gjk@talcott.UUCP (Greg Kuperberg) (01/05/85)
There seems to be considerable skepticism about my proposition that a non-linear system is a sufficient condition. Well, here is a simple model of a non-linear system in which there is exactly *zero* predictability. In this "universe", time goes in discrete jumps, as in Conway's Game of Life. In fact, one could easily picture this universe as a cellular automaton. Anyway, the universe at any point in time consists of a non-terminating sequence of 0's and 1's. After each time unit, the sequence has its first binary digit (bit, I guess) removed. Thus, the whole sequence shifts right by one. The evolution of the universe would progress like this: Time 0: 1101110011... Time 1: 1011100111... Time 2: 0111001110... Time 3: 1110011101... Time 4: 1100111010... And now, the twist. The *observable* universe is only the first bit of the sequence at each point in time. So what is the state of the observable universe as it progresses with time? Simply any sequence of 0's and 1's. Utterly unpredictable, even though the entire state of the universe "existed" at the beginning of time! Well, someone might object, "but it's only because of QM that we are limited in our measurements, so we still need QM." But I say that QM is not our only limitation. Even without QM, the thickness of the doping on our silicon chips is limited to the size of the atom, the speed that the signal in them propagates is limited to the speed of light, and the size of our computers is limited to the size of the universe (and probably the size of our galaxy). These restrictions probably apply to *any* computing device in our universe. So, if we're lucky, we may get 10^200 megaflops tops. By the (exponentional) nature of non-linear systems, this would give a predictive power of, say, 200 days of weather. Less than one year of weather predicted, even with a computer as big as the universe! Furthermore, it would necessarily disrupt the weather patterns if we were to take 10^200 data samples. Thus, the exact weather for next year is forever beyond our reach, without all those theories that the Nobel Laureates have cooked up. --- Greg Kuperberg harvard!talcott!gjk " " -Charlie Chaplin, for IBM
guy@rlgvax.UUCP (Guy Harris) (01/06/85)
(Totally unpredictable universe proposed; you are given a hypothetical infinite sequence of bits, and the state of the universe at time T (T an integer, so that time goes in discrete jumps) is the value of the Tth bit of the sequence. This in response to the claim that the non-linearity of a system can be a sufficient condition for unpredictability; this system is described as "a simple model of a non-linear system" by the original author.) Objection, Your Honor; the defense's claim is irrelevant. The system isn't unpredictable because its equations of motion are non-linear; the system is unpredictable because it *doesn't have* any equations of motion. This system is "non-linear" only in a trivial sense. (Discussion of the restrictions of real live physical computing devices, and of the fact that taking enough measurements to find the initial state of a system will disrupt that state enough that perfect prediction is in practice impossible). Again, irrelevant. We're not discussing whether enough Crays can be constructed to produce real printouts and graphs predicting the future state of the Universe. We're discussing whether the future state of the Universe is not predictable *in principle*. If the model we construct of the universe is based on the real numbers, the size of a silicon atom doesn't even get a chance to enter into it, because the initial state consists of C (where C is the cardinality of the continuum, a nice infinite number) points, each one of which is specified with infinite precision (and would require log2(C) bits to represent, which is still a nice infinite number). We all *K*N*O*W* that in practice the universe won't be perfectly predictable. Please don't waste our time informing us of the detailed reasons why; I, for one, knew them before this discussion began and don't really give a damn about hearing them again. > Thus, the exact weather for next year is forever beyond our reach, > without all those theories that the Nobel Laureates have cooked up. Yes, we all know that; only an idiot would think otherwise (i.e., think that you could compute the exact value of the temperature, to infinite precision, at each one of the C locations on the earth). Guy Harris {seismo,ihnp4,allegra}!rlgvax!guy
stekas@hou2g.UUCP (J.STEKAS) (01/08/85)
> Again, irrelevant. We're not discussing whether enough Crays can be > constructed to produce real printouts and graphs predicting the future > state of the Universe. We're discussing whether the future state of the > Universe is not predictable *in principle*. The concept of "predicable, in principle" is usefull if and only if it has some connection with reality. If 10^70 Cray's calculating for 10^10 years cannot predict next year's weather then next year's weather is unpredictable. To argue that the weather is predictable "in priciple" is equivalant to saying that "My theory predicts something which is IMPOSSIBLE to calculate, but if we could I'm sure it would be correct!" It wasn't too long ago that any angle could be trisected, pi could be expressed as the ratio of 2 integers, particles positions and momentum could be determined simultaneously, ... In principle, of course. Jim
guy@rlgvax.UUCP (Guy Harris) (01/09/85)
