We will never know the two prime factors of a composite 10^10^500 digit number.
It’s not that it’s unknowable in the abstract philosophical sense, only that it would take more computational power than could be found in the observable cosmos.
(And the joy of it is, if I’m wrong, I only need to raise 10 to that power once more, which I can do a lot more often than anyone else can factor numbers bigger than Scrooge McDuck’s wealth in dollars.)
An interesting comment in the context of this discussion, because applying more computational power than can be found in the observable cosmos may be closer than you think! If we can build a modestly powerful quantum computer, Shor’s algorithm would allow it to factorize your number in a few hours – a task that a conventional supercomputer would still be working on when the universe ended. The intriguing thing about this is that the computing resources to do it are nowhere in evidence in the observable universe. Some like Peter Deutsch believe the computational resources come from the many-worlds associated with an objective-reality interpretation of the quantum wavefunction (the physical reality of Hugh Everett’s many-worlds hypothesis).
Okay, that’s scary… I knew that quantum computing had a lot of promise, but didn’t know it had THAT much power!
(I can still cheat by raising ten to an increasingly grotesque level of exponents, but that’s kind of philosopically dishonest and also hugely inelegant!)
(I do have a crackpot theory that, in Heaven, mathematical discoveries will be a kind of “medium of exchange,” or objects of value. It’s the only thing that no realm can ever run out of.)
I’m sorry, but if you think any scientific theory is graven in stone, you’re the one who needs to go back to class. Please don’t tell me what I know or don’t know about science, or try to ‘educate’ me about it. I’ve taken more hard science classes than you can shake a stick at.
This is, of course, trivial.
The two prime factors of 10^10^10^500 are 2 and 5.
That’s basically what I meant since we can only deal with what we can perceive. I just meant we can’t directly know what is ‘out there.’
You see, the use of the word ‘mechanical’ and ‘material’ indicates to me a reluctance to accept a non-deterministic, non-local view of reality, something that is not necessary in interpreting QM. All you are doing is trying to extend materialism to non-materialistic reality.
It doesn’t matter what kind of experiment or particular interpretation is being considered, the knowledge gained has no meaning unless experienced by a human observer. You seem to be suggesting that knowledge somehow has an independent existence and can even generate itself! This is illogical because knowledge has to be processed by a human brain. There is ample evidence of this in the experiments that have been performed by human researchers and how could a scientific experiment possibly happen without somebody conducting it? You’re wrong when you say the observer isn’t an implicit part of an experiment because at some point the results of an experiment have to be ‘experienced’, something the ‘hardware’ of an experiment is unable to do. If somebody writes a book, stranded on a desert island, the information in the book will never have any meaning until somebody who discovers it some time later, perhaps in the sea, reads it. So information in itself is meaningless unless decoded by an agency that is conscious.
With macro objects, one could bring in the Many World Interpretation as a solution to what happens when observed so you can take your pick!
It’s not a question of being right or wrong but what seems to work. Bohmian mechanics has its strengths but also its weaknesses and today is considered a minor approach but may come to prominence again, who knows? I think what this shows is that trying to make classical notions of reality ‘fit’ the data is highly problematical and naive. You have not resolved anything about entanglement other than saying ‘it’s nothing that can’t be explained’, which is just untrue. In fact, you point out that it is the measurement process that acts to cause such phenomena so you are really admitting the role of an observer is all important.
Well, here again, The Many Worlds can account for this by saying we determine the kind of reality we exist in by making a choice.
*EDIT:
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Many interpretations of QM are materialistic in the strictest sense. Bohmian mechanics, Nelsonian statistics, Girardi-Rimini-Weber/Penrose-Diosi collapse theories all essentially don’t need any sort of departure from classical mechanistic or material notions. Neither do most many worlds-type interpretations, and for most others, the issue is debatable at best.
You see, the use of the word ‘mechanical’ and ‘material’ indicates to me a reluctance to accept a non-deterministic, non-local view of reality, something that works very well in interpreting QM. All you are doing is trying to extend materialism to non-materialistic reality.
That’s the opposite of what I was saying. The ‘no miracles’-argument argues for scientific realism, i.e. the stance that our best scientific theories do tell us what’s ‘out there’, while your stance rather sounds like scientific anti-realism. Which is problematic, since as I pointed out, if we can’t know what’s out there, then in particular, we can’t know that the observer plays any role, since that’s a claim about what’s out there.
And again, very few interpretations are ‘non-materialistic’ in your sense. Many worlds, for instance, is perfectly local and deterministic. All that I’m saying is that you claim that a certain view of reality is implied by quantum mechanics, while there are counterexamples, i.e. ways of making sense of QM without introducing notions of consciousness, knowledge, and the like.
