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Mathematical immortality? Name that theorem

作者：单载 时间：2019-03-07 04:18:00 人气： ℃

By Jacob Aron During my time as an eager undergraduate mathematician, I’d often wonder what it would feel like to prove a truly new result and have my name immortalised in the mathematical history books. I thought that dream had died when I gave up maths to become a science writer, but Aron’s theorem is now a reality – and I’ve got the certificate to prove it. While most mathematical theorems result from weeks of hard work and possibly a few broken pencils, mine comes courtesy of TheoryMine, a company selling personalised theorems as novelty gifts for £15 a pop. Its automated theorem-proving software can churn out a theoretically infinite number of theorems for customers wishing to join the ranks of Pythagoras and Fermat. “We generate new theorems and let people name them after themselves, a friend, a loved one, or whoever they want to name it after,” explains Flaminia Cavallo, managing director of TheoryMine, based in Edinburgh, UK. You may think this is an elaborate scam, or that you’ll just end up with an obscure equation copied from some long-forgotten textbook, but TheoryMine claims to have far more validity than superficially similar companies selling star names. “We’re inventing totally novel theorems, and the tradition is you have the right to name these theorems,” explains Alan Bundy, professor of automated reasoning at the University of Edinburgh and another member of the TheoryMine team. “There are 10 star companies out there, and none of them have any affiliation to the International Astronomical Union.” He’s got a point. Automated theorem proving is a well-respected mathematical field, used by manufacturers to guarantee that the algorithms in computer processors will work correctly. Bundy and his colleagues have worked in this area for a number of years, and Cavallo came up with the idea for TheoryMine during her final year of an undergraduate degree in artificial intelligence and mathematics at the University of Edinburgh, where she wrote a program to generate novel theorems for her dissertation. From its library of mathematical knowledge, the program generates a set of mathematical axioms, then combines them in different ways to produce a series of conjectures. It then uses the library to discard a portion of these on the basis that there are already counter-examples, showing they can’t be true. Overly complex conjectures are also ignored. Then it applies a technique known as “rippling”, in which it tries out various sequences of logical statements until one of these sequences turns out to be a proof of the theorem. “It’s a clever idea,” says Lawrence Paulson, a computational logician at the University of Cambridge and the creator of Isabelle, a theorem prover that Cavallo’s program uses. He is more interested in the theory behind the new program though, adding that “some of the technology here is quite impressive, and I would hope that it finds other applications apart from selling certificates”. It may well do. Lucas Dixon, another TheoryMiner, is investigating the possibility of using the same techniques to elucidate the rules of algebra in quantum computing systems, which follow different mathematical rules to classical systems. Don’t prepare your Fields medal acceptance speech just yet though, as TheoryMine’s theorems are unlikely to break drastically new ground. “We can’t say that we’ll never do that, but having looked at the things that come out, they’re not typically things that are going to change the world,” says Dixon. If not revolutionary, your theorem will at least be unique, since the TheoryMine software ignores trivial variants of already proved theorems. And in case you’re wondering, Aron’s theorem, delivered as a pdf by email yesterday, says something about the order in which you can add a new variety of numbers. The TheoryMine team explained the logic behind the theorem to me, but I’m not quite sure I fully understand it – a common criticism of machine-aided proofs. Turning over proving responsibilities to a computer also means I haven’t really captured that exciting feeling of discovering something new. Mathematical proof is about more than just racing to the end result: it’s a creative process in its own right. Sorry,