> > Again, irrelevant. We're not discussing whether enough Crays can be > > constructed to produce real printouts and graphs predicting the future > > state of the Universe. We're discussing whether the future state of the > > Universe is not predictable *in principle*. > > The concept of "predicable, in principle" is usefull if and only if > it has some connection with reality. The problem is that the connection of any discussion of predictability with reality is tenuous, at best. If you're discussing it in terms of whether you can know what the thermometer is going to read tomorrow, the practical impossibility of making that prediction is important. If you're arguing the philosophical question of whether the universe is a deterministic machine or not, it is *totally irrelevant* whether you can construct a machine which models the universe with 100% accuracy out of real materials. Unless somebody out there has something useful to add to the discussion of the question of whether the universe is a deterministic machine, let's drop the discussion; pointing out that there's nothing out there with which a machine that models the universe can be constructed adds nothing to this particular discussion. Guy Harris {seismo,ihnp4,allegra}!rlgvax!guy
breuel@harvard.ARPA (Thomas M. Breuel) (01/09/85)
> The concept of "predicable, in principle" is usefull if and only if > it has some connection with reality. If 10^70 Cray's calculating for > 10^10 years cannot predict next year's weather then next year's weather > is unpredictable. To argue that the weather is predictable "in priciple" > is equivalant to saying that "My theory predicts something which is > IMPOSSIBLE to calculate, but if we could I'm sure it would be correct!" > > It wasn't too long ago that any angle could be trisected, pi could > be expressed as the ratio of 2 integers, particles positions and momentum > could be determined simultaneously, ... In principle, of course. Sorry, you're wrong. The technical term for "predictable in principle" is computable. This is a sensible notion of great interest and significance in modern information science. What you're aiming at is the notion of "computational complexity". But even if a problem is not computable, or if a problem is NP complete, not all hope is lost. You may not be able to find an analytical solution for it, or to find such a solution efficiently, but you may be able to find good heuristics. Indeed, many heuristics are so good that we don't even notice that they are only approximations. "In principle", the path of a bullet is not predictable. Nevertheless, everybody uses Newtonian mechanics to predict it. Just a few remarks about your concluding remarks: the trisection of the angle and the rationality of pi were *unsolved* mathematical problems. It has now been shown that these problems are unsolvable (although you can graphically trisect an angle to any desired degree of accuracy and express pi to any desired degree of accuracy as a rational number). Whether particle positions and momenta can or cannot be determined simultaneously is still an unsolved question of modern physics. The fact that quantum mechanics gives answers that are in good agreement with experimental data does not imply that quantum mechanics is by any means "correct". Indeed, it has some highly questionable points and is in disagreement with other theories that show equally good argeement with (other) experimental data. Thomas. breuel@harvard
rosen@sunybcs.UUCP (Jay Rosenberg) (01/10/85)
(long quotes at end of article) [And now, for the BIG MONEY, is the universe ... Random?] Since it is highly unlikely (do you have any idea how high I mean) that the matter will be resolved by anything approaching direct testing. The question on whether the universe is fundamentally random, or only so in New Jersey, is most likely moot (as far as physics is concerned) unless it can be argued that somehow the choice of one view would yield different predictions that could either be supported or lambasted. I seriously doubt that either system couldn't completely account for the other, and no test (including Occum's disposable razor) could cast a deciding vote. Does that mean that it doesn't matter which philosophical bent you (the physics community) choose? Of course not. I personaly think that the frame of mind that settles for the "well, all in all, its random, let's go home I'm hungry" approach is not a good one. Once you accept randomness as the underlining mechanism, there is no longer a drive to study it. Just like it is not uncommon for people to think that human behavior is random (don't believe me, sit in on a good intro to AI class and listen to some comments). The upshot is that saying its random, I believe, can easily induce the attitude of "Why bother?". This is a dangerous axiom to hold as the underlining and most basic principle of physics. Of course, I'm probably wrong. Jay Rosenberg > > Again, irrelevant. We're not discussing whether enough Crays can be > > constructed to produce real printouts and graphs predicting the future > > state of the Universe. We're discussing whether the future state of the > > Universe is not predictable *in principle*. > > The concept of "predicable, in principle" is usefull if and only if > it has some connection with reality. If 10^70 Cray's calculating for > 10^10 years cannot predict next year's weather then next year's weather > is unpredictable. To argue that the weather is predictable "in priciple" > is equivalant to saying that "My theory predicts something which is > IMPOSSIBLE to calculate, but if we could I'm sure it would be correct!" > > It wasn't too long ago that any angle could be trisected, pi could > be expressed as the ratio of 2 integers, particles positions and momentum > could be determined simultaneously, ... In principle, of course. > > Jim
josh@topaz.ARPA (J Storrs Hall) (01/10/85)