No. I’m saying that knowledge doesn’t play a role in how quantum phenomena come about, or at the very least, that quantum mechanics in its present form does not imply that it does.
Again, either you’re making an argument that doesn’t rely on quantum mechanics at all, and verges on triviality—that all experiments include an experimentalist—, or what you’re saying is that the experiment somehow has a causal role to play in bringing about the outcome of a given measurement—which is something quantum mechanics simply doesn’t imply. (It would help, however, if you could cite concrete experiments you refer to.)
Indeed, something like a ‘Consciousness Causes Collapse’-interpretation, strictly speaking, would require modifying quantum theory in order to give consciousness (or knowledge, or what have you) a privileged role. Think about a Wigner’s friend-like setup: we have the usual box with the cat, but, surrounding that, a sealed laboratory perfectly isolated from outside influence, containing also Experimenter A. Outside of the laboratory, we have Experimenter B. At some point, after some amount of time has elapsed, A checks whether the cat’s still alive, but does not pass on this information to B.
Now, quantum mechanics makes the following prediction: the entirety of the sealed lab will be in a superposition of states, namely ‘dead cat, sad A’ and ‘live cat, happy A’. This is something that, in principle, can be checked: B could perform an experiment on the entirety of the sealed lab to confirm that it was indeed in a superposed state (although this would be impractical). Quantum mechanics gives a probability distribution for the experiment’s outcome which differs from every classical distribution.
However, if consciousness/knowledge indeed is responsible for realizing a definite possibility, for collapsing the wave function, then the prediction will differ: upon opening the box, A’s observation collapses the cat into a definite state, and A will likewise be in a definite state. So, after A opens the box, the state of the lab will be either ‘dead cat, sad A’ or ‘live cat, happy A’, and not a superposition of both.
Consequently, the predictions of a theory in which consciousness/knowledge is responsible for collapse differ (in principle) observably from those of standard quantum mechanics. So, not only does standard quantum mechanics not give us any indication towards a special rule for consciousness, if we want it to play such a role, we have to modify QM! Since there is, however, no good evidence that such a modification should be needed, one generally prefers sticking to vanilla QM.
In the technical sense, information is exactly that: meaningless. It’s simply a pattern of differences; the number of differences—e.g. different states—some substrate admits, tells us the amount of bits of information we can store. A system that can be in one of two states carries a bit of information, in one of four states two bits, in one of eight states three, and so on. This is the sense in which information is used in technical literature—which has no reference to a conscious observer.
Observation plays no role at all in many worlds. Branching occurs whenever there are two or more possibilities: so, an unstable atom will evolve into a superposition of ‘decayed’ and ‘undecayed’ states, which in many worlds correspond to different branches, all on its own. That’s the main attraction of the MWI: that it allows you to get rid of observation altogether.
Among people thinking about the foundations of quantum mechanics (which excludes most working physicists), Bohmian mechanics isn’t really that minor; however, the kind of interpretation you’re advocating for is basically not represented at all.
No. On Bohmian mechanics, the measurement process is just a completely ordinary physical interaction, having no special status at all. You have to take that interaction into consideration, but it’s exactly of the same kind as particles interacting with one another in the absence of any observer whatsoever. In BM, measurement is completely indistinguishable from all other kinds of interactions. Moreover, BM is one of the few interpretations that actually do explain how the effects of entanglement come about: via the so-called quantum potential, which exerts a non-local force on distant particles.
This is not something the MWI says at all. Neither does our choice determine our reality, nor does branching occur when we make a choice. Indeed, one could argue that on the MWI, we never really make choices at all, since all outcomes will occur in their respective branch worlds.
Again, branches occur whenever there is more than one possible future state for an object—any given object whatever. The observer, the making of choices, and so on, simply doesn’t figure into it; even measurement isn’t anything special. That’s one of the strengths of that interpretation.
Well, while quantum computers do yield an exponential speedup, numbers of this magnitude are still a tall order—you need about 2log[sub]2/sub qubits in order to factor a number N, so that’d be on the order of 10[sup]500[/sup] qubits. Compare that to there being about 10[sup]80[/sup] particles (here meaning protons, neutrons, and electrons) in the universe, and I’m not holding my breath.
Note that that’s still an incredible gain—for instance, a 2048-bit RSA key, corresponding to a number of order ~10[sup]616[/sup], needs ‘only’ 4096 qubits. That’s still far beyond current capabilities, but give it twenty years, and I wouldn’t be so sure anymore. (Of course, I’m also talking about ideal qubits here—for any real implementation, you’d have to do error correction, which increases the number of qubits by at least a factor of ten, and likely more.)
