Celebratio Mathematica

In Ju­ly 1996, the As­so­ci­ation for Wo­men in Math­em­at­ics (AWM) held the Ju­lia Robin­son Cel­eb­ra­tion of Wo­men in Math­em­at­ics at the Math­em­at­ic­al Sci­ences Re­search In­sti­tute in Berke­ley as part of AWM’s year long 25th An­niversary cel­eb­ra­tion. In Au­gust 1997, an­oth­er con­fer­ence to cel­eb­rate wo­men math­em­aticians in num­ber the­ory and ana­lys­is was held at Berke­ley. And there are more.

Thirty years ago, such cel­eb­rat­ory iden­ti­fic­a­tion would have soun­ded the pro­fes­sion­al death knell. Lest you had any doubt, if you were a fe­male gradu­ate stu­dent at the time you may very well have been told in “jest” by one of your pro­fess­ors: “There’ve only been two wo­men math­em­aticians. One wasn’t a wo­man and one wasn’t a math­em­atician.” And if your name was iden­ti­fi­ably fe­male, you would not have been ad­mit­ted to the gradu­ate pro­gram in math­em­at­ics at Prin­ceton. It was not un­til 1968 that wo­men were al­lowed.

How did we get from there to here? While the 1960’s might be char­ac­ter­ized as a time of na­iv­eté, deni­al, and ly­ing low, the early 1970’s be­came a time of con­scious­ness rais­ing. But then, quickly, the com­munity of wo­men math­em­aticians moved in­to a pro­act­ive stance and de­veloped con­struct­ive pro­grams to in­crease the par­ti­cip­a­tion of wo­men in math­em­at­ics. These yiel­ded pos­it­ive res­ults, re­spect, and self-con­fid­ence that set the stage for achieve­ment, re­cog­ni­tion, and cel­eb­ra­tion. Hon­or­ing the “two wo­men math­em­aticians,” AWM in­aug­ur­ated the first of its an­nu­al Emmy No­eth­er Lec­tures in 1980 and, in 1985, its Sonya Ko­va­levsky High School Days [e15].

Ju­lia, A Life in Math­em­at­ics [e18], em­in­ently en­ga­ging and ac­cess­ible, evokes such a per­spect­ive, im­pli­citly and ex­pli­citly. A com­pil­a­tion of four pre­vi­ously pub­lished art­icles (start­ing with a beau­ti­fully format­ted “Auto­bi­o­graphy of Ju­lia Robin­son” by Con­stance Re­id, then art­icles by Lisl Gaal, Mar­tin Dav­is, and Yuri Mati­jaseivich), it proves to be a great deal more than the sum of its parts.

The four art­icles re­in­force as well as com­ple­ment each oth­er, some­times re­lat­ing the same story from dif­fer­ent vant­age points. In­deed, the evol­u­tion of the solu­tion to Hil­bert’s tenth prob­lem, as seen through the eyes of the key prot­ag­on­ists, is a won­der­ful, ex­cit­ing, and very hu­man story of math­em­at­ic­al pro­gress with all its ups and downs.

Here is a tele­scop­ic view.1 Hil­bert’s tenth prob­lem (HTP), as posed in 1900 at the In­ter­na­tion­al Con­gress of Math­em­aticians in Par­is, is to find an ef­fect­ive meth­od to de­term­ine if a giv­en Di­o­phant­ine equa­tion is solv­able in in­tegers.

Dur­ing the 1930’s lo­gi­cians made pre­cise the in­form­al no­tion of “ef­fect­ive meth­od,” and raised the pos­sib­il­ity (of show­ing) that no such meth­od ex­ists. A strategy for demon­strat­ing such a neg­at­ive res­ult would be to show that a known “un­de­cid­able” prob­lem could be re­stated in terms of the solv­ab­il­ity of cer­tain Di­o­phant­ine equa­tions. Such would be the case if all re­curs­ively enu­mer­able sets of nat­ur­al num­bers (non­neg­at­ive in­tegers) were Di­o­phant­ine. Think of a re­curs­ively enu­mer­able (r.e.) set as a set listable by a com­puter pro­gram. These com­puter pro­grams are in turn listable, so each r.e. set A can be as­so­ci­ated with a (Gödel) num­ber, which we de­note by \( [A] \).

