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Some Things *ARE* Unknowable By: ScottN on 10/21/2001; 5:59 PM WARNING -- THIS ARTICLE IS KIND OF TECHNICAL Over on NitCentral, there have been some debates about the proposition that "G-d is unknowable". This is not part of those debates. Instead, I am going to continue my writings about science in general. In that vein, let us discuss the concepts of uncertainty and undecidability. There are three major items that I will talk about in this context. The Halting Problem, Godel's Undecidability Theorem, and The Heisenberg Uncertainty Prinicple. The Halting Problem essentially states that it is impossible to write a computer program (A) that takes a second computer program (B) as its input, and determines if B completes processing. This is a fundamental theorem of Computer Science, and has been known since the early days of automata theory. Thus, there are definite limits on the knowledge that we can derive from a mechanistic approach. OK, mechanistic reasoning is out, what about a more intuitive approach? Surely pure math, wherein sometime brilliant leaps of logic are made doesn't have these limitations! Alas, it does. In 1931, Kurt Godel dropped a bomb on the mathematical world, as follows:
To every w-consistent recursive class k of formulae there correspond recursive class-signs r such that neither v Gen r nor Neg (v Gen r) belongs to Flg (k) (where v is the free variable of r).Did you understand that? Good. I didn't expect you to either. In English, it essentially says that "All consistent axiomatic formulations of number theory include undecidable propositions" (Hofstadter D., Godel, Escher Bach: An Eternal Golden Braid, p.17). So how do we interpret that? A consistent system is one where it is impossible for a statement and its negation to be true simultaneously. In other words, "2 + 2 equals 4" and "2 + 2 is not equal to 4" cannot both be true. If they are both true, the system is inconsistent. So if we have a consistent system, it is possible to write a statement that is true, but cannot be proven. Godel's genuis was to discover that by encoding the symbols of number theory into numbers themselves, and creating rules to manipulate those numbers, it is possible to write statements of number theory that have two meanings. The first is the literal meaning of the symbols themselves. However, if the numbers have been properly manipulated, there is a second "meta-meaning" to the statement, which can be obtained by reversing the encoding. Using this scheme, Godel was able to construct a statement (G) which had the meta-meaning "G has no proof". Since statment G was constructed using the proper formalisms and encoding on the literal level, the meta-meaning must be true. So it is possible to construct a statement which is true, but has no proof. Therefore in the mathematical world, there are unknowables as well. Now that we've determined that the abstract worlds of computers and mathematics have unknowables, what about the real world? Sadly, the foundation of quantum mechanics is the fact that the physical world is unknowable as well. Werner Heisenberg discovered the principle that bears his name: The Heisenberg Uncertainty Principle. In essense, it says that for any entity, certain complementary properties cannot be known more accurately than a certain value. Specifically, the product of the uncertainties in these properties must be greater than Planck's constant h. Generally, the properties discussed are position (x) and momentum (p). Writing it out mathematically, we get Dx * Dp > h Putting that into plain English, "the uncertainty in position multiplied by the uncertainty in momentum is greater than Planck's Constant". Now, h is incredibly small, somewere on the order of 6 * 10-34 erg-seconds, so the Uncertainty Principle is unnoticable for macroscopic objects such as a grain of sand, you, me, your computer, the Empire State Building, etc... When you are standing still, you think you are not moving, and you have an absolute position. In actuality, you are bouncing around a sub-microscopic circle. But since you are so big, compared to these fluctuations, you don't notice it. On the other hand, for microscopic objects such as atoms, subatomic particles, molecules, or a politicians brain, the effect is quite large and noticable. This means, for example, if you could confine a single atom within a small (molecule sized) box, the atom would blur out, and you wouldn't be able to tell how fast or which direction it was moving. By confining the atom, you have reduced/limited the uncertainty in its position, so it has a corresponding increase in the uncertainty of its momentum. The mathematics of this are simple and elegant, but I'm not going to post them here. So, we have the fact that there are unknowable things in the world... Are there others? I don't know...
