The Weil Conjectures: A tale of mathematics, philosophy, and art

riemannhypo
The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 1/2. The first non-trivial zeros can be seen at Im(s) = ±14.135, ±21.022 and ±25.011. The Riemann hypothesis, a famous conjecture, says that all non-trivial zeros of the zeta function lie along the critical line.

For some, mathematics much more than a matter of solving problems. It transcends abstraction and intellectual pursuit into a way of determining meaning from life. For a brother and sister, it can mean a relentless search for truth that reads like a Romantic fable. A history that consists of settings across time and space punctuated by individual actions and events, the novelist creates a narrative that sheds light on a new meaning of truth. Truth may be elusive, especially in a post-truth society, but, in a metamodernist manner, it’s closer to reality – an authentic, original reality – than it seems.

In Karen Olsson’s The Weil Conjectures: On Math and the Pursuit of the Unknown, she intertwines the stories of French brother and sister André and Simone Weil during a Europe in the midst of World War II. The former, a mathematician known for his contributions to number theory and algebraic geometry, and the latter, a philosopher and Christian mystic whose writing would go on to influence intellectuals like T.S. Eliot, Albert Camus, Irish Murdoch, and Susan Sontag. Hearkening back to the childhood of the siblings, we follow their stories studying poetry, mathematics, tragedies, and other artists and scientists. Between these glimpses of their lives, Olsson throws in her personal anecdotes studying mathematics as an undergraduate at Harvard. She describes a “euphoria” from thinking hard about mathematics such that, while knowledge itself is the goal, it’s a disappointment to reach it. You lose your pleasure and sensation in seeking truth once you find it. André characterizes his own search for happiness through this search for truth. Drawing parallels between herself and the siblings, their stories depend less on the context that surrounds them and more on the similarities in their narratives. 

With multiple stories happening at once, the reader feels a sense of timelessness in the writing. The plot has less to do with one event happening after another, but more with a grand narrative carrying each part of the story with one another. A mix of elements of modernism and postmodernism together, Olsson’s book serves as a sign of the next step: metamodernism. In separate directions, mathematics and philosophy, the two venture for truth that seems to lie just outside their reach. Olsson tells the narratives through letters between the siblings, the notebooks upon which Simone scribbled her thoughts – philosophic, mathematical, and religious. On the purposes of mathematics and philosophy, Olsson questions how mathematics had become disconnected from the world around them. So focused on attacking problems in an abstract, self-referential setting, the field’s myopic focus on truth had strayed from meaning, she believed. Simone’s story through working in factories and a Resistance network with a wish to free herself from the biases of her own self would lead to her death by starvation in solidarity towards war victims. 

If the labor of machinery is so oppressive, Simone wondered how to create a successful revolution technological, economic, and political. The pain she sought through suffering made her who she was. It humanized her as she wrote about the German army defeating France. The evil in the world was God revealing, not creating, the misery inside us. Simone sought to achieve a state of mind that liberated herself from the material pursuits of the world through philosophy and Christian theology. She wanted an asceticism to provide she could a morally principled life on her self-imposed rules. This included donating money during her career as a teacher so that she would earn the same amount as the lowest-paid teachers. 

D. McClay, senior editor of The Hedgehog Review, wrote that Simone’s own struggle with Catholicism partly had to do with her anti-Semitism in his essay “Tell Me I’m OK.” “Though Weil was herself Jewish, she did not identify as Jewish in any significant sense, and her sense of solidarity with the oppressed did not extend to other Jews,” McClay said. Feminist philosopher Simone de Beauvoir who, according to her memoir, didn’t get along with Weil when they met, offers a contrast to Weil in how to live a good life. 

In Beauvoir’s The Ethics of Ambiguity, she argued existentialist ethics are rooted in recognition of freedom and contingency, McClay said. Beauvoir wrote, “Any man who has known real loves, real revolts, real desires, and real will knows quite well that he has no need of any outside guarantee to be sure of his goals; their certitude comes from his own drive…. If it came to be that each man did what he must, existence would be saved in each one without there being any need of dreaming of a paradise where all would be reconciled in death.” Beauvoir’s atheism created friction with Weil, McClay said. They also define a reality of what we do in the world that defines their own “drive,” which seems like a response to the threats of existential nothingness. 