A set \( A \) of nat­ur­al num­bers is Di­o­phant­ine (or ex­ist­en­tially defin­able) if there is a poly­no­mi­al \[ P_A(x, y_1, \dots,y_n) \] with in­teger coef­fi­cients with the prop­erty that \( a \) be­longs to \( A \) if and only if \[ P_A(a, y_1, \dots,y_n) = 0 \] is solv­able in in­tegers, i.e., \begin{equation} \label{star} a \in A \quad\text{if and only if}\quad (\exists y_1 \dots \exists y_n \in \mathbf{Z})(P_A(a, y_1,\dots, y_n)=0). \end{equation} Us­ing stand­ard ar­gu­ments, we can stip­u­late solv­ab­il­ity in the nat­ur­al num­bers \( \mathbf{N} \) in­stead of in \( \mathbf{Z} \).2 Di­o­phant­ine re­la­tions are sim­il­arly defined.

It is easy to see that Di­o­phant­ine sets are re­curs­ively enu­mer­able. On the oth­er hand, the Di­o­phant­ine­ness of all r.e. sets seemed a bit pre­pos­ter­ous since this would im­ply the ex­ist­ence of a uni­ver­sal \( d, N, \) and Di­o­phant­ine equa­tion \[ P(u, x, y_1,\dots,y_N) = 0 \] of de­gree \( d \) in the vari­ables \( y_1,\dots,y_N \) that define (as above) every r.e. set \( A \) (after re­pla­cing \( u \) with \( [A] \)). Hence, for ex­ample, the set of prime num­bers would be so defined by \( P \).3 It would also fol­low that every Di­o­phant­ine equa­tion \[ D(a_1,\dots, a_m, y_1,\dots, y_n) = 0 \] with para­met­ers \( a_1,\dots,a_m \) would be equi­val­ent (in terms of solv­ab­il­ity) to one with the same para­met­ers and of de­gree \( d \) in \( N \) un­knowns (the uni­ver­sal \( d \) and \( N \)).4

Ju­lia star­ted work­ing on HTP after re­ceiv­ing her Ph.D. in 1948. Al­fred Tarski, her thes­is ad­visor, had sug­ges­ted to Raphael Robin­son, her hus­band, that the set of powers of 2 was not Di­o­phant­ine.5 Ju­lia tried to show it was. This she was un­able to do, but in­stead showed [2] that

1. the re­la­tion \( y = x^z \) is ex­ist­en­tially defin­able in terms of any re­la­tion of “ex­po­nen­tial growth.”

Here \( x \), \( y \), and \( z \) are nat­ur­al num­bers.

A re­la­tion \( J(u, v) \) is of ex­po­nen­tial growth if it “grows faster than a poly­no­mi­al but not too ter­ribly fast.” That is, let \[ J = \{(a, b) \in \mathbf{N}^2 \mid J(a, b) \text{ holds}\} .\] Then

1. for each \( k > 0 \) there ex­ists \( (a, b) \in J \) with \( b > a^k \) and
2. for each \( (a, b) \in J \),  \( b < a^a \).

For \( y = x^z \) to be ex­ist­en­tially defin­able in terms of \( J \) means: for all \( a \), \( b \), \( c \in \mathbf{N} \), \begin{equation} \label{twostar} b = a^c \quad\text{if and only if}\quad (\exists w_1 \dots \exists w_n \in \mathbf{N}) (Q(a,b,c,w_1,\dots,w_n,J)) \end{equation} where \( Q(x, y, z, w_1,\dots, w_n, J) \) is a for­mula built up from the vari­ables \( x \), \( y \), \( z \), \( w_1 \),…, \( w_n \); the math­em­at­ic­al sym­bols 0, 1, \( + \), \( \times \), \( = \); lo­gic­al con­nect­ives & and \( \vee \); and the re­la­tion \( J \).

Mar­tin Dav­is also had been think­ing about HTP since his gradu­ate stu­dent days at Prin­ceton. He was able to show [e1] that every re­curs­ively enu­mer­able set could be defined in a way that seemed al­most Di­o­phant­ine.6 Meet­ing in 1950 at the first post-war In­ter­na­tion­al Con­gress at Har­vard, Ju­lia and Mar­tin com­pared their dis­tinct at­tacks on HTP.7 Build­ing on their com­bined ap­proaches, they later pub­lished a pa­per with Hil­ary Put­nam [3] show­ing that

(ii)  every re­curs­ively enu­mer­able set of nat­ur­al num­bers is ex­po­nen­tial Di­o­phant­ine.

