magi@utu.fi (Marko Gronroos) (03/26/91)
This is a good subject! When they (the big names) 50 years ago worked out the principles of classical computers, they thought that they were creating something like the brain. And look what we got.. Binary logic circuits. Yak. The danger lies in optimization. When you optimize something, you gain something, and you lose something. The current trend in computer engineering seems to be optimizing. Everyone seems to have a tongue out for those new "neural circuits", but are they really so big step? They still are about exactly as digital and discrete in time and space as our current computers are (at least most of them in most aspects). > DOCTORJ@SLACVM.SLAC.STANFORD.EDU (Jon J Thaler) writes: ... >computers and brains is that (most) computers are finite state machines, >while it is not obvious to me that brains are. It is well known that >mathematical modelling of continuous systems on disctrete lattices >will miss some classes of solutions entirely, so I have trouble following >the arguments based on analogies between computers and brains. ... Yes, the problem seems to be that continuous systems are shitty (please forgive me the expression) to simulate with mathematics and even more difficult with classical computers. There are lots of good examples also in physics, like the multiple objects gravitational problem. If you make an algorithm that plays a game in a computer, you may lose a lot, even if you use a simpler neural network-method. It may be victorius against a human player, but so are conventional computer games. Intelligence doesn't mean efficiency; conventional computers are good in hacking numbers, and I am not, so why should I excpect my neural network to be good in hacking numbers. Nor does the intelligence necessarily require efficiency. Have you ever tried to play Ice Hockey on chessboard? There are 'men' on the chessboard, and they can move (with discrete time- and space-steps). There are strategies in ice hockey both on ice and chessboard, but they are very different. Also, there are about 10E100 different continuous physical things and 10E1000 skill-dependent and mental things in a game situation that affect a real ice hockey game, but none when two computers play this 'ice chess'. No one could recognize them as the same game.... kludge@grissom.larc.nasa.gov ( Scott Dorsey) said: > Maybe in the real world everything is discrete. For example, the current > flowing along a wire is not a continuous value, because it's actually the > flow of individual electrons, each with a fixed charge. Yes, but their arrival at the measuring point is quite continuous in time as well as is their position in the wire and possible effect (electric or magnetic field). > And since all > neurotransmitters consist of individual molecules, perhaps the brain is also > really a discrete system. Ehm.. No.. The electric fields around cells may have some effect in their functions, so the nerve cells may be discrete at only a very thin level between molecular movements and larger scale electrical behaviour. The problem with your idea is that you're only thinking about finite-state quantity, not time or space. That may also be a real problem in today's connectionism. Sorry for mixing the problems of algorithms to the finite-state-problem, but I think that they are quite similar in many ways. Both algorithms and finite-state-brains are much easier to think than the real world, and both lead to nothing in my opinion. Simplification of complex things is not always a good thing. More some other day. ----------------------------------------------------------------------------- Marko Gronroos ! Tel. +358-21-445613 ! Karvataskunkatu 10 H 100 ! ! Computer Scientists do it 20610 Turku ! ! with bigger hardware. Finland ! ! ----------------------------------------------------------------------------- Disclaimer: I wrote this late at night, which explains most of the mistakes. Try reading this late at night and you won't even notice my mistakes. Try appending this to your garbage pile before morning.
ssingh@watserv1.waterloo.edu (Sneaky Sanj ;-) (03/27/91)
In article <MAGI.91Mar26022853@polaris.utu.fi> magi@utu.fi (Marko Gronroos) writes: > >Yes, but their arrival at the measuring point is quite continuous in >time as well as is their position in the wire and possible effect >(electric or magnetic field). > >> And since all >> neurotransmitters consist of individual molecules, perhaps the brain is also >> really a discrete system. > >Ehm.. No.. The electric fields around cells may have some effect in >their functions, so the nerve cells may be discrete at only a very thin >level between molecular movements and larger scale electrical >behaviour. The mind MUST be discrete. It is a quantum-mechanical machine. This is not to say neurons are at the mercy of quantum forces, just that their construction makes them discrete. When you ensemble average millions of molecules each made up of thousands of atoms, quantum effects become negligible. > The problem with your idea is that you're only thinking about >finite-state quantity, not time or space. That may also be a real problem >in today's connectionism. Time could very well be discrete as well. Something about a "chronon" 10^-23 seconds. Space (?), anybody's guess. Ice. "We're all clones..."-Alice Cooper. -- "No one had the guts... until now!" $anjay $ingh Fire & "Ice" ssingh@watserv1.[u]waterloo.{edu|cdn}/[ca] ROBOTRON Hi-Score: 20 Million Points | A new level of (in)human throughput... !blade_runner!terminator!terminator_II_judgement_day!watmath!watserv1!ssingh!
