#### by Lenore Blum

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

- 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

- for each
__\( k > 0 \)__there exists__\( (a, b) \in J \)__with__\( b > a^k \)__and - 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.

#### Postscript

*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.