Ex­po­nen­tial Di­o­phant­ine sets \( A \) are defined as in \eqref{star} but now the poly­no­mi­al \[ P_A(x, y_1,\dots, y_n) \] may have ex­po­nents that are vari­ables.

Thus by (i), the neg­at­ive res­ol­u­tion to HTP, a metamathem­at­ic­al res­ult, would fol­low from show­ing a purely num­ber the­or­et­ic res­ult, namely,

(J.R.)  there is some Di­o­phant­ine re­la­tion of ex­po­nen­tial growth,

as hy­po­thes­ized in Ju­lia’s 1952 pa­per and later dubbed “J.R.” by Mar­tin.

Thus one need only ex­hib­it a Di­o­phant­ine equa­tion \[ B(u, v, x_1,\dots, x_m) = 0 \] with the prop­erty that the set of para­met­er pairs \[ J = \{(a, b)\mid (\exists x_1 \dots \exists x_m \in \mathbf{N})(B(a, b, x_1 ,\dots, x_m)=0)\} \] is a re­la­tion of ex­po­nen­tial growth.

Many math­em­aticians thought this ap­proach was mis­guided and, as far as I know, vir­tu­ally no one else pur­sued this dir­ec­tion. In Math­em­at­ic­al Re­views, Georg Kre­isel [e2] was par­tic­u­larly sharp:

These res­ults [3] are su­per­fi­cially re­lated to Hil­bert’s tenth prob­lem. […] The proof of the au­thors’ res­ults, though very el­eg­ant, does not use re­con­dite facts in the the­ory of num­bers nor in the the­ory of r.e. sets, and so it is likely that the present res­ult is not closely con­nec­ted with Hil­bert’s tenth prob­lem. Also it is not al­to­geth­er plaus­ible that all (or­din­ary) Di­o­phant­ine prob­lems are uni­formly re­du­cible to those in a fixed num­ber of vari­ables or fixed de­gree, which would be the case if all r.e. sets were Di­o­phant­ine.

In the 1960’s when he would give talks on HTP, Mar­tin would em­phas­ize the im­port­ant con­sequences of a proof or dis­proof of J.R. To the in­ev­it­able query on how did he think mat­ters would turn out, he had an ever-ready reply: “I think that Ju­lia Robin­son’s hy­po­thes­is is true, and it will be proved by a clev­er young Rus­si­an.”

Yuri Mati­jaseivich was a sopho­more at Len­in­grad State Uni­versity in 1965 when he star­ted think­ing about HTP. Dis­suaded at first from look­ing at the work of the “Amer­ic­ans” (“So far [they] have not suc­ceeded, so their ap­proach is most likely in­ad­equate”), it was not un­til 1968 that he sought out their pa­pers in trans­la­tion. Im­me­di­ately he star­ted spend­ing all his free time look­ing for a Di­o­phant­ine re­la­tion of ex­po­nen­tial growth, but got nowhere. Em­bar­rassed to con­tin­ue work­ing on a fam­ous prob­lem with no suc­cess, he steeled him­self from think­ing more about it, even re­fus­ing to look up Ju­lia’s new pa­per when it came out in 1969. Luck­ily (“there must be a god or god­dess of math­em­at­ics”), Yuri was ob­liged to re­view it for the So­viet coun­ter­part of Math­em­at­ic­al Re­views.

In this pa­per [5], Ju­lia at­temp­ted to prove J.R. by con­sid­er­ing solu­tions to a spe­cial form of Pell’s equa­tions \[ x^2 - (a^2 - 1)y^2 = 1,\quad a > 0, \] a form she had already used in her 1952 pa­per. Re­in­spired, Yuri star­ted con­sid­er­ing sim­il­ar equa­tions whose solu­tions are the Fibon­acci num­bers. Shortly after New Year’s day 1970, he fin­ished con­struct­ing a Di­o­phant­ine equa­tion \[ B(u, v, x_1,\dots, x_m) = 0 \] ex­ist­en­tially de­fin­ing the re­la­tion \( v = F_{2n} \), where \( F_k \) is the \( k \)th Fibon­acci num­ber [e7]. In early Feb­ru­ary, Grig­orii Tseitin presen­ted a talk on the work in Nov­os­ibirsk. John Mc­Carthy was present and sent his rough notes to Ju­lia when he re­turned to the States. Im­me­di­ately, Ju­lia re­con­struc­ted the proof and a few days later wrote Yuri:

…now I know it is true, it is beau­ti­ful, it is won­der­ful. If you really are 22, I am es­pe­cially pleased to think that when I first made the con­jec­ture you were a baby and I just had to wait for you to grow up.

