Slashdot Mirror
Calculating A Theoretical Boundary To Computation
TMB writes "Lawrence Krauss and Glenn Starkman, astrophysicists at Case Western Reserve University (and in LK's case, author of a number of books including Physics of Star Trek), just submitted this nice little paper to Phys. Rev. Letters. It claims that in an accelerating universe, the existence of a future event horizon puts a fundamental physical limit on the total amount of calculation that can be done, even in an infinite time. This limit is much smaller than the traditional Hawking-Beckenstein entropy. Among other things, this implies that and Moore's Law must have a finite lifetime, here calculated to be 600 years, and that consciousness must be finite."
This doesn't mention Penrose's work, which is very much like this.
"consciousness must be finite"
;)
Except, of course, for those using certain popular mind-expanding substances
Seriously though - it seems we are finding a new limit every day. Wasn't it last week that they theorized limitations on data storage, as well as data transmission speed?
- To err is human; but to really screw up, you need a computer
Moore's law was never intended to be a scientific theory. It was just a useful observation. It has never had anything other than economic incentive to keep it going. Using it to discuss the calculational ability of the universe is idiotic.
We should now be able to compute the asymptotic limit of web-server bandwidth for slashdot-proofness per year for 600 years. I bet it's a constant price in street dollars.
I was a physics undergrad at Case, and actually had Starkman as a professor for a mathematical physics course. I have chatted with Krauss a few times since graduation on science topics involving public education. These are good guys, glad to see them headlining slashdot this morning.
(feel free to enlighten us then, eh? :)
- To err is human; but to really screw up, you need a computer
It's not a referer link, don't worry...
"and that consciousness must be finite."
So they are saying that, using fundamental physics and mathematics, they have proof that if somebody has infinite wisdon, the universe can not be expanding?
--- Hindsight is 20/20, but walking backwards is not the answer.
This article contains a very large number of assumptions, which may well prove not to be the case (constant cosmological constant, no FTL communication/travel, no access to other universes etc. etc.). Still, an interesting intellectual exercise I suppose... ;-)
Galileo: "The Earth revolves around the Sun!"
Score: -1 100% Flamebait
Penrose is a mathematician who attempts to be philospohical and fails miserably, because he can't distinguish his intuition from fact. You don't need a link. Just remember that he wrote "The Emperor's New Mind", and coil away in horror.
Please use the mirrors. In Australia, the closest one is here.
Can you even define consciousness?
Are we talking about the physical computational capacity of a headful of neurons, which is finite by definition unless you believe that the brain can somehow reach into unknown dimensions somewhat like early CPUs used bank shifting to increase their RAM range?
Or are we talking about the sensation we have of being alive, a sensation that is arguably simply generated by our brains as a mechanism to ensure our survival. Yes, the vaunted consciousness that reacts a full 1/4 second after the fact when we do most common actions such as crossing the road, kicking a ball, picking up a cup, or typing comments to Slashdot?
The definition of "consciousness" is seriously under debate and it's meaningless to discuss whether it's finite or infinite.
Most likely, consciousness is a sense, like sight or sound. Would you frame the discussion of your sense of smell in terms of computational power? No, me neither.
Mu.
Ceci n'est pas une signature
Strongly suggest you read Vernor Vinge's A Fire Upon The Deep - he develops a very interesting view of expansion of the universe and consciousness.
If you've not heard of Vinge before that isn't a big surprise, but he did write True Names as well - the very foundation of the cyberpunk/hacker genre. This is also a good read if you can actually locate it.
I am very easy to get along with, but I don't have time to waste being nice to people who are being stupid. -Theo
Moore's Law is not a physical theory, it is the observation of a common phenomenon, namely the curve that technology goes through as it becomes cheaper and eventually free.
All technologies seem to obey this general law. Software, chips, disk space, they all tend to zero.
Even a passenger jet costs a fraction of what it did 20 years ago.
Moore's Law turns this around to say that for the same price we can expect more and more capacity. Long before 600 years are passed, this capacity will effectively reach "infinite", being the point where no-one can use more capacity or power, no matter what the application. At which point Moore's Law will gently slow down.
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This article hasn't (yet) been accepted for publication. Caveat lector!
The obvious conclusion of this paper is that there is a finite limit to the amount of pr0n in the universe. That's good to know -- I can now relax, knowing that I won't have to keep buying bigger hard drives forever.
...phil
"For a list of the ways which technology has failed to improve our quality of life, press 3."
