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Maths | Science

Everything we know about maths could be wrong

Is maths best described as an art or a science? A top academic fears that modern theorems are riddled with errors. Computers could fix them. But if they did, would the result still be maths? When Professor Kevin Buzzard opened the 10th Interactive Theorem Proving Conference before an audience of expert mathematicians the other day, little did they know that he was about to tell them something that would turn their world upside down. "I'm suddenly concerned that all of published maths is wrong," explained the number theorist. "Because mathematicians are not checking the details." The problem, he said, lies in mathematical proofs. A proof is a mathematical statement which logically demonstrates that a theorem is true. However, Buzzard believes that proofs have become so complex that no one can understand them - or really knows if they are correct. A senior mathematical expert might cite 20 established proofs in one paper. When the paper is published, other mathematicians take it for granted that these proofs are correct without really checking, particularly when a single proof might contain 1,000 pages of dense reasoning. Buzzard uses the example of Fermat's Last TheoremIn 1637, Pierre de Fermat wrote in the margin to one of his books that he could prove there was no set of numbers that would satisfy the equation xn+ yn= zn. .He died without writing out the proof, and the theorem then went unproved for the next 350 years., once listed by the Guinness Book of World Records as the world's "most difficult maths problem". "I believe that no human, alive or dead, knows all the details of the proof of Fermat's Last Theorem. But the community accepts the proof nonetheless," he says. Because theorems depend upon each other like building blocks, one tiny mistake in the foundations could bring the whole edifice crashing down. So, how could we know if there was a mistake? Scientists have developed a proof verification software called Lean. The computer programme can follow the logical steps in a proof conclusively to determine if it is true. Computer scientists are working on "a general automated theorem prover", which could create its own proofs and develop its own theorems. In the future, maths could be entirely automated. But Professor Michael Harris of Columbia University worries that such a machine would take the creativity out of his field. And without creativity, it would not really count as maths. Mathematics is "wildly creative", says Professor Anthony Bonato of Ryerson University in Toronto. "There are elements of both art and science in the field, but it isn't a subset of either." Granted it isn't either. But which best describes the essential process: art or science? Doesn’t add up? Sciences use logic and rationality to tell us about the world around us, and this is exactly what maths does. While numbers do not physically exist, we use them to describe the natural world and the intricate patters found in it. This is why many mathematicians view the field as a discovery, rather than a human invention. Like all science, mathematics is a body of provable knowledge. But as G.H. Hardy wrote in his Apology, maths - like the greatest art - is concerned with symmetry and beauty. "A mathematician, like a painter or a poet, is a maker of patterns [...]. Mathematics, rightly viewed, possesses not only truth, but supreme beauty - a beauty cold and austere [...], sublimely pure, and capable of a stern perfection such as only the greatest art can show." A mere computer would not get far with this. KeywordsFermat's Last Theorem - In 1637, Pierre de Fermat wrote in the margin to one of his books that he could prove there was no set of numbers that would satisfy the equation xn+ yn= zn. .He died without writing out the proof, and the theorem then went unproved for the next 350 years.

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