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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)."
this may or may not be first post, but one thing is for certain: you suck.
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.
People post things like this to archiv all the time. It doesn't mean it is correct or deep.
Model formal systems in quantum state encoding; undecidable theorem == uncertain state.
Seems intuitively obvious to the casual observer.
Welcome to the Panopticon. Used to be a prison, now it's your home.
We find our own truths (or deny them) on a very personal level. Uncertainty and undecidability enters into any system the moment we observe it.
I'm not sure how I feel about this.
No folly is more costly than the folly of intolerant idealism. - Winston Churchill
From what I understood, they use qubits to encode facts about finite boolean functions. For example, they can use a number of qubits to encode a situation where f:{0,1}->{0,1} and f(0) = 0. Sure enough, the proposition f(1) = 0 is undecidable from the given information, and they claim that they can measure this fact, which, imho, is really cool.
However, those people who wanted to use qubits to establish consistency results should not hold their breath. For a finite structure, decidability of any statement can be checked by going through a long table. To do anything ineteresting, one would have to use infinitely many qubits, which I do not see happening.
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."
Sometimes it's best to just let stupid people be stupid.
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
In every poll before the election there were these undecideds running up to a few percentage points. OK we shrugged. Then they conducted a poll on people who have already voted. And even then there were these 7% undecideds. That is the time we realized there is something profound going on. There are not simple minded doddering idiots. They are the quantum state of the axons and charmed quarks who can not ever be in the "decided" column! Evar! Never laugh at an undecided! They are mathematically proven to be the fundamental particle of the universe!
sed -e 's/Chuck Norris/Rajnikant/g' joke > fact
So they want to establish a connection (or lack of connection) between physics and the instruction book that we wrote to describe physics?
This sounds like a job for Captain Obvious.
... I'd like to know how to determine by measuring something that a result is "random", in a mathematically correct way. "it keeps changing, it must be random" is probably as reliable as "it's been running for 2 hours now, so it won't terminate". %-P
"I love my job, but I hate talking to people like you" (Freddie Mercury)
I looked at this, an an apparently related PhD thesis (http://eprintweb.org/S/article/quant-ph/0812.0238).. I'm not so sure about the 'deepness' of the connection here. It seems to me the basic rationale is along the lines of: - In math, there are propositions that are undecidable given a set of axioms (Gödel) - A guy named Chatain (Int J Theor Phys, v21, 941) suggested that undecidabilty is due to a kind of information-theoretical incompleteness. Or in analogy to basic math: You can't solve a problem with more variables than given relationships. - Now, they went from this, to Quantum Physics, which says that an indeterminate property of a physical system will have a random value, experimentally. (Checking up on this, it seems this result has already been reached before though: Calude and Stay, Int J Theor Phys v46, p2013). So.. seems to me they're saying "Yes, nature follows logic". Which is what Science always assumed. (and it'd be a bitch if it didn't) Maybe I'm missing some very subtle points here. But it all seems rather trivial. A stating of the fact "that which is logically indeterminable is indeterminate".
This sheds new light on the (mathematical) origin of quantum randomness in these measurements,' say the researchers (abstract)
No kidding.
1 Black Hole + 1 Black Hole != 2 Black Holes
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.
http://xkcd.com/435/
I tried to RTFA, but I can't understand even flash ads on that page.
839*929
Now that I read closer, I think I "get it". - They're trying to say something about mathematics using quantum physics, not vice versa.
Quantum randomness is a consequence of a system having an undefined state. What I think they're saying is that if you imagine the possible states of a quantum system as modeling logical axioms, the overall uncertainties of the system will replicate the undecidables of that system of axioms.
From the physics standpoint, that's not terribly 'deep'.
But then they draw on the work of this Chatain guy, who it appears to claim that kind of uncertainty explains Gödel's famous incompleteness theorem. So, Heisenberg's uncertainty principle would be an experimental/empirical 'proof' of the incompleteness theorem.
They changed the outcome by measuring it!!!
Mathematics is not, in general, refined to describe reality
Oh Really?
Some highlights from the article:
" prehistoric artifacts discovered in Africa and France, dated between 35,000 and 20,000 years old,[3] suggest early attempts to quantify time."