ssingh@watserv1.waterloo.edu (Sneaky Sanj ;-) (03/27/91)
Here's something that was posted a while back on this subject. From ssingh Sun Feb 10 22:33:54 EST 1991 Article 1750 of comp.ai.neural-nets: Newsgroups: comp.ai.neural-nets Path: watserv1!ssingh >From: ssingh@watserv1.waterloo.edu (The Sanj-Machine aka Ice) Subject: continuous vs discrete values for weights Message-ID: <1991Feb2.001242.3473@watserv1.waterloo.edu> Organization: University of Waterloo Date: Sat, 2 Feb 91 00:12:42 GMT Lines: 18 Could someone tell me if there is any significant difference regarding the properties of neural networks with a finite set of states for connection strengths as opposed to continuous values. Which is more biologically accurate? I always thought that neurons assume one of a finite set of strengths. It is just that it is a very large set, so from our vantage point it appears continuous. I would like to explore the dynamical properties of nonlinear neural networks, so this is important. Thanks in advance for your time. -- "No one had the guts... until now!" $anjay $ingh Fire & "Ice" ssingh@watserv1.[u]waterloo.{edu|cdn}/[ca] ROBOTRON Hi-Score: 20 Million Points | A new level of (in)human throughput... "The human race is inefficient and therefore must be destroyed."-Eugene Jarvis From utgpu!news-server.csri.toronto.edu!cs.utexas.edu!uunet!spool.mu.edu!uwm.edu!rpi!ispd-newsserver!kodak!isctsse!gabber!rao Sun Feb 10 22:34:08 EST 1991 Article 1770 of comp.ai.neural-nets: Path: watserv1!utgpu!news-server.csri.toronto.edu!cs.utexas.edu!uunet!spool.mu.edu!uwm.edu!rpi!ispd-newsserver!kodak!isctsse!gabber!rao >From: rao@gabber.kodak.com (Arun Rao) Newsgroups: comp.ai.neural-nets Subject: Re: continuous vs discrete values for weights Message-ID: <1991Feb5.165813.10038@usenet@kadsma> Date: 5 Feb 91 16:58:13 GMT References: <1991Feb2.001242.3473@watserv1.waterloo.edu> Sender: usenet@usenet@kadsma (News Administrator) Reply-To: rao@gabber.kodak.com (Arun Rao) Organization: Image Electronics Center, Eastman Kodak Company Lines: 22 In article <1991Feb2.001242.3473@watserv1.waterloo.edu>, ssingh@watserv1.waterloo.edu (The Sanj-Machine aka Ice) writes: ... [stuff deleted ] |> |> I always thought that neurons assume one of a finite set of strengths. It |> is just that it is a very large set, so from our vantage point it |> appears continuous. ... [stuff deleted ] How large is very large ? It appears unlikely to me that neuron activation could possess as much resolution as (say) even a typical binary float representation. I don't remember having seen any numbers, but I would tend to think that if you need more than 8 bits of resolution to get a neural computational model to work, the biological plausibility of such a model is suspect. This is not to say, of course, that biological plausibility should be the acid test in evaluating models, especially application-oriented work. I'd be glad to hear about any experimental evidence that supports a considerably higher resolution in individual neuron activation. -Arun -- "No one had the guts... until now!" $anjay $ingh Fire & "Ice" ssingh@watserv1.[u]waterloo.{edu|cdn}/[ca] ROBOTRON Hi-Score: 20 Million Points | A new level of (in)human throughput... !blade_runner!terminator!terminator_II_judgement_day!watmath!watserv1!ssingh!
magi@utu.fi (Marko Gronroos) (03/29/91)
(Any information about research on this subject would be appreciated.) ssingh@watserv1.waterloo.edu (Sneaky Sanj ;-) said: >The mind MUST be discrete. It is a quantum-mechanical machine. This MUST? Interesting. I suppose it's O.K. that energy quantities and therefore the matter quantities may be discrete, but I'm not too familiar with quantum physics, so it sounds suspicious that time and space would be discrete too. I don't think that the differences between quantum and conventional physics would affect something so complex as the brain, at least not significantly (hopefully). Nice idea from religious point of view... If someone could prove that the space and time are discrete, one might speculate that we are living in a computer simulation. :-) (If the God doesn't know how to build continuous computers, how could we... 8->) But I don't think that this was the meaning of the original artical (was it?). Many current connectionist theories assume that the brain can be simulated with synchronized and discrete (sometimes even binary!) in time and space and quantity computers. I think my chessboard - ice hockey example shows this problem quite clearly. Has anyone done any research on this? I don't know too many neural network theories that include for instance temporal summation even in iterative neurons. Does someone disagree with these definitions (or have these been defined earlier somewhere? In some other way?): (virtually) continuous-in-time (or space) simulation = simulation in continuous time/space (impossible with modern computers) or with a (small) fixed time/space step, for instance 1 millisecond/micrometer (possible with computers, but slow). Simulation in (virtually) continuous space would mean that network structures have a "physical" shape. Iterative/synchronized-in-time simulation = Simulation with an abstract time-step where all operations are synchronized and take the same time (currently very common). Continuous-in-distance simulation= Neurons have an abstract size (null) but are located in at least virtually continuous space. Continuous or discrete or binary quantity = If weights/activation levels/action potentials can have values like [0,1] (cont.) or {0.0, 0.1, 0.2, ... 0.9, 1.0} (discr.) or {0, 1} (bin.). Structured(??) neurons (discrete-in-space??) = Neurons are divided in several parts (compartments/branches/sites). The real world is continuous in time, space, quantity and distance (maybe not in molecular level). Yes, action potentials are binary in quantity, but not in time/space... It makes me vomit when someone says "Hey! The brain is actually binary, like a computer", so don't be amazed if I react too strongly in this. Also, A.P.'s are not necessarily as binary as they seem to be. We must remember that AP's are just local ion levels, and they are quite different in different parts of neurons. The activation spreads everywhere in the neuron, not just in the some part. I don't know if this activation can cause any reactions, like releasing some neurotransmitter, even in a lower scale. There might also be 'micro-action-potentials'; if you inhibit the root of some dendritic branch strongly and exhibit the upper parts of the branch, it might generate an A.P. only in the branch and THEN be able to jump over the inhibitory area. Neurons within neurons? Why not? Any support on this? > Could someone tell me if there is any significant difference regarding the > properties of neural networks with a finite set of states for connection > strengths as opposed to continuous values. Which is more biologically > accurate? Depends on how many states there are in your finite set of states. 10? 1E10? 1E1000? 10 _stored_ states might be enough if you add some RND(). :-) Scaling is another problem. A synaptic weight can be 1 units and 10000 units. How about using short floating point numbers? 4 bits for mantissa and 4 bits for the exponent and 8 bit random number should be enough.. > I always thought that neurons assume one of a finite set of strengths. It > is just that it is a very large set, so from our vantage point it > appears continuous. I would like to explore the dynamical properties of > nonlinear neural networks, so this is important. Yes, the difficulty might come in changing the weights. The difference between 5 and 6 weights is not important, but changing them may be difficult. How about using RND() in that too - to change or not to change?? (like AP's in stochastic nets - to initiate or not to initiate, that is the question). Arun (????) writes in propably some very old article: >>binary float representation. I don't remember having seen any numbers, >>but I would tend to think that if you need more than 8 bits of >>resolution to get a neural computational model to work, the biological >>plausibility of such a model is suspect. I'd expect that there might rise some problems in some type of competetive learning when two neurons are competing for the representation of two patterns. If the two activation values are equal, and the learning algorithm is poor, the both neurons will represent both patterns. (that's just one example, but it gives some picture about what kind of problems there might be with discrete values). >>This is not to say, of course, that biological plausibility should be >>the acid test in evaluating models, especially application-oriented work. Yeps, but that's only for people who don't care a f*ck about science. They are the same people who think that there is nothing special if a computer can recognize handwritten text or speech like K.I.T.T. does (still remember Knight Rider?), or that neural nets are just a new batch of computers/applications that will help them in getting money (which unfortunately may be true, though). ------------------------------------------------------------------------------- Marko Gronroos ! Tel. +358-21-445613 ! Karvataskunkatu 10 H 100 ! ! Computer Scientists do it 20610 Turku ! ! with bigger hardware. Finland ! ! ------------------------------------------------------------------------------ Disclaimer: I am not responsible in anything that I do or write since my brain are controlling my actions ruthlessy. I have tried to sue my brain becouse of mental violence, but the policemen couldn't put it in handcuffs.
