Deep in the deluge of knowledge that poured forth from science in the 20th century were found ironclad limits on what we can know. Werner Heisenberg discovered that improved precision regarding, say, an object’s position inevitably degraded the level of certainty of its momentum. Kurt Gödel showed that within any formal mathematical system advanced enough to be useful, it is impossible to use the system to prove every true statement that it contains. And Alan Turing demonstrated that one cannot, in general, determine if a computer algorithm is going to halt.
David H. Wolpert, a physics-trained computer scientist at the NASA Ames Research Center, has chimed in with his version of a knowledge limit. Because of it, he concludes, the universe lies beyond the grasp of any intellect, no matter how powerful, that could exist within the universe. Specifically, during the past two years, he has been refining a proof that no matter what laws of physics govern a universe, there are inevitably facts about the universe that its inhabitants cannot learn by experiment or predict with a computation. Philippe M. Binder, a physicist at the University of Hawaii at Hilo, suggests that the theory implies researchers seeking unified laws cannot hope for anything better than a “theory of almost everything.”
Wolpert’s work is an effort to create a formal rigorous description of processes such as measuring a quantity, observing a phenomenon, predicting a system’s future state or remembering past information—a description that is general enough to be independent of the laws of physics. He observes that all those processes share a common basic structure: something must be configured (whether it be an experimental apparatus or a computer to run a simulation); a question about the universe must be specified; and an answer (right or wrong) must be supplied. He models that general structure by defining a class of mathematical entities that he calls inference devices.
The inference devices act on a set of possible universes. For instance, our universe, meaning the entire world line of our universe over all time and space, could be a member of the set of all possible such universes permitted by the same rules that govern ours. Nothing needs to be specified about those rules in Wolpert’s analysis. All that matters is that the various possible inference devices supply answers to questions in each universe. In a universe similar to ours, an inference device may involve a set of digital scales that you will stand on at noon tomorrow and the question relate to your mass at that time. People may also be inference devices or parts of one.
Wolpert proves that in any such system of universes, quantities exist that cannot be ascertained by any inference device inside the system. Thus, the “demon” hypothesized by Pierre-Simon Laplace in the early 1800s (give the demon the exact positions and velocities of every particle in the universe, and it will compute the future state of the universe) is stymied if the demon must be a part of the universe.
Researchers have proved results about the incomputability of specific physical systems before. Wolpert points out that his result is far more general, in that it makes virtually no assumptions about the laws of physics and it requires no limits on the computational power of the inference device other than it must exist within the universe in question. In addition, the result applies not only to predictions of a physical system’s future state but also to observations of a present state and examining a record of a past state.
The theorem’s proof, similar to the results of Gödel’s incompleteness theorem and Turing’s halting problem, relies on a variant of the liar’s paradox—ask Laplace’s demon to predict the following yes/no fact about the future state of the universe: “Will the universe not be one in which your answer to this question is yes?” For the demon, seeking a true yes/no answer is like trying to determine the truth of “This statement is false.” Knowing the exact current state of the entire universe, knowing all the laws governing the universe and having unlimited computing power is no help to the demon in saying truthfully what its answer will be.
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21 Comments
Add CommentVery nice article.
Reply | Report Abuse | Link to thisIsn't this "proving the negative" - that knowing it all cannot be done?
Reply | Report Abuse | Link to thisI thought proving the negative was not possible?
This article stresses the problems of scientific realism. Is there really a way to confirm that science corresponds directly to the world without being able to step outside the universe? Is science more a descriptive yet very succesful endeavor? It seems to me that it is mostly a metaphysical problem that is being discussed here.
Reply | Report Abuse | Link to this"It seems to me that it is mostly a metaphysical problem that is being discussed here."
Reply | Report Abuse | Link to thisIf you ignore metaphysics, and just dismiss it as worthless and not of any use, then you are setting yourself up for failure. For shame, we as a society spit on philosophy as an academic endeavor and then worship science without questioning its meaning and purpose for humanity. Science has a real hard time answering for itself. Science can not scientificaly deduce that science has all the answers. Metaphysics does answer some problems that science can not, and epistomology is something that science can not even handle.
Wolpert's paper is on arXiv.org, entitled "Physical limits of inference" (arXiv.0708.1362v2) and should be readable by most people who can handle math or physics at a graduate level. He includes discussion of philosophical implications and and addendum with detailed proofs.
Reply | Report Abuse | Link to thisBon appetit!
Doesn't this then prove a omnesant GOD does not exist
Reply | Report Abuse | Link to thisOnly if this omniscient god is within the universe that he/she/it supposedly knows everything about. :-) Of course believers will argue that their god is somehow "outside" the universe and/or that the normal rules of logic or proofs do no apply.
