How Julia Robinson helped define the limits of mathematical knowledge

Every December 8 for years,
Julia Robinson blew out the candles on her birthday cake and made the same
wish: that someday she would know the answer to Hilbert’s 10th problem. Though
she worked on the problem, she did not care about crossing the finish line
herself. “I felt that I couldn’t bear to die without knowing the answer,” she
told her sister.

In early 1970, just a couple of months after her 50th birthday, Robinson’s wish came true. Soviet mathematician Yuri Matiyasevich announced that he had solved the problem, one of 23 challenges posed in 1900 by the influential German mathematician David Hilbert.

Matiyasevich
was 22 years old, born around the time Robinson had started thinking about the
10th problem. Though the two had not yet met, she wrote to Matiyasevich shortly
after learning of his solution, “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!”

The
conjecture Robinson was referring to was one of her contributions to the
solution to Hilbert’s 10th problem. Matiyasevich put the last piece into the
puzzle, but Robinson and two other American mathematicians did crucial work
that led him there. Despite the three weeks it took for their letters to reach
each other, Robinson and Matiyasevich started working together through the mail
in the fall of 1970. “The name of Julia Robinson cannot be separated from
Hilbert’s 10th problem,” Matiyasevich wrote in an article about their
collaboration.

Robinson was the first woman to be elected to the mathematics section of the National Academy of Sciences, the first woman to serve as president of the American Mathematical Society and a recipient of a MacArthur Fellowship. She achieved all of this despite not being granted an official faculty position until about a decade before her death in 1985.

Robinson
never thought of herself as a brilliant person. In reflecting on her life, she
focused instead on the patience that served her so well as a mathematician,
which she attributed in part to a period of intense isolation as a child. At
age 9, while living with her family in San Diego, she contracted scarlet fever,
followed by rheumatic fever.

Diagnosed with scarlet fever and then rheumatic fever at age 9, Julia lived away from home with a nurse. In the image above, she has returned home for a visit.C. Reid, Courtesy of Neil Reid

Penicillin
had just been discovered and was not yet available as a treatment. Instead, she
lived at the home of a nurse for a year, missing two years of school.

Even
after she rejoined her family, attended college and married, complications from
rheumatic fever led to lifelong health problems, including the inability to
have children. After a much-wanted pregnancy ended in miscarriage, doctors told
her another pregnancy could kill her. She had a heart operation when she was
around 40 years old that improved her health, but she was never able to have
the family she deeply desired.

Despite
her accomplishments, Robinson was reluctant to be in the spotlight, only
consenting to tell her story for publication near the end of her life. The
quotes attributed to Robinson in this article come from that record, an “autobiography”
written by her older sister, Constance Reid, in close consultation with
Robinson.

The 10th problem

Hilbert issued the first of his 23 challenges to the mathematics community during a lecture in Paris at the 1900 International Congress of Mathematicians. The questions, which helped guide the course of mathematics research for the next century and through the present day, spanned several disciplines in mathematics, probing everything from the logical foundations of various branches of mathematics to very specific problems relating to number theory or geometry.

The 10th problem is a deep question about the limitations of our mathematical knowledge, though initially it looks like a more straightforward problem in number theory. It concerns expressions known as Diophantine equations. Named for Diophantus of Alexandria, a third century Hellenistic mathematician who studied equations of this form in his treatise Arithmetica, a Diophantine equation is a polynomial equation with any number of variables and with coefficients that are all integers. (An integer is a whole number, whether positive, negative or zero.)

Examples
of Diophantine equations include everything from simple linear equations such
as 5x+y=7 (the variables are x and y, and their coefficients are 5 and 1) to the
Pythagorean distance formula a2+b2=c2 (the
variables are a, b and c, and their coefficients are all 1) to towering
monstrosities in googols of variables.

Mathematicians
are interested in whether Diophantine equations have solutions that are also
integers. For example, Pythagorean triples — sets of numbers such as 3, 4 and 5
or 5, 12 and 13 — are solutions to the equation a2+b2=c2.
Some Diophantine equations have integer solutions, and some do not. While a2+b2=c2
has infinitely many integer solutions, the similar equation a3+b3=c3
has none (except for solutions including zeros, which mathematicians consider
uninteresting).

If
an equation does have integer solutions, you do not need to be particularly
clever to find them — you just need to be patient. A brute-force search will
eventually give you numbers that work. (Of course, being cleverer may mean you
can be less patient.) But if you do not know whether the equation can be solved
in integers, you will never know whether your failure to find a solution is because
none exists or because you have not been patient enough.

Earlier this fall, mathematicians Andrew Booker of the University of Bristol in England and Andrew Sutherland of MIT announced that they had used a mix of clever algorithms and a powerful supercomputer to find that 42 = −80,538,738,812,075,9743 + 80,435,758,145,817,5153 + 12,602,123,297,335,6313. In other words, the Diophantine equation x3+y3+z3=42 has an integer solution.

This
is one case of the more general question of which integers n can be written as
the sum of three integer cubes: x3+y3+z3=n.
Forty-two was the last two-digit number for which mathematicians didn’t know
whether there was a solution, but infinitely more numbers await integer
solutions, if they exist.

What
Hilbert wondered in his 10th problem was how to tell whether an equation has
integer solutions or not. Is there an algorithm — a terminating process
yielding a yes-or-no answer — that can determine whether any given Diophantine
equation has such a solution?

A large part of the appeal of the 10th problem and related questions is sheer curiosity. Do these often very simple polynomials have integer solutions? Why or why not? The answers generally do not have concrete practical applications, but the area of research is related in deep ways to theoretical computer science and the limits of what computer programs can do.

