My Father’s Theorem

The Rockmore theorem made its first—and perhaps only—named appearance in print in 1977, in the journal Physics Letters, Volume 72B, No. 4. A photocopy of the journal page hung on my father’s office door, at Rutgers University, in New Jersey. The author of the article—one B. L. Birbrair, of the Academy of Sciences of the U.S.S.R.—wrote that, in 1959, my dad had demonstrated that the “interaction between quasiparticles exactly compensates” for “the difference between the effective and real mass” of an “infinite non-superfluid fermi system.” Since then, he went on, “the statement J(Δ = 0) = Jrig” has been known as “the Rockmore theorem.”

My dad had underlined those last three words in bold, black pen. Other professors on the same hallway displayed conference posters on their doors, in which they were featured as keynote speakers in exotic locales. My dad couldn’t produce such a poster, but the photocopy, which hung amid New Yorker cartoons, preprints, and departmental announcements, showed something arguably more valuable.

Theorems are mathematical facts justified by proofs—logical, airtight arguments made of other facts and deductions. Usually, a proof consists of statements taking the form “if this, then that”; this chain of “ifs” is grounded in bedrock mathematical truths. In high-school geometry, you probably saw some simple proofs—two-column accounting ledgers, with statements on one side and justifications on the other. Many proofs by professional mathematicians are baroque in comparison; they contain twists, turns, and surprise appearances, with intermediate steps, known as propositions, claims, or “lemmas,” that require proofs of their own. A theorem is only published if it’s been deemed interesting by journal editors and successfully “refereed” by external evaluators. Mathematicians prove thousands each year, but of the many published theorems only a few are named.

The eponymous theorems are the most famous, but are they rightly named? There’s Pythagoras’ well-known beauty, a fact about the three sides of a right triangle: the sum of the squared lengths of its two shorter sides equals the squared length of its hypotenuse (a2 + b2 = c2). A proof for the Pythagorean Theorem can be found among Euclid’s “Elements,” the thirteen-volume treatise on geometry that many see as an exemplar of Greek mathematics. The Roman statesman and philosopher Cicero connected the theorem to Pythagoras some five centuries after his death, though the attribution is hardly a slam dunk. Part of the beauty of Pythagoras’ theorem is that it admits revelation through a broad range of proofs. So far, there have been more than a hundred. Even Albert Einstein set one down. It seems likely that some other early geometer could have been the first.

You may have also heard of Fermat’s Last Theorem. It’s a fact about other kinds of numerical “triples”: it says that, if you make a small change to the Pythagorean equation, and replace the exponents with larger whole numbers, then there are no solutions where a, b, and c are all whole numbers. “It is impossible to separate a cube into two cubes, a fourth power into two fourth powers,” the mathematician Pierre de Fermat explained, around 1635. Fermat, a jurist by day and a mathematician during his off-hours—the mathematician and writer E. T. Bell called him the “Prince of the Amateurs”—put down his theorem in the margin of a book, and added that he had “discovered a truly marvellous demonstration” of it, “which this margin is too narrow to contain.” He never explained himself elsewhere. It was arguably the most famous mathematical cliffhanger of all time.

Fermat was a genius, and for years we mathematicians gave him the benefit of the doubt. But his theorem’s proximity to the readily proved Pythagorean theorem, and the mathematicians’ credo of “truth means proof,” meant that its lack of resolution couldn’t be ignored. Over the centuries, the failed proofs accumulated, and Fermat’s Last Theorem became the mathematician’s version of “will they or won’t they?” The techniques developed to attempt its proof grew in scope and ingenuity, becoming a body of thought far beyond anything Fermat could have imagined.

Finally, in the mid-nineteen-eighties, progress on various aspects of algebraic geometry suggested a way forward to the mathematician Andrew Wiles. He devoted roughly seven years to proving Fermat’s Last Theorem. When he was pretty sure that he had a proof, he brought in a few close mathematician friends for some sanity checks, and rumors began to spread that he would soon make a historic announcement. During a lecture on June 23, 1993, before a standing-room-only audience, Wiles proved Fermat. Or did he? Whereas some proofs of the Pythagorean theorem can be understood by those with little or even no mathematical background, the adjudication of Wiles’s proof required the convening of a group of absolute experts—a Supreme Court of mathematics. A painstaking refereeing process revealed some holes in the proof, and Wiles went back to the blackboard, this time with the help of a friend and colleague, Richard Taylor. Together, they plugged the holes, making Fermat’s Last Theorem into a real theorem. Three and a half centuries had passed. In a sense, the theorem was no longer Fermat’s.