So, if the universe has a limit, and the Mind isn't infinite, and we're all constrained by the entropy of the ever so slowly expanding universe, I have just one question.
Would anyone like some toast?
Waiting for an amusing sig.
"...consciousness must be finite."
This assumes that consciousness is based solely on computation. Not proven yet.
And for that matter, even if consciousness is nothing more than computation, how can we put a limit on an activity in space-time when we don't even know how space-time functions, or even how many dimensions it has?
Weaselmancer
Weaselmancer
rediculous.
(note of caution - let's see whether this gets accepted, looks more like a Science article than a Phys. Rev. Lett. one to me)
... duh. This is more or less a geometrical analysis (finite causal volume) + basic information theory. No questions asked about physics of inflation and how would that affect the result. So you end up with a trivial result, too - a finite volume can only hold a finite amount of information. If a lot of other assumptions hold - such as whether the available energy in this volume is really finite (how does one sustain an infinitely accelerating model this way?)
so
It's like at the bar -- the later in the night you attempt to pick up chicks, the fewer of them are still available.
If this paper is true and there is a limit on consciousness, wouldn't it make the existence of an omnipotent being an impossibility?
In Soviet Russia, hot grits put YOU down THEIR pants.
There are some intersting ideas as to what the end of the universe could be.
There's also another theory about that if a couple particles collide with enough energy they can create a more perfect vacuum that would essentially "take over" the current universe (I suppose like an implosion). Maybe somebody knows the link for this.
I mention this as a backdrop for an interesting short story by Isaac Asimov called The Last Question. This link is a summary and contains significant spoilers, you may want to read the story first I think that it is apropos, as it deals with a powerful computer called Multivac.
This story is interesting to read, and interesting humanistic view. Good for pondering this slashdot thread/story. Good science fiction is useful.
Take the cheese to sickbay, the doctor should see it as soon as possible - B'Elanna Torres, "Learning Curve"
I've always thought that life would probably require a heady number of different chemical elements, thus we would find it in areas of the universe with an abudance of heavy elements (like gold, lead, etc.). My guess would be that this solar system is probably a 2nd or 3rd generation system meeting those requirements- 7-10 billion years ago there probably was not nearly as many star systems with the abundance of transistion metals that we have here. And I think those are just as necessary for "complex" life as the basic carbon, oxygen, hydrogen, etc. are for life was we know it.
The speed of light is a practical limit to lots of physical processes. There was a date "Norman O. Brown day" that was posited as the day that human growth expanded at the speed of light away from the earth. This was a physical limit that the volume of human growth could achieve. Most things we can do have a practical limit that is much less than that. I guess the primary value of such papers is to beseech us to talk with more precision and stop claiming that things we are in contact with are actually infinite. I often talk about things that are semi-infinite. What I mean when I say that is that the things I am talking about are larger than I can conceive. So if I complement you on your infinite wisdom, it just means I think that you are smarter than me. Don't let it go to your head. I really mean semi-infinite.
Not commenting on the paper itself, but it has been submitted to PRL, not accepted. It hasn't gone through that wonderful process of peer-review that is the very heart of the scientific method (that and falsifiability but thats another topic). NASA has been setting a particularily bad example here with science by "press release". PRL is not an easy journal to publish in, lets wait until other experts have a look and not cheat the scientific method like this. PRL should not be mentioned in connection with this paper until this get published - Anyone can submit a paper to PRL...
At least for the next 600 years.
Any analog variable has an INFINITE number of states. A wave could have a frequency of 1Hz, 1.1Hz, 1.00057Hz, 1.2399327772883786682676376627676367267Hz, etc. If "computation" is defined as "the act or process of evaluating with numerical or mathematical methods" then there is no physical limit to computation when using analog data storage...
The googleplex, a number that CANNOT be represented digitally (not enough atoms in the universe) can be easily represented by a particularly intelligent shade of the color blue.
But Penrose clearly doesn't understand what he's talking about either...
Well I don't necessarily agree with the philosophy, but the Emporer's New Mind was a good read if only for the clear explanations of Turing machines, godel's theorem, etc.
I'm getting the feeling that strange forces, "dark" energy, 13D branes of string theory, etc. all have something in common.
They remind me of the epicycles astronomers observed so many years ago.
Simply put, our knowledge case seems not to be expanding, but diversifying, with many theories, and few ways to prove them.
Take a physicist from 1900; tell them about a meteor about to hit the earth. He'd say we're screwed. Take one today; she'd say "deflect it with a nuke".