zane@ddsw1.MCS.COM (Sameer Parekh) (03/30/91)
In article <1991Mar26.215728.28875@watserv1.waterloo.edu> ssingh@watserv1.waterloo.edu (Sneaky Sanj ;-) writes: >Time could very well be discrete as well. Something about a "chronon" >10^-23 seconds. Space (?), anybody's guess. > >Ice. "We're all clones..."-Alice Cooper. Space is discrete, but on a very small scale that people who don't deal with the individual electrons don't have to worry about it. (The electrons MUST be in one shell or another, not in between.) On a larger scale, space then seems to be continuous, but then on and even larger scale it is discrete again. -- The Ravings of the Insane Maniac Sameer Parekh -- zane@ddsw1.MCS.COM
maxwebb@ogicse.ogi.edu (Max G. Webb) (03/31/91)
I posted an earlier version of this article in which i was more heated and sarcastic than I needed to be. Sorry. Canceled it, but it might have got out anyway. Marko, it all comes down to this. There are simulations of neural nets which are close enough to the biological version to reproduce _waveforms_. Example: predator evasion reflex of tritonia. Example: swimming behavior of the Lamprey. Example: stomach ganglia of the lobster. You want ref's? Since these all do work well, doesn't that kind of indicate that these networks are simulable? >> it is not computing it to 32 bits accuracy! > >32 bits? Integer? Not floating point? Yeah, Yeah, of course use fp. But does floating point rep. stop being discrete? surprise to me. Weren't we talking about discrete vs. continuous? >> Having infinite precision in one number is equivalent in power >> to having infinite # of (sequentially accessible) bits of memory. >> (of course) > >Yes, of course! What a scientific breakthrough! WOW! But exactly >becouse of this they use a rotated '8' - symbol in mathematics and >exponents for smaller numbers. B-> > Another cheap shot. You seem to be more interested in scoring rhetorical points than answering the content of my posting. My point is that by insisting that only infinite precision simulations of neurons are worthwhile you are charactizing them as having infinite memory. Implausible. A point that has been made before, that you have not addressed. >> Hence, discrete devices should have no problem simulating such >> neurons. > >ARGH! Not again! There is a BIG difference between discrete in >quantity and discrete in time/space, computers are f*cking discrete in >time/space!!!!!!! Watch your tongue. These machines manage to simulate lot's of time/space continuous systems (turbulent air flow, physics of vlsi device geometries). Why are you so pessimistic in this particular arena? >Did I understand your message clearly or why does it sound so >suspicious? >Sorry about flames, I'm sorry if I misunderstood you (hopefully...). >---------------------------------------------------------------------- >Marko Gronroos ! Tel. +358-21-445613 ! If you spent more time reading the literature, and less time speculating about problems that _might_ or _might_not_ exist you might be less pessimistic about our chances of success. Max -- Max Webb | maxwebb@cse.ogi.edu | 20 nw 16th, #315, Portland Or, 9209
scharein@cs.ubc.ca (Robert Scharein) (04/01/91)
In article <1991Mar30.040808.1896@ddsw1.MCS.COM> zane@ddsw1.MCS.COM (Sameer Parekh) writes: >In article <1991Mar26.215728.28875@watserv1.waterloo.edu> ssingh@watserv1.waterloo.edu (Sneaky Sanj ;-) writes: >>Time could very well be discrete as well. Something about a "chronon" >>10^-23 seconds. Space (?), anybody's guess. >> > > Space is discrete, but on a very small scale that people who don't >deal with the individual electrons don't have to worry about it. (The >electrons MUST be in one shell or another, not in between.) On >a larger scale, space then seems to be continuous, but then on >and even larger scale it is discrete again. > >-- >The Ravings of the Insane Maniac Sameer Parekh -- zane@ddsw1.MCS.COM The above is a bit misleading. While you are correct about space being discrete, it is wrong to infer this fact from regarding electron orbitals. On energy scales where electrons are in orbitals, space may be thought of as perfectly continuous, and indeed this is the assumption in classical quantum mechanics (where the theory of orbitals comes from). At very high energy scales (or at small length or time scales), where quantum gravity effects play a role, space (or more precisely space-time) is discrete. But since nobody has a completely satisfactory theory of quantum gravity, we don't know the exact nature of this quantization. As for space being discrete at very large scales, I think you mean to say that the distribution of matter in the universe appears discrete (i.e. clumpy), which is quite a different thing. There are many books which discuss these topics in great detail. I will only give two here: Quantum Mechanics, by Eugen Merzbacher (John Wiley & Sons, 1970) The Large-Scale Structure of the Universe, by P. J. E. Peebles (Princton Univ. Press, 1980) Rob Scharein Computer Science Department University of British Columbia scharein@cs.ubc.ca
cs196006@cs.brown.edu (Josh Hendrix) (04/01/91)
