Scott Taylor: The mystic and the mathematician
The Babylonian mathematical tablet, dated to 1800 B.C.
‘Waterville, Maine
Like most mathematicians, I hear confessions from complete strangers: the inevitable “I was always bad at math.” I suppress the response, “You are forgiven, my child.”
Why does it feel like a sin to struggle in math? Why are so many traumatized by their mathematics education? Is learning math worthwhile?
Sometimes agreeing and sometimes disagreeing, André and Simone Weil were the sort of siblings who would argue about such questions. André achieved renown as a mathematician; Simone was a formidable philosopher and mystic. André focused on applying algebra and geometry to deep questions about the structures of whole numbers, while Simone was concerned with how the world can be soul-crushing.
Both wrestled with the best way to teach math. Their insights and contradictions point to the fundamental role that mathematics and mathematics education play in human life and culture.
André Weil’s rigorous mathematics
André Weil, pictured here in 1956, was a prominent mathematician in the 20th century. Konrad Jacobs via Wikimedia Commons, CC BY-SA
Unlike the prominent French mathematicians of previous generations, André, who was born in 1906 and died in 1998, spent little time philosophizing. For him, mathematics was a living subject endowed with a long and substantial history, but as he remarked, he saw “no need to defend (it).”
In his interactions with people, André was an unsparing critic. Although admired by some colleagues, he was feared by and at times disdainful of his students. He co-founded the Bourbaki mathematics collective that used abstraction and logical rigor to restructure mathematics from the ground up.
Nicolas Bourbaki’s commitment to proceeding from first principles, however, did not completely encapsulate his conception of what constituted worthwhile mathematics. André was attuned to how math should be taught differently to different audiences.
Tempering the Bourbaki spirit, he defined rigor as “(not) proving everything, but … endeavoring to assume as little as possible at every stage.”
The Bourbaki congress in 1938. Simone, pictured at front left, accompanies André, obscured at back left. Unknown author via Wikimedia Commons
In other words, absolute rigor has its place, but teachers must be willing to take their audience into account. He believed that teachers must motivate students by providing them meaningful problems and provocative examples. Excitement for advanced students comes in encountering the unknown; for beginning students, it emerges from solving questions of, as he put it, “theoretical or practical importance.” He insisted that math “must be a source of intellectual excitement.”
André’s own sense of intellectual excitement came from applying insights from one part of mathematics to other parts. In a letter to his sister, André described his work as seeking a metaphorical “Rosetta stone” of analogies between advanced versions of three basic mathematical objects: numbers, polynomials and geometric spaces.
André described mathematics in romantic terms. Initially, the relationship between the different parts of mathematics is that of passionate lovers, exchanging “furtive caresses” and having “inexplicable quarrels.” But as the analogies eventually give way to a single unified theory, the affair grows cold: “Gone is the analogy: gone are the two theories, their conflicts and their delicious reciprocal reflections … alas, all is just one theory, whose majestic beauty can no longer excite us.”
Despite being passionless, this theory that unifies numbers, polynomials and geometry gets to the heart of mathematics; André pursued it intensely. In the words of a colleague, André sought the “real meaning of every basic mathematical phenomenon.” For him, unlike his sister, this real meaning was found in the careful definitions, precisely articulated theorems and rigorous proofs of the most advanced mathematics of his time. Romantic language simply described the emotions of the mathematician encountering the mathematics; it did not point to any deeper significance.
Simone Weil and the philosophy of mathematics
On the other hand, Simone, who was born three years after André and died 55 years before him, used philosophy and religion to investigate the value of mathematics for nongeniuses, in addition to her work on politics, war, science and suffering.
Simone Weil, pictured here in 1925, was a prominent philosopher. Anonymous via Wikimedia Commons
All of her writing – indeed, her life – has a maddening quality to it. In her polished essays, as well as her private letters and journals, she will often make an extreme assertion or enigmatic comment. Such assertions might concern the motivations of scientists, the psychological state of a sufferer, the nature of labor, an analysis of labor unions or an interpretation of Greek philosophers and mathematicians. She is not a systematic thinker but rather circles around and around clusters of ideas and themes. When I read her writing, I am often taken aback. I start to argue with her, bringing up counterexamples and qualifications, but I eventually end up granting the essence of her point. Simone was known for the single-minded pursuit of her ideals.