But is anything interesting unknowable By: Brian Carnell on 7/14/2000; 2:20 PM There are some other philosophy issues relating to foundational notions (whether or not perception actually gives accurate information, for example, has to by necessity be an axiomatic proposition -- there is no way for us to have knowledge about the accuracy of our perceptions.) But aside from that, there doesn't seem to be much that is both a) interesting (or worth knowing) and b) unknowable. If it had turned out, for example, that it was impossible to know whether or not a solution existed for some interesting math problem that would be startling (for example, speculation before it was solved that Fermat's Last Theorem might fall into this category). Same thing with quantuum issues. It's fascinating stuff but most of the things it deals with are decidedly uninteresting on a human scale. BTW, is the set of possible consistent number systems finite, or is one of the implications of Godel's theorem that there are an infinite number of unkowable things? Ooops -- I should correct myself. There *is* one thing that is interesting that in our current understanding is unknowable and related directly to the God question. Based on our present knowledge it is impossible to know what happened before the beginning of the universe since information can't pass the singularity boundary. The question "did something exist before our universe?" is unanswerable in principle if current cosmological theories are correct.
RE: But is anything interesting unknowable By: ScottN on 7/14/2000; 2:57 PM Same thing with quantuum issues. It's fascinating stuff but most of the things it deals with are decidedly uninteresting on a human scale. In fact, the Heisenberg Principle is what causes the Sun to shine. I'd say that's rather interesting to us humans... Another thing, which relates DIRECTLY to your comment about the origin of the Universe: Energy and Time are another of those "complementary" properties to which the Uncertainty Principle applies. It has been proposed that the Universe is actually a quantum fluctuation of zero-point energy, which got rather out of hand. Now, if this is indeed the case, the Heisenberg Principle can give us some interesting information about the lifetime of the Universe. Current estimates of the total amount of energy in the Universe are so close to zero as not to make a difference. Yes, those stars shining give off lots of energy, as is the energy of all the mass of the universe. However, the energy of the gravitational attraction of all that mass is negative, and so counteracts the mass energy. If, in fact, the Universe is a zero-point fluctuation, and the net energy of the Universe is very small (close to 0), then the Universe MUST exist for a hell of a long time before the fluctuation collapses (remember delta E * delta T > h). So, I'd say that it does apply to something that concerns us humans on a macroscopic scale.
Re: But is anything *interesting* unknowable By: Mark Morgan on 7/14/2000; 10:18 PM >But aside from that, there doesn't seem to be much that is both a) >interesting (or worth knowing) and b) unknowable. This is a bit different than quantum mechanics and Godel, but if I understand it correctly choas theory implies that the climate is ultimately unpredictable. I'd say that's something both interesting and unknowable... A side note: gentlemen, in the future you might want to avoid HTML coding in your message subjects. It works fine for the title, but I think it looks just odd on the browser window bar...
RE: But is anything interesting unknowable By: Brian Carnell on 7/14/2000; 11:43 PM Oh quantum stuff is very fascinating and relevant, but I think the claims about unknowable stuff isn't quite so shocking because most of the things that are unknowable aren't particularly interesting. I mean the fact that the position of electrons under certain circumstances may be indeterminate is incredible, but it doesn't seem to have much of an effect at human scales (i.e. the the things that are true about electrons don't hold for my hand). If on the other hand you had an important, human scale problem that was indeterminate, now that would be a bit more frightening. Unless I misunderstand the logic, for example, there is know way to know if a problem satisfies Godel's proof (i.e. if Fermat's theorem had remained unproven for another few hundred years, we could certainly speculate that it might be true but unprovable, but we would have no way of proving that it belonged to this class of problems. Is this correct or am I off base?) Mark in another post mentioned chaos theory and suspicions about dynamic systems such as weather, but I'm not convinced that either chaos theory or our knowledge about weather systems is advanced enough to warrant belief in that claim ... at least not to the same degree that we can be sure about uncertainty in quantum physics. (BTW, does this uncertainty issue a feature only of the Copenhagen interpretation or is it pretty much a problem regardless)?