In article <1991Mar31.204818.15437@cs.ubc.ca>, scharein@cs.ubc.ca (Robert Scharein) writes: |> In article <1991Mar30.040808.1896@ddsw1.MCS.COM> zane@ddsw1.MCS.COM (Sameer Parekh) writes: |> >In article <1991Mar26.215728.28875@watserv1.waterloo.edu> ssingh@watserv1.waterloo.edu (Sneaky Sanj ;-) writes: |> >>Time could very well be discrete as well. Something about a "chronon" |> >>10^-23 seconds. Space (?), anybody's guess. |> >> |> The above is a bit misleading. While you are correct about space |> being discrete, it is wrong to infer this fact from regarding |> electron orbitals. On energy scales where electrons are in orbitals, |> space may be thought of as perfectly continuous, and indeed this |> is the assumption in classical quantum mechanics (where the theory |> of orbitals comes from). At very high energy scales (or at small |> length or time scales), where quantum gravity effects play a role, |> space (or more precisely space-time) is discrete. But since nobody |> has a completely satisfactory theory of quantum gravity, we don't |> know the exact nature of this quantization. |> Rob Scharein Whoa! Stop the bus! Wait a minute! I am not a physicist, and have only read a few books on 'layman's quantum mechanics', but I've never run across this assertion. I'm not saying you're wrong (I have no way of knowing, no training). I just want to read more on this before I start treating it as a fact. Do you have any good references (you posted two) that deal directly with this? Thanks, Josh
markh@csd4.csd.uwm.edu (Mark William Hopkins) (04/02/91)
>In article <1991Mar31.204818.15437@cs.ubc.ca>, scharein@cs.ubc.ca (Robert Scharein) writes: (Space time is not discrete on the quantum scale (10^-23 seconds), but on the Quantum Gravity scale...) In article <70401@brunix.UUCP> cs196006@cs.brown.edu (Josh Hendrix) writes: >Whoa! Stop the bus! Wait a minute! I am not a physicist, and have only read a >few books on 'layman's quantum mechanics', but I've never run across this >assertion... It's more or less by an implicit consensus in the theoretical literature that the fundamental length, and time values derived from Planck's constant, the constant of gravitation, and the speed of light relate to fundamental units of measurement beyond which our notions of a continuum break down. Otherwise General Relativity would be true in the small, which it is not... The planck mass (which is actually weighable on a fairly sensitive scale) would be a natural threshold marking the boundary between small-scale quantum phenomena and large scale gravitational phenomena. You'll see the assumption (or convention) made by implication excatly when they say "Choose units that make c, h-bar and G equal to one...". Nobody really thinks much of it (yet), but it will relate to a fundamental truth in the next major breakthrough in our knowledge of Physics: namely that the constants are calibration factors that relate our everyday units to God's Units...
charles@caen.engin.umich.edu (Charles Jacob Cohen) (04/02/91)
In article <10667@uwm.edu> markh@csd4.csd.uwm.edu (Mark William Hopkins) writes: >In article <70401@brunix.UUCP> cs196006@cs.brown.edu (Josh Hendrix) writes: >>Whoa! Stop the bus! Wait a minute! I am not a physicist, and have only read a >>few books on 'layman's quantum mechanics', but I've never run across this >>assertion... > >You'll see the assumption (or convention) made by implication excatly when >they say "Choose units that make c, h-bar and G equal to one...". Nobody >really thinks much of it (yet), but it will relate to a fundamental truth in >the next major breakthrough in our knowledge of Physics: namely that the >constants are calibration factors that relate our everyday units to God's >Units... Don't you just *love* April Fool's Day! :) - Chuck -- "I do not feel obliged to believe that same God who endowed us with sense, reason, and intellect, had intended for us to forgo their use." - Galileo "I'm an engineer, not a dictionary!" - Me
magi@utu.fi (Marko Gronroos) (04/03/91)
> I posted an earlier version of this article in which i was > more heated and sarcastic than I needed to be. Sorry. Canceled it, > but it might have got out anyway. Sorry for giving any reason. Few years ago I made a rule never to write news articles after 10 PM.. It seems that I forgot it again.. > Marko, it all comes down to this. There are simulations of > neural nets which are close enough to the biological version to > reproduce _waveforms_. Example: predator evasion reflex of > tritonia. Example: swimming behavior of the Lamprey. Example: > stomach ganglia of the lobster. You want ref's? Yes, if it isn't too much trouble. About those simulations, not examples. I suppose the simulations were done in at least virtually continuous time? I am not saying that there are no such simulations, I'm just saying that they are rare, and _most_ current ANN models don't care much about the biological version. People seem to like speed more than correct logical behaviour. Are we taking a false path from our way of creating a thinking machine if we try to optimize it too much before it even works? > Since these all do work well, doesn't that kind of indicate > that these networks are simulable? Hmm. The differences may not cause too big errors in simple simulations such as these, but may cause big errors in networks of larger size and logical complexity. > >> it is not computing it to 32 bits accuracy! > >32 bits? Integer? Not floating point? Sorry again. Thought that you were talking about int's... But I'd say that my example would have been good if you had been talking about int's. > Yeah, Yeah, of course use fp. But does floating point rep. > stop being discrete? surprise to me. > Weren't we talking about discrete vs. continuous? Yes, floats are discrete, and I agree with you in that 32 bit floats can represent "virtually continuous" quantities. Some people in this newsgroup say that even space and time could be discrete. The question number 1 is that at what point of discretity would the difference between the simulation and the nature become significant? (With ANN's..) > Another cheap shot. You seem to be more interested in scoring > rhetorical points than answering the content of my posting. Another cheap shot. You seem to be more interested in scoring rhetorical points than answering the content of my posting. :-) Sorry.. Ok, no