Despite the discomfort her viewpoints provoke, they are worth engaging. Although her childhood was largely happy, her whole life she felt stupid in comparison with her brother. She channeled her feelings of inadequacy into an exploration of how to experience a meaningful existence in the face of oppression and affliction. Over her life, she developed an interpretation of beauty and suffering intertwined with geometry.
Along with her lifelong mathematical discussions with André, her views were influenced by one of her first jobs as a teacher. In a letter to a colleague, she described her pupils as struggling because they “regarded the various sciences as compilations of cut-and-dried knowledge.” Like André, Simone saw the ability to motivate students as the key to good teaching. She taught mathematics as a subject embedded in culture, emphasizing overarching historical themes. Even those students who were “most ignorant in science” followed her lectures with “passionate interest.”
For Simone, however, the primary purpose of mathematics education was to develop the virtue of attention. Mathematics confronts us with our mistakes, and the contemplation of these inadequacies brings the ability to concentrate on one thing, at the exclusion of all else, to the fore. As a math teacher, I frequently see students grit their teeth and furrow their brow, developing only a headache and resentment. According to Simone, however, true attention arises from joy and desire. We hold our knowledge lightly and wait with detached thought for light to arrive.
For Simone, the “first duty” of teachers is to help students develop, through their studies, the ability to apprehend God, which she conceptualized as a blending of Plato’s description of the ultimate Good with Christian conceptions of the self-abnegating God. A true understanding of God results in love for the afflicted.
Simone might even locate the lingering anxiety and frustration of many former math students in the absence of attention paid to them by their teachers.
Authors grapple with the Weil legacy
Recently, others have wrestled with the Weil legacy.
Sylvie Weil, André’s daughter, was born shortly before Simone’s death. Her family experience was that of being mistaken for her aunt, ignored or demeaned by her father and not being acknowledged and appreciated by those in her orbit.
Similarly, author Karen Olsson uses Simone and André to explore her own conflicted relationship with mathematics. Her forlorn quest to understand André’s mathematics eerily reflects Simone’s desire to understand André’s work and Sylvie’s desire to be seen as her own person, to not be in Simone’s shadow. Olsson studied with exceptional math teachers and students, all the while feeling out of place, overwhelmed and intimidated by her fellow students. Most painfully, in the process of writing her book on the Weil siblings, Olsson asks a mathematician, who had been a student with her, for help in understanding some aspect of André’s mathematics. She was ignored. Both Sylvie Weil and Karen Olsson are living witnesses to Simone’s observation that each of us cries out to be seen.
Christopher Jackson, on the other hand, gives testimony to how mathematics can live up to Simone’s vision. Jackson is incarcerated in a federal prison but found a new life through mathematics. His correspondence with mathematician Francis Su is the backbone of Su’s 2020 book “Mathematics for Human Flourishing,” which uses Simone’s observation that “every being cries out silently to be read differently” as a leitmotif. Su identifies aspects of mathematics that promote human flourishing, such as beauty, truth, freedom and love. In their own ways, both Simone and André would likely agree.
Scott Taylor is a professor of mathematics at Colby College in Waterville, Maine.
Disclosure statement
He receives funding from the National Science Foundation and the John and Mary Neff Foundation.
Don Pesci: Simone Weil’s faith amidst the world’s agonies
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Simone Weil (1909-1943)
VERNON, Conn.
When you were very young, in college, you ran across some authors, almost by accident, who took your breath away, so lucid and creative were their minds. And you promised yourself – someday I will return to your table and feast again on your wisdom. Will it be as nourishing then. you wondered, as it is now?
Simone Weil (pronounced VEY), whom Albert Camus thought was the most courageous person among the writers of his day, some of them, including Camus, literary giants, was hauled into the Christian faith by the following George Herbert poem:
Love (III)
Love bade me welcome. Yet my soul drew back
Guilty of dust and sin.
