Quanta Magazine

Calculus is a powerful mathematical tool. But for hundreds of years after its invention in the 17th century, it stood on a shaky foundation. Its core concepts were rooted in intuition and informal arguments, rather than precise, formal definitions. Two schools of thought emerged in response, according to Michael Barany, a historian of math and science at the University of Edinburgh. French mathematicians were by and large content to keep going. They were more concerned with applying calculus to problems in physics — using it to compute the trajectories of planets, for instance, or to study the behavior of electric currents. But by the 19th century, German mathematicians had begun to tear things down. They set out to find counterexamples that would undermine long-held assumptions, and eventually used those counterexamples to put calculus on more stable and durable footing. One of these mathematicians was Karl Weierstrass. Though he showed an early aptitude for math, his father pressured him to study public finance and administration, with an eye toward joining the Prussian civil service. Bored with his university coursework, Weierstrass is said to have spent most of his time drinking and fencing; in the late 1830s, after failing to get his degree, he became a secondary school teacher, giving lessons in everything from math and physics to penmanship and gymnastics. Weierstrass didn’t begin his career as a professional mathematician until he was nearly 40. But he would go on to transform the field by introducing a mathematical monster. Find out what makes the Weierstrass function so monstrous: https://lnkd.in/gD8wrPwb

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The truth about the proton is too bizarre to fully capture with words or images. That hasn’t stopped us from trying. Explore Quanta’s visual explainer from the archive: https://lnkd.in/ecACp9zz

May you never confuse BPP and BQP again. (From the archive) https://lnkd.in/e58U9vuh

How could the brain have one or more concept cells for each of the countless concepts we engage with over a lifetime? It’s possible that these neurons take on different identities based on the task at hand, “like a Swiss Army knife,” suggested neuroscientist Cory Miller. https://lnkd.in/epK5Qj4K

Just learning the basics of elliptic curves will take you on a journey through several regions of the mathematical world and touch on some of its biggest open questions. In this week's Fundamentals newsletter, staff writer Joseph Howlett examines one of mathematicians' favorite specimens.

“There’s no animal model of language,” said computational linguist Isabel Papadimitriou (center). “Language models are the first thing that we can experiment on in any interventional way.” https://lnkd.in/eCrXZPmQ

For the theoretical quantum fields seen here, magnetic order never melts away. Even at infinity degrees. https://lnkd.in/euZw3rKh

Given a central sphere, how many identical spheres can “kiss” it without overlapping? Arrange six pennies around a central penny, and you’ll find they fit snugly into a daisylike pattern. But in higher dimensions, the problem gets much harder. https://lnkd.in/e7fNKt_s

In the late 1950s, Noam Chomsky spearheaded an ambitious attempt to uncover the universal structure underlying all languages. It remained the dominant strain of theoretical linguistics for half a century. Then came the machine learning revolution. https://lnkd.in/eCrXZPmQ