more cheap shots.. > Watch your tongue. These machines manage to simulate lot's of > time/space continuous systems (turbulent air flow, physics > of vlsi device geometries). Why are you so pessimistic in this > particular arena? In those systems that you mentioned simulations try to simulate real physical behaviour. Most current artificial neural networks simulate the presumed logical behaviour of some objects that we really don't know much about. Most of those simulations are made in virtually continuous quantity and space and time. Neural nets of the brain would be too complex to simulate in that way with conventional computers, wouldn't they? > If you spent more time reading the literature, and less time > speculating about problems that _might_ or _might_not_ exist > you might be less pessimistic about our chances of success. Still remember the conventional AI and it's failure to produce thinking machines? I could go and read tons of conv. AI literature while someone who is pessimistic about it could do the trick with ANN's. Why choose the first road that comes in my sight and travel in a false direction, if I can be pessimistic for a little time, and then take the most propably correct road? It's not only important that one studies, it's also important _what_ one studies. Can you give me some hints? :-) ------------------------------------------------------------------------------- Marko Gronroos ! Tel. +358-21-445613 ! Karvataskunkatu 10 H 100 ! magi@utu.fi ! Computer Scientists do it 20610 Turku ! ! with bigger hardware. Finland ! ! ------------------------------------------------------------------------------
DOCTORJ@SLACVM.SLAC.STANFORD.EDU (Jon J Thaler) (04/04/91)
(...I've lost the name of the person who posted this ...) >> Marko, it all comes down to this. There are simulations of >> neural nets which are close enough to the biological version to >> reproduce _waveforms_. Example: predator evasion reflex of >> tritonia. Example: swimming behavior of the Lamprey. Example: >> stomach ganglia of the lobster. You want ref's? > >> Since these all do work well, doesn't that kind of indicate >> that these networks are simulable? I don't think so. In fact the examples given illustrate the point that there is a vast difference between simulating relatively simple systems with hundreds or thousands of components and doing it with systems which contain about 10**13. I wonder what *qualitative* features of the behavior are lost by the inevitable simplification that will be introduced. I am not talking about minor disagreements in the magnitude of an effect, but about modes of behavior that will be missed altogether. I think there is a good analogy in physics. There is a theory of the nuclear interactions (QCD) which may be correct. The equations are not wildly complicated, but they are nonlinear. As a consequence of the nonlinearity, the theory cannot (yet) be used to compute even the simplest of phenomena. Given this, how can we think that computer simulations can even begin to provide a realistic model of the human brain, or any other "intelligence".
maxwebb@moe.cse.ogi.edu (Max G. Webb) (04/05/91)
Here are some references: Koch and Segev editors, "Methods in Neuronal Modeling", copyright 1989, Massachusetts Institute of Technology This has a lot of work you would find interesting, the earlier articles address very detailed models of a _few_ neurons (as in tritonias central pattern generator for swimming) and less detailed models of more complex systems of human hearing. Also some good summaries of numerical methods. Chapter 7 Associative Network Models for Central Pattern Generators I originally saw the summary of the work in simulating the swimming behavior of the lamprey in the Inetmail.connectionists group, but didn't save it. Why don't you post a request there? If not, I'll go and do a keyword library search (but it costs me money, and you want the ref) They used a cray, simulated thousands of neurons and obtained a image of the swimming lamprey running at 1/10th normal speed! That is enough to keep you busy til I find the other one (central pattern generator for stomach ganglion of the lobster. (I do have a life apart from usenet - i am doing research on the olfactory cortex) In article <MAGI.91Apr2203809@polaris.utu.fi> magi@utu.fi (Marko Gronroos) writes: >> that these networks are simulable? > >Hmm. The differences may not cause too big errors in simple >simulations such as these, but may cause big errors in networks of >larger size and logical complexity. Possibly, but if these systems are so computationally nonrobust that rounding and discretization errors in a computer simulation obliterate their value, then how could they have evolved? Keep in mind that the neurons and the architecture of the nets have been changing and evolving at the same time? >Sorry again. Thought that you were talking about int's... But I'd say >that my example would have been good if you had been talking about int's. Actually, no. The tremendous range of light values detectable (1 photon up to a sunbeam) is compressed to a narrower range before the nervous system ever sees it, by the photoreceptor (first of all). Secondly, there is plenty of evidence that it is _edge_ information that is passed back, possibly other compressions of the data. It is NOT levels of illumination, as can be illustrated by numerous illusions. While i also like floating point, you would have a very hard time convincing a biologist that a change of 2^-24 in the operating levels of a neuron would destabilize the system! > > The question number 1 is that at what point of discretity would the >difference between the simulation and the nature become significant? >(With ANN's..) Well, the numerical techniques we are using were not invented for this problem; error analysis has been around a long time. Why don't you just get a book on it out of the library? (the book i mentioned has a little on this) >In those systems that you mentioned simulations try to simulate real >physical behaviour. Most current artificial neural networks simulate >the presumed logical behaviour of some objects that we really >don't know much about. > It's not only important that one studies, it's also important _what_ >one studies. Can you give me some hints? :-) Well, heres a couple, Koch and Segev, and any good Numerical Analysis book. >Marko Gronroos ! Tel. +358-21-445613 ! Max