McClay continued to compare the two Simones to provide an account for how to live a moral life, involving abandoning the idea of a “good person” in favor of goodness without regard to how others judge us. “It might mean living more like Weil—taking what you need, and giving away the surplus—”, McClay said, “with the caveat that one takes what one actually needs.” Beauvoir and Weil, moral philosophers that describe how “we are always, simultaneously, together and alone,” may even be guides for the crises of our age. Living together and alone, through the community of one another and the isolation of intellectual work, we can live like Weil intended. McClay’s writing also shows this mix of modernity’s unified, centralized identity withothers with postmodernism’s decentered self. 

Interspersed in Olsson’s book are stories about Archimedes’ having “eureka” moments, René Descartes’ search for the “unknown” (x in algebra), L. E. J. Brouwer’s work in topology, and even the mathematician Sophie Germain who studied mathematics in secrecy and corresponded with male mathematicians under a pseudonym. Tracing the foundations of mathematics, language, and other tenets of society to the Babylonians, she carefully compares the methods of problem solving and invention using language to reveal deeper nature of the phenomena (“Negative numbers infiltrated Europe during the Middle Ages” making mathematics seem deceptive or insidious) or method in discovery (“Are numbers real or not? Were they discovered or invented? We pursue this question for a couple of minutes.”). The figures comment on their own judgements on the deeper meaning and purpose in their work such as George Cantor saying “I see it, but I do not believe it.” Olsson drops these quotes and glimpses of history in between moments of trials of other characters.

When the early 20th-century Jewish-born mathematician Felix Hausdorff set the grounds for modern topology, an anti-semitic mob claiming they would send him to Madgascar where he could”teach mathematics to the apes” gathered around his house. Olsson then switches to her perception that she always read André and Simone Weil’s last name as “wail,” despite it actually pronounced as “vay.” Then, Olsson returns to Hausdorff’s story of taking a lethal dose of poison after failing to find a way to escape to America. In a farewell letter to his friend Hans Wollstein, who would later die in Auschwitz, Hausdorff wrote “Forgive us our desertion! We wish to you and all our friends to experience better times.” Olsson’s juxtaposition of the “wail” last name alongside the Kristallnacht, a systematic attack on Jews, compares the personal struggles of André and Simone as inseparable from the Nazi’s persecution of Jews – as though the siblings were “wailing” in response to their persecution. It also emphasizes Olsson’s own perception of the siblings that, no matter how hard she tries, she still has her own take on the story. Even when she shares the rise of Nazis in Europe, Olsson’s limited perspective preserves the postmodern disunity of culture alongside a modern master narrative. The art of narration is both a process of Olsson’s own struggles to share and an authenticated, objective authority of knowledge that can forgive Hausdorff’s suicide and provide a better future for everyone. 

With Descartes’ discovery of the “unknown,” he also introduced methods of standard notation of mathematics that would let researchers use superscripts (x² as “x squared”) and subscripts (x as “x naught”). Olsson demonstrates the similarities between the methods of reasoning that let mathematical invention become the same engine underneath the creation of science, art, and literature, as French mathematician Jacques Hadamard explained. Hadamard’s interest of what goes on in a mathematician’s mind as they do what they do was also in response to the crisis of modernity having witnessed the horrors of both world wars. The mathematician frequently seek new ways of looking at problems in mathematics as researchers came and visited during seminars twice a week. The pieces of each story come together in a flow that uses a variation in style, length, and meaning to create a multidimensional work of art that is the book. Each passage flows seamlessly in the interplay between exposition and narrative, description and action, showing and telling.

