Jordana Cepelewicz in Quanta:
In 2012, the mathematician Shinichi Mochizuki claimed he had solved the abc conjecture, a major open question in number theory about the relationship between addition and multiplication. There was just one problem: His proof, which was more than 500 pages long, was completely impenetrable. It relied on a snarl of new definitions, notation, and theories that nearly all mathematicians found impossible to make sense of. Years later, when two mathematicians translated large parts of the proof into more familiar terms, they pointed to what one called a “serious, unfixable gap” in its logic — only for Mochizuki to reject their argument on the basis that they’d simply failed to understand his work.
The incident raises a fundamental question: What is a mathematical proof? We tend to think of it as a revelation of some eternal truth, but perhaps it is better understood as something of a social construct.
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A recent rise in activism in Iran has added a new chapter to the country’s long-standing history of murals and other public art. But as the
In Consciousness as Complex Event: Towards a New Physicalism, Craig DeLancey argues that what makes conscious (or “phenomenal”) experiences mysterious and seemingly impossible to explain is that they are extremely complex brain events. This is then used to debunk the most influential anti-physicalist arguments, such as the knowledge argument. A new “complexity-based” way of thinking about physicalism is then said to emerge.
The opening sentence of Jacob Mikanowski’s sweeping history of Eastern Europe—“a place,” he asserts, “that doesn’t exist”—calls to mind a seminal 1983 essay by the recently deceased Czech-French novelist, Milan Kundera. In A Kidnapped West: A Tragedy of Central Europe, Kundera reminded his readers that Poland, Hungary, and Czechoslovakia were historically and culturally closer to the West than to the Soviet East, and should therefore be thought of as central rather than eastern European. Alas, his appeal fell on deaf ears, and the region remains “eastern,” shorthand for a place where, rumor has it, nobody smiles and the smell of burned cabbage wafts through the corridors of charmless, concrete apartment blocks. Indeed, western prejudice was never more in evidence than during the early days of Russia’s invasion of Ukraine, when many commentators painted the war as just another incomprehensible scuffle in the other Europe, among the denizens of lands once obscured by the Iron Curtain.
An artificial-intelligence system can describe how compounds smell simply by analysing their molecular structures — and its descriptions are often similar to those of trained human sniffers.
Over the last year, AI large-language models (LLMs) like
One hundred and fifty years ago, John Stuart Mill died in his home in Avignon. His last words were to his step-daughter, Helen Taylor: “You know that I have done my work.”
We often think of proteins as immutable 3D sculptures. That’s not quite right. Many proteins are transformers that twist and change their shapes depending on biological needs. One configuration may propagate damaging signals from a stroke or heart attack. Another may block the resulting molecular cascade and limit harm.
It’s fitting that choreographer George Balanchine is experiencing a cultural moment around the 40th anniversary of his death. In the ephemeral realm of dance, the longevity of his influence is unique and it shows no signs of waning. Balanchine’s ballets—beloved for their sophisticated abstraction and musicality—have become staples of the repertoire for ballet companies in the United States and beyond, and they have been performed more often since his death than during his lifetime. Two institutions jointly founded by Balanchine and impresario Lincoln Kirstein—the New York City Ballet (NYCB) and the School of American Ballet (SAB)—continue to cultivate his artistic legacy on the stages and in the classrooms of Lincoln Center. Among his other accomplishments, Balanchine made The Nutcracker a Christmas classic in the United States, and for many ballet enthusiasts it would be as difficult to conceive of ballet without Balanchine as it would be to spend the holidays without the Mouse King and the Sugar Plum Fairy.
In March, 1929, when the twenty-four-year-old Chinese American actress
Among the many dolls mentioned in Greta Gerwig’s film Barbie there is one associated with time and memory and literally named after Marcel Proust. It didn’t sell well. Perhaps Mattel got the wrong writer. They could have gone for the same Marcel, but as a comedian, a French, philosophical, disguised partner of Dickens. Critics have been finding Proust funny since 1928—he died in 1922. Christopher Prendergast’s