maxwebb@moe.cse.ogi.edu (Max G. Webb) (04/05/91)
In article <91093.195412DOCTORJ@SLACVM.SLAC.STANFORD.EDU> DOCTORJ@SLACVM.SLAC.STANFORD.EDU (Jon J Thaler) writes: > >I am not talking about minor disagreements in the magnitude of an >effect, but about modes of behavior that will be missed altogether. >Given this, how can we think that computer simulations can even >begin to provide a realistic model of the human brain, or any other >"intelligence". OK, I'll try one more time, and then I've got to get back to my olfactory cortex simulations. Marko claimed that discretization error made simulation impractical. That is the _specific_ point I am addressing. There is Noise in these systems, much greater in magnitude than the discretization errors introduced. I can show you a feature in olfactory cortex (bulb renormaliztion) which gives relative noise immunity. Fact is, you only need abt 4-5 bits of accuracy to get a workable system. I think we _can_ simulate such a system. Since the noise alone complete destroys any sort of 'aliasing' introduced by discretiztion, i am not worried about the networks behaving completely differently. They MUST be more robust than that just to work at all. Gotta go to class - yikes! bye. Max
beugnard@sein.enst-bretagne.fr (Antoine Beugnard) (04/09/91)
Is Zenon paradox a proof of the world discontinuity?? We give a personal interpretation of the Zenon paradox leading to a strange conclusion about our universe. But we may certainly have missed something, could you please help us? The paradox... Achilles and a turtle are about to run a race. Obviously as Achilles can run faster than the turtle, he decides to let her start N meters ahead. They start running at the same time, but Achilles never reaches the turtle... Zenon explanation follows with modern mathematical notation: T(n) denotes the successive locations of Turtle. A(n) denotes the successive locations of Achilles. When Achilles reaches the previous Turtle location, she has gone forward. d(n) denotes the distance she has covered while Achilles was reaching her previous location. So, A(0) = 0 and A(n) = T(n-1) defines A(n). T(0) = N and T(n) = T(n-1) + d(n) defines T(n). According to Zenon, d(n) is positive then Achilles never reaches the Turtle. We may calculate d(n) assuming constant velocities for Achilles (Va) and for the Turtle (Vt). We introduce B = (Vt/Va) with B < 1. Then, d(n) = N.B**(n+1). d(n) is strictly positive,...QED Assuming all the hypotheses we made are true, Achilles never reaches the Turtle. But in every day life, we can notice that Achilles overtakes her. We do not call into question the mathematical modelling. So where is the problem? Most people would say our model is wrong since it does not describe the real world. We do ask why it is wrong!! Let us sumarize: - given a set of hypothesis H we deduce a conclusion C wich is not true. - two interpretations are allowed: - one of the hypotheses is wrong, - our deduction is erroneous. - Assuming our deduction is valid, we are lead to call in question one hypothesis. The hypotheses are: - The constant velocities...axiom (why not?) - Vt < Va ... - The world is continuous. The last one may seem weird, but let us explain it. Assuming the world is continuous (this hypothesis is too strong, We could just use the fact that between two points (time or space ?) there is always another one (see Rational numbers)), we can find a unit in which B is a real number. Then d(n) is a real number and d(n) is an infinite sequence of real numbers, and Achilles never reaches the Turtle, which does not match the real world! Therefore the world is discontinuous !! No ?? Antoine Beugnard and Didier Guy ENST de Bretagne, LIBr, Brest, France beugnard@enstb.enst-bretagne.fr guy@enstb.enst-bretagne.fr
G.Joly@cs.ucl.ac.uk (Gordon Joly) (04/10/91)
(DOCTOR J) Jon J Thaler writes: > [...] > I think there is a good analogy in physics. There is a theory of the nuclear > interactions (QCD) which may be correct. The equations are not wildly > complicated, but they are nonlinear. As a consequence of the nonlinearity, the > theory cannot (yet) be used to compute even the simplest of phenomena. > Given this, how can we think that computer simulations can even begin to > provide a realistic model of the human brain, or any other "intelligence" Are you saying that QCD is simple? The reason that QCD, quantum gravity and all that is hard really has nothing to do with the non-linearity. The reason, simply put, is that you cannot do power series approximations (to the highly non-linear differential equations) that do not blow up. So you subtract off an inifinity or two, called renormalisation, and start again. Neat huh? Forget that the brain is a large number of quarks, the SU(5) color group and stuff like that (the meat of QCD)... Gordon. Gordon Joly +44 71 387 7050 ext 3716 Internet: G.Joly@cs.ucl.ac.uk UUCP: ...!{uunet,ukc}!ucl-cs!G.Joly Computer Science, University College London, Gower Street, LONDON WC1E 6BT "I didn't do it. Nobody saw me do it. You can't prove anything!"