But quick-eyed Love, observing me grow slack
From my first entrance in,
Drew nearer to me, sweetly questioning
If I lacked any thing.
A guest, I answered, worthy to be here:
Love said, You shall be he.
I the unkind, ungrateful? Ah my dear,
I cannot look on thee.
Love took my hand, and smiling did reply,
Who made the eyes but I?
Truth Lord, but I have marred them: let my shame
Go where it doth deserve.
And know you not, says Love, who bore the blame?
My dear, then I will serve.
You must sit down, says Love, and taste my meat:
So I did sit and eat.
Weil, whose parents were of Jewish background, was strongly tempted to join the Catholic Church but resisted, she wrote in letters to Father Jean-Marie Perrin, a member of the Dominican order, because she felt her place was outside formal structures. She could best serve from the outside, looking in. And she loved what she saw.
Five years after her soul had been besieged by Christ, she knelt for the first time at the shrine in Assisi and persuaded herself at last to say the Pater Noster daily with such attention that she felt at each repetition Christ himself had “descended and took her.”
Her’s was a hard mind and brilliant, like diamonds. Her religious experiences did not – could not – separate her from the agonies of daily life. She was driven by the wild horses of her personality never to separate herself, in any fashion, from the misfortunes of others. For this reason, she refused for a time to leave Paris, then occupied by Nazis. She eventually joined Gustave Thibon, a lay theologian in charge of an agricultural colony in the south of France. Later, she would petition Charles De Gaulle to parachute her into Paris. He refused.
She worked in the vineyards with peasants until her health, always fragile, broke down. At first Thibon mistrusted her motives. She had already attained a reputation in Paris as a radical intellectual, and here she was perversely “returning to the soil,” eating what peasants ate, sleeping as they slept, clothed in poverty, and giving lectures to them on the Upanishads.
In time, Thibon became closely attached to her – as did many others with whom over the years she had established close contact. It was to Thibon that she entrusted her journals and occasional jottings, which he decided to publish after her death, caused in part by her refusal to eat any more than French soldiers had at their disposal. She died in exile, in London, in 1943.
Fr. Perrin, the anvil upon which she hammered out her thoughts on Christ, was a great friend and confidant. In the introduction to Waiting on God, a invaluable collection of Weil’s letters and essays, Leslie Fielder writes, “One has the sense of Simone Weil as a woman to whom ‘sexual purity’ is as instinctive as breath; to whom indeed any kind of sentimental life is scarcely necessary. But a few lines in one of her absolutely frank and unguarded letters to Father Perrin reveal a terrible loneliness which only he was able to mitigate to some degree and vulnerability which only he knew how to spare: ‘I believe that, except for you, all human beings to whom I have ever given, through my friendship, the power to harm easily, have sometimes amused themselves by doing so, frequently or rarely, consciously or unconsciously, but all of them at one time or another.…’”
For Weil, who suffered from crippling migraines for a good portion of her life, suffering is not a nullity. Nor was it a nullity for Christ pinned to a cross: “…persevering in our love, we fall to the point where the soul cannot keep back the cry ‘My God, my God, why hast though forsaken me?’ If we remain at the point without ceasing to love, we end by touching something that is not affliction, not joy, something that is the central essence, necessary and pure, something not of the senses, common to joy and sorrow – the very love of God… Extreme affliction…is a nail whose point is applied at the very center of the soul, whose head is all necessity spreading throughout space and time… He whose soul remains ever turned towards God, though pierced with a nail, finds himself nailed to the center of the universe…at the intersection of creation and its Creator…at the intersection of the arms of the Cross.”
It is nearly impossible for the postmodern, materialistic mind to believe that such airy effusions are other than poetry, and dangerous poetry at that. But if postmodernism wishes to argue the point with Weil, its arguments will lack conviction; for Weil’s larger point is that the believing Christian is, in truth, a slave to the truth, a convict of Godly love. To attend is to love.
It may shock the shriveled, inattentive mind of postmodern academics that Herbert’s poem, which opened to Weil a window on the splendor of Christ the criminal, is not poetry either. It is the life-giving breath of God.
Don Pesci is a Vernon-based columnist.