> > The concept of "predicable, in principle" is usefull if and only if > > it has some connection with reality. If 10^70 Cray's calculating for > > 10^10 years cannot predict next year's weather then next year's weather > > is unpredictable. ... > Sorry, you're wrong. The technical term for "predictable in principle" > is computable. This is a sensible notion of great interest and > significance in modern information science. What you're aiming at is > the notion of "computational complexity". Let's try to keep something straight here: there are lots of things that a mathemetician assumes trivially, like adding two real numbers, that are not computable--they're infinitely long, remember? So something that's computable in principle to a mathemetician or a very theoretical physicist has little to do with what we can actually compute. Predicting the motion of a perfectly deterministic Newtonian system is impossible *because you can't "know" the initial conditions: they entail an infinite number of bits*. One way to look at QM uncertainty is to say that it, or something like it, is necessary merely to avoid having to say that each particle in the universe represents an infinite amount of information. --JoSH
gwyn@brl-tgr.ARPA (Doug Gwyn <gwyn>) (01/11/85)
> One way to look at QM uncertainty is to say that it, or something like > it, is necessary merely to avoid having to say that each particle in > the universe represents an infinite amount of information. No, this is not what quantum uncertainty is about (although perhaps this is a consequence of it). Even quantum mechanics assumes that quantities can be expressed as sets of real numbers, each of which is essentially infinitely precise. The uncertainty principle is that every measurable quantity (well, that's stretching it a bit) has a conjugate quantity such that the product of the precision to which the conjugate variables can be simultaneously determined is no less than some universal constant (something like 10^-27 erg-sec as I recall). There is no injunction against any quantity being measured to as many "bits" of accuracy as one may desire, although its conjugate becomes fuzzier as the measurement is made more precise. I would like to hear a discussion of the implications of a closed and/or bounded (not the same thing) universe on such issues. I think there is something very interesting to be discovered about this..
gwyn@brl-tgr.ARPA (Doug Gwyn <gwyn>) (01/11/85)
I think Jay has raised a valid and important point, which is (paraphrasing) that one's accepted philosophy has a significant influence on one's actions and motivations. It really should matter even to a pragmatist (which I am not) whether the underlying structure of the universe is inherently random, whether it follows laws laid down arbitrarily by a supreme being, whether it is possible for humans to gain a real understanding of their world or not, and so forth. I think one of the reasons that modern physics has gotten simultaneously silly and boring (in my opinion, obviously; I used to be a physicist) is that most physicists have intellectually defaulted on this matter, leaving the crucial issue of their fundamental beliefs to be determined by the mystics that dominate the field of philosophy (including religion of course) in today's culture. `Tis a pity.
stekas@hou2g.UUCP (J.STEKAS) (01/11/85)
Randomness and predictability are two different things. Randomness is sufficient for unpredictability but not the only source. In linear types of systems, like Newtonian orbital mechanics, one can easily show that intitial states which are infinitesimally different at t=0 will be infinitesimally different at t=T. For many non-linear systems, infinitesimally different states at t=0 can be arbitrarily "far apart" at t=T. When such things occur it becomes impossible, even in principle, for a computer to predict the future because one does not even know the precision to which the calculation must be carried. One cannot guarentee that the truncation error on even a 10^10 bit computer would not introduce unacceptable error. Jim ihnp4!hou2g!stekas
rivers@seismo.UUCP (Wilmer Rivers) (01/12/85)
In article <1027@sunybcs.UUCP>, rosen@sunybcs.UUCP (Jay Rosenberg) writes: > Once you accept randomness as the underlining mechanism, there is no > longer a drive to study it. I don't buy that. After all, there are many reasons for studying a system other than being able to predict exactly its future (or past) behavior. Just because a system must be regarded as being stochastic rather than deterministic doesn't mean that you have no interest in determining the structure of that randomness - if you have a black box which is spewing out random numbers, the first thing you would like to do is determine whether those random numbers have a uniform, gaussian, chi-squared, etc., distribution. An extraordinary amount of effort is expended on the analysis of random phenomena such as turbu- lence in fluids or noise in electric and mechanical systems. Ultimately, the knowledge gained in these investigations cannot be used to predict future behavior exactly, but it can be used to express that behavior probabilistically. There is thus considerable motivation for studying phenomena which one regards as being random; in fact, one might even take the opposite viewpoint of regarding the study of deterministic phenomena as being somehow less exciting, because all possible answers that you might work out for the system's future behavior are "predestined" by the initial conditions (in the absence of future perturbations.) It would be rather boring to continue to make measurements and plot points on a graph, knowing that they would all continue to fall on a parabola. Stochastic systems are at least fun to observe precisely because you don't know exactly what's going to happen next. (If you don't believe that, ask the inventor of the "lava lamp" how much he's made from that little gizmo.)