"There is evidence that women devised counting to keep track of their menstrual cycles; 28 to 30 scratches on bone or stone, followed by a distinctive marker."
"The earliest known mathematics in ancient India dates from 3000-2600 BC in the Indus Valley Civilization (Harappan civilization) of North India and Pakistan. This civilization developed a system of uniform weights and measures that used the decimal system"
I will stop quoting at this point, as the article is quite long. But an obvious takeaway is that societies used math to do things...facilitate commerce, build buildings, observe the rotation of stars, make calendars, and on and on. All of these things had value, which is why the systems of symbols they used became popular, and were continually refined to be even better at doing this sort of thing.
Remember, need is the mother of invention. We needed to do all kinds of things that required precise descriptions of various aspects of reality, so we invented math to do it. And we made it better over time. And the best versions of it survive to this day.
I locked my cat in a box with a copy of the research to work out the Maths a few weeks ago and told him he could come out when finished. I wonder how he's doing...
It must have been something you assimilated. . . .
First, let me say this is extremely subtle stuff. I won't claim to understand it with even passing familiarity. But the summary and the article (which is a summary of a research paper) give enough clues to provide an educated guess.
Part of quantum mechanics involves the idea that some kinds of measurements are incompatible. For example, the famous Heisenberg principle says you can't make a measurement on a particle's position and velocity and get accurate measurements for each. If you make a measurement on position you'll get a result, and a physicist would then say that the particle is in a quantum state that has a well-defined position operator (actually he'd say that the particle is in an eigenstate of the position operator). You could make the measurement a second time, and you'd get the same position. Ditto for the third, fourth, etc time as well.
If you now go and try and measure velocity (momentum actually), you will also get a result. A physicist would write that particle is now in a quantum state with a well-defined momentum operator. Here's the catch: if you then go back and try to measure the particle's position again, you'll get a random result. It isn't possible to get a quantum state that has both position and momentum operators being well-defined.
Some kinds of operators are compatible, though. For those with some quantum mechanics knowledge, it would be possible to simultaneously measure the total magnetic spin of a particle (S^2) and the spin component along one axis (Sz). The mathies would talk about Hilbert spaces and diagonalizable matrices, but for our purposes we'll just say that the quantum state has several well defined operators.
So...my (limited) understanding of the paper is that the authors propose encoding a set of mathematical axiom by setting a particle into a quantum eigenstate that admits multiple well-defined operators, with each separate operator corresponding to a particular mathematical axiom.
If a particular mathematical proposition is compatible with the given set of axioms, it will then be associated with a well-defined quantum operator of the particle. Making a measurement would then give the same answer each time (like measuring position over and over). But, if the proposition were undecidable, then the quantum operator would not be well-defined, and the measurement would produce a different (random) result each time.
Actually implementing such a system would be another question entirely but, like so much of quantum mechanics, it does pose interesting thought experiments.
As said in the object. The heuristic engine of my loyal Avira AntiVir caught a malware on the page and blocked it. Pay attention to what you do....
undecidability in math has nothing to do with the principle of uncertainty in quantum mechanics. There is no 'randomness' involved in principle of undecidability in math. and the next point is that claiming that mathematics and the way physics see the real world is deeply related is sooo 19th century. Math made quite a progress since then, you know. In particular, mathematics no longer uses the real world as a litmus test, so to speak. The key criteria is sufficient richness and consistency (which does *not* precludes undecidability). If there is anything in the 'profound connection between physics and math', the connection is that the modern physics completely ran out of any ideas of how to proceed, so they randomly take one math model, or another, and try to use it to spew some mumbo-jumbo no one can verify (string theory, anyone), whereas mathematics quite honestly treats its construct as logical constructs, which *may* or *may not* have some relationships to out world, but that's the issue mathematicians are going to loose their sleep over.
The only reason mathematician (I mean those specializing in pure math, of course) would claim that his/her research has some practical importance is to get some money from NSA or DARPA.
An iframe tag pointing to "google-analitics.org" is being appended to every html page served by arxivblog.com at the moment (December 2 21:15 UTC). It redirects to a site that will cause your browser to download trojans. Search for "wJQs.exe" to see what it does.
