THE INTELLECTUALLY TOWERING French mathematician André Weil, whose life spanned most of the 20th century, spent the latter half of his career as a professor at the imposing Institute for Advanced Study (IAS) in Princeton, New Jersey. If you visited his faculty profile page on the IAS website any time before June 2017, you would have seen only one honor listed: “Member Poldavian Acad of Sci and Lett.”

Don’t bother looking up Poldavia on a map or its academy in a directory of learned societies. The country is fictional. It was invented by a right-wing journalist in 1929 to provoke and mock political opponents whose sympathies extended to Eastern Europeans devastated by the Great War. Poldavia’s academy came about in the next decade, when a renegade collective of French mathematicians launched a project to rewrite the foundations of modern mathematics. At Weil’s suggestion, they adopted a collective pseudonym, Nicolas Bourbaki, and endowed him with a backstory as a Poldavian refugee.

Weil contributed prolifically to both Bourbaki’s mathematics and his lore, in 1948 going so far as to complete an application in Bourbaki’s name to join the American Mathematical Society. At some unknown point, Poldavia and other Bourbaki fictions (like the portmanteau city of Nancago from Bourbaki’s early post–World War II homes of Nancy, France, and Chicago, Illinois) migrated from Bourbaki’s curriculum vitae to Weil’s. In town for my PhD graduation, I paid a visit to the IAS archives and came across a Weil CV that included Poldavia and Nancago. I shared the find with an archivist, who was not so charmed by the farce. Shortly thereafter, Poldavia disappeared from Weil’s web profile.

Poldavia appears only fleetingly in Karen Olsson’s genre-defying *The Weil Conjectures*. But the entangled motifs and motives Poldavia represents — mathematics, fiction, speculation, brilliance, biography, hardship, mockery, intimidation, solidarity, generosity, and moral and theoretical imagination — run throughout the book, making it one of the most insightful meditations on modern mathematics I have ever read. With startling originality, Olsson confronts the problem of knowing mathematics from the outside. In the process, she vividly portrays the human dimensions of mathematical creativity.

Olsson’s muses are André and his younger sister Simone Weil, a philosopher and mystic who died at 34 in 1943. In her very different way, she was also a defining mind of the 20th century. To close chapter one, Olsson imagines how their story would begin in a fairy tale: “Once there were a brother and sister who devoted themselves to the search for truth. A brother who spent his long life solving problems. A sister who died before she could solve the problem of life.”

The book introduces them as alarmingly precocious children, conversing in ancient Greek and competitively reciting verse, and, more whimsically, swearing off knee socks. Part One follows their lives through adolescence to the outbreak of World War II: Simone propounds inscrutable philosophies and quixotically pursues a moral program of radical empathy in coal mines and factories and on the front lines of the Spanish Civil War, while André emerges as a formidable mathematician, learns Sanskrit, decamps to India, returns to France, and begins conspiring under the sign of Bourbaki. Parts Two, Three, and Four take the story through André’s imprisonment for failing to report for duty during the war, and then his flight to the United States; Simone’s dogged attempts to comprehend her brother’s mathematics and her own wartime moral and philosophical obsessions; André’s eponymous conjectures that used daring analogies to “span the gap between the continuous (space) and the discrete (whole numbers)”; and Simone’s death in 1943, partly from limiting her diet to what she imagined were the wartime rations of children in France. Part Five considers Simone’s legacy and then moves the main arc of the story to Japan, where André would become associated with another conjecture developed by Yutaka Taniyama and Goro Shimura. He returned there near the end of his life to accept the Kyoto Prize.

The Weils’ twinned lives, recounted in fragments assembled in part from André’s memoir and Simone’s writings, structure a compellingly digressive text that includes historical vignettes, philosophical musings, and Olsson’s own experiences as a student of mathematics and as a novelist returning to the subject in adulthood, watching lectures on YouTube and discussing the meaning of numbers with her young son. The author is ever-present, always reacting and speculating. “Let’s say,” “or so the story goes,” she oft interjects to remind us not to take her tales literally.