rpw3@redwood.UUCP (Rob Warnock) (01/13/85)
+--------------- | For many non-linear systems, infinitesimally different states at t=0 can | be arbitrarily "far apart" at t=T... | Jim | ihnp4!hou2g!stekas +--------------- In fact, it should be possible to construct a system which is "arbitrarily non-linear everywhere", to coin a phrase. That is, given some time T and some epsilon E > 0 (no matter how small) and delta D (no matter how large), one should be able to construct systems S(s0,t) for which, given two initial states s1 and s2 such that the initial states are at least E apart (according to whatever convenient metric ||x-y|| you choose): || S(s1,0) - S(s2,0) || > E the states at time t > T are farther apart than D: || S(s1,t) - S(s2,t) || > D I remember some constructions we did in math in college many (many!) years ago with functions that were "discontinuous everywhere". Has the above sort of thing been done with physical systems? Is this the sort of stuff Catastrophy Theory is supposed to deal with? Rob Warnock Systems Architecture Consultant UUCP: {ihnp4,ucbvax!dual}!fortune!redwood!rpw3 DDD: (415)572-2607 USPS: 510 Trinidad Lane, Foster City, CA 94404
stekas@hou2g.UUCP (J.STEKAS) (01/14/85)
>> The concept of "predicable, in principle" is usefull if and only if >> it has some connection with reality. > >Sorry, you're wrong. The technical term for "predictable in principle" >is computable. You mean "predictable in principle" (i.e. computable) is a useful concept even when it has NO connection with reality?! I wasn't aiming at a definition of "computability" only an understanding of the usefullness of the concept. Even so, I would think that most of modern physics is NOT computable! The point I was trying to make was that if the universe is governed by a collection of non-linear equations (of sufficient ugliness) then the question of whether the universe is predictable may moot because we might not be able to calculate with them. So we couldn't verify the correctness of our theories, or determine the future even if they were correct. Is QED a "computable" theory? QCD? General Relativity? Have we verified them to sufficient accuracy to believe them? Are they deterministic theories? Are they deterministic approximations of a non-deterministic reality? Are these questions of more concern to a physicist, philosopher, or theologian? Jim
carlc@tektronix.UUCP (Carl Clawson) (01/15/85)
In article <386@hou2g.UUCP> stekas@hou2g.UUCP (J.STEKAS) writes: >Randomness and predictability are two different things. Randomness >is sufficient for unpredictability but not the only source. > >In linear types of systems, like Newtonian orbital mechanics, >one can easily show that intitial states which are infinitesimally >different at t=0 will be infinitesimally different at t=T. Sorry, but Newtonian mechanics is not necessarily linear. (I'm assuming you're referring to the equation of motion F=ma.) For Newtonian gravity F is proportional to 1/(r**2), thus the non-linear equation 2 2 2 d r/dt = constant x 1/(r ) Linearity is not necessary for "predictability" as described in the referenced article. Carl Clawson ...tektronix!carlc
anand@utastro.UUCP (Anand Sivaramakrishnan) (01/16/85)
>In article <386@hou2g.UUCP> stekas@hou2g.UUCP (J.STEKAS) writes: >Randomness and predictability are two different things. Randomness >is sufficient for unpredictability but not the only source. > >In linear types of systems, like Newtonian orbital mechanics, >one can easily show that intitial states which are infinitesimally >different at t=0 will be infinitesimally different at t=T. >>Sorry, but Newtonian mechanics is not necessarily linear. (I'm >>assuming you're referring to the equation of motion F=ma.) . . >>Carl Clawson I append the following note... In point of fact, Newtonian gravity is not only non-linear but also solutions of the equations of motion are very often 'Sensitively Dependent on Initial Conditions' (SDIC). Frequently higher energy 'orbits' (trajectories) in many Newtonian systems diverge away from each other in any neighbourhood through which they pass (i.e. they have at least one positive 'Liapunov Exponent'). Typical examples of this phenomenon are found in various models for the behaviour of (massless) bodies moving under the gravitational influence of two massive bodies in Keplerian (elliptical or circular) motion around each other. This SDIC is so prevalent that it is now a buzzword amongst us nonlinear dynamicists. It is the hallmark of 'chaos in deterministic systems'. This SDIC is found in most nonlinear differential equations. Anand Sivaramakrishnan
jss@sftri.UUCP (J.S.Schwarz) (01/21/85)
> > I remember some constructions we did in math in college many (many!) years > ago with functions that were "discontinuous everywhere". Has the above > sort of thing been done with physical systems? Is this the sort of stuff > Catastrophy Theory is supposed to deal with? > > > Rob Warnock > Systems Architecture Consultant > No. The subject is nonequilibrium thermodynamics. A good introduction is the book From Being to Becoming: Time and Complexity in the Physical Sciences by Ilya Prigogine W. H. Freeman & Co. ISBN 0-7167-1108-7 pbk (Prigogine won a Nobel prize for work related to the contents of the book, but this book is readable by those with a year or two of undergraduate physics.) Jerry Schwarz