When Mar­tin met Yuri that sum­mer at the In­ter­na­tion­al Con­gress in Nice, he was fi­nally able to tell him that “he had been pre­dict­ing his ap­pear­ance for some time.”

In a real sense, Yuri was the gradu­ate stu­dent Ju­lia nev­er had (more on this later). They be­came great friends and col­lab­or­at­ors. In his con­tri­bu­tion to Ju­lia, there is a de­light­ful ac­count of the math­em­at­ic­al tri­als and tribu­la­tions of their joint work that re­duced the uni­ver­sal \( N \) (for solv­ab­il­ity in the nat­ur­al num­bers) from 200 to 13 [6] (later, Yuri was able to fur­ther re­duce \( N \) to 9)8 and glimpses in­to life be­fore e-mail and be­fore the end of the cold war.

The work on HTP, prob­ably the most fa­mil­i­ar of Ju­lia’s math­em­at­ic­al con­tri­bu­tions, is part of a body of work that lies in the in­ter­face between lo­gic and al­gebra, and might be cat­egor­ized as ap­plied lo­gic. This area was the theme of Ju­lia’s 1980 Amer­ic­an Math­em­at­ic­al So­ci­ety Col­loqui­um talks, “Between lo­gic and arith­met­ic.”9 It had already been an un­der­ly­ing theme of her thes­is [1]. There, us­ing the­or­ems of Hasse on quad­rat­ic forms, Ju­lia had shown that the in­tegers are arith­met­ic­ally defin­able in terms of the ra­tion­als and hence, as a con­sequence of the land­mark case of the in­tegers, the arith­met­ic the­ory of the ra­tion­als is not de­cid­able. (A set \( A \) is arith­met­ic­ally defin­able if in \eqref{star} some of the ex­ist­en­tial quan­ti­fi­ers are al­lowed to be uni­ver­sal. A dis­cus­sion of Ju­lia’s dis­ser­ta­tion is in Lisl Gaal’s con­tri­bu­tion.)

My first in­tro­duc­tion to Ju­lia’s work was in the 1960’s when I was a gradu­ate stu­dent at M.I.T. I had been fas­cin­ated by emer­ging work of Jim Ax and Si­mon Kochen [e3], [e4], and [e5] solv­ing prob­lems about \( p \)-adics us­ing (non-stand­ard meth­ods of) mod­el the­ory. (I had been able to get some re­lated res­ults about dif­fer­en­tial fields [e6].) Ju­lia’s beau­ti­fully writ­ten pa­per, “The de­cision prob­lem for fields,” was a con­stant ref­er­ence [4]. It provided an over­view of what was known at the time, presen­ted a wealth of in­ter­est­ing ideas, and dis­cussed sev­er­al open prob­lems, some of which have sub­sequently been re­solved.10 It’s a pa­per I dis­trib­uted to my lo­gic class at Berke­ley sev­er­al years later and, over the years, have had oc­ca­sion to refer to from time to time. A faded purple mi­meo­graphed copy is with me as I write this in Hong Kong. Luck­ily, the Amer­ic­an Math­em­at­ic­al So­ci­ety has pub­lished The Col­lec­ted Works of Ju­lia Robin­son [7] where one can find all her pa­pers and also a fine mem­oir by Sol Fe­fer­man.

“The Auto­bi­o­graphy of Ju­lia Robin­son,” writ­ten by her sis­ter Con­stance Re­id dur­ing the last month be­fore Ju­lia died of leuk­emia in 1985, con­veys a real sense of Ju­lia’s life and spir­it. It is a de­light to read.