This book is not a fable, but it pithily prods the moral dimensions of legends. Nor is it a biography or memoir, though it probes the Weils’ lives and the author’s own, weaving them into an account of what it means to grapple with mathematics. It is not history, but it mines the past for evocative stories. Nearing the end of my first read-through, reflecting on Olsson’s preoccupation with memories and legacies, I was almost ready to call the work a composite elegy. But some pages later, as often happened in my reading, I found that Olsson had anticipated the thought and interrupted it with uncanny precision: “Is that what I am writing, I wonder, some sort of elegy for math, or for my own entanglement with math? At times it feels that way, but I don’t think that’s what this is.”

Olsson ultimately makes the case for conjecture as its own literary genre. “The word *conjecture* derives from a root notion of throwing or casting things together,” she offers at the start of chapter four, adding that “it has referred to prophecies as well as to reasoned judgments, tentative conclusions, whole-cloth inventions, and wild guesses.” Mathematicians’ conjectures, she explains, are “ideas that have taken on weight but haven’t been proved” and that tie together “what can be firmly established and what might turn out to be the case.” Conjecture, she later notes, is a relatively recent genre in mathematics, at least in the form wielded to such effect by André Weil. Conjectures can be thought of as using analogies and imagination to build an architectural view of mathematics, a grand plan of how things ought to fit together that can guide the hard work of engineering and of creating edifices of theory. “A conjecture forges a trail,” chapter 10 begins, to a destination that “is not yet in sight and might well be unreachable.” Conjectures, she suggests, combine faith, optimism, unease, and regret. They give the mathematician the confidence to go off in one direction but cannot guarantee the direction will turn out to be right, nor can they account for the directions not taken.

Olsson’s own conjecture casts together all of these senses of the term, sketching a gestalt picture of modern mathematics through a trail of fragments and connections. She combines creative reimaginings of present and past realities with speculative projections of possible and necessary futures. The word “conjecture,” in this sense, lends a surprising unity to the siblings’ stories. They conjecture in their different ways: the mathematician and moral philosopher alike must make up worlds, impose coherences upon them, and follow those fictions wherever they lead with whatever consequences.

In life and number theory, André chased what he termed “these slightly adulterous relationships” linking theoretical realms like topology and number theory, as well as people and places. Bourbaki was a kind of social conjecture: a radical team of authors put their heads together to convince the world of mathematics to change its basic patterns of thought. André’s marriage was its own kind of conjecture, starting with a literally adulterous relationship in the early days of Bourbaki with the wife, Eveline, of one of the group’s co-founders.

In life and moral philosophy, Simone chased the deeply personal implications of empathetic suffering and solidarity. Without the aid of much prior knowledge or experience, she imagined herself as a laborer, farmhand, Spanish Republican, battlefield nurse, or starving child, then resolutely followed the path where that imagination led, which was often into harm’s way. Her philosophical writings, meanwhile, drew challenging connections from often-heterodox reimaginings of her subject matter. “Each had the run of an elaborate mental (or mental-spiritual) universe,” Olsson observes, and “each subjected perceptions to a ruthless accounting.”

A couple of pages into chapter seven, the book’s virtuosic fulcrum, she recounts a dream of André’s. He is on a ship and gradually realizes it is full of members of the Bernoulli family, “that sprawling, backstabbing mathematical dynasty” that towered over more than a century of mathematics starting in the late 17th century. Somewhere on that dream-ship André might have encountered Jacob Bernoulli, author of one of the founding books of the mathematical theory of probability, the *Ars Conjectandi*, or *Art of Conjecturing* (1713). To gain numerical mastery over the unruly wages of chance, Bernoulli asked his readers to imagine the process of picking out stones from an urn. Seen this way, probability could be reduced to a kind of accounting, tallying up combinatorial configurations of things thrown together. Jacob and his book do not appear in Olsson’s narrative, but her evocations certainly rattle across her pages like stones in an urn, mixing and echoing and reverberating. In their combinatorial proliferation, they afford a measure of mastery over the lifeworlds of mathematics.

Olsson confesses a “weakness for juxtaposition,” but this is also the book’s strength. Juxtapositions, in her book, do what a cold progression of facts and deductions could not do. They open new perspectives and suggest overarching harmonies born of fecund connections and entanglements. Such connections, Olsson makes clear, would define 20th-century mathematics. As she tentatively summarizes: “[I]dentity became less important than relatedness.”