This is a pretty much nonsensical statement. It may be true that, if we lived in a classical universe, such processing power wouldn’t exist (although even that, we don’t know for certain—there could be classical algorithms equaling the performance of quantum algorithms that simply haven’t been found yet); but we don’t. So, the computing power that exists in the universe is just that computation we can perform with the resources we have—thus, if we can factor such large numbers, then our universe trivially has that sort of computing power.
This is fundamentally wrong because all we can ever accomplish is to construct subjective models of the data we gain from scientific experiments. All of the interpretations you have so far referred to are mere self-referential ‘schemas’ of data. But the data is not the actual phenomena because it has undergone a transformation from origin to measurement and this includes the interpretation made by human ‘interference.’ You have still not made the leap of insight that there is no real separation between what we observe and us. What we observe comes about as a result of a reorganization of our mental picture of whatever experiences we have had; it does not actually ‘change’ in any way whatever it is that is causing such experiences, only our ‘version’ of it. So, in a way, the study of science is really the study of ourselves.
Yet again, you are asserting things like the many worlds, etc., exist as a separate and independent phenomenon. None of these ‘interpretations’ are real in the sense of being able to live outside our ideas of them. Having said that, once we have ‘adopted’ a particular interpretation and pursue it, we empower it to transform into something we are able to function within, and indeed brings about a particular kind of reality. However, such a reality is not an ‘out there’ reality but an ‘in here’ reality. But that will depend on whether we can attribute the many solutions to the many problems of experiments. Entanglement is one example of something that has been repeatable due to many scientific tests and that has become part of our reality. It never existed before that because no measurement were ever made of it. This characterizes the process of ‘reality building’ that goes on over time and will continue onwards to greater heights.
This seems a very odd statement. Why? Becuase without knowledge of events there is no quantum phenomena.
Of course all experiments include an experimentalist otherwise how are they an experiment? And how can such experiment be interpreted unless an experimenter is doing the interpretation? You can’t just throw a random collection of scientific instruments into a room and expect all the parts to assemble themselves into a pre-meditated arrangement and expect something to happen. It takes human beings to put everything together in order to test some well thought-out hypothesis.
You see, no offence, but you keep assuming there is a pre-existing ‘reality’, out there waiting for us to stumble across it. I don’t think it works this way. *We *are part of the process, so are implicit in ‘defining it.’
The many-worlds interpretation can account for this where whatever state we find the cat in (dead or alive), a branching of the universe takes places creating a ‘new reality.’ So, the observer is crucial here.
All these are just proposals to try to account for the behaviour we see of quantum objects and no more or less real than each other. You do tend to talk about some of these as if they were ‘real’ They might become real one day if they are able to satisfy the many question that remain about quantum behaviour. But, so far at least, Bohmian mechanics has no real validity and remains as an interesting footnote in the history of physics. The reason is that it does not tell us or predict anything more than alternative interpretations do.
The problem is, though, that we then have no explanation as to why our models correctly predict experiment. This is easily explained if these models at least track what goes on in the world; but if they fail to do so, then why can they predict, or explain, anything? This would seem to be miraculous otherwise: we’d need to believe that our models are fundamentally false, and yet, for some unknown reason, their predictions happen to come true.
And again, if you actually held to the antirealist position you claim, then all the metaphysical claims you’re making would be for naught—since they’re themselves just more models that don’t track anything in the world. IOW, if our theories are truthful, then we don’t need to appeal to the observer; if they’re not, then we can’t make any claims about the role of the observer. In both cases, your position is groundless.
That’s a bold claim; unfortunately, you haven’t produced any argument in its favor. Neither quantum mechanics nor the philosophy of science supports it.
Again, this is just a self-contradictory claim: either, something like this happens; then it’s an objective fact about reality, and thus, reality is not dependent on our subjective perceptions. Or, there are no objective facts about reality, at least in so far as we could grasp them; then, you can’t claim that something like the above happens.
What makes you say that? If this were true, quantum mechanics would simply be wrong. Entanglement, as a phenomenon, is ubiquitous, and quite independent from our understanding or knowledge thereof. For instance, even in empty space, across any arbitrary dividing line, what is to one side of that line will be close to maximally entangled with what is to the other side—whether there’s an observer or not.
Entanglement is a physical phenomenon; it’s completely independent of our description thereof.
Well, you can of course hold such a solipsistic/idealistic point of view. However, quantum mechanics does not point to this view, anymore than classical Newtonian mechanics does.
But again, there’s nothing in QM that depends on interpretation at all. And all you say here can be equally well said about Newtonian mechanics. So appealing to quantum mechanics here is at best extraneous.
Because that’s the simplest explanation for the way we seem to find a preexisting reality out there. And it’s certainly what we should believe if we trust our theories. And if there’s no objective reality out there, then where do we come from? Is then my existence also not an objective fact? If so, then there does not seem to be any grounds for me having any experiences whatever. But if I exist (and if we remember our Descartes, the fact that I’m asking these questions seems strong grounds to assert this), then there’s at least one thing that objectively exists.
