Celebratio Mathematica

My pa­ternal grand­fath­er was named Crom­well Percy Kirby, and therein lies some fam­ily lore. Ap­par­ently a Percy fought for Oliv­er Crom­well in the Eng­lish Civil Wars (1642–1651) and was re­war­ded with land. The next eight gen­er­a­tions of old­est sons died in battle, fight­ing either for or against the king. Much later in time (and still on Eng­lish shores), a Percy daugh­ter mar­ried a Hack­shaw and a Hack­shaw daugh­ter mar­ried my great-grand­fath­er. The couple left Eng­land (per­haps for bet­ter air) in 1870 to homestead in Neb­raska. My grand­fath­er, one of nine chil­dren, was born there in 1873. They lived in a sod house, dug part way in­to a hill­side, a full day’s wag­on ride from the nearest town. Crom­well be­came a small town Baptist min­is­ter, and even­tu­ally my fath­er, Bern­ard Crom­well Kirby, was born in In­di­ana­pol­is in 1907. He aimed to be­come a min­is­ter and went to Den­ison Col­lege (a Baptist school) where he met my moth­er Pau­line Ro­bi­on.

Mom was born in Chica­go in 1907 to a French im­mig­rant fath­er, Ab­ram Ro­bi­on, and a Ger­man im­mig­rant moth­er, Edith Fish­er. Her fam­ily lived in Oak Park and she also even­tu­ally went to Den­ison — a year ahead of my fath­er, who had worked for a year after high school to help sup­port his fam­ily.

Dad worked for a com­pany mak­ing sax­o­phones in Elkhart In­di­ana, for a steel foundry, and had a string of those old-fash­ioned gad­gets in­to which you put a penny and got a hand­ful of pea­nuts. He said he bought a car for \$100 in or­der to make the rounds re­plen­ish­ing pea­nuts (I’ve nev­er un­der­stood the eco­nom­ics of this).

My moth­er taught Eng­lish for a year in the coal min­ing town of Blue­field, West Vir­gin­ia, and then my par­ents mar­ried in 1930. They both got jobs in so­cial work in Chica­go and took a grad course or two at the Uni­versity of Chica­go’s School of So­cial Work. My Dad had de­cided not to pur­sue the min­istry, to his fath­er’s deep re­gret, but to save the world through so­cial ac­tion. In the depths of the De­pres­sion, my par­ents were in­formed that the city could not em­ploy more than one per­son in a fam­ily, and as my moth­er was prob­ably more pro­fi­cient as a so­cial work­er, my fath­er left and star­ted or­gan­iz­ing for the CIO, the Con­gress of In­dus­tri­al Or­gan­iz­a­tions. They’d get him a job, he’d start or­gan­iz­ing, he’d get fired, he’d get an­oth­er job. In his free time he ran for the Chica­go City Coun­cil as a So­cial­ist, and got more votes than he ex­pec­ted be­cause the Chica­go ma­chine liked the Com­mun­ists even less that the So­cial­ists and switched some Com­mun­ist votes his way, or so the story goes.

I was born in 1938, and though Mom went back to work, she fairly soon de­cided she’d rather stay home and raise her son. Dad tried his hand at writ­ing fic­tion for a liv­ing — short, boy-meets-girl, boy-loses-girl, boy-gets-girl Sat­urday Even­ing Post-type stor­ies, but al­though they wer­en’t bad, they didn’t sell. So he took the “Go west, young man” route, got a job over­see­ing wel­fare in the north half of Idaho, and we found ourselves on Mil­it­ary Drive in Coeur d’Alene, Idaho in 1940. Dad drove com­bines dur­ing the wheat har­vest­ing sea­son in the Pal­ouse coun­try. He had quite a vari­ety of moon­light­ing jobs dur­ing his life, in­clud­ing chan­ging tires in a truck stop, but­cher­ing rab­bits, col­lect­ing bills, driv­ing a taxi and oth­ers I can’t re­mem­ber.

World War II began and Dad was in danger of be­ing draf­ted. He was a quiet, con­scien­tious ob­ject­or, not a pop­u­lar po­s­i­tion, so he lost a few jobs when his views were dis­covered. We lived in Walla Walla, Wash­ing­ton (a county job with a de­fer­ment) where my broth­er Douglas was born in Oc­to­ber 1943, then two months in Yakima in early 1945, and then Spokane with the War Hous­ing Au­thor­ity.