At one point, Simone and André’s reading habits are interrupted by the narrator of Clarice Lispector’s Agua Viva proclaiming mathematics as the “madness of reason.” The rational, coherent, commonsense nature of mathematics would seem to contradict the foolish wildness of madness. But, as an interruption to Simone’s love of Kant and Chardin as a child and André’s interest in the Bhagavad Gita in college, this “madness of reason” becomes more apparent. In Why This World: A Biography of Clarice Lispector, Benjamin Moser wrote: 

My passion for the essence of numbers, wherein I foretell the core of their own rigid and fatal destiny,” was, like her meditations on the neutral pronoun “it,” a desire for the pure truth, neutral, unclassifiable and beyond language, that was the ultimate mystical reality. In her late works, bare numbers themselves are conflated with God, now without the mathematics that binds them, one to another, to lend them a syntactical meaning. On their own, numbers like the paintings she created at the end of her life, were pure abstractions, and as such connected to the random mystery of life itself. In her late abstract masterpiece Água Viva she rejects “the meaning that her father’s mathematics provide and elects instead the sheer “it” of the unadorned number: “I still have the power of reason-I studied mathematics which is the madness of reason-but now I want the plasma-I want to feed directly from the placenta.

The Renaissance depiction of madness as an intrinsic part of man’s nature is found in the literature and philosophy of the time period. An imbalance, or excess, of reason could lead to the madness that seeks this mysterious, “pure truth” that transcends language itself. Much the same way Simone and André seek the essence through different forms of this “madness.” Simone’s personal battles with health and existential issues seem more alike a mathematician’s search for reason. Olsson later mentions the “madness of reason” as she narrates her own lonely experience “trying to demonstrate small truths” an undergraduate in her lonely dorm room on a cold, wintery day. It’s a localized truth that Olsson finds in her work, but still remains part of a grander narrative that connects their stories. The interjecting quote from Lispector’s text highlights this search for truth in the stories of Simone, André, and Olsson herself. 

According to Olsson, Descartes used “x” to refer to the unknown because the printer was running out of letters, but there may have been an aesthetic choice in addition to the pragmatic use. “x “ would come to mean that which we don’t know in other contexts such as sex shops and invisible rays. Olsson continues her personal story asking the question “What is my unknown? My x?” She narrates her venture back to mathematics after writing novels in her time since she graduated from Harvard University. 

Olsson emphasizes Simone’s inferiority complex to her brother as one of the primary causes for this perspective on the world. Simone’s own desire to be a boy, use the name “Simon,” and absence of any lover while André proposed the Weil Conjectures, married, and had children show these contexts. She found truth in this suffering and disregard for material pleasures – even chasing states of mind in which she could perceive the world in a state of purity and without any biases of her own self. The conjectures would become the foundation for modern algebra, geometry, and number theory. 

When Olsson took a course under Harvard mathematician Barry Mazur, she didn’t dare speak to him. The conjecture, Mazur explained, would lay down the basis of a theory, expectations believed to be true, driven by analogy. Olsson still recalls her feeling of awe when she first learned and geometry and the power of understanding the world without memorizing it. After André was arrested while on vacation in Finland in 1939 on suspicion of spying, he barely missed execution when a Finnish mathematician suggested to the chief of police during a dinner before the day of the execution to deport him instead. While André is forced by train to Sweden and England, Olsson returns to her childhood excitement in middle school learning about “math involving letters.” She then recalls moments teaching her two-year-old daughter how to count as the child asks “Where are numbers?” When Olsson returns to André’s story, now as he’s transferred to a prison in France and requires unidle intellectual activity, she comments that escaping France was a more pressing problem than anything in mathematics. 

As André longs for an ability to engage in research even in the cloistered sepulchre of a prison cell, he writes to Simone comparisons of mathematics to art. Simone is allowed to visit him for a few days a week, and the two rassure each other that they’re okay. André tells Simone he told an editor to send page proofs of his article to her so she can copyedit them. The writing between the two goes into stories of Babylonians and Pythagoreans reminiscent of the dialogue the two siblings had as children. Olsson’s own story intertwined with the communication between Simone and André serves as a parallel to demonstrate that she, too, can make mathematics accessible to the common person the same way Simome did with André’s work. André’s colleagues would even start to envy the quiet solitude of prison in which he could produce work undisturbed.  Comparing mathematics to art, though, André described the material essence of a sculpture that limit a mathematician’s objectivity while remaining an explanation in and of itself. In this sense, it has both objective and subjective value the same way a mix of modern and postmodern story would. Simone doubted this, though. Works of art that relied on a physical material didn’t directly translate to a material for the art of mathematics. Though the Greeks spoke of the material of geometry as space, André’s work, Simone argued, was an inaccessible system of previous mathematical work, not a connection between man and the universe. 