RE: But is anything interesting unknowable By: Mark Morgan on 7/15/2000; 12:31 AM I'm sure Scott has books on Godel on his shelf. I've read The Emperor's New Mind by Roger Penrose recently enough to note that it discusses Godel, makes a yoeman's effort and making it accessible, and lost me utterly at some point. Are quantum effects important on macroscopic scales? I'd say the answer is, we don't know. And I don't think we will know until a theory of quantum gravity arrives, the so-called "theory of everything" that unites quantum mechanics and relativity. It may not seem interesting right now, but there's no reason I know of to assume that will always be the case. We simply don't know yet. (An aside, for those who don't follow nitcentral, the "God is unknowable" ranting I've been doing is not to dismiss that as an idea, but as an excuse to not have to think through the paradoxes and complications of religious ideas. Why did God destroy the world with a flood, then cover up all the evidence so it looks like it never happened? "God is unknowable." That's a cheat.)
RE: But is anything interesting unknowable By: Brian Carnell on 7/15/2000; 6:06 PM Mark wrote: "Are quantum effects important on macroscopic scales? I'd say the answer is, we don't know." Maybe it's unknowable!!! :) Since you mentioned God, there are two common macro-level applications of quantum effects that I think are weasel-y. Obviously I've heard theists claim that since some things are unknowable in principle, one of those things might be God which I don't think makes a lot of sense for a whole host of reasons aside from quantum stuff (if God is in principle unknowable, then we atheists are off the hook even if he does exist.) The God is unknowable claim actually creates far more problems for the theists than the atheists. Second, I have heard a lot of speculation that quantum effects prove free will, on the view that thoughts are influenced by quantum effects and therefore in some way what I am going to think of next is in principle unknowable. I think we're still so far from understanding human cognition that this is extraordinarily speculative. It's kind of funny how people take findings from physics and try to relate them to other things which they really have nothing to do with. You would not believe the number of people I have read who claim that the Einstein's claims about relativity are postive proof that there are no *moral* absolutes as well.
RE: But is anything interesting unknowable By: ScottN on 7/15/2000; 11:36 PM I don't know, I still say that the lifetime (and potential origin as well) of the Universe is a macroscopic Heisenberg effect that I think we might just be interested in... The theory goes like this: The universe resulted from a zero-point energy quantum fluctuation. As such, the total amount of energy in the Universe is the uncertainty in the fluctuation's energy. Since energy and time are complementary, the smaller the net energy of the Universe, the larger the time necessary to satisfy the Uncertainty principle. Since the net energy of the Universe is close to 0 (see my previous post), the amount of time that the universe can last must be a fairly large number. Incidentally, the Uncertainty Principle applies to all quantum mechanics, not just the Copenhagen interpretation. Here's a quick rundown of the derivation of the Uncertainty Principle. It follows from the fact that anything can be described as a wave as well as a particle. Also, here's some more and even more that I've written about Godel. Hope this makes things as clear as mud!
Re: Some Things *ARE* Unknowable By: Matthew Patterson on 10/25/2000; 8:14 AM Peter wrote: > > Msg #: http://www.voicesofunreason.com/fullthread$981#VU981 > Site URL: http://www.voicesofunreason.com/ > ---------------------------------- > > Is this a genuine attempt at a discussion or informative essay, Scott, or are you just trying to show off? > > Peter. Is this a genuine attempt at a reply, or are you just being rude? (Sorry, but I, at least, get most of it.) ==== Sign you watch too much anime #6: When taking a Spanish test dealing with relationships, one of your answers is "Mi mejor amiga se llama Megumi Hayashibara" and progresses from there. __________________________________________________ Do You Yahoo!? Talk to your friends online with Yahoo! Messenger. http://im.yahoo.com
Re: Some Things *ARE* Unknowable By: Mark Morgan on 10/25/2000; 8:38 AM Editor's note: Peter appears to have deleted his post.
Re: Some Things *ARE* Unknowable By: ScottN on 10/26/2000; 6:40 PM Frankly, I don't give a S**T about what Peter thinks. I write about what interests me. If he doesn't like it, that's his problem. Cthulhu for President! Why vote for the lesser evil?
Re: Some Things *ARE* Unknowable By: Mark Morgan on 10/26/2000; 6:51 PM Not to call this a tempest in a teapot, but again I note for the record that Peter deleted his post. I'm going to change some settings so that a post doesn't go out so quickly in e-mail, so that a poster has a little longer to rethink.
