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Quantum Test Found For Mathematical Undecidability
KentuckyFC writes "Philosophers have long wondered at the profound link between mathematics and physics, but how deep does this connection go? Pretty deep according to the results of a quantum experiment exploring the nature of mathematical undecidability. Here's how: any logical system must be based on axioms, which are propositions that are defined to be true. A proposition is logically independent from these axioms if it can neither be proved nor disproved from them; mathematicians say it is undecidable. In the experiment, researchers encoded a set of axioms as quantum states. A particular measurement on this system can then be thought of as a proposition which, if undecidable, yields a random result — which is what they found. 'This sheds new light on the (mathematical) origin of quantum randomness in these measurements,' say the researchers (abstract)."
They found a way to physically encode a mathematical "axiom" into quantum states. They set up a particular axiom as a quantum state machine, then measure the system. The measurement is done in such a way that it is equivalent to asking "is X true given this axiom?" where X is any mathematical "proposition". The answer to that question can be "yes", "no", or "not enough information". If the latter is the case, the results from the physical quantum experiment will show a random distribution.
So, if I have a mathematical proposition and I'm not sure if it is supported by a certain axiom, I could actually build the axiom into a quantum state machine and measure it in a way that tests my particular proposition. If the results after multiple runs are distributed randomly, then it means that the axiom can not prove or disprove the proposition.