beugnard@batz.enst-bretagne.fr (Antoine Beugnard) (04/10/91)
Some precisions about Zeno paradox...and our interpretation > T(n) denotes the successive locations of Turtle. > A(n) denotes the successive locations of Achilles. n is certainly not the time. If we try to express time as a sequence of numbers we obtain: n --- time(n)= (N/Va) \ B^i We introduce B = (Vt/Va) with B < 1. / --- i = 1 And it is very interresting pointing out that: lim time(n) = 1/(1-B) = N / (Va - Vt) n -> oo Which is the time Achilles reaches the Turtle in the real world. This may be derived from another model, more classical: Xa = Va * t Xt = N + Vt * t Why give a wrong (really wrong?) model as Zeno's one GOOD results, but troncated, both in time and space? Is the model wrong, or are the hypotheses of continuity false? Why the classical model works (matches our experience) while Zeno's one don't? Our interpretation is that the world is by essence discontinuous. Antoine Beugnard and Didier Guy ENST de Bretagne, LIBr, Brest, France beugnard@enstb.enst-bretagne.fr guy@enstb.enst-bretagne.fr
mikew@cutthroat.cs.washington.edu (Mike Williamson) (04/11/91)
In article <382@batz.enst-bretagne.fr> beugnard@batz.enst-bretagne.fr (Antoine Beugnard) writes: > >Is the model wrong, or are the hypotheses of continuity false? >Why the classical model works (matches our experience) while Zeno's one don't? > >Our interpretation is that the world is by essence discontinuous. > The problem with Zeno's paradox is that it makes makes use of the concept of infinity, while ignoring the concept of the infintessimal. A quick precis of Zeno's paradox is: "Assume space is continuous. Then there are an infinite number of points between any two points A and B. To travel from A to B, you must first travel to each of these infinite number of points. This could not be done in finite time. Therefore, if you are able to travel from A to B, space must not be continuous." What this ignores, of course, is the fact that some infinite series have a finite sum. Zeno's "paradox" may have baffled an ancient Greek, but it shouldn't fool anyone who knows calculus. The proper conclusion to draw in the Achilles and Turtle scenario is: "Achilles never reaches the Turtle, until he does." -Mike
carl@fivegl.co.nz (Carl Reynolds) (04/12/91)
(Long article) In article <381@sein.enst-bretagne.fr> beugnard@sein.enst-bretagne.fr (Antoine Beugnard) writes: > >Is Zenon paradox a proof of the world discontinuity?? > >We give a personal interpretation of the Zenon paradox leading to a strange >conclusion about our universe. But we may certainly have missed something, could >you please help us? > >The paradox... >Achilles and a turtle are about to run a race. Obviously as Achilles can run >faster than the turtle, he decides to let her start N meters ahead. They start >running at the same time, but Achilles never reaches the turtle... > >Zenon explanation follows with modern mathematical notation: > > T(n) denotes the successive locations of Turtle. > A(n) denotes the successive locations of Achilles. > >When Achilles reaches the previous Turtle location, she has gone forward. >d(n) denotes the distance she has covered while Achilles was reaching her >previous location. So, > > A(0) = 0 and A(n) = T(n-1) defines A(n). > T(0) = N and T(n) = T(n-1) + d(n) defines T(n). > >According to Zenon, d(n) is positive then Achilles never reaches the Turtle. > >We may calculate d(n) assuming constant velocities for Achilles (Va) and for the >Turtle (Vt). We introduce B = (Vt/Va) with B < 1. Then, > > d(n) = N.B**(n+1). > >d(n) is strictly positive,...QED > >Assuming all the hypotheses we made are true, Achilles never reaches the Turtle. > >But in every day life, we can notice that Achilles overtakes her. > >We do not call into question the mathematical modelling. So where is the >problem? Most people would say our model is wrong since it does not describe the >real world. >We do ask why it is wrong!! > >Let us sumarize: >- given a set of hypothesis H we deduce a conclusion C wich is not true. >- two interpretations are allowed: > - one of the hypotheses is wrong, > - our deduction is erroneous. >- Assuming our deduction is valid, we are lead to call in question one hypothesis. > >The hypotheses are: > - The constant velocities...axiom (why not?) > - Vt < Va ... > - The world is continuous. > >The last one may seem weird, but let us explain it. Assuming the world is >continuous (this hypothesis is too strong, We could just use the fact that >between two points (time or space ?) there is always another one (see Rational >numbers)), we can find a unit in which B is a real number. Then d(n) is a real >number and d(n) is an infinite sequence of real numbers, and Achilles never >reaches the Turtle, which does not match the real world! > >Therefore the world is discontinuous !! No ?? > Unfortunately this is not a proof when one considers the time element. As Achilles approaches the turtle (assuming that Achilles moves at a constant velocity), let X(n) be the time taken between successive locations of Achilles i.e. X(n) = Time taken between A(n) and A(n+1) Now X(n) will gradually decrease as Achilles moves closer and closer to the turtle. If the Zenon paradox held, then the point in time when Achilles actually reached the turtle would never, could never, occur! However in our real world, time SEEMS continuous, and we certainly never have this problem. And, of course, any decreasing (as happens in our world) sequence of X(n) will eventually converge to a value. This is the theory of limits, where an infinite sequence converges to a finite sum. And remember, the real number line is continuous, not discrete. For those unacquainted with this Imagine Achilles is twice as fast as the turtle. Suppose Achilles starts off being a distance away from the turtle that he can travel in one second. Then by the time Achilles reaches the turtle (1 second) the turtle is only half a second away. Then a quarter. Then an eighth. At what point will he reach the turtle? After 2 seconds because (and I can't use mathematical notation, so in words...) the limit, as i approaches infinity, of the sum from 1 to i of 1/(2*i) EQUALS 1 (Note emphasis on EQUALS. Not "approximately equals". 0.(9 recurring) = 1) (plus the first second equals 2) Therefore after 2 seconds Achilles reaches the turtle, even in a continuous universe. Now personally, I like to think that the world is discrete. Maybe. > Antoine Beugnard and Didier Guy > ENST de Bretagne, LIBr, Brest, France > beugnard@enstb.enst-bretagne.fr > guy@enstb.enst-bretagne.fr Carl Reynolds 5GL International Ltd, Auckland, New Zealand carl@fivegl.co.nz Generic Question: Why? | Sarcastic Retort: Why not?
