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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)."
Didn't Rush have a song about this?
No, really, they're serious.
The rules of math (which weren't so much invented as identified) seem oddly linked to the underlying physics. TFA mentions the unreasonable effectiveness of mathematics -- it's not so much that we can count the physics with the math, it's that the math predicts things which should be true, and are subsequently proven to be. The existence of things like a negative square root in an equation have predicted the existence of things like anti-particles, and those particles have been found experimentally.
It's precisely the fact that the math isn't independent of the physics that is at issue here That's a very startling proposition because it goes well beyond simply counting what is, it means the same rules which define the math in the first place underly the physical mechanisms.
Cheers
Lost at C:>. Found at C.
Okay, disclaimer: I suck at math. ^_^ That said -- how does this actually prove anything? How do they know that the way they set the system up isn't the reason why its creating random results and another system could be created that has all those axioms in it and doesn't produce a random result? Put another way -- how do they know amongst all the possible configurations that there isn't one?
I've always looked at math as more of a language than a discipline, so in my own way I guess what I'm saying is how do they know they're asking the question right?
#fuckbeta #iamslashdot #dicemustdie
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."
I'm glad you're so sure of yourself. However, the connection between *counting* (ring of integers) and, say, complex conjugation isn't so obvious. If you'd like to compete with Dirac (for example) and argue that he was dumb for taking so long to recognize antiparticles' existence, or that Green should have "obviously" recognized that there must be such things as evanescent waves because the Helmholtz equation has some complex roots for the wavenumbers, then be my guest.
I don't know what your background is, but such connections between mathematics and the "real world" are NOT always obvious, and it is a continued source of delight and puzzlement when one explores some neglected branch-cut in the maths, and it turns out to have real impact on the physics. Please, explain to all of we poor physicists how bananas can point us to truth.
instruction book that we wrote to describe physics?
There's the thing that you don't understand. We didn't create mathematics to describe physics, yet mathematics always seems to do the job, and ussually much more simply than you would expect.
How many of us sat through algebra in middle school thinking "I'll never use this". Then sat through calculous in high school thinking "Nobody would ever use this". Then took our first calc based physics course in high school and thought, "No way, this is actually how the universe works?".
As far as we can determine, mathematics is the universal language of the universe, it certainly isn't something that we created. The fact that we are near to describing the infinately complex universe with a handful of equations would seem to indicate that mathematics is a part of the very stucture of the universe.
1 Black Hole + 1 Black Hole != 2 Black Holes
Mathematics is an abstract game of counting, built up into great complexity.
Mathematics is a game of abstraction, played out in a wide variety of directions, counting being just one of them. The assumption that mathematics is just counting is rather frustrating. Yes, you can reduce mathematics to arithmetic, but then you can also reduce it to set theory, or to topos theory/category theory, and so on. The ability to express things in a particular way does not that that is what the the things are, especially given the profusion of different mutually interpretable "reductions" available.
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.
Actually you can dream up universes where 1+1=2 doesn't hold. It can fail to hold for a variety of reasons. The various hypothetical universes vary with those reasons from completely uninteresting and trivial, through to, well, in this case, still relatively uninteresting. Of course there are other "fundamental truths" that you can drop (the law of excluded middle, for example, or DeMorgan's laws, which are both conceivably more fundamental than 1+1=2) and end up with remarkably rich and interesting universes. The absolute universality of mathematical truth is on rather shaky ground; certainly the mathematics we use seems pretty solid for our universe, but that doesn't make it universal over all possible universes.
We use mathematics to quantify physics, but there is no "connection" between the two
There is a connection to the extent that ideas developed in the abstract for purely mathematical reasons have often had surprising, unseen, and unlooked for applications to physics. It is the surprising aspect of that that makes philosphers question the apparently unreasonable effectiveness of mathematics.
Craft Beer Programming T-shirts
You can't say this and also have previously said "We use mathematics to quantify physics, but there is no "connection" between the two.
Well, you can, but only one can be true.
It's true that the our understanding of physics is tied to the math, but for the math to accurately imply the existence of new phenomena which haven't previously been conceived of speaks more to the fact that the "real" physics obeys the same rules of math that have been observed.
