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
I don’t see any sentence of any undecidable logical theory in that paper. All statements seem to reduce to propositional expressions which are decidable by their truth tables. There is no undecidability involved in what they are showing, IMHO. If any undecidable expression could be decided by repeated quantum measurements as implied, then quantum computers would be super-Turing, which they are not.
I see that Johannes Kofler’s thesis, arXiv:0812.0238, has the conceptual foundations of this paper out of Zeilinger’s group as its Chapter 4, “Mathematical undecidability and quantum randomness”.
I don’t get it. First, everything discussed in the paper
seems to concern finite systems defined by finite truth tables. Every such thing is true, or false, and ALWAYS this truth or falsity is provable by a finite proof (namely, exhaustive examination). There is NEVER
an undecidable statement even possible in such a framework.
So I believe all the “profundity” inferred here is garbage.
Here’s a little thought experiment. Please tell me how to make a quantum system which will demonstrate or refute that P=NP, or show this to be an undecidable question. I’m waiting.
They didn’t make undecidable propositions decidable, that’s impossible. They showed that quantum systems can detect undecidability under certain axiomatic systems. The point is not “super-turing” computers, the point is that quantum systems do real mathematical computations. If anything this MIGHT throw a wrench in the halting problem, as a quantum calculation might be able to detect the decidability of some class of computations.
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?
One could say the same thing about mathematics since most of the cutting edge research (um, for the past few centuries) is about discovering tautologies in made up abstract structures. But these abstract structures help describe models with real value. And discovering and proving the tautologies take considerable effort.
Tim,
Even standard turing machines can solve the halting problem for “some class of computations”. All it says it that a turing machine cannot decide this for ALL computations”. If you’re limiting the set of computations, of course there are turing machines that can decide this.
Example: Construct a turing machine to decide whether or not the following computation halts: The computation to be decided is described by the following turing machine –> Immediately halt on all inputs.
If a quantum turing machine (or quantum calculation as you called it) could throw a wrench into the halting problme, then by definition QTMs would disprove the church turing thesis, which it has already been shown they do not. maybe some other computational model can disprove it, but QTMs are equivalent computationally to standard TMs.
How do you define “random result”? How do you handle systems with infinitely many axioms?
if the word “mathematical” = decidable
then
nothing more
mathematical undecidability is the proposition itself already
proofing something running systematic = outter set is random?
who knows…….if and only if there is no grey
Zach – “If a quantum turing machine (or quantum calculation as you called it) could throw a wrench into the halting problme, then by definition QTMs would disprove the church turing thesis, which it has already been shown they do not.” ….
do you have a reference? I’ve suspected otherwise (but not in the current formulation of a quantum computer) and I’ld love to read a paper. sydspoetry@hotmail.com
look in the bible 🙂
*Quantum Computation and Quantum
Information* Nielsen & Chang.
See for example p. 126
What would be interesting is how a quantum system would determine a FORMALLY undecidable (but true if you assume the consistency of your system) proposition like Gödel’s. If I read Zach’s comment correctly, it has already been shown to flag it as undecidable. And what would the implications be (of either result) for Penrose’s speculations on AI?
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