Ju­lia’s abil­ity to work in­de­pend­ently throughout her life — and in her own way — was cer­tainly shaped by events early on: the death of her moth­er when she was a tod­dler; a bout with scar­let fever when she was nine, fol­lowed by rheum­at­ic fever, which pre­ven­ted her from at­tend­ing school for two years. Her fas­cin­a­tion with num­bers was ap­par­ent at an early age:

One of my earli­est memor­ies is of ar­ran­ging pebbles in the shad­ow of a gi­ant saguaro, squint­ing be­cause the sun was so bright. I think that I have al­ways had a ba­sic lik­ing for the nat­ur­al num­bers. To me they are the one real thing.

The “Auto­bi­o­graphy” em­braces Ju­lia’s right­ful im­port­ance in the his­tory of wo­men in math­em­at­ics; yet at the same time it ap­pears to dis­count, or per­haps ig­nore, lim­it­a­tions placed on her math­em­at­ic­al ca­reer merely be­cause she was a wo­man. Ju­lia was part of the Berke­ley lo­gic com­munity from her stu­dent days un­til her death; yet there is no ques­tion­ing of that com­munity’s role in her lack of of­fi­cial stature at Berke­ley for much of her pro­fes­sion­al life.

That said, the “Auto­bi­o­graphy” in Ju­lia is con­sid­er­ably more re­veal­ing than its ori­gin­al in­carn­a­tion (in [e11] and later in [e13]). In­cluded now on al­most every page are pre­vi­ously un­pub­lished pho­to­graphs and mem­or­ab­il­ia found among Ju­lia’s things. It’s a won­der­ful treat for math­em­aticians, for math­em­at­ics teach­ers and stu­dents, and even for non­mathem­aticians.

Prob­ably the most fas­cin­at­ing doc­u­ment for me is a hand­writ­ten draft note, ap­par­ently writ­ten to Paul Co­hen, ex­hib­it­ing a feisti­ness not usu­ally part of Ju­lia’s pub­lic per­sona. After he showed the in­de­pend­ence of the con­tinuum hy­po­thes­is in 1963 (thus solv­ing Hil­bert’s first prob­lem), Paul turned his at­ten­tion to the Riemann Hy­po­thes­is (RH), per­haps with the thought that it too might be in­de­pend­ent. In the first para­graph of her note, Ju­lia chides:

Paul, you should know that if RH is un­de­cid­able from PA [Peano Arith­met­ic] then it is true! and for heav­en’s sake there is only one true num­ber the­ory! It’s my re­li­gion that’s why it is so ex­cit­ing to prove any­thing at all about it.

Any­one wish­ing to get a hint of Ju­lia’s can­did views on lo­gic might want to care­fully ex­am­ine the rest of this in­triguing note (on p. 49), both for what is left in and what is crossed out.

At the end of the book there is a one-page cur­riculum vitae. Read without ref­er­ence to the times, the quantum leap in status in 1976, and sub­sequent re­cog­ni­tion, it ap­pears totally bizarre. Read in his­tor­ic­al con­text, it speaks volumes:

Born, Decem­ber 8, 1919, St. Louis, Mis­souri;
Gradu­ated, San Diego HS (1936); San Diego State (1936–39);
AB (1940), MA (1941), PhD (1948), UC Berke­ley;
Ju­ni­or Math­em­atician, RAND Cor­por­a­tion (1949–50); ONR (1951–52);
Lec­turer, UC Berke­ley (Spring 1960);
Lec­turer, UC Berke­ley (Fall 1962);
Lec­turer, UC Berke­ley (1963–64);
Lec­turer, UC Berke­ley (Spring, Fall 1966);
Lec­turer, UC Berke­ley (Fall 1967);
Lec­turer, UC Berke­ley (1969–70);
Lec­turer, UC Berke­ley (Spring 1975);
Elec­tion, Na­tion­al Academy of Sci­ences (1976);
Pro­fess­or, UC Berke­ley (1976–1985);
Dis­tin­guished Alum­nus, San Diego State (1978);
Elec­tion, AAAS (1978);
Vice-Pres­id­ent, AMS (1978–79);
Hon­or­ary De­gree, Smith Col­lege (1979);
Col­loqui­um Lec­turer AMS (1980);
No­eth­er Lec­turer, AWM (1982);
Ma­cAr­thur Found­a­tion Prize (1983);
Pres­id­ent AMS, (1983–84);
Elec­tion, AAAS (the oth­er one) (1985);
Died, Ju­ly 30, 1985, Oak­land, Cali­for­nia.