Even the Rockmore theorem is subtly misnamed. My father’s original family name was Rochmovich, or Rochemovitz, or one of any number of variations on that theme—it depended on who was writing the name on a form, beginning with the clerk who processed my great-grandfather through Ellis Island, in 1881, when the name was Rochomovyitz, or possibly Rochomowitz. According to my father, my great-grandfather came from Ukraine. He might have been fleeing conscription or the frequent pogroms for which the region became well known. In 1908, the name became Rockmore—at some point, I was told that it was chosen for its “strong American sound.” The page that records the change in the records of the New York State Legislature is littered with a host of similar transformations, marking the first stages of American assimilation for many Jewish families.

Origins, whether of family names or ancient Greek theorems, can be tricky to uncover. Although the many variations on Rochomovich share a derivation from the Yiddish word for compassion (“rachmones”), Rockmore is culturally hard to place. (My father has never forgotten the department chair who told him, “We didn’t know you were Jewish when we hired you.”) Rockmore is an uncommon name, and family lore has it, semi-ironically, that all Rockmores are related. I know for a fact that I’m related to the painter Gladys Rockmore Davis and her painter son, Noel Rockmore, famous for his portraits of the Preservation Hall jazz greats; there are more distant connections to the theremin virtuoso Clara Rockmore, and to a war hero or two. Once, checking into a hotel, I shared a laugh with the African-American woman behind the desk, also a Rockmore, as we tried to find a common ancestor in one of my family’s thoroughly assimilated Christian branches. But, in truth, I don’t know much of my family tree; my father’s father was one of at least twelve children, but there was a big family rift long ago, and my father’s side has always seemed to begin and end with his parents and brother, now all gone.

My father’s father was always moving from job to job, either flush or deep in the hole. For a while, the family lived in Oceanside, on the South Shore of Long Island, where my dad, a bookish kid, could nurture his interests in birds and stars. In 1925, ten thousand members of the Ku Klux Klan had rallied in Oceanside; in 1940, partly because of anti-Semitism, the family moved back to Brooklyn, where my grandmother worked as a bookkeeper in her brother’s mirror-making business. My father spent two years at the yeshiva next door and two years in high school, and then won the New York City Latin competition. His “plan,” according to the caption of his high-school yearbook, was to go to Columbia University. He applied, and my grandmother told him that she had learned, from an admissions officer, that his application had been among those diverted owing to “regional quotas”—systems used by Ivy League institutions to limit the number of Jewish students. He never did see the rejection letter; it may have just been the money.

He attended Brooklyn College, majoring in mathematics and commuting by subway while living at home. After graduation, he was accepted to Columbia’s doctoral program in physics. He started graduate school a little after the Second World War, during a golden age of research built on the momentum of the Manhattan Project. A steady succession of discoveries in quantum mechanics seemed to be moving relentlessly toward a grand unified theory; experimental physics was diving ever deeper into the atom to reveal a wild jungle of subatomic particles, whose interactions required models of ever-greater precision and sophistication. A small stipend as a teaching assistant supported him, and from time to time allowed him to send some money downtown to his parents and brother. (Their finances and support would always be a concern.) He took classes with the Nobel laureate I. I. Rabi, and his thesis adviser was Robert Serber, Oppenheimer’s right-hand man on the Manhattan Project. From Ukraine to Columbia, an unlikely career as a scientist was emerging.

The work that would become the Rockmore theorem was contained in a paper that my dad published a few years after he graduated. The theorem settles a conjecture about a version of a “many-body problem”—a scenario in which many things orbit one another in an intricate, hard-to-predict dance. In this case, the bodies are subatomic particles called fermions. It had been conjectured that, under certain conditions, when you accounted for all of the particles’ mutual pushes and pulls, the total inertia of the system—that is, its over-all tendency to continue to whirl about—would add up to zero. “Vanishing to all orders,” is how we say it. My father was able to reveal a previously unseen symmetry which manifested itself in a perfect accounting of contributions and withdrawals from the system’s inertial bookkeeping. It’s a dense piece of work, somewhere in the middle of the range of complexity bookended by the Pythagorean theorem on one side and Fermat’s Last Theorem on the other; its six pages are mainly equations, with a little bit of interpolative and explanatory text.