Again, this is simply not true. In many worlds, universes branch, whether there are any observers or not. It’s the fact that universes branch that explain our observations within them; so, because the universe branched into two, containing a dead and a living cat respectively, I observe a living cat, if I’m in that branch of the wave function. That’s the exact opposite of the observer being crucial—and indeed, getting rid of the mysterious ‘observation’ is exactly what prompted Everett to come up with this interpretation.
Yes, thank you, quite so. I got a little carried away recalling David Deutsch making the claim years ago that a quantum computer could summon 10[sup]500[/sup] times the computing resources of the observable universe, not that it could prime factorize a number like that suggested above. (Not to mention, with Peter Shor and his quantum algorithm on the brain, I referred to Deutsch as “Peter” instead of “David”.)
Here’s a Nature article on some of the presumed requirements – RSA-768 could be factorized with 1154 qubits, and a 20,000-bit number with 30,002 qubits.
Deutsch so firmly believes in the objective reality of many-worlds that he originally proposed quantum computers years ago, not as a practical computational technology, but as putative evidence of this underlying hypothesis. It’s admittedly a fringe view, but no more so than many-worlds itself which has become regarded more seriously in recent years than when Hugh Everett first proposed it. I wouldn’t call it “nonsensical”, I’d just say that it’s more plausible that it’s not necessary to appeal to many-worlds but rather just properties of QM. But Deutsch does keep hammering away at this point, “where are these computing resources? Why can’t we see them?” If and when we can actually do this these questions may become a lot more salient.
See, my point is: we can see these resources just as well as we can see the resources of classical computing. They’re qubits and quantum gates, or their physical carriers. The argument of Deutsch requires a commitment to the belief that there can’t be computational resources outstripping those of classical mechanics, so that if we find something that outperforms classical computation, those resources must come from ‘somewhere else’. But ultimately, it’s just that we can do more with qubits than with classical bits; there’s no mystery here, that’s just an empirical finding about the nature of the world.
Deutsch’s argument has another obvious counter: if we were ‘borrowing’ computational power from other branches of the wave function, then it wouldn’t be there in those other branches anymore; but these branches can carry out computations of equal sophistication, so that the total amount of computation across all branches would still outstrip the naively, classically possible computation across those branches.
Gosh, to think that you appeared on the Dope less than a month ago without a firm grasp on the notion of “integers” and here you are parsing the qualities of quantum mechanics. Our little boy has grown up so quickly!
Grin! You didn’t address what I actually said – that was the number of digits in the insanely large number. But I do like the answer!
Is this certain? Maybe those other branches can’t carry out the same computations. Maybe “it takes a village,” i.e., it might be valid that the computation is executed by a whole bunch of universes, which are somehow (?) committed to the task and can’t perform the same task “for themselves.”
There isn’t any evidence for this…but it seems to be a possibility. (Like Many Worlds itself, or string theory, or runaway inflation, or other really cool ideas we can’t examine critically at this point in time.)
(If, in our own near future, there is a nasty nuclear war with Korea, and we never actually build a 200 qubit Q.C., would this be “evidence” that our universe is one of those that can’t “do it for ourselves,” but has ceded its computing power to some other universe?)
(Echoing Hans Moravec’s whimsical idea on how to solve a math problem: simply guess at the answer, and if it’s wrong, destroy the universe. In some universe, the answer will be right, and our “point of view” only exists there.)
If the same laws of quantum mechanics are valid in those universes, then yes, they will be able to also carry out the computation. After all, once you set up the computer, all that happens is the usual quantum mechanical evolution; so if you somehow can’t perform the computation because another universe is hogging your computing resources, then it follows that you must observer a deviation from the usual quantum mechanical laws—which is obviously not something quantum mechanics can predict, so in particular, it’s not something that follows from the many worlds interpretation.
It certainly doesn’t follow…but is it absolutely contradicted? What would a world look like if some portion (not all!) of its quantum computing capability were co-opted by some other world? We’ve got a hell of a lot of “world” here in this one cosmos; maybe an entire supercluster of galaxies is busy helping some other-worldly NSA break codes!
(Obviously, I’m only saying this in a spirit of fun; it’s a meaningless and moot question. I just think there needs to be some room for the possibility, and that it can’t be dismissed easily.)
If QM is valid, then yes; obviously, we can’t say that for certain. So I guess it’s a logical possibility, in the same sense that pink unicorns living on the dark side of the moon are—but besides random flights of fancy, we have no reason to believe in it.
Alternate approach to brute force is a lucky random guess.