I learned arith­met­ic while play­ing games with my moth­er and read­ing as she read to me. Mom got me in­to first grade at five and a half, but after a few weeks the teach­er told her that it wasn’t work­ing, that I was not par­ti­cip­at­ing and just look­ing out the win­dow. But they real­ized I was just bored, already know­ing the ma­ter­i­al, so with ex­tra ma­ter­i­al it worked out ok. I had already got­ten in­to the habit of not listen­ing to the teach­er, a habit that con­tin­ues to this day, a habit that means I rarely learn much from lec­tures.

I was a quiet child, but per­haps with a re­bel­li­ous streak. For ex­ample, in Janu­ary 1945, when I was not quite sev­en, I at­ten­ded a Yakima pub­lic school where I got lunch which I was re­quired to fin­ish. I hated ma­car­oni and cheese, served every Thursday, so on one Thursday I went out to the school bus but didn’t get on. In­stead I went through the trees and played in the nearby creek un­til the bus re­turned and I came home. No one found out, but sev­en hours in Janu­ary without lunch prob­ably cured me of play­ing hooky.

When I was 8, my grand­fath­er vis­ited and taught me the rules of chess. I bugged him in­cess­antly to play more games, so got re­stric­ted to one per day.

The war ended and Dad heard of an op­por­tun­ity to teach at the newly foun­ded Far­ragut Col­lege and Tech­nic­al In­sti­tute at the site of the de­com­mis­sioned Far­ragut Nav­al Base on Lake Pend Or­eille in north­ern Idaho. Over Thanks­giv­ing, 1946, we moved and he took over, mid-semester, four courses in the so­cial sci­ences and hu­man­it­ies. As we ar­rived, Mom de­veloped a weak­ness in her legs and even­tu­ally a hos­pit­al in Spokane de­cided she had polio — at age 39! She came home Christ­mas Eve to a house with a wood stove, an over­worked hus­band and two boys, 8 and 3, and was bedrid­den.

Some 18 months later, Mom and Doug took the train (I’m not sure how, but surely a wheel­chair was in­volved) back to Chica­go where they stayed with Mom’s sis­ter while she was “re­hab­il­it­ated”, get­ting a full brace on one leg and a half brace on the oth­er, and learned to walk with crutches and even drive a spe­cially out­fit­ted car. But after driv­ing across the coun­try to Spokane, her old re­flexes took over in the midst of traffic and she crashed in­to a store’s plate glass win­dow, in­jur­ing no one, for­tu­nately. She nev­er drove again.

I was in a three-room school­house in Far­ragut, the best school I went to, be­cause while in 4th grade I listened to the 5th graders and then skipped that class. After school and dur­ing sum­mers I headed out­doors, un­su­per­vised, and learned how to ne­go­ti­ate the woods and lake by my­self or with a friend or two. A great grow­ing-up ex­per­i­ence (later I told my chil­dren they were dis­ad­vant­aged grow­ing up in Berke­ley, by com­par­is­on).

In Far­ragut, when I was nine, Dad got me a job at the Col­lege lib­rary shelving books, three morn­ing hours three days a week, for the princely sum of \$.25/hour. I got bored and asked Dad if I could quit. He re­marked that he had to work to give me a very mod­est al­low­ance, so what did it mean that I didn’t want to work to en­hance my spend­ing money? I got the mes­sage, a good one.

Dad dis­covered he liked teach­ing and de­cided to go to gradu­ate school in so­ci­ology at the Uni­versity of Wash­ing­ton. We settled in­to Ed­monds, Wash­ing­ton where I spent grades 7–12. We were poor, liv­ing on a TA’s salary and Dad’s moon­light­ing. When Dad passed his Ph.D. or­als, we cel­eb­rated by split­ting a pint of cheap Neapol­it­an ice cream. But it was a good life. I worked four sum­mers in a nurs­ery, sav­ing for col­lege, and learned a bit about garden­ing. I also had to do the house clean­ing for those six years, a full clean­ing (dust­ing, va­cu­um­ing, floor wash­ing) once a week and a half-clean mid­week. This led to a fair num­ber of dis­putes since my moth­er’s stand­ards were much high­er than mine. It was very frus­trat­ing for her to be un­able to get things done as she would have done them.