That seems to indicate a more coherent coupling between what we've learned about math, and what we're in the middle of learning about how things actually work.
Cheers
Lost at C:>. Found at C.
It's precisely the fact that the math isn't independent of the physics that is at issue here That's a very startling proposition
The word "math" refers to a huge collection of symbolic rule sets. These rule sets were not all invented at once by some magical mathematician in the past. They were produced over thousands of years of refinement.
One important point to note here is that many of these refinements were made specifically for the purpose of giving math a higher level of practical value. For example, the number zero, and subsequently the negative numbers, were added by most cultures only after they realized that they could derive a useful model of some aspect of reality by using these numbers.
I don't see why it would be surprising at all that a language which has been refined, over time, to describe reality would wind up describing reality.
I will further suggest that the truths of mathematics that seem intuitively obvious to us seem so only because our brains are structured such that these truths will seem intuitively obvious. What gave our brains this structure? Refinement-after-refinement due to the process of natural selection. So the reality which is being modeled by mathematics happens to be the same reality in which the inventors of mathematics (ie our brains) evolved. Who would have ever guessed that there would be some correspondence here?
I think the surprise only comes about when we forget the true origins of mathematics, and the true origins of the brains that understand mathematics and use it to represent reality.
Not prove in the mathematical sense, but show that the statements are true with arbitrarily high probability. It is akin to determining the area of the circle using Monte Carlo method. The law of large numbers guarantees that you will get the correct result if you invest infinite time.
1 + 1 = 2 will be true in any universe, under any god(s), in any circumstances.
Not true. It is often 0.
After all, I am strangely colored.
> It's entirely possible and reasonable we can determine universal laws without having the faintest idea of *why* they are that way.
> 1 + 1 = 2 will be true in any universe
Really? I find the opposite is true.
You need to know "why the laws hold" in order to know if the laws are applicable at all.
Take one liter of water and add one liter of alcohol and mix together. I guarantee you won't get two liters of the mixture. Ditto with one liter of matter and one liter of antimatter.
You might say, that you have to be referring to the same substance, so I'll counter with one ball of mud plus another ball of mud is just one ball of mud.
You might counter that if both balls of mud have the same mass (i.e. 1 kg), then the total will have 2 kg of weight. Fine. Then I can point you to the Banach Tarski paradox ( http://en.wikipedia.org/wiki/Banach_Tarski_paradox ) which shows that it should be possible to cut a two kilogram ball into finite number of non-overlapping pieces and put together to give two two kilogram balls, so 2=2+2.
You might counter that you can't divide a real world solid the way you can divide a mathematical solid. But in that case, you've shown that the real world is not 100% mathematical in every sense, so all the free variable are interchangeable without consequence. T
his is precisely the point and why "quantum test" is genuinely something new as opposed to "an obvious fact that was known for ages". It provides us more information on math-like the universe is. When math corresponds to reality in a nonobvious way, it is important. For instance, I'd be extremely surprised if the Banach Tarski paradox held in real life, though I'm sure someone who believes in multiverses will try to prove me wrong on that.:-)
So how does 1+1=2 in the real world? As an approximation. Most of the time the approximation is very good. But often times it's not, which is why we regularly add in fudge factors in real life (e.g. You've asked for 10 apples, but since my apples are smaller than the typical apple, I'll though in an extra one, so 1+1+1+1+1+1+1+1+1+1+1=10). People are natural engineers (as opposed to mathematicians), so they don't blink when a fudge factor is added.
That's why it's natural to distrust statistics or metrics. You can't just know the numbers and formulas involved. You need to know the nature of what's being counted.
If you claim you don't distrust statistics, then you would not ask questions if your manager (or teacher) measured your performance based on a set of formulas you trusted but didn't tell you where the numbers that plug into the formula come from.
It's actually not an ad hominem argument. The plea was to "not get too excited" and the reason given was the track record of the source. No claim about the accuracy of the paper was made, either way. Before anybody opens up some 12 year old scotch, that author of the paper must successfully defend it.
Fascism trolls keeping me up every night. When I starts a preachin', he HITS ME WITH HIS REICH!
He didn't assert that it must be incorrect, he said that it may not be correct. It's not an ad hominem to be suspicious of a source.