The brother responded with the role of analogy in mathematics far beyond a mental activity. It was something you felt, a version of eros, “a glimpse that sparks desire,” Olsson wrote. Going through the history of mathematics from the nineteenth-century watershed time in which questions of numbers were solved using equations, the mathematician feels “a shiver of intuition” in connecting different theories. Simone would imagine societies built upon mathematics, mysticism, and existential loneliness. Through this, all of Olsson’s jumping between stories becomes clear. She had set the reader up to view mathematics as an art the way André did and, through the world Simone created, something the general audience could understand. Olsson continues with her personal experience as an undergraduate being recommended by a professor to write about mathematics for a general audience as a career alongside dream sequences of André and Simone. 

In 1938, Simone attended a Bourbaki conference, a group of French mathematicians that André had initiated with the purpose of reformulating mathematics on an abstract and formal, yet self-contained basis. The mathematicians would sign their names collectively as “Bourbaki” on papers as they attempted to unify contemporary mathematics with a common language just as Euclid did centuries ago. While the group members would yell at one another with hard-hitting questions, even threatening at times, Simone began to believe that mathematics should be made more accessible to a mass audience. The Bourbaki group’s vision lead them to describe hundreds of pages of set theory before defining the number 1. They sought to create an idea of mathematics as a system of maps and relationships that were more important than the intrinsic qualities of numbers and other mathematical objects themselves. Scientific American would call André “the last universal mathematician.” This method of universalizing while still emphasizing relationships among objects shows a modernist tendency, the former, interacting with a postmodernist one, the latter.

Olsson’s own stories through studying mathematics as a student and teaching her children She explains the highlight of her mathematics career was finding the answer to a course problem before one of her classmates did. Her humility and sense of humor make her writing all the more approachable and relatable.

The book’s weakness is that the individual stories feel abbreviated at times. Olsson switches back and forth between many narratives that may leave the reader feeling confused or even frustrated that desires and beliefs of the characters are unexpanded. It can make it difficult to get committed to the story events or feel connected to characters when their moments are so brief and spread out across the book. The short snippets of stories across time and space alongside Olsson’s juxtaposition of them with one another make the reading easy to understand for anyone without a strong background in either mathematics or philosophy. Still, much the same way Olsson describes the search for truth, it leaves the reader in a perpetual search. We get a feeling of excitement that we are bound to get to the correct answer to a problem or find meaning in research while still never quite achieving it. 

Olsson’s book serves as a beacon of the power of evidence and justification in a post-truth world. Olsson addresses the constant searches for truth and meaning in our current society by capturing opposites and extremes in her writing. The empirical, hypothesis-driven mathematics and speculative, argument-driven philosophy contrast one another on the meandering search for truth. The isolation of intelligence for both André and Simone in their work contrast the warmth of community and social engagement the two find in their respective environments. Truth becomes less of something that we must obtain by being on one side or the other and more of finding appropriate methods of addressing problems. It’s objective in that it lies in the techniques of various disciplines, but constructed because it comes from the individual’s choice. In a typical mix of modernism and postmodernism, Olsson’s personal story to find the answers to her personal curiosities by turning back to mathematics demonstrates this mix of the personal with the impersonal. 

Like postmodern stories, Olson’s book is non-linear and reveals truth as a series of localized, fragmented pieces. Like modernism, we find greater purposes and narratives between the different stories as a testament to the power of science and technology. It switches between the progressive, exalted story of André with the melancholic, tragedy of Simone with parallels between the stories together. The grand themes of the power and style of mathematics and philosophy dictate the rules and principles that set the foundation for the stories. André’s story and Simone’s may even be treated with the former as a modernist tale of the triumph of science and the latter, a postmodern warning of society’s so-called “progress.” In regular metamodernist fashion, Olsson uses elements of both modernism and postmodernism in her book. In metamodernist fashion, the two searches for truth become one and the same. Philosophy may ask “Why?” but, for mathematics, the question is “y?”

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