I loved play­ing games: sports, chess, and the pop­u­lar card games of those days (any­one re­mem­ber Rook or Flinch?). But there was al­ways a short­age of op­pon­ents (we were liv­ing in a semirur­al area on a dirt road and the nearest friends were two miles away), and a ten­nis match meant find­ing an old slightly bent rack­et, some half dead balls, and a court with grass in the cracks and a sag­ging net. In ret­ro­spect, this still seems bet­ter to me than be­ing driv­en to ten­nis les­sons on a snazzy court with the best equip­ment where you spend half your time listen­ing to a coach.

There were few chess play­ers around, but oc­ca­sion­ally I ended up at the Uni­versity of Wash­ing­ton Chess Club, which was worth a story in the Seattle pa­pers.

High school wasn’t hard, and I ended up as va­le­dictori­an and wrote and mem­or­ized a speech on the “Chal­lenge of non­con­form­ity”.

When it came time to ap­ply to col­lege, I knew that I didn’t like to write and liked math best of my aca­dem­ic sub­jects, so I de­cided that Cal­tech was the place I’d like to go. I also ap­plied to Reed and to the Uni­versity of Chica­go be­cause of my par­ents con­nec­tions with it (how oth­er­wise would I have known of its ex­ist­ence as it did not have a foot­ball team?). But only Chica­go gave me a schol­ar­ship (\$1200, yet with tu­ition only \$690, I could make ends meet) so there I went.

It was won­der­ful. I’d been starved for boys to play games with and sud­denly I was in a dorm full of boys wait­ing to be lured in­to games (par­tic­u­larly chess) and small-time sports (they had a great in­tra­mur­al pro­gram).

At that time, in 1954, Chica­go offered 14 year-long courses with the yearly grade be­ing de­term­ined by a six-hour ex­am at the end of the year. The idea was to give the stu­dents the free­dom to choose what to fo­cus on at any giv­en time, but en­sure that by the end of the year they had learned the sub­ject. The ex­ams were very well writ­ten but ten­ded to fa­vor the smart rather than the hard work­er. There were three courses in each of the phys­ic­al sci­ences, so­cial sci­ences and hu­man­it­ies, as well as one course each in philo­sophy (called OMP, or­gan­iz­a­tion, meth­ods, and prin­ciples of know­ledge), Eng­lish, his­tory, a lan­guage of one’s choos­ing, and a gen­er­al math course. There were place­ment ex­ams which you took upon en­ter­ing, and it was pos­sible to do well enough in all the ex­ams to be im­me­di­ately giv­en a BA. I was ex­emp­ted from four, math, eng­lish, and the first of the nat­ur­al and so­cial sci­ences.

There were quite a few early entrants, some young­er than 16, and too many didn’t handle the aca­dem­ic free­dom well, so the Uni­versity even­tu­ally transitioned to a more tra­di­tion­al course and ex­am sys­tem. I found my­self im­mersed in games, learned some aca­dem­ics, but got more Fs than As. For ex­ample in my third year I signed up with a friend for an elec­tri­city and mag­net­ism lab course, which re­quired ten three-hour labs in a quarter. After five weeks, we figured two labs per week was doable. That no­tion died by the end of the eighth week when five labs per week was ob­vi­ously not go­ing to hap­pen.

I joined a fra­tern­ity, Psi Up­si­lon, in my third year. That seems odd now, but Chica­go un­der Robert Maynard Hutchins had not just quit foot­ball but also banned fra­tern­it­ies for un­der­gradu­ates. Some fra­tern­it­ies con­tin­ued to ex­ist as res­id­ences for grad stu­dents with some fra­tern­ity tra­di­tions main­tained, e.g., new mem­bers be­ing chosen by the old ones. A bunch of my friends joined Psi U, and I fol­lowed suit, be­ing ad­mit­ted to an older group of GIs, grad stu­dents and older un­der­grads. They were a good bunch. Our fra­tern­ity house was loc­ated dir­ectly across the street from Bart­lett Gym, and I ima­gine ped­es­tri­ans must oc­ca­sion­ally have been startled by the sight of a scantily clad young man ra­cing across the street through winter snow dodging slush-puddles to get to or from the gym.