Olsson’s conjecturing is so effective in part because she reckons so frankly with what it means to comprehend mathematics as a nonmathematician, an endeavor she finds by turns thrilling and fascinating, frustrating and alienating. As a Harvard undergraduate, she tells us she was reasonably good at math but never saw herself as an exceptional talent destined for mathematical glory. During adulthood and her career as a writer, she felt lingering admiration for the subject, a sense of her own limitations and insecurities, and an anthropological sensibility toward, even affinity for, the mathematical life. Hence her affinity, too, for Simone Weil’s lifelong desire to know what and how her brother knows, to understand his mathematical conjecturing, and her palpable disappointment in falling short (perhaps, in her mind, its own kind of failure of empathy).

Olsson does not try to build up a mathematical point of view by explaining specific concepts. Nor does she take at face value mathematicians’ attempts to articulate in lay terms and with ready metaphors just what is going on in their heads. Rather, she stresses the struggle to understand such concepts and to convey them to those who do not. She emphasizes the pathos of trying, and rarely succeeding, to join one partial understanding to another in the lucid reverie of “a mathematical fugue state.”

She thus conjures up her own kind of empathy for the mathematical life. She writes, “I wonder whether mathematicians and fiction writers might be people for whom the lure of alternate worlds is particularly strong.” Her conjectural method of narration makes the point by leaning on myth, fiction, speculation, and wishful reimagining. For instance, she summons Pierre de Fermat, the 17th-century jurist who turned from “the dreary business of regional administration” to “another convoluted system of rules, searching for laws governing the whole numbers.” Elsewhere, she pictures André’s “lawyer as a harried sad sack, exhaling” as he opens a long letter in which Simone relays what she has learned from André of Fermat and the origins of number theory: “He blinks and sets the letter down on his desk.” The stories Olsson draws from history books, memoirs, personal recollections, and other sources are indeed more conjecture than hard reality. They are invitations to imagined worlds governed by their own logics and resonances, applicable to the Weils’ worlds as to ours.

It is an approach André Weil might recognize, even as Olsson speculates that he “would have something acid to say about all this.” André is notorious for his biting confrontations with historians of math who tried to imagine the past in its own terms instead of as the long prologue to the present. Though he wouldn’t see it this way, André’s approach to history prioritized the fruitfulness of misremembering the past. In a famously scathing review of a historian’s book on Fermat, André counters the historian’s assertion — regarding a theorem whose only known proofs “lay totally beyond the realm of Fermat’s mathematics” — with his own modern proof and an assertion that “it is a rather easy exercise for a first course in elementary number theory.” That is, anything so easy for Weil must have been easy for Fermat, historical context be damned.

Seeing Fermat’s mathematics as fully continuous with his own, Weil accorded to his own understanding of numbers a sense of historical inevitability. The mathematicians who came between Fermat and himself simply offered “inspired commentary” on Fermat’s prescient ideas. As for Olsson, she vividly shows how the past can be mined not just for facts but also for idealizations that, while perhaps departing from the authentic sense of a bygone event, are all the more capable of inspiring understanding. Her conjectures put a human face on mathematics by sketching its stories in caricature, playing up the fantastical and the absurd. For both André and Olsson, this kind of revisionism-with-a-purpose isn’t a matter of misremembering. It is what it means to remember well.

In erasing the Poldavian Academy from André Weil’s profile, is the Institute for Advanced Study doing him justice? Is it remembering him well? More than just about any mathematician in the 20th century, Weil stood for the power of mathematicians to make their own worlds, to insist that others join them there, and to pursue those worlds until they harmonized with our own. This kind of radical imagination, Olsson shows, is not just mathematical but also moral, and it offers a key to the intellectual history of the 20th century. In Weil’s life, Poldavia, with all its contradictions and double-edged implications, was more real than many of the signifiers now populating Weil’s CV: Kyoto and Wolf prizes, academies of science in Washington, DC, and Paris. I say, bring back Poldavia.

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