Re: Some Things *ARE* Unknowable By: ScottN on 10/26/2000; 9:34 PM For those who wish to understand the uncertainty principle a tad better, George Gamow wrote a classic book called Mr. Tompkins in Wonderland, wherein Mr. Tompkins, an ordinary man encounters relativistic and quantum effects by going to strange places where they can be observed in the ordinary world. For example, to discuss relativity, we find Mr. Tompkins in a world where c=50mph. Or we find him trying to play billiards with balls made from ivory from elephants that live in an area where Planck's Constant is = 1. There is very little math, IIRC, and the content is accessible to anyone. Highly recommended.
RE: Some Things *ARE* Unknowable By: Brian Carnell on 10/27/2000; 10:59 AM Scott wrote: >For those who wish to understand the uncertainty principle a tad better... Scott, did you see the recent reports about claims that the electron can be split?
RE: Some Things *ARE* Unknowable By: Mark Morgan on 10/27/2000; 11:10 AM Regarding the electron, here's the New Scientist article.
RE: Some Things *ARE* Unknowable By: ScottN on 10/27/2000; 5:15 PM Yes, I've seen it. It appears to be a variant of the Schroedinger's Cat paradox. Gribbin discusses this particular variant in "Schroedinger's Kittens". Of course, I could be wrong.
Re: Some Things *ARE* Unknowable By: Matthew Patterson on 10/27/2000; 5:27 PM Mark Morgan wrote: > > Msg #: http://www.voicesofunreason.com/fullthread$1028#VU1028 > Site URL: http://www.voicesofunreason.com/ > ---------------------------------- > > Regarding the electron, here's the New Scientist article (http://www.newscientist.co.uk/features/features.jsp). My chemistry teacher loved it. Actually mentioned it in class the other day. _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at http://mail.yahoo.com
Re: Some Things *ARE* Unknowable By: ScottN on 10/31/2000; 2:56 PM Over on NitCentral, I see Peter has a bug up his rear about this thread again, just because I write about what I like... Well, that's his problem, not mine. I happen to like Quantum Mechanics and find it interesting. I also find esoteric math (that's maths to you, Peter) interesting, but due to the difficulty of writing the notation in HTML, it's not worth posting any essays/whatever about it here. Heck, I might just post an article about the Complementarity Principle and the Copenhagen interpretation, especially since an experiment has called that into question! And you know what, Peter? If you don't find it interesting...
Re: Some Things *ARE* Unknowable By: Mark Morgan on 10/31/2000; 3:17 PM Editor's note: I agree with this sentiment. If a flame war erupts over this, expect components of this discussion to vanish, then move. I won't make regular visitors suffer through flame wars and personal attacks. --------------------------- Mark Morgan: mark_morgan@yahoo.com http://www.VoicesOfUnreason.com A resource for writers and readers of all stripes. ----- Isaac Asimov: "The most exciting phrase to hear in science, the one that heralds the most discoveries, is not 'Eureka!' (I found it!) but 'That's funny....'"
RE: Some Things *ARE* Unknowable By: Brian Carnell on 10/31/2000; 3:32 PM At 07:12 PM 10/31/00 +0000, Scott wrote: >Well, that's his problem, not mine. I happen to like Quantum Mechanics and find it interesting. I also find esoteric math (that's maths to you, Peter) interesting I like this stuff too. Keep it coming.
Re: Some Things *ARE* Unknowable By: Matthew Patterson on 10/31/2000; 6:03 PM ScottN wrote: > Over on NitCentral, I see Peter has a bug up his rear about this thread again, just because I write about what I like... > > Well, that's his problem, not mine. I happen to like Quantum Mechanics and find it interesting. I also find esoteric math (that's maths to you, Peter) interesting, but due to the difficulty of writing the notation in HTML, it's not worth posting any essays/whatever about it here. > > Heck, I might just post an article about the Complementarity Principle and the Copenhagen interpretation, especially since an experiment has called that into question! And you know what, Peter? If you don't find it interesting... > > THEN YOU BLOODY WELL DON'T HAVE TO READ IT! Hear, hear. Just for the record, I like it, even though I don't understand a lot of it. I get enough for it to be interesting. Pray continue. _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at http://mail.yahoo.com
Re: Some Things *ARE* Unknowable By: juli k on 10/31/2000; 10:23 PM I'm with Matthew. Scott (belated Happy Birthday, by the way), I didn't understand more than about 10 percent of your essay. Frankly, as a math-impaired adult, the topic is not something that I would normally seek out and read for enjoyment. I read it purely because I know you (in a cyberspace sort of way). I trusted you to write about something interesting, and I was curious to see what your interests were. As a result, I was exposed to some ideas I might not have thought to expose myself to otherwise, and I think that's great. Don't stop! I like the fact that we can pool our various interests and specialties together here at Unreason and broaden our horizons in a fun and friendly way. Anyone who is threatened by the fact that somebody else might be a little smarter than him in a certain field, simply doesn't belong here.
