November 10, 2010
NO, TO UNDERSTAND THEM WE MUST UNDERSTAND THAT THERE'S NO DIFFERENCE:
Large cardinals: maths shaken by the 'unprovable': A shocking discovery has unsettled the world of numbers (Richard Elwes, 11/10/10, The Telegraph)
In the esoteric world of mathematical logic, a dramatic discovery has been made. Previously unnoticed gaps have been found at the very heart of maths. What is more, the only way to repair these holes is with monstrous, mysterious infinities.
To understand them, we must understand what makes mathematics different from other sciences. The difference is proof.
Other scientists spend their time gathering evidence from the physical world and testing hypotheses against it. Pure maths is built using pure deduction.
But proofs have to start somewhere. For all its sophistication, mathematics is not alchemy: we cannot conjure facts from thin air. Every proof must be based on some underlying assumptions, or axioms.
And there we reach a thorny question. Even today, we do not fully understand the ordinary whole numbers 1,2,3,4,5… or the age-old ways to combine them: addition and multiplication.
Over the centuries, mathematicians have arrived at basic axioms which numbers must obey. Mostly these are simple, such as "a+b=b+a for any two numbers a and b". But when the Austrian logician Kurt Gödel turned his mind to this in 1931, he revealed a hole at the heart of our conception of numbers. His "incompleteness theorems" showed that arithmetic can never have truly solid foundations. Whatever axioms are used, there will always be gaps. There will always be facts about numbers which cannot be deduced from our chosen axioms.
Gödel's theorems showed that maths meant that mathematicians could not hope to prove every true statement: there would always be "unprovable theorems", which cannot be deduced from the usual axioms. Most known examples, it's true, will not change how you add up your shopping bill. For practical purposes, the laws of arithmetic seemed good enough.
However, as revealed in his forthcoming book, Boolean Relation Theory and Concrete Incompleteness, Harvey Friedman has discovered facts about numbers which are far more unsettling.
The Enlightenment ultimately ends up right back at ineffability.
Posted by Orrin Judd at November 10, 2010 6:08 AM