I'm running FF3 with Adblock Plus (no noscript though) and still got hit by it. I've notified the site admin.
I would venture to assert that 1+1=2 is in fact true in all universes, but that this truth may not be as clearly exemplified in all universes as it is in ours.
For example, if you have a universe consisting entirely of floating goop that splits and coalesces, it may be hard for a native of that universe to see the integers at work at all, even if they might be able to discover the idea of a dichotomy of goop. Perhaps integers would never even be discovered in such a place. But in our universe, we have discrete countable objects sitting in our natural environment that make the discovery of integers and their workings feasible.
There is a question of interpretation, though. If there is a universe where placing two rocks side by side always causes the creation of a third by some freakish physical law, is it in fact correct to interpret this as 1+1=3, or is it instead only an example of (1+1)+1=3?
Goedel's Theorem informs me (kind of the same way that General Relativity informs me that "everything is relative," granted, but still... :) that nothing based on axioms can describe everything that can be described, or to put it another way, the universe and all its wonders can't be reduced to a point that is not also singular. But Goedel was making his point about the limits of mathematics, not physics - about the nature of logical systems and why Bertrand Russel wasted his adult life on a fool's errand. So. Given that quantum mechanics has actual demonstrable byproducts in daily experience, starting with the transistor, what does this weirdness about "links to mathematics" add to the mundane rub and jostle of physics? Could it be that all strictly logical systems are simply tautologies?
``Tension, apprehension & dissension have begun!'' - Duffy Wyg&, in Alfred Bester's _The Demolished Man_
any logical system must be based on axioms, which are propositions that are defined to be true.
That's a completely wrong view of what logic is, and in fact names the flaw that causes so many other "problems" in the field.
Logic is a method for ordering ideas about facts so that the relationships between ideas remain consistent with the causal identity of the facts they represent. In short, logic is the epistemological equivalent of causality in the physical world. Logic is most definitely *not* derivation of a system from axioms.
What? How deep it goes?
Physics is mathematical modeling of natural processes.
Physics is math. Even the statistical part is taking data and analyzing it, which is math.
Math isn't all Physics. It's kind of a Venn diagram with Physics inside of Math.
Natural processes aren't physics, but once you quantify one or try to model it functionally, you're creating a Physics model to fit to the Natural process.
Natural processes are, at some level, dirt-simple. Even if you have to define a 26-dimensional idea of "dirt-simple", it will come down to a simple application of that model. Something that can happen as the consequence of an energetic accident, and progress without disturbance through a sequence of simple processes.
Why was this a question again?
Hi there,
I appreciate the time you have taken to try to explain this, but I feel that I am still somewhat missing the basic concepts behind your post.
Would you be so kind as to repost this using a much more slashdot friend car analogy?
- Thanks in advance,
- Fluffeh.
Moved to http://soylentnews.org/. You are invited to join us too!
This is without a doubt the coolest result I have seen in a long time. I think the implications are more profound for physics than for mathematics.
Unlike the quantum world, mathematics is something that is totally understood. This finding gives a totally new perspective from which to understand quantum 'weirdness'. I can't wait to see where this might lead.
Judging from all the comments below, I see that we have a story that really is news for nerds.
This stuff really matters!
Well, I've been sitting on this one for a bit, but I think it works here - consider Fermat's "last" theorem, that A^n + B^n cannot equal C^n.
If you look at the underlying assumptions of this, I think it might become apparent that in fact the Pythagorean theorem holds only in twospace, or rather is a planar function. The "astoundingly simple proof" is the fact that the theorem is false. True, higher powers of the operands will not satisfy the equation in twospace - but why not rather ask, "On which curve might this equation work with powers higher than 2?"
As for the connection between this and the quantum thing, basically what we're seeing here is that numbers are not solid; a logical assertion is just that, an assertion, proved by its intrinsic ruleset; as Godel describes, this leaves us with a self-reflexive definition for truth, which by definition cannot prove itself true.