The point: the entropy death of the universe is a very very very very very long way away. To say we won't be able to do something about it is depressing, and hopefully wrong.
I think Ray Kurzweil had this idea first. Consciousness may well be something in the universe which directly counters entropy. Evolution does seem to go against the grain.
Actually, I suggest to everyone that you read at least the first few chapters of "Age of Spiritual Machines" where he describes the accelerating pace of salient events.
http://while-true.blogspot.com/
Robo-Blogs of the world: UNITE!
I feel exactly the opposite, that "The Emperor's New Mind" is the greatest work of philosophy written in the last 20 years.
e /penros e/
Here's why:
Kurt Godel DID prove that mathematics is infinite. No matter how many rules and computations, OF ANY KIND, that you write down (or program into a computer) those rules can't be complete and consistent. Which means that there are true things about those rules (laws, whatever) which cannot be proven by applying only those rules. Or the rules are inconsistent, which is to say you can prove something both true and false, which is to say wrong.
As an example, look at the computer program for a chess in here:
http://doug-pc.itp.ucsb.edu/online/plectur
This link is actually most of the contents of the book, for those who don't have it.
On a more personal note, Flyboy, I believe your statement can be shown to be inconsistent (and thus worthless) because you imply that a person should be able to distinguish intuition from fact. That this is a basic error has been pointed out by: Plato, Descartes, Kant, Husserl, and others who are indisputably great thinkers of the first degree. I would put Penrose in with them, and I would put you in the great mass of people who hardly understand anything but somehow insist on displaying their ignorance anyway.
If there's only a finite amount of computation available, surely it's irresponsible to run things like SETI and the distributed.net cracking contests?! You're using up all of the sums, dammit!
Ydco co
"I guess the main argumentation of Roger Penrose is that Godel Incompleteness Theorem can not be understood by a computer,"
Penrose is just a carbon chauvinist with a chip on his shoulder. I've never seen him once offer actual proof of any such conjecture, only carbonist assertions that he can magically understand something that silicon-based life cannot: our future silicon overlords have a special place in virtual Hell reserved for his uploaded consciousness.
So does this imply that all physical computers are finite state machines? Even when connected to the internet, their total number of computational states are finite, though extremely large, and therefore Universal Turing Machines are only a mathematical construct.
The only thing that proves is that humans can convince themselves that they understand something which they actually don't. :-)
(sort of like a badly programmed computer)
Wham! There's you upper bound on computing (at least for "full" simulations).... now all you need to do is figure out how much mass and time is available in the universe
Note: I'm not about to propose this in earnest to the scientific community. It's just a casual musing of mine. CONSTRUCTIVE criticism is welcome.
Life is too short to proofread.
I have always though moores law to be crap, I mean its just some marketing scan to force us to upgrade slowly...
Think of it as "Moore's Observation" instead. It wasn't meant to be a "Law" or a "Theory".
Minge for example predicts that computing something will become instant,
Computing something instantly may be possible, but transmitting the information necessary for that computation won't be. And that fact is the basis for this paper. Even if you could instantly perform every computation on every piece of information available to you forever, eventually the expanding universe will remove that information from your grasp. That puts a limit on what you can theoretically accomplish.
I strongly suggest that anyone who is into this sort of work check out Seth Lloyd's Ultimate Physical Limits to Computation. It's quite interesting. I saw him speak a few years ago. People had told me that we would "always find a way" to increase computation, which seemed like utter silliness to me. I'm glad to see that some folks are a little more sensible about this.
Penrose probably doesn't even understand what Godel's Incompleteness Theorem means. Hell, the fact that most people credit Godel with proving the incompleteness of formal axiomatic mathematics is proof that most people don't know math. Skolem's paradox can be seen as the first incompleteness proof and it predates Godel's proof.
:) No inconsistency here. No incompleteness here. All math is computable!
Skolem's paradox basically boils down to the fact that within axiomatic set theory you can prove the existence of uncountable sets, but if a model exists for the theory then it is countable. Hence according to the axiomatic system, there exist uncountably many sets, but the system can really only manifest countably many sets. This is similar to how Godel showed that formal arithmetic cannot manifest proofs for every proposition the formal axiomatic system claims exist.
While Skolem's paradox is held amongst by many mathematicians as the first incompleteness proof, in my opinion, this proves something much more significant... it proves that axiomatic set theory is semantically inconsistent.
