It was the physicist Eugene Wigner who discussed the “unreasonable effectiveness of mathematics” in a now famous paper that examined the profound link between mathematics and physics.

Today, Anton Zeilinger and pals at the University of Vienna in Austria reveal this link at its deepest. Their experiment involves the issue of mathematical decidability.

First, some background about axioms and propositions. The group explains that any formal logical system must be based on axioms, which are propositions that are defined to be true. A proposition is logically independent from a given set of axioms if it can neither be proved nor disproved from the axioms.

They then move on to the notion of undecidability. Mathematically undecidable propositions contain entirely new information which cannot be reduced to the information in the axioms. And given a set of axioms that contains a certain amount of information, it is impossible to deduce the truth value of a proposition which, together with the axioms, contains more information than the set of axioms itself.

These notions gave Zeilinger and co an idea. Why not encode a set of axioms as quantum states. A particular measurement on this system can then be thought of as a proposition. The researchers say that whenever a proposition is undecidable, the measurement should give a random result.

They’ve even tested the idea and say they’ve shown the undecidability of certain propositions because they generate random results.

Good stuff and it raises some interesting issues.

Let’s leave aside the problem of determining whether the result of particular measurement is truly random or not and take at face value the groups claim that “this sheds new light on the (mathematical) origin of quantum randomness in these measurements”.

There’s no question that what Zeilinger and co have done is fascinating and important. But isn’t the fact that a quantum system behaves in a logically consistent way exactly what you’d expect?

And if so, is it reasonable to decide that, far from being fantastically profound, Zeilinger’s experiment is actually utterly trivial?

Ref: http://arxiv.org/abs/0811.4542: Mathematical Undecidability and Quantum Randomness