Dur­ing the sum­mers of 1956–1959, I had sum­mer jobs or in­tern­ships at the Navy Elec­tron­ics Lab on Point Loma in San Diego. They had a state-of-the-art com­puter and I was asked to write a pro­gram in which the com­puter would learn how to play tic-tac-toe. I did this in ma­chine lan­guage (For­tran ex­is­ted but I don’t re­mem­ber what else). The boss sug­ges­ted I sub­mit it to the Journ­al of the As­so­ci­ation for Com­put­ing Ma­chinery. They ac­cep­ted it (to my great sur­prise — maybe they were des­per­ate in those days) but asked for re­vi­sions. But I was lazy, hated to pol­ish, and didn’t think much of what I’d done, so I nev­er made the re­vi­sions. And thus missed my chance to really get in­to com­puters in the very early days.

But my time at the Lab non­ethe­less ef­fected my life quite deeply: a cowork­er at the Lab in­tro­duced me to rock climb­ing — and Tahquitz Rock — and an­oth­er cowork­er or­gan­ized a nine-day hike in the Si­er­ras, so my sum­mers at the Lab marked the be­gin­ning of years of rock climb­ing, hik­ing and moun­tain climb­ing.

While at Chica­go, I ma­tured as a chess play­er, and played first or second board on our col­lege team, along­side Lester Franken­stein, Mike Robin­son and Mitch Sweig. We won the In­ter­col­legi­ate Cham­pi­on­ship in Decem­ber of 1956 and again in 1958 (it was only held every oth­er year). Typ­ic­ally we played sev­en rounds in those com­pet­i­tions. You could de­term­ine the win­ning team by adding up total points, count­ing \( .5 \) for draws (28 max) or by adding up match wins (max 7). In 1956 the tour­na­ment win­ner was de­cided by total points, and so we won, though we would have been second if the tally had been made by match points. We were up­starts, and the east­ern power­houses were dis­pleased, so they switched to a match point tally for the 1958 con­test. On that oc­ca­sion, we won the match point total and so the tour­na­ment, and would have lost if the tally had been done by total points. In those days, Sports Il­lus­trated thought chess was a sport, and we got a nice write-up of our 1956 vic­tory in the Janu­ary 14, 1957 is­sue.

I con­tin­ued to play chess un­til 1968, but didn’t study the game much after col­lege. I did get ranked as high as 25th in the US at one point, but dis­covered that math was a much bet­ter game. I nev­er played Bobby Fisc­her, but did have a few games against in­ter­na­tion­al grand­mas­ters and didn’t do badly. I was Illinois state cham­pi­on while an un­der­grad, and that may have helped de­flect at­ten­tion for a time from my poor grades.

I lost my schol­ar­ship after three years (no sur­prise) and also flunked Ger­man (a full-year course with the grade de­term­ined by an end-of-the-year ex­am). I took it again in my fourth year and was lucky; I flunked it again (it is hard to pass an ex­am with a ser­i­ous or­al por­tion if you have nev­er gone to class). I say “lucky” be­cause it meant I had to go back for a fifth year, dur­ing which I took sev­en of the eight grad courses in math which were covered by the Mas­ters Ex­am. And passed Ger­man with a D. So I had a BS by June 1959.

I don’t re­mem­ber wor­ry­ing, or even think­ing much, about the fu­ture, but I did ap­ply to Chica­go for grad school in math (I did not ap­ply any­where else for who would take me with my re­cord), ar­guing that I had already taken sev­en of the eight courses and was thus on the verge of be­ing ready for the Mas­ters Ex­am. I was ad­mit­ted pro­vi­sion­ally with the pro­viso that I get grades closer to B than C. In the fall quarter I re­ceived a B, a C and a Pass, not closer to B than C but no one ob­jec­ted.