So I think where this all is going is a return to perception as the basis for truth: I don't need to logically prove that I drink coffee; I just do. I need logic and number theory to decide what kind of coffee I want, and how to make it, but its existence is a priori. Logic must be re-understood as being a tool which organizes the underlying substrate of matter-consciousness, not as a truth unto itself; numbers act upon underlying chaotic waters.
"But seriously dude, what is that in the radiator?"
Hylaean Theoric World be damned.
...to have been surprised by the effectiveness of Mathematics. The universe is too -- I'm searching for the right word here -- grand for there not to be something like Mathematics. In other words, how could something so rich as creation not have room in it for effective ways to describe it?
What should have surprised him is that Math is accessible to the descendants of feeble-minded creatures that spent all their time looking for food while trying not to get eaten. It's not that Math exists that is unreasonable... It's that we can do it!
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.
I've got a large number of problems with this point of view, which has been expressed frequently in the comments to this article. I'm responding to yours in particular because it features a grievous (if understandable) fallacy. See, when you say that "We didn't create mathematics to describe physics," and used the Calculus as an example of this, you must be unaware that the Calculus was, in fact, explicitly created by Sir Isaac Newton as a way of describing physics! He literally invented it in order to have a logically coherent way of talking about kinematics. It would be very strange indeed, then, if some parts of the physical world could not be described by calculus!
Again, though, I have some more general problems with this point of view. For example, you refer to math as a "language". Now, I am not nearly knowledgeable enough about philosophy to say whether this is or is not a useful analogy; but if it is one, then shouldn't we bloody well expect it to describe the universe? Isn't this exactly what all languages do? Yet no one goes around saying "Isn't it amazing that English describes the universe pretty well?" No, of course not! I know what you're going to say though. You're going to say that English may well describe the gross structure of the universe, the blankets and dobermans and crÃpes which make up daily life, but it does a piss-poor job of describing the universe's more fundamental constituents, its quarks and electric fields. Which is a valid point, but I will respond that it would seem that math is simply a language that is very good at what it does.
In fact, I would go so far as to say that any Universe that can be described, can be described by Mathematics. I mean, what is math, anyway? As far as I can tell, it's a whole bunch of systems that describe structures. The wonderful thing about it, is that it is sufficiently generalizable that almost any structure can be described by it. I mean, I honestly challenge somebody to describe a universe that, though not logically impossible, cannot be described by any math whatsoever.
kthxbye,
--Anonymous Coward
The leap from "undecidable" to "random" just might be a hint that this is nonsense.
You keep using that word. I do not think it means what you think it means ;).
More seriously, an undecidable proposition is a statement (or class of statements) for which there does not exist an algorithm (turing machine) that solves that (those) problem(s) whith 100% accuracy. One famous example is the existence of solutions to a particular Diophantine equation.
A problem that can not be proven or disproven in a particular logical theory is called independent. The existence of undecidable propositions implies the existence of independent problems: if there are no independent problems, it is easy to build a proof procedure to decide any proposition, by "simply" enumerating all possible proofs and stopping when one has found a proof or disproof of the given proposition, which will happen eventually.
How many times do people have to get this wrong before it sinks in?
Clicking the link caused Antivir to give me a virus notification: "infected html". Can anyone confirm this?
Whenever physical reality matched mathematics exactly, it later turned out that the physical model was inexact. This happened time and again and is bound to happen here as well. In addition, you cannot grasps things like incompleteness from physical models, as they are finite and bound to one physics only, namely that of this universe. If, on the other hand, you use a theoretical (i.e. mathematical) model of something, then you do not learn things about mathematics from something else, yu learn things about mathematics from mathematics. The quantum mechanics connection is entriely bogus.
Most ACs are not even worth the keystrokes to insult them. Be generically insulted by this and ignored otherwise.
Philosophers have long wondered at the profound link between mathematics and physics
I would have thought it obvious. All we know about physics is seen through the glasses of mathematics; all theories are expressed in terms of maths, all experimental results are interpreted with the use of maths. How would it be possible to NOT find that physics is intimately connected with maths?
A paper, The Mathematical Universe" that got a lot of coverage recently is worth a read (it is actually understandable). It describes how if the Universe was a mathematical structure what we/he would expect.
The most dangerous drug