In the “Auto­bi­o­graphy,” Ju­lia re­lays events sur­round­ing her elec­tion to the Na­tion­al Academy of Sci­ences:

When the Uni­versity press of­fice re­ceived the news, someone there called the math­em­at­ics de­part­ment to find out just who Ju­lia Robin­son was. “Why, that’s Pro­fess­or Robin­son’s wife.” “Well,” replied the caller, “Pro­fess­or Robin­son’s wife has just been elec­ted to the Na­tion­al Academy of Sci­ences.” Up to that time I had not been an of­fi­cial mem­ber of the Uni­versity’s math­em­at­ics fac­ulty, al­though from time to time I had taught a class at the re­quest of the de­part­ment chair­man. In fair­ness to the Uni­versity, I should ex­plain that be­cause of my health, even after the heart op­er­a­tion, I would not have been able to carry a full-time teach­ing load. As soon as I was elec­ted to the Academy, however, the Uni­versity offered me a full pro­fess­or­ship with the duty of teach­ing one-fourth time — which I ac­cep­ted.

I re­mem­ber when I first came to Berke­ley in 1968 I was quite sur­prised to find that Ju­lia was not a mem­ber of the Berke­ley fac­ulty; I like many oth­ers had al­ways as­sumed she was. In re­sponse to my quer­ies, I was giv­en vague reas­ons al­lud­ing to her health, nepot­ism rules, and her rank­ing on some lin­early ordered list of lo­gi­cians. As to her health, an of­fer of a po­s­i­tion with re­duced teach­ing would have been ap­pro­pri­ate, as her state­ment sug­gests. I ac­tu­ally looked up the nepot­ism rule and found that it ex­pli­citly al­lowed for ex­cep­tions to be made on the re­com­mend­a­tion of the de­part­ment chair.

What about Ju­lia’s rank­ing in lo­gic? Re­call, the 1960’s were glory days for pure math­em­at­ics and ab­strac­tion. Ap­plied math­em­at­ics was viewed some­what dis­dain­fully, per­haps as not deep or hard enough. Ap­plied lo­gic was sim­il­arly slighted by lo­gi­cians ob­sessed with set the­ory and high­er re­cur­sion the­ory. To a great ex­tent Ju­lia was af­fected by these pre­ju­dices.11

Lo­gic missed out. Math­em­at­ics missed out. Berke­ley missed out. Had Ju­lia been a reg­u­lar mem­ber of the Berke­ley fac­ulty she would surely have su­per­vised some bright math­em­at­ics gradu­ate stu­dents. Cer­tainly, she would have sug­ges­ted a try at the hy­po­thes­is J.R. There is a good chance Hil­bert’s tenth prob­lem would have been solved at Berke­ley. The pieces were all there. Who knows what oth­er gems would have been un­covered? And per­haps lo­gic would not have be­come so isol­ated from the rest of math­em­at­ics.

Ju­lia’s elec­tion to the Na­tion­al Academy of Sci­ences in 1976 opened the way for in­sti­tu­tions and as­so­ci­ations to award her po­s­i­tion and re­cog­ni­tion long over­due. And, clearly these events helped Ju­lia move in­to the lime­light. By ac­cept­ing the pres­id­ency of the Amer­ic­an Math­em­at­ic­al So­ci­ety (AMS), Ju­lia knew she would be thrust in­to the role of pub­lic per­son, and she rose to the oc­ca­sion. But also, on a more per­son­al level, this pub­lic re­cog­ni­tion, I be­lieve, helped Ju­lia ac­know­ledge the sig­ni­fic­ance of her own con­tri­bu­tions to math­em­at­ics.

As I wrote in [e10], in this re­gard, I feel priv­ileged to have been able to wit­ness a side of Ju­lia that I think many oth­er math­em­aticians rarely had a chance to see. One par­tic­u­lar en­counter stands out vividly.

