The Conversation
André Weil was a mathematician. His sister Simone Weil was a philosopher. They both thought deeply about the nature and value of mathematics and …
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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.
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.”
A black and white photo of one woman and six men standing in front of a doorway. 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.
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
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