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Mathematician Reveals 'Equals' Has More Than One Meaning In Math
"It turns out that mathematicians actually can't agree on the definition of what makes two things equal, and that could cause some headaches for computer programs that are increasingly being used to check mathematical proofs," writes Clare Watson via ScienceAlert. The issue has prompted British mathematician Kevin Buzzard to re-examine the concept of equality to "challenge various reasonable-sounding slogans about equality." The research has been posted on arXiv. From the report: In familiar usage, the equals sign sets up equations that describe different mathematical objects that represent the same value or meaning, something which can be proven with a few switcharoos and logical transformations from side to side. For example, the integer 2 can describe a pair of objects, as can 1 + 1. But a second definition of equality has been used amongst mathematicians since the late 19th century, when set theory emerged. Set theory has evolved and with it, mathematicians' definition of equality has expanded too. A set like {1, 2, 3} can be considered 'equal' to a set like {a, b, c} because of an implicit understanding called canonical isomorphism, which compares similarities between the structures of groups. "These sets match up with each other in a completely natural way and mathematicians realised it would be really convenient if we just call those equal as well," Buzzard told New Scientist's Alex Wilkins. However, taking canonical isomorphism to mean equality is now causing "some real trouble," Buzzard writes, for mathematicians trying to formalize proofs -- including decades-old foundational concepts -- using computers. "None of the [computer] systems that exist so far capture the way that mathematicians such as Grothendieck use the equal symbol," Buzzard told Wilkins, referring to Alexander Grothendieck, a leading mathematician of the 20th century who relied on set theory to describe equality. Some mathematicians think they should just redefine mathematical concepts to formally equate canonical isomorphism with equality. Buzzard disagrees. He thinks the incongruence between mathematicians and machines should prompt math minds to rethink what exactly they mean by mathematical concepts as foundational as equality so computers can understand them. "When one is forced to write down what one actually means and cannot hide behind such ill-defined words," Buzzard writes. "One sometimes finds that one has to do extra work, or even rethink how certain ideas should be presented."
"It turns out that mathematicians actually can't agree on the definition of what makes two things equal, and that could cause some headaches for computer programs that are increasingly being used to check mathematical proofs," writes Clare Watson via ScienceAlert. The issue has prompted British mathematician Kevin Buzzard to re-examine the concept of equality to "challenge various reasonable-sounding slogans about equality." The research has been posted on arXiv. From the report: In familiar usage, the equals sign sets up equations that describe different mathematical objects that represent the same value or meaning, something which can be proven with a few switcharoos and logical transformations from side to side. For example, the integer 2 can describe a pair of objects, as can 1 + 1. But a second definition of equality has been used amongst mathematicians since the late 19th century, when set theory emerged. Set theory has evolved and with it, mathematicians' definition of equality has expanded too. A set like {1, 2, 3} can be considered 'equal' to a set like {a, b, c} because of an implicit understanding called canonical isomorphism, which compares similarities between the structures of groups. "These sets match up with each other in a completely natural way and mathematicians realised it would be really convenient if we just call those equal as well," Buzzard told New Scientist's Alex Wilkins. However, taking canonical isomorphism to mean equality is now causing "some real trouble," Buzzard writes, for mathematicians trying to formalize proofs -- including decades-old foundational concepts -- using computers. "None of the [computer] systems that exist so far capture the way that mathematicians such as Grothendieck use the equal symbol," Buzzard told Wilkins, referring to Alexander Grothendieck, a leading mathematician of the 20th century who relied on set theory to describe equality. Some mathematicians think they should just redefine mathematical concepts to formally equate canonical isomorphism with equality. Buzzard disagrees. He thinks the incongruence between mathematicians and machines should prompt math minds to rethink what exactly they mean by mathematical concepts as foundational as equality so computers can understand them. "When one is forced to write down what one actually means and cannot hide behind such ill-defined words," Buzzard writes. "One sometimes finds that one has to do extra work, or even rethink how certain ideas should be presented."
Read more of this story at Slashdot.