In those days you could get four grades on the Mas­ters Ex­am: (i) pass with fin­an­cial sup­port; (ii) pass but with no fin­an­cial sup­port; (iii) pass with the re­com­mend­a­tion that you go else­where; and (iv) fail. I passed at the low­est level, but ig­nored the re­com­mend­a­tion that I go else­where, and bor­rowed a bit of money to con­tin­ue at UC in the fall of 1960. For sup­port, I ap­plied for teach­ing at Roosevelt Uni­versity, a night school in down­town Chica­go. Saun­ders Mac Lane wrote a re­com­mend­a­tion and I got a job (three courses, all pre­cal­cu­lus, at three hours/week each), enough to pay my way.

But dur­ing the sum­mer (1960) I met my fu­ture wife, In­grid. She had fin­ished her fresh­man year at Berke­ley in Ger­man Lit­er­at­ure and we met in San Diego where our par­ents lived. I was headed back to Chica­go that fall, but de­cided I would trans­fer to Berke­ley for the ob­vi­ous reas­on. I ap­plied to Berke­ley fig­ur­ing that an MS from Chica­go would get me in. I packed up my car and drove out to Berke­ley only to dis­cov­er I wasn’t ad­mit­ted. A pe­ti­tion didn’t help.

So I went back to Chica­go in the fall of 1961 hav­ing loitered through the pre­vi­ous year un­der the as­sump­tion I was leav­ing. Now I had to get ser­i­ous and think about the Ph.D. qual­i­fy­ing ex­am, a two-hour or­al ex­am on two top­ics. I chose to­po­logy, with top­ics in Hu’s book Ho­mo­topy The­ory, and fi­nite group the­ory, with some of the chapters in Mar­shall Hall’s book The The­ory of Groups. They wanted breadth, and ho­mo­lo­gic­al al­gebra was not far enough from al­geb­ra­ic to­po­logy.

I took the ex­am in the fall of 1961, my second year of grad school. Of course I failed. But there was a second chance. I got good ad­vice: to start talk­ing math, join the math com­munity and pick up folk­lore. I hadn’t taken a mock qual or any­thing re­motely like that. My friends were jocks and out­side the math com­munity. I didn’t have an of­fice where I would nat­ur­ally meet math grads, but I star­ted go­ing to tea.

Nor­man Steen­rod was vis­it­ing that year, and a not­able event for me oc­curred at tea where I eaves­dropped on his ex­plan­a­tion of the Hopf map, \( S^3 \to S^2 \), where the fibers are the unit circle in the \( xy \)-plane and the \( z \)-ax­is uni­on in­fin­ity, as well as in all the \( (1,1) \)-tor­us knots in the com­ple­ment. The map \( (z,w) \to [z,w] \) nev­er en­lightened me, but this mar­velous pic­ture did.

One of my qual ex­am­iners was a postdoc named George Mc­Carty (he later wrote a text book on al­geb­ra­ic to­po­logy) who was friendly and asked if I’d be in­ter­ested in his Ph.D. thes­is on the ho­mo­topy groups of homeo­morph­isms of a man­i­fold.

I took the Ph.D. qual again in the spring of 1962 and passed, but was told that my per­form­ance in to­po­logy was weak and I shouldn’t choose that sub­ject for a thes­is. So I dropped in on Mac Lane to ask him about be­ing my ad­viser. He asked what I was in­ter­ested in and I said group the­ory, ho­mo­lo­gic­al al­gebra and to­po­lo­gic­al man­i­folds (Mc­Carty’s thes­is). Mac Lane wisely ad­vised me to go home for the sum­mer and think about the three and talk to him again in the fall. I nev­er touched the first two, and thought some about to­po­lo­gic­al man­i­folds.

In the fall, now mar­ried, I ap­proached El­don Dyer ask­ing him to be my ad­viser. Dick Lashof would have been the more geo­met­ric and nat­ur­al choice, but he’d been on my qual com­mit­tee and pre­sum­ably had made the neg­at­ive com­ment about work­ing in to­po­logy, where­as Dyer had been away on sab­bat­ic­al. Dyer did not an­swer as I ex­pec­ted, for I figured he’d want to look at my file. I peri­od­ic­ally dropped in on Dyer with math ques­tions, but nev­er asked him for an an­swer; I’d asked, and now the ball was in his court. Later I found out from an­oth­er math­em­atician that Dyer even­tu­ally real­ized that I had be­come his in­form­al, hence form­al, stu­dent.

