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February 19, 2006
At the Cutting Edge, Mathematicians May Only Have Evidence Themselves
I've noted a couple of times that only mathematicians and logicians have proofs and that the rest of us only have evidence. Well things may be changing.
A mathematical proof is irrefutably true, a manifestation of pure logic. But an increasing number of mathematical proofs are now impossible to verify with absolute certainty, according to experts in the field."I think that we're now inescapably in an age where the large statements of mathematics are so complex that we may never know for sure whether they're true or false," says Keith Devlin of Stanford University in California, US. “That puts us in the same boat as all the other scientists.”
As an example, he points to the Classification of Finite Simple Groups, a claimed proof announced in 1980 that resulted from a collaboration in which members of a group each contributed different pieces. "Twenty-five years later we're still not sure if it's correct or not. We sort of think it is, but no one's ever written down the complete proof," Devlin says.
The whole article in New Scientist raises some very intriguing issues. Can the logicians be far behind?
The issues facing mathematicians are somewhat different from those facing archaeologists, anthropologists, biologists, or anyone else for that matter. The problems that mathematicians face are problems of complexities compounded to the point that no mathematician or any team of mathematicians can be certain that the "proof" is correct. There remains the theoretical possibility of a proof. The algorithm either does or does not provide a proof. It's just nearly impossible to determine with certainty if it does or not. Outside of mathematics and logic, the issues are different. Proofs in the mathematical sense are just not possible. Some things are extremely likely (that thing that looks like a door over there will provide an exit path from this room) and other things are extremely unlikely (that thing that looks like a door is a painting) and many things are in between (the cat knocked that pencil on the floor) but nothing is proven. At least this is true until I walk through that door or smash my face trying. I'll never be sure about the cat and the pencil.
Posted by Duane Smith at February 19, 2006 1:28 PM | Read more on Science - General |
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