Re: Some Things *ARE* Unknowable By: ScottN on 12/19/2002; 3:55 PM Thanks, all... Here is a bit more detail on the math, without actually going into it... Joseph Fourier developed a bit of math called Fourier Analysis (imagine that!). Fourier analysis is what makes digital music possible (among other things). Of course, had Fourier known that his work would eventually lead to Britney Spears CDs, he would have burned all his notes and joined a monastery, but that's beside the point... Fourier discovered that it is possible to create ANY wave form by superposing sine waves of various frequencies and amplitudes. You can even create a wave that is zero at all points, except for one spot, like this: .... ________|```|________ .... Now, if we look at the QM description of a particle, it's wave function looks very much like that shown above. It's a well known fact of classical physics (that applies to QM as well), that a wave has a momentum which is determined by its energy and wave velocity: p = E/c. But in QM, energy is a linear function of frequency: E = hn (n is the greek letter "nu", denoting frequency, and h is Planck's Constant). Now, let's get back to Fourier. To create a waveform like that above, we need to add waves with smaller and smaller wavelengths until they cancel themselves out except in the limited area we want. We can make the area as narrow as we desire, but to do that, we need a smaller wavelength. Smaller wavelengths (l) imply a higher frequency (n) (c = ln, where c is the constant velocity of propagation). So let's put this together. We can localize our QM wave function by adding more and more waves with shorter wavelengths/higher frequencies. BUT... When we add those higher frequencies, we are increasing the energy (E = hn). And... since momentum is derived from energy, we then have an increase in momentum (actually, it's the uncertainty in the momentum - I'm simplifying things). Therefore, the uncertainty in location/momentum thing (Dx * Dp > h)is not an artifact of observation, as it is often described, but an inherent part of the wave/particle duality. (Author's note: Edit was to prettify the formatting on the "diagram")
RE: Some Things *ARE* Unknowable By: ScottN on 2/6/2002; 12:36 AM Just a note on the above derivation. I recently obtained a copy of Heisenberg's lectures that he delivered at the University of Chicago, and that is the argument he uses to propound the Uncertainty Principle. I did simplify it immensely, though...
RE: Some Things *ARE* Unknowable By: ScottN on 7/31/2004; 2:02 PM Going back to Brian Carnell's comment that the unknowable stuff doesn't affect us in the real world... You know that computer you used to post your comments? If it wasn't for the Uncertainty Principle, the silicon chips used to power it wouldn't work. Is that good enough? :-)
RE: Some Things *ARE* Unknowable By: Mark Morgan on 8/1/2004; 2:04 PM Lord, I had forgotten about the Peter-related silliness in this thread. (I was hit by a car while riding a bicycle and have forgetten most of that incident as well. Perhaps a related phenomena.) Here's another way to think about Brian Carnell's thoughts on interesting unknowable things. Quantum mechanics is hard. "If you think you understand quantum mechanics you haven't studied it enough." Why? Because our brains evolved on macroscopic scales and quantum effects do not affect our ability to produce grandchildren. We have to think in terms of metaphors and math and brain-hurting things because we have no direct experience with quantum effects. The fact that things are both waves and particles depending on how they are measured is irrelevant to me avoiding being eaten by a tiger on the way home from gathering wood for the fire. We can successfully predict the outcome of Mark versus the tiger without worrying about Schroedinger's poor cat.