I now began to be a ser­i­ous math­em­atician, dis­cov­er­ing that math was a bet­ter game than all my oth­ers. Re­search ap­pealed to me, something like sort­ing out a dif­fi­cult end game in chess. I got to know Wal­ter Daum, a fel­low Dyer stu­dent who had ac­com­plished in four years what had taken me sev­en to do. We ran a two-man hour-long sem­in­ar meet­ing every day. Pre­par­a­tion was frowned upon and one of us would stand at the black­board, writ­ing down the next the­or­em in Zee­man’s notes or a Stallings pa­per or Brown’s proof of the Schoen­flies the­or­em, which we would try to prove on our own be­fore look­ing for a hint, pro­ceed­ing slowly but learn­ing a lot. This sem­in­ar was prob­ably the most cru­cial edu­ca­tion­al ex­per­i­ence in my young life. (Later, it was learn­ing math with my gradu­ate stu­dents, both while they were stu­dents and then later on.)

My son Rolf was born in March, 1968, and then Kate came along in April 1971. They have been a great joy in my life. Rolf is a med­ic­al doc­tor and Kate a bio­s­tat­ist­i­cian, and both are very much out­doors people. We’ve hiked, skied, rock climbed and run to­geth­er many times.

My mar­riage to In­grid ended in the late 1970s and for a few years I was a single fath­er with cus­tody of both Rolf and Kate. They were such sens­ible kids it was pretty easy. In 1981 I met Linda and we mar­ried in April, 1982. She had two daugh­ters of sim­il­ar ages, Kara and Erika, so we had an in­ter­est­ing time mer­ging the two fam­il­ies, with great res­ults.

Linda liked to travel, so we six spent the sum­mer of 1981 in Cam­bridge; the fall of 1982 at IMPA in Rio de Janeiro; the spring of 1983 in Wash­ing­ton D. C. (Uni­versity of Mary­land); and the sum­mer of 1984 again in Cam­bridge. By then it be­came harder to get high school-age kids to travel.

Linda’s broth­ers, Gerry and Bob Slav­in, were al­ways fun and we hiked with them many times in the Canyon­lands and Grand Canyon (in­clud­ing our hon­ey­moon for six days be­low the south rim of the Grand Canyon).

My par­ents died in 1991, Mom on May 23, Dad on June 1. Mom had had a stroke sev­en years earli­er, leav­ing her com­pletely bedrid­den, only able to eat with her left fin­gers, un­able to an­swer a yes–no ques­tion, but re­cog­niz­ing us. Her life was as un­lucky as mine has been lucky. Dad got non-Hodgkins can­cer, and when he spent his last month in the hos­pit­al, sep­ar­ated from my Mom, the light seemed to go out in her eyes. Two days be­fore Dad died, he asked me to help him get up from bed. It was ob­vi­ously pain­ful and I asked him if he really wanted to get up. He replied, “I’m not yet ready to have taken my last steps”. As I write this in Septem­ber, 2021, after not hik­ing last sum­mer in the High Si­er­ras, I won­der if I have already taken my last mul­ti­day hike there, after 60 splen­did years of do­ing so.

I men­tioned above hik­ing in the Si­er­ras in 1959, and I want to re­turn to that ex­ped­i­tion for a mo­ment. My broth­er Doug, still 15, joined me on that trip and we in­ten­ded to climb the clas­sic route on the east face of Mount Whit­ney. In those days you got a climb­ing rope by writ­ing to the Seattle Coop (now REI) for a nylon 120 foot rope that ten­ded to kink; you got climb­ing shoes by draw­ing an out­line of your foot on pa­per and send­ing it to the Ski Hut in Berke­ley (which re­turned a pair of not-too-snug shoes); and you got a pack by writ­ing to Kelty in Burb­ank. No store in San Diego sold such stuff! (Ten years later every kid had a down jack­et be­cause it was cool, but not in 1959).