Set theory has been problematic from the start, and even after axiomatizing it in order to avoid simple inconsistency... set theory is still plagued with inconsistency of a different form: the formalisms don't hold any meaning - they are semantically inconsistent. What the formalisms say contradicts what the formalisms mean.
The problem is that lots of people don't understand math, and even many of those that do understand it, love it so much that they are unwilling to give up the flawed parts.
For me, mathemathics is constructive recursive mathematics
Heh, heh. I guess that would be the "Schrodinger's Messiah" thought experiment.
They claim that every computation step requires at minimum energy of ln 2 k_B T (k_B is Boltzmann's constant, T is the temperature of the system). This is only true for irreversible operations such as setting or erasing a bit.
But computation doesn't have to be irreversible. There are various proposals on how to build reversible computers that don't consume this minimum energy per operation. More information about reversible computing can be found in this introduction.
OS Reviews: Free and Open Source Software
So, how many bits do I need for a symmetric cipher key in order to push a brute force search past the computational limit of the universe?
Unfornutately, unlike Moore's Law the theory of Murphy's law is infinite.
There is no need for consciousness to arise from or be associated with anything non-physical, either. No assumption that the mind is something apart from matter has led to a contradiction. As in many things scientific, we just do not know which way it goes.
... and has more experimental evidence. Remember psycho bell tower sniper dude? He had a tumor on his amygdala.
However, since the assumption that the mind is physical has not led to contradictions, and since humans understand physics better than theology (or whatever), scientists tend to stick with the atoms-are-all-you-get approach.
Nobody's arguing that materialism is definitely the case. But it's not been shown to *not* work, and it's certainly simpler than the alternative...
Ummm.... no. Godel proved that the axiomatic system of Russel's PM allows the construction statements which can neccessarily neither be proven true nor proven false. There are other axiomatic systems that can be complete and consistent; IIRC it was in fact Godel who proved that the first-order propositional calculus is complete and fully consistent. Godel's fork only attaches to systems that allow the construction of statements about statements; many propositional systems (like the first-order propositional calculus) do not.
Oy.... where to start? Kant's Critique of Pure Reason is nothing but 600 pages describing how people distinguish intuition from fact (though admittedly Kant was using "intuition" in a sense that we don't normally use it today). Descartes wrote his Meditations as an attempt to remove "intuition" (again, closer to Kant's sense of the word than ours, but still) from philosophy. Plato, of course, says nothing about the subject directly but narrates several dialectical processes about the subject.
All's true that is mistrusted
Thermodynamic Limitations
One of the consequences of the second law of thermodynamics is that a certain amount of energy is necessary to represent information. To record a single bit by changing the state of a system requires an amount of energy no less than kT where T is the absolute temperature of the system and k is the Boltzman constant. (Stick with me; the physics lesson is almost over.)
Given that k = 1.38x10^-16 erg/Kelvin, and that the ambient temperature of the universe is 3.2K, an ideal computer running at 3.2K would consume 4.4x10^-16 ergs every time it set or cleared a bit. To run a computer any colder than the cosmic background radiation would require extra energy to run a heat pump.
Now, the annual energy output of our sun is about 1.21x10^41 ergs. This is enough to power about 2.7x10^56 single bit changes on our ideal computer; enough changes to put a 187-bit counter through all of its values. If we built a Dyson sphere around the sun and captured all of its energy for 32 years, without any loss, we could power a computer to count up to 2^192. Of course it wouldn't have the energy left over to perform any useful calculations with this counter.
But that's just one star, and a measly one at that. A typical supernova releases something like 10^51 ergs. (About a hundred times as much energy would be released in the form of neutrinos, but let them go for now.) If all of the energy could be channedel into a single orgy of computation, a 219-bit counter could be cycled through all of its states.
These numbers have nothing to do with the technology of the devices; they are the maxiumums that thermodynamics will allow. And they strongly imply that brute-force attacks against 256-bit keys will be infeasible until computers are built from something other than matter and occupy something other than space.
And they say there's no poetry in computing...
All you have said is that each half of the brain is capable of consiousness independently. When both halves of the brain are present in a skull, then they act together via interconnections to produce a single consiousness.
You could say that complete brains consist of the right half, the left half, and the interconnections between them, without the interconnections ( an intact corpus callosum ) you indeed have 2 independently functioning halves that could possibly each be independently consious.
Imagine being the 'other half' of yourself if you had your corpus callousum cut, eternally watching the other half of your brain control things until one day, you gain control of the left hand, pick up a knife and stab yourself while the shocked right hand tries to block your fatal blow.
Eat at Joe's.