Some­time after Ju­lia be­came pres­id­ent of the AMS, we ar­ranged to meet for lunch with Nancy Kre­in­berg of the Lawrence Hall of Sci­ence to dis­cuss ways to en­cour­age girls and wo­men in math­em­at­ics. I re­mem­ber ar­riv­ing at the Wo­men’s Fac­ulty Club at Berke­ley a bit early, just as the noon­time bells were be­gin­ning to ring. Ju­lia was already there. Sit­ting in one of those high-backed Vic­tori­an chairs, wear­ing a bright flor­al dress, she looked quite majest­ic. I sensed a spe­cial oc­ca­sion. As I came up to greet her, she broke in­to one of her broad imp­ish grins. “Guess what?” she said ex­citedly. “I have just been awar­ded the Ma­cAr­thur prize — for my part in solv­ing Hil­bert’s tenth prob­lem!”

To cel­eb­rate, we ordered a bottle of cham­pagne. I can’t re­mem­ber now the de­tails of our con­ver­sa­tion. But I do re­mem­ber clearly the warmth of three wo­men thor­oughly en­joy­ing each oth­er’s com­pany, shar­ing feel­ings of ex­cite­ment, tri­umph, and plans for the fu­ture. And, by the end of the meal, after sev­er­al rounds of toasts, we had very nearly fin­ished the bottle of cham­pagne. Later, when I men­tioned this lunch to a friend in the Berke­ley Math­em­at­ics De­part­ment, he pro­tested, “But don’t you know? Ju­lia Robin­son nev­er drinks!”

I ap­pre­ci­ate that Ju­lia in­cluded me in her joy. It’s a way in which I will al­ways re­mem­ber her.

A pre­lim­in­ary ver­sion of this re­view was cir­cu­lated to mu­tu­al friends and col­leagues. Their com­ments add per­spect­ive both to Ju­lia and to my re­view. With re­gard to the isol­a­tion of lo­gic, Sol Fe­fer­man pro­poses an­oth­er ex­plan­a­tion:

I don’t think “ap­plied lo­gic” was looked down on by lo­gi­cians in the 1960’s, at least not lo­gic ap­plied to al­gebra via mod­el the­ory. And I think the isol­a­tion of lo­gic from math­em­at­ics is a long-term phe­nomen­on that has little to do with its ap­plic­a­tions or non-ap­plic­a­tions to math­em­at­ics, and more to do with the nature of its no­tions and meth­ods, which are not for the most part con­tigu­ous with main­stream math­em­at­ics. That’s a great pity, be­cause, not­with­stand­ing that sep­ar­a­tion, there is so much of both found­a­tion­al and math­em­at­ic­al in­terest that has come out of the work in lo­gic in the last 50 years that ought to be bet­ter known and ap­pre­ci­ated.

Mar­tin Dav­is of­fers af­fec­tion­ate in­sight in­to “the full hu­man be­ing Ju­lia was:”

Ju­lia’s sens­ib­il­ity was not ca­reer ori­ented in a way hard for people nowadays to fathom. This was partly true be­cause she had in­tern­al­ized the so­cial defin­i­tion of a wo­man’s role, only be­gin­ning to think about cre­at­ive math­em­at­ics as something she might do after a mis­car­riage and very much on Raphael’s prompt­ing. But it was also the mod­est style of both of them. After Yuri’s won­der­ful res­ult, she in­sisted that the solu­tion of HTP be cred­ited en­tirely to Yuri. Yuri and she had this in­cred­ible dance after he had re­duced their \( N = 13 \) uni­ver­sal equa­tion to \( N = 9 \). He re­fused to pub­lish un­less she signed on as co-au­thor, she re­fused to sign on be­cause she’d con­trib­uted noth­ing new. Very re­fresh­ing!

As you know, she had in­ten­ded to de­vote her re­tir­ing AMS pres­id­en­tial ad­dress to Raphael’s work. She was con­vinced that he was the stronger math­em­atician. She was little at­trac­ted to the idea of a reg­u­lar po­s­i­tion, and told me that she had ac­cep­ted the part-time pro­fes­sion­al po­s­i­tion be­cause it would be good for wo­men in math­em­at­ics, not be­cause she was es­pe­cially eager for it. She told me that it was a priv­ilege that she could do math­em­at­ics when she wished, and drop it to work for Ad­lai Steven­son (a cous­in) when that seemed more im­port­ant.

She was mod­est but not at all dif­fid­ent. This came out in work­ing with her, but also, for ex­ample, in the way she dealt with So­viet, math­em­aticians re­gard­ing anti-Semit­ism in the So­viet Uni­on.