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)."

Comments

Sheesh

[Reality Master 101]

Philosophers have long wondered at the profound link between mathematics and physics, but how deep does this connection go?

What an utterly meaningless bit of drivel. Any philosopher wondering this ought to turn in his license.

"Physics" is (to simplify) the scientific study of what rules the universe operates under. It's entirely possible and reasonable we can determine universal laws without having the faintest idea of *why* they are that way. It's observed truth that might even be totally different in a different part of the universe (we assume it's not, but that's just an assumption).

Mathematics is an abstract game of counting, built up into great complexity. 1 + 1 = 2 will be true in any universe, under any god(s), in any circumstances. And all of mathematics is built up from that. It's universal truth.

We use mathematics to quantify physics, but there is no "connection" between the two, except in the sense that we can count *anything* and say there's a connection. It's like saying, "How deep does the connection go between mathematics and bananas when I observe there are 10 bananas, and I add two more, and then I observe 12 bananas."

Re:Sheesh

[againjj]

Actually you can dream up universes where 1+1=2 doesn't hold. It can fail to hold for a variety of reasons.

And down the relativistic shoot we go. Would those reasons be related to physics by any chance? Honestly, I can't see how such an abstract concept such as math could conceivably even hint how 1+1 would not equal 2. If it equals something else, then congratulations, you have created an operator which does not have the quality that simple addition does.

You fail to have enough imagination. As a trivial example of when 1+1=2 doesn't hold, what if addition did not exist? This is not an interesting example, nor can I come up with one that is interesting, but that is what GP said too.

Equally, special numbers such as Pi and e will always output the same pattern of digits in any multi/quasi/supro-uno universe (given a particular base to start with - it doesn't have to be 10 of course).

Ah, something more interesting! Pi only has its familiar value in Euclidean space (which is the space we live in, not so coincidentally). Imagine hyperbolic space, and you have a value for pi that is larger than standard, exactly how much bigger depending on the curvature. Imagine spherical space and don't get anything meaningful at all. If you want a space where the parallel postulate holds, imagine a torus (donut). In all these cases, Pi is different or non-existent, and I imagine e would be too.

Re:Umm

[physicsphairy]

I suppose you could think of it as testing "computability." If your proposition is understandable by the quantum system you set up, it will spit out an answer. And you'll always get that answer.

But if it is not understandable by the quantum system you set up, then no operation is performed, and whatever comes out is simply the result of quantum randomness.

Re:My take on it

[jeffasselin]

The feeling I get from reading this is that it might be possible to offer an interpretation of the Universe as a huge decidability-machine. It's a leap, of course, but might be interesting to explore.

Re:Umm

[CorporateSuit]

Can someone please explain in layman's terms how this results in a decision, for those of us who aren't quantum mathematicians? I somewhat get the whole "indecision results in a decision" thing but seems to be a hard idea to wrap my brain around so to speak.

They're saying that no one orders lobster at McDonald's -- not because people don't like lobster, but because it's not on the menu. You can't base how the general population feels about lobster by asking McDonald's how many lobsters they sell compared to how many hamburgers.

So instead of looking to see what people feel about lobster, they're asking restaurants how many lobsters they sell in order to determine if lobster is even on the menu. Once that's set in stone, THEN they can start testing the demographics of how many people prefer lobster to what.

At least that's how I interpreted what they're doing... :\

Re:My take on it

[melikamp]

Interesting. I think you are onto something here. We can think of a universe as an encoding of a particular axiomatic system, and then there are "facts" in that universe which come up to surface with high probability. To an observer they look like "laws". Moreover, there may be some undecidable propositions which, to an observer, appear like sheer randomness. Also, if the number of qubits in the universe is infinite, it is quite possible that the universe "knows" everything.

Re:Umm

[Eli Gottlieb]

Does this also mean we could also prove theorems by physical experiment?

Re:Umm

[LoyalOpposition]

Can someone please explain in layman's terms how this results in a decision, for those of us who aren't quantum mathematicians? I somewhat get the whole "indecision results in a decision" thing but seems to be a hard idea to wrap my brain around so to speak.

I immediately thought of Euclid's five postulates. For years people thought that the fifth, parallel, postulate could be derived from the other four. That held for about 2100 years until a couple of boffins found used two different negations of the fifth to derive entire geometries. Applying that to this, I would suppose that if it were possible to encode Euclid's first four postulates into quantum states, and ask whether there was exactly one line parallel to another through a point not on the second line, then the result would sometimes be yes and sometimes no.

-Loyal

Re:Theory versus reality

[morgan_greywolf]

Perhaps there is no randomness. Perhaps all things behave according to some order.

Of course, now we just left physics and mathematics and entered the realm of philosophy... ;)

Re:Umm

[Anonymous Coward]

No. Theorem proving is undecidable in anything stronger than PA.

The Many-Worlds interpretation of quantum physics "stipulates"[1] that the universe is a super-position of "possible worlds". These possible worlds are mathematically modelled in terms of "models". A basic result in mathematical logic is that if there are distinct models for a set of axioms in which a proposition A is true in one and false in another, then there can be no proof of A from those axioms. The latter two together imply that there must be propositions about quantum states that cannot be proved, even in principle -- what has classically been called "quantum uncertainty".

[1] It is merely an interpretation of the physical phenomenon.

Re:Don't get too excited

[Garridan]

Peer-reviewed journals print things like this all the time. It doesn't mean it is correct or deep.

There... fixed that for you. You aren't incorrect, but your statement indicates a bias against information based on its source. That's an ad hominem argument, and is logically unsound. If you spot a problem in the paper, point it out.