In 1854, a young mathematician named Bernhard Riemann delivered a lecture in which he laid out a theory for how to study geometric spaces in any dimension. But most mathematicians found Riemann’s ideas too vague and abstract to be of much use. “Many scientists and philosophers were saying, ‘This is nonsense,’” said José Ferreirós, a philosopher of science at the University of Seville in Spain. And so, for decades, the work was largely ignored. Riemann’s lecture didn’t appear in print until 1868, two years after his death.
But by the end of the 19th century, mathematical greats like Henri Poincaré had recognized the importance of Riemann’s ideas. And in 1915, Albert Einstein used them in his general theory of relativity, bringing them out of the realm of philosophical abstraction and into the real world. By the middle of the 20th century, they had become a mathematical staple.
Riemann had introduced a concept that could encompass all possible geometries, in any number of dimensions. A concept that would change how mathematicians view space.
A manifold.
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覺得AI浪潮最大驚奇是：以往的舊思維都不管用了……
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