Reply | Report Abuse | Link to thisLanguage, including mathematics, is such a hopeless tool when it comes to explaining the cosmic order. As it happens there is a non-linguistic way to delineate the cosmic order that embraces all possible structural varieties of experience and meaning. Anyone can intuitively see this. It has never been explored before. This One System subsumes an open ended series of discrete Higher Systems nested within it such that the Lower Systems subsume and transcend the Higher Systems. It is not a TOE. It is a universal methodology than complements traditional approaches to the physical, biological, and social sciences. It is the intuitive picture on the cover of the puzzle box needed to meaningfully assemble the empirical jigsaw pieces. There are various articles about it at www.cosmic-mindreach.com including Unified Theories, Fantasy, & Cosmic Order that is a good introduction.
Reply | Report Abuse | Link to thisThese proofs assume a true/false universe.
Reply | Report Abuse | Link to this"These proofs assume a true/false universe."
Reply | Report Abuse | Link to thisHow do you plan on dealing with the excluded middle?
A true or not true statement is meaningless, and a true and not true statement is a contradiction.
If you could know everything, say when you die, how would you know that it really is everything?
Reply | Report Abuse | Link to thisquite interesting
Reply | Report Abuse | Link to this"The definition/description of 'the complete universe', at any moment, is the sum total of all Conscious Perspectives (us)!" - Book of Fudd (4:20)
Reply | Report Abuse | Link to this'The First Law of Soul (Conscious Perspective) Dynamics'; "For every Perspective, there is an equal and opposite Perspective!" - (op cit)
By definition, a Perspective is, to one extent or another, limited. Each and every Perspective is a (unique!) 'reality/truth' (if to no one other than the perceiver), and the sum total of all Perspectives is the complete tapestry of existence.
So, no, one single Perspective cannot even imagine the complete momentary tapestry of existence/universe but as one very small (yet integral) 'reality', his.
'Integral', in that without any one Perspective, the entire universe must be different! (Think Butterfly effect!)
To fully define anything, context is necessary. To exist, context is necessary.
Ultimately, the entire universe, at the moment of definition, is necessary for a complete definition of 'you' or 'me' or a 'hamburger' or a 'thought', etc...
Perceiver and perceived are truly one!
Nice to see this 'reality' being 'discovered' by such diverse disciplines.
Genuinely successful theories interconnect information from previously disparate areas of experience, said Adolf Gr�nbaum, the Andrew Mellon Professor of Philosophy at the University of Pittsburgh.
A negative can be proven in Set Theory when all members of a finite-member set are accounted for. For example, "No rats in cage #3 are black."
Reply | Report Abuse | Link to thisNegatives can be proven in Set Theory and Meta-Mathematics, but not necessarily about objects in the real world. There are exceptions, however, even to this rule. One might for instance prove that there are no elephants in a small box simply by doing the math.
Reply | Report Abuse | Link to thisMaybe what the article is really saying is that we cannot know all the answers...as frustrating as that truth may be, it just simply is...we cant step outside and look "in", we are part of how the "in" works, we need to be present for it to be what ot is, if we did move out in oreder to look in, then the truth will get altered.
Reply | Report Abuse | Link to this"Negatives" are proved all the time. You can prove that the set of all sets does not exist, that certain classes of equations have no general solution, that the set of real numbers is not denumerable, that the set of complex numbers is not well-ordered, and that the proverbial Village Barber who shaves all those who do not shave themselves does not exist.
Reply | Report Abuse | Link to thisThis stuff is so cool. If anyone has suggestions on how to attain the proficiency in logic, mathematics, and physics one would need to understand this proof, please post it! Thanks!
If I were asked the question "Will the universe not be one in which your answer to this question is yes?", my answer would be "That question is inherently circular, and therefore logically invalid." If pressed, I would insist that it is a logical error to answer the question either yes or no, and indeed a logical error to ask the question in the first place. Like quantum mechanics, this situation represents a limit on what quantities exist, not a limit on what quantities we can know.
Reply | Report Abuse | Link to thisTEST
Reply | Report Abuse | Link to thisThis theory reminds me of W. Ross Ashby's more general Law of Requisite Variety. For any system A that controls (or comprehends, I would suppose) another system B, A must have more states available to it than B.
Reply | Report Abuse | Link to thisI found David Wolpert's study very interesting, although I do not have a math/physics background. My interest was piqued after coming across a reference to this research in an intriguing blog post titled 'The Role of Psychological Distance in Creativity and Innovation'. Not sure if I can insert a link here, so suggest you Google the above and have a read. (Link: http://thelaughingbuddha.wordpress.com/2010/06/14/the-role-of-psychological-distance-in-creativity-and-innovation/)
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