just didn’t in­spire me. When I de­veloped an in­terest in­stead in a prob­lem on non-com­mut­at­ive rings posed by a vis­it­or of Her­stein, the young Chris Rob­son, Saun­ders could eas­ily have washed his hands of the pro­ject. He did not: though it was far from his cur­rent area of in­terest, he wel­comed what I had done, and painstak­ingly read draft after draft of my thes­is.

Saun­ders’ mode of in­struc­tion in thes­is-writ­ing bears men­tion. I had writ­ten a couple of pa­pers, jointly with Rob­son, of which my thes­is res­ults were par­tially an ex­tract. Rob­son cared a lot about ex­pos­i­tion, and so (learn­ing from Saun­ders among oth­ers) did I. We’d gone through many drafts, and I thought the writ­ing pretty pol­ished. Saun­ders did not. He began at the be­gin­ning and worked his way through the thes­is un­til he’d com­piled a list of ex­actly 25 sub­stant­ive sug­ges­tions. Then he stopped, and re­turned the doc­u­ment to me for an over­haul. When I had fin­ished mak­ing the cor­rec­tions he’d flagged and all their ana­logues, I gave it back to him, eager to be done. But… after a week or so I got a second list of ex­actly 25 more sug­ges­tions. The third list was a bit short­er, and Saun­ders al­lowed the pro­cess to con­verge be­fore I got too frus­trated.

It must be clear by now: over these forty years I’ve learned many les­sons from Saun­ders. I’m deeply grate­ful to him.