Among the frontiers identified are time, the cosmos, consciousness and God, but aren’t swaths of knowledge concerned with meaning rather than scientific fact?

Scientists like to see themselves as modern counterparts of the great explorers, sailing off into the unknown and coming back with marvellous tales of adventure and discovery. But the heroic age of exploration lasted no more than 500 years: after the so-called conquest of the poles there was not much terra incognita left to conquer. Does a similar fate await the sciences? Will nature yield up its last secret one day? Will our questions all be answered? Will scientists abandon their laboratories and take up poetry, painting or tap dancing instead?

These are the questions raised by an engaging new book in which Marcus du Sautoy promises to lead us to “the edges of knowledge”. He begins by recalling a speech given by the physicist Lord Kelvin at the end of the 19th century. “There is nothing new to be discovered in physics now,” Kelvin said, “all that remains is more and more precise measurement.” Albert Einstein soon proved him wrong, but scientists carried on dreaming of the day when they could declare mission accomplished. In September 1930, for instance, the distinguished mathematician David Hilbert addressed a meeting in his honour in Königsberg. Nothing could hold out against the progress of science, he said: “We must know – and we shall.” Unluckily for him, a young logician called Kurt Gödel had demonstrated the exact opposite in a paper delivered in the same city on the previous day. Every conceivable system of mathematics, Gödel showed, must contain statements that cannot be proved, so the idea of scientific closure was a quixotic fantasy.

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