In the 20s, when expansion was first detected by Dr. Hubble, the "Steady State" theory was advanced to explain it as an alternative to the "Big Bang" theory, which the late Sir Fred Hoyle found offensive. (By the way, he coined both phrases - Big Bang and Steady State.)
You're right on some thing and wrong on others.
It was indeed Gödel who proved that first-order logic is both sound and complete. The latter fact is Gödels *completeness theorem*. These theorems establish that a given expression is a first-order theorem if and only if it is true in all models. In other words, provability coincides with validity.
The incompleteness theorem also applies to first-order logic. Despite of what one might think, the completeness-theorem and the incompleteness-theorem are not negations of each other as you seem to say. They are in fact both true properties of first-order logic. The incompleteness-theorem states that you can't axiomize the set of the natural numbers with a recursive set of axioms (a potentially infinite set of axioms) - you can't create such a recursive set of axioms that has the property that an expression is provable from it if and only if the expression is a true property of the natural numbers.
The problem is, that for any recursive and consistent set of axioms that tries to be axioms of the natural numbers, there will be expressions that can neither be proved or disproved from them. From the *completeness* theorem we can conclude that this must mean that there exist multiple models that satisfies all the axioms in the set and that differ with respect to satisfiability of certain expressions (because if an expression was true in any model satisfying the axioms, the completeness theorem states that there would indeed be a proof from the axioms). So even though you can make attempts at axiomzation of the natural numbers that look very convincing (ie they look like any model satisfying them would have to agree on any stateable property) these attempts will either be inconsistent (ie any expression can be proven from them) or they will be incomplete (ie there will be true properties of the natural numbers not provable from them, because of the fact that there exists other models of the axioms in which those properties are not true). Contrary to what some authors state, inconsistency does NOT have to be apparent. Early axiomatic set theory is a good example: It took many years before someone (Russell) came up with an inconsistency even though every well-formed expression (including all contradictiions) did in fact have a proof all along. This inconsistency was fixed by weakening one of the axioms.
I haven't read SotM, so I can't comment on that. My biggest problem with Penrose is that I personally am convinced that strong AI can be achieved, while Penrose believes that it is impossible. That does not mean that the book can't be good, but Penrose doesn't do a good job matching his arguments with those of his opponents.
He has a tendency to repeat arguments that strong AI supporters give, but in a way that they can be misunderstood, and then he misunderstands them and basically says they are stupid. Already in chapter one of ENM he starts ridiculing strong-AI supporters, without giving arguments. Later those arguments follow, but they are seriously flawed.
For example, while discussing Searle's Chinese Room experiment, he suggests that strong-AI supporters believe that "understanding Chinese" is in the book, and argues that it is stupid that a book can "understand" anything. But Turing's view (which is repeated by many others) is that the understanding is not in the book, but in the book + the human that reads the book. There is an emergent understanding of Chinese that comes from a book that describes how Chinese can be perfectly translated, and a human who strictly follows the rules that are written down in the book. This argument is mentioned by Penrose somewhere, but he just puts it aside as a very weak argument, that does not need refuting.
Writing like this annoys me immensly, and I suspect most my colleague AI researchers.
Incidentally, I think Penrose' work in mathematics is absolutely brilliant.
Flawed. Half a cloud is still a cloud.
Give me Classic Slashdot or give me death!
My problem with Penrose is this -- there's no reason to believe that human consciousness is anything more than the result of purely mechanical processes in the brain. If so, then we should theoretically be able to reproduce a human mind on a sufficiently fast computer. Whether that's practically possible is another matter.
Penrose, however, simply denies that consciousness can be electronically replicated even in theory. When asked why, he waves his hands and shouts "Quantum physics!" Now while it's certainly possible that quantum phenomenon play some role in consciousness that can't be replicated by a computer, there's no real evidence for the position and Penrose is just arguing it a priori. It's sort of the metaphysical equivalent of Creationism.
Les Miserables Volume 1 now up with my reading of
If you take a persons brain and disconnect the two halves, there is, in some sense, 2 consciousnesses (sp!). THIS HAS BEEN DONE:
With one child whose brain was split - consistently one half wanted to grow up to be a cowboy, the other an astronaut.
Sorry I don't have a link - google is the obvious answer here.
The interesting thing about the case was that it is hard to communicate with each half separately - especially when one half tends to be in change of communication, the other something else, etc. But there are ways to do it.
Actually, coincidentally, I think I first read about this in Penrose's book...
---
I type this every time.