Doug and I car­ried this gear (plus piton ham­mer and pitons) and 200 feet of ma­nila hemp rope in case we were forced to rap­pel off Whit­ney. We went over Kearsarge Pass, headed south over For­est­er’s Pass, and went past Lake Tu­lainyo (the lake at the highest alti­tude in the US) around to East Face Lake be­low Mt. Whit­ney. We had car­ried a canned ham (freeze dried food was rare) but were too tired to eat it. The next day’s as­cent was straight­for­ward (only a few piton place­ments and it was my broth­er who suffered car­ry­ing the rap­pel rope), in­clud­ing the des­cent down the Moun­tain­eer’s Route.

That duress of that ex­per­i­ence could have killed Doug’s in­terest in moun­tains but 25 years later we were to­geth­er again for an­oth­er big ad­ven­ture: an as­cent of Mt. McKin­ley. In those days, one of every 300 people at­tempt­ing to sum­mit McKin­ley died. Those are not good odds. The pre­vi­ous year a fam­ous Ja­pan­ese climber so­loed McKin­ley in the winter, ra­di­oed from the top and was nev­er seen again. The oth­er death was that of a guide who, des­cend­ing on skis dur­ing a white-out, had come to a stop just in time at the edge of a cre­vasse only to fall in when the edge gave way. Well, we wer­en’t go­ing to solo in the winter nor ski in a white-out; in fact we packed 21 days of food and fuel (used only half), ad­vanced slowly so as to ac­cli­mate well, and then had good luck on our sum­mit day for the wind was mild. It took 14 days, two for the des­cent.

Later Doug and I tried twice to climb Mt. Rain­i­er and were turned back by bad weath­er (which could kill climbers); Doug and Kate sum­mited on a sub­sequent at­tempt.

In 2010 Doug talked ten of the fam­ily in­to climb­ing Kili­man­jaro. The ten were my cous­in Kay, 72, me, Linda, Doug and wife Gail and son Camer­on, Rolf and wife Jan­nell, Kate and then hus­band Da­mon. There are rules for climb­ing Kili, one be­ing that you must hire loc­als as guides and port­ers, so we had around 36 who guided, put up tents, car­ried gear, brought fresh eggs, chick­en and wa­ter­mel­on, etc. It was not a wil­der­ness ex­per­i­ence. The com­par­is­on with the Si­er­ras is in­ter­est­ing. Most traffic in the High Si­er­ras oc­curs in Au­gust and Septem­ber and then the Si­er­ras have ten months to re­cov­er. Kili can be climbed year-round and the total num­ber of people in any giv­en party is pretty much triple the num­ber of “climbers”, be­cause of the port­ers. Kili is a vol­cano and most pre­cip­it­a­tion sinks in­to the ground, so there are few streams and lakes. The Si­er­ras are a new gran­ite range dot­ted by beau­ti­ful lakes and streams. So, while Kili was an in­ter­est­ing ex­per­i­ence, I can’t heart­ily re­com­mend it.

Doug wanted to climb the highest moun­tain in the world, meas­ured by the dis­tance from the cen­ter of the earth. Be­cause the earth is not round, but wider at the Equat­or, the highest such moun­tain is not Everest, but Chim­borazo in Equador, at 6,263 meters (20,548 feet). I wasn’t suf­fi­ciently in­ter­ested, so he went there in Decem­ber 2012 with his son, Camer­on, a friend and two guides. On Decem­ber 22, he was des­cend­ing from a warm-up climb on Coto­paxi (5897 meters) at dawn, be­fore the sun hit the treach­er­ous slopes. He sat down with his guide on a rock out­crop­ping to have a drink of wa­ter. He looked at the beau­ti­ful view and ex­claimed, “Isn’t life won­der­ful!”, then slumped over and died. It was a glor­i­ous way to go, but it was much too soon for the rest of us.

Twice, in 1992 and 2002, Rolf and I hiked for two-week-long stretches on the north slope of the Brooks Range in Alaska. Dur­ing the first trip, we spent twelve days with no sign that hu­mans had ever ex­is­ted, no hikers, no trails, no planes over­head, just hun­dreds of cari­bou and a fleet­ing glance at a wolf. It felt really primev­al.