RE: Some Things *ARE* Unknowable By: Brian Carnell on 8/1/2004; 2:38 PM I don't really understand ScottN's statement about computers (then again, I barely understand quantuum mechanics, so perhaps that's understandable). Perhaps he could elaborate. I find it difficult to understand how things that could effect us directly could be unknowable since as soon as they have any sort of effect they are, by definition, knowable. Take the singularity (the one that started the universe, not the transhumanist one).What occurred "before" the singularity (if that is even a coherent idea) is unknowable and hence has no effect on our universe even if our universe is dependent upon what that unknowable "before" was. So if a computer is dependent on an unknowable quantuum effect, that doesn't necessarily mean that something unknowable has a direct effect at the macroscopic level. Rereading this thread, however, my main point was against pseudo-scientific interpretations of the uncertainty principle. I am just so sick and tired of reading New Age crap that invokes the uncertainty principle as an explanation for allegedly noncausal or nonlocal macroscopic behavior. Writing a book about ESP and including a few paragraphs about Heisenberg does not an explanation make.
RE: Some Things *ARE* Unknowable By: Brian Carnell on 8/19/2004; 8:56 PM An interesting example of a quantum effect that may end up having important real world consequences. Temperature is an emergent property that doesn't exist once you get to individual atoms, but due to quantum statistical fluctuation it also appears likely to be nonexistent among very small clumps of atoms, such as in carbon nanotubes. Researchers at a UK university claim that a 10-micrometer carbon nanotube might never reach a thermal equilibrium which might pose a big problem (or not) for efforts to build nanomachines -- the temp. at each end of the nanotube can be different, might never reach equilibrium, and may be inherently unpredictable. Nanotubes may have no 'temperature'.
RE: Some Things *ARE* Unknowable By: ScottN on 8/26/2004; 3:47 AM Brian, I don't really understand ScottN's statement about computers. Your computer works off of semiconductors. Semiconductor behaviour is intimately tied to the Uncertainty Principle and quantum behaviour. Hence, your computer wouldn't work without it.
RE: Some Things *ARE* Unknowable By: TomM on 10/30/2004; 8:59 AM Note: this is an extremeley oversimplified, non-mathematical explanation, offered more to give Brian a feel for how quantum uncertainty can have macrocosmic effects, than to supplement Scott's essays. >>Take the singularity (the one that started the universe, not the transhumanist one).What occurred "before" the singularity (if that is even a coherent idea) is unknowable and hence has no effect on our universe even if our universe is dependent upon what that unknowable "before" was. So if a computer is dependent on an unknowable quantuum effect, that doesn't necessarily mean that something unknowable has a direct effect at the macroscopic level. Take a singularity (also known as a black hole). What occurs inside the event horizon is unknowable, since no information can escape. But the gravity of those unknowable particles can still affect us. Likewise, although normally both properties in the Heisenberg equation tend toward the "average" (I'm not going to try to figure if I mean the mean, the median or the mode, it's not important to my point), keeping both δx and δp fairly small, but sometimes one will briefly become extremely small, allowing the other to become large enough to have a macrocosmic effect. You still don't know exactly what is going on with the affected particles, but you see a sudden, unexplained, but extremely short-lived, increase in the momentum, or looked at another way, in the energy. and that energy can affect things macroscopically before it disappears again. One example Scott mentioned twice it the very existence of the Universe may simply be a quantum fluctuatuion. If that is true, the singularity you referred to, the one before the universe, did not exist, or perhaps only existed momentarily before resulting in the Big Bang. Under this theory, before that moment nothing would have existed. BTW, Scott, I remember long ago reading an essay by Isaac Asimov which he whimsically titled "I'm looking over a four-leaved clover." It was written before he learned the theories that predict the existence of singularities (black holes), and I don't believe that he mentions Heisenberg, either. (It's been well more than thirty years since the last time I saw that essay, so I may be misremembering.) And yet, somehow he, by starting with the parity/time/charge problem, came up with the idea that the Universe might be a momentary flicker, what you called a zero-point fluctuation.
RE: Some Things *ARE* Unknowable By: ScottN on 10/30/2004; 6:39 PM I'll have to look for that one, Tom. Thanks.
RE: Some Things *ARE* Unknowable By: TomM on 10/31/2004; 3:19 PM I've checked on it. It's the third of a three-part essay which was first printed in his column in Analog in July, August and September of 1966. (Which is where I read them.) The other two installments are titled "Balancing the Books" and "BB or not BB." They were later included in one of the books of collections of the Analog essays that Doubleday published: Science, Numbers and I, (1968).
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