Julia Robinson submitted this “neat little problem” to amuse a colleague on his 60th birthday.C. Reid, Courtesy of Neil Reid

Unknowability

Robinson’s interest in Hilbert’s 10th problem started fairly early in what was an atypical mathematical career. She married Raphael Robinson, a mathematician at the University of California, Berkeley, not long after graduating from the university with a bachelor’s degree in mathematics. UC Berkeley’s antinepotism rules prohibited her from working in his department. (Her situation was not uncommon for women in academia in the 1940s and 1950s.) After earning her Ph.D. in math in 1948, also at UC Berkeley, she worked in industry and outside her field for a few years and volunteered for Democratic candidate Adlai Stevenson’s presidential campaigns. She also worked as an unofficial member of the UC Berkeley math department, using Raphael’s office and occasionally teaching classes.

Although
she did not have the stability or salary of an official faculty position, she
published in mathematics journals, both individually and with collaborators,
and presented her work at conferences, often bringing a bicycle along. She’d
become an avid cyclist after her heart surgery, delighted by her ability to
exercise after years of being perpetually short of breath.

Shortly after getting her bachelor’s degree from UC Berkeley, Julia married mathematician Raphael Robinson. They are shown here beside their first home.C. Reid, Courtesy of Neil Reid

When
she was elected to the National Academy of Sciences in 1976, the university
press office had to call the mathematics department to ask who Julia Robinson
was. UC Berkeley quickly made her a full professor. Robinson writes, “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.”

Shortly after she graduated with her Ph.D., her adviser, Alfred Tarski, mentioned a problem to Raphael, who in turn told Julia. This particular problem involved Diophantine sets, groups of integers that when substituted for one variable in some Diophantine equation would allow integer solutions in the other variables. Consider the equation c−x2=0, which has integer solutions for x only when c is a perfect square. Thus the perfect squares form a Diophantine set. The problem Raphael told Julia about was to determine whether the powers of 2 — 2, 4, 8, 16 and so on — form a Diophantine set. Through her work on that question, she found her way to the 10th problem.

Robinson
first met Martin Davis, then an instructor at the University of Illinois at
Urbana-Champaign, in 1950. “It started with our working on the same problem but
from absolutely opposite directions,” says Davis, now age 91. Both researchers
had been looking at Diophantine sets. Davis was starting generally, trying to
show that all sets with a particular property called listability were
Diophantine. Robinson was starting from the particular, trying to show that a
few special sets — including prime numbers and the powers of 2 she had been
working on — were Diophantine.

In
1959, Robinson and Davis started working together. With Hilary Putnam of
Princeton University, they kept pushing on the problem. Eventually they showed
that all they needed was what Davis describes as a “Goldilocks” equation. “The
solutions aren’t supposed to grow too fast, and they aren’t supposed to grow
too slowly,” he says. But that equation eluded them for almost a decade.

In the U.S.S.R., Matiyasevich had tried to tackle Hilbert’s 10th problem as a college student but abandoned it around the time he graduated in 1969. Then a new paper from Robinson sucked him back in. “Somewhere in the Mathematical Heavens there must have been a god or goddess of mathematics who would not let me fail to read Julia Robinson’s new paper,” he wrote.

He
was asked to review it — a mere five pages about the relative growth of
solutions to certain Diophantine equations in two variables. Her ideas
immediately sparked new ideas for him, and he was able to produce the needed
“Goldilocks.”

“It’s
such a romantic thing — in the wider sense of the word romantic — that the four
of us, such different people with different backgrounds, all together produced
this piece of work,” Davis says.

Together,
they had shown that no all-purpose algorithm exists to determine whether an
arbitrary Diophantine equation has integer solutions.

In response to a questionnaire that asked whether she had ever faced discrimination as a student or professional, Robinson shared this response, describing her experience as a woman in math.C. Reid, Courtesy of Neil Reid

But
that isn’t the end of the story. Building on the work of Robinson and her
colleagues, mathematicians continue to probe the boundary between knowability
and unknowability. “Her work is still very relevant today,” says Kirsten
Eisenträger of Penn State, a number theorist whose research is related to the
10th problem.

If
Robinson were still alive on her 100th birthday this December, what problem
would she be thinking about as she blew out her candles? The fact that there is
no general algorithm for all Diophantine equations leaves many tantalizing
questions open. For example, does an algorithm exist for Diophantine equations
of a certain form, say, multivariable cubic equations?

A woman of firsts, Julia Robinson (pictured here a couple of weeks before her death) helped answer one of the 20th century’s grand mathematical questions.Julia Reid, C. Reid, Courtesy of Neil Reid

Mathematicians are also looking at what happens if you change the types of solutions sought for Diophantine equations. One change is to ask the question for rational numbers: Is there a way to determine whether a polynomial equation with integer coefficients has any solutions that are rational numbers? (A rational number is the ratio of two whole numbers; 1/2 and −14/3 are two examples.) Most experts believe that the answer is no, but mathematicians are far from a proof. One potential path to a solution involves building on work Robinson did in her Ph.D. thesis over 70 years ago.

In 1984, during her term as president of the American Mathematical Society, Robinson was diagnosed with leukemia. During a remission the next spring, while cycling with her sister, Robinson decided that Reid would write her life story, “The autobiography of Julia Robinson.” Weeks later, the cancer had returned. Reid finished writing the record of Robinson’s life as her sister’s health deteriorated. Robinson died on July 30, 1985, at age 65.

“What
I really am is a mathematician,” Reid writes on behalf of Robinson on the
closing page. “Rather than being remembered as the first woman this or that, I
would prefer to be remembered, as a mathematician should, simply for the
theorems I have proved and the problems I have solved.”

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