We did have a mem­or­able en­counter with a grizzly bear and her cub. We were walk­ing at the edge of a wide (\( \sim \)25 yards), shal­low, half-dry stream bed when we saw the grizzly emerge from low bushes on the far shore. I took three pho­tos as she drew op­pos­ite us. As I put my cam­era away in my shirt pock­et, she charged. Ima­gine in foot­ball a wide re­ceiv­er catch­ing a pass on the 25-yard line and then ar­riv­ing in the end zone in no time; the bear is faster. We were sur­prised and even at the time could not re­mem­ber how we re­acted. We were lucky that it was a bluff, for about 10 yards from us she turned away and her cub scrambled to catch up.

A few years later at a Geor­gia to­po­logy con­fer­ence, Cliff Taubes was lec­tur­ing and half-way through he said he was go­ing to wake up his audi­ence by telling a story be­fore go­ing back to math. I was speak­ing in the next slot, so I did the same, telling the grizzly bear story. No one, in­clud­ing me, re­mem­bers my math talk, but every­one re­mem­bers the grizzly bear story!

I did most of my rock climb­ing and kayak­ing with Den­nis John­son, and touch on some of those ex­ped­i­tions in my brief bio­graphy of him for Cel­eb­ra­tio.

In 1983, I saw a let­ter to the ed­it­or by Steph­en Salt­er in the journ­al Nature; it de­scribed a way of re­du­cing nuc­le­ar arms us­ing the idea of cut­ting a cake fairly so that all cake eat­ers are happy.1 I found the idea in­ter­est­ing and wrote an elab­or­a­tion which can be found here.

My ver­sion is to al­low the US and the USSR to as­sign val­ues to each of their own nuc­le­ar arms, with the total val­ues adding up to an agreed upon con­stant N. Then, the US would choose arms be­long­ing to the USSR whose val­ues (already as­signed by the USSR) ad­ded up to a con­stant M and these would be des­troyed. Sim­il­arly, us­ing the same M, the USSR would choose US arms to be des­troyed.

Be­cause the val­ues of a giv­en arm would likely dif­fer between the two coun­tries, the US would be able to des­troy USSR arms that we con­sidered more valu­able than the USSR did, and con­versely. Thus, as with a chocol­ate cake, both parties should be pleased with the out­come, and con­tin­ue with fur­ther it­er­a­tions.

For ex­ample, I like icing bet­ter than you, you like cher­ries on the icing more than I do, and we equally like the cake it­self. Then, if I am the cut­ter, I will try to cut roughly equal pieces but with one hav­ing more icing and the oth­er more cher­ries. You choose the slice with more cher­ries and I then am happy with the ex­tra icing. For two people this is simple2 but for \( n \) people, it gets much more com­plic­ated (as Erica Klar­reich ex­plains in “How to cut cake fairly and fi­nally eat it too,” Quanta magazine, Oc­to­ber 6, 2016).

Such schemes ap­peal to the math­em­at­ic­ally minded, but seem not to go far in the real world.

I over­lapped with Camer­on Gor­don and Justin Roberts on sev­er­al vis­its to War­wick (thanks to Colin Rourke for mak­ing them pos­sible) and en­joyed listen­ing to them play­ing acous­tic gui­tars, usu­ally cov­er­ing old rock songs. I offered en­thu­si­asm and the ti­ni­est bit of or­gan­iz­ing and soon both were play­ing at vari­ous birth­day fests: mine in 1998, 2008 and 2018 (the lat­ter was mostly for Abby Thompson and her ad­viser Schar­le­mann), for Schar­le­mann’s 60th, for Gor­don’s 60th, for Mike Freed­man’s 60th and a few oth­ers. Camer­on and Justin were of­ten joined by Eli Grigsby, an ex­cel­lent vo­cal­ist, Taiyo Inoue on bass, and vari­ous drum­mers.

I’ve left to the last a top­ic/pro­ject which took a con­sid­er­able amount of time and thought and which grew out of years of dis­cus­sions with my fath­er and broth­er about how to “save the world”. I wrote the first words in Septem­ber 2007 just be­low Bish­op Pass in the Si­er­ras while wait­ing for Doug and the oth­ers to wake up. Over the next ten years I would stop while driv­ing some­where and write a few pages. After a while I put these bits in some or­der and star­ted a \( \mathrm{\TeX} \) file. Even­tu­ally it turned in­to a manuscript. Some friends read it and made com­ments, usu­ally to tone it down so as not to of­fend. It still has not yet gone to a pub­lish­er.