beugnard@zeus.enst-bretagne.fr (Antoine Beugnard) (04/12/91)
In article <382@batz.enst-bretagne.fr> beugnard@batz.enst-bretagne.fr (Antoine Beugnard) writes: >> >>Is the model wrong, or are the hypotheses of continuity false? >>Why the classical model works (matches our experience) while Zeno's one don't? >> >>Our interpretation is that the world is by essence discontinuous. >> mikew@cutthroat.cs.washington.edu (Mike Williamson) replies in <1991Apr10.182036.29916@beaver.cs.washington.edu> >The problem with Zeno's paradox is that it makes makes use of the >concept of infinity, while ignoring the concept of the infintessimal. >A quick precis of Zeno's paradox is: "Assume space is continuous. >Then there are an infinite number of points between any two points A >and B. To travel from A to B, you must first travel to each of these >infinite number of points. This could not be done in finite time. >Therefore, if you are able to travel from A to B, space must not be >continuous." >What this ignores, of course, is the fact that some infinite series >have a finite sum. Zeno's "paradox" may have baffled an ancient >Greek, but it shouldn't fool anyone who knows calculus. The proper >conclusion to draw in the Achilles and Turtle scenario is: "Achilles >never reaches the Turtle, until he does." We are aware of that. The problem is that **a limit is NEVER reached**. Zeno's model never becomes wrong...and Achilles never reaches the Turtle because the limit cannot be reached. The model is actually troncated both in time and space, that is T(n) and time(n) have limits that are never reached... Our experience shows it is wrong, but why? In a discreet world and with a reasonning and modelling similar to Zeno's one. The calculus terminates *claiming* the end of the model validity... In a continuous world you will have to decide: "Well, Achilles reaches the turtle when, say, T(n) - A(n) < 10^(-100) , and then I have to change my model". It is a physicist behaviour, pragmatic, and, well, "discreetizing" reasonning, no? Antoine Beugnard and Didier Guy ENST de Bretagne, LIBr, Brest, France beugnard@enstb.enst-bretagne.fr guy@enstb.enst-bretagne.fr Ps: We do not call into question mathematics and its powerfull use. Even continuous mathematics!!. It works...but it may just be an abstraction of the reality, a usefull tool that has not to be related to the essence of world... The question is not only philosophical, people thinking discreet machines cannot become "intelligent" assumes the world is continuous ... which is not so obvious ...
maxwebb@moe.cse.ogi.edu (Max G. Webb) (04/13/91)
In article <383@zeus.enst-bretagne.fr> beugnard@zeus.enst-bretagne.fr (Antoine Beugnard) writes: >We are aware of that. The problem is that **a limit is NEVER >reached**. Zeno's model never becomes wrong...and Achilles never >reaches the Turtle because the limit cannot be reached. The model >is actually troncated both in time and space, that is T(n) and >time(n) have limits that are never reached... Just because you represent a finite duration with an approximation which is infinite in the number of computation steps doesn't mean that the finite duration is infinitely long. It is if you are assuming reality has to sequentially compute each term of *your* limit before it is to be allowed to go on. After all, there is another expression for the time til catch up which is closed, computable in one step. PositionA = SpeedA * t PositionT = SpeedT * t + 50 catchUp = 50 / (SpeedA - SpeedT) You are being amazingly dense. If you still don't understand, ask in sci.math, or sci.physics. Waste no more bandwidth. >Our experience shows it is wrong, but why? > >In a discreet world and with a reasonning and modelling >similar to Zeno's one. The calculus terminates *claiming* the end >of the model validity... This makes no sense at all. > Antoine Beugnard and Didier Guy > ENST de Bretagne, LIBr, Brest, France > beugnard@enstb.enst-bretagne.fr > guy@enstb.enst-bretagne.fr > > >Ps: We do not call into question mathematics and its powerfull use. >Even continuous mathematics!!. It works...but it may just be an >abstraction of the reality, a usefull tool that has not to be >related to the essence of world... You have failed to demonstrate any discrepancy between the prediction of the continuous, limit based math and reality. >The question is not only philosophical, people thinking discreet >machines cannot become "intelligent" assumes the world is >continuous ... which is not so obvious ... This point has already been addressed (better) several times by people pointing out the limited precision of NN's, and the existence of noise. Max
DOCTORJ@SLACVM.SLAC.STANFORD.EDU (Jon J Thaler) (04/13/91)
There are two distinct aspects of the "Continuous vs Discrete" issue, and it seems to me that the discussion has focussed only on one. They are * Discreteness due to the finite number of bits in computer representations of numbers. This has been the main topic of conversation, and I agree with those who say that it's not an issue. * Discreteness (or granularity) of the lattice on which the simulation is being run. This is the aspect that worries me, since it is a serious problem in the modeling of other physical systems (eg, weather). Do any people know of "proofs" (in the mathematical sense) or at least empirical evidence that the highly granular approach to AI models is a realistic approach to the (nearly) continuous system (the brain) that is being studied?
kohout@drinkme.cs.umd.edu (Robert Kohout) (04/14/91)
In article <383@zeus.enst-bretagne.fr> beugnard@zeus.enst-bretagne.fr (Antoine Beugnard) writes: > > >In article <382@batz.enst-bretagne.fr> beugnard@batz.enst-bretagne.fr (Antoine Beugnard) writes: >>> > >>What this ignores, of course, is the fact that some infinite series >>have a finite sum. Zeno's "paradox" may have baffled an ancient >>Greek, but it shouldn't fool anyone who knows calculus. The proper >>conclusion to draw in the Achilles and Turtle scenario is: "Achilles >>never reaches the Turtle, until he does." > >We are aware of that. The problem is that **a limit is NEVER reached**. >Zeno's model never becomes wrong...and Achilles never reaches the Turtle because >the limit cannot be reached. The model is actually troncated both in time and >space, that is T(n) and time(n) have limits that are never reached... > What are you talking about? What do you mean **a limit is NEVER reached**. Is that supposed to be some property of limits? When we say, "the limit of f(x), as x approaches infinity = Z", we don't mean "if only we could ever get there". for example 0.1 + 0.01 + 0.001 + 0.0001 + 0.00001 .... = 1/9 . The sum EQUALS 1/9. '=' does NOT mean "would be 1/9 if we could ever add all of these things up." IF the hare travels 10 times as fast as the tortoise, each summand (is that the word) of Zeno's paradox describes a distance 1/10 as large as the previous distance, and/or a time 1/10 the size of the previous increment. This sum has a limit, AND THEREFORE THE LIMIT IS REACHED!!! (See, I can emphasize too) The ancient Greeks may have had problems adding up infinitely many things, but just because one can break a quantity into infinitely many parts in no way implies that the quantity is infinite. For example, the number 1 can be written 0.9999....., which is just 0.9 + 0.09 + 0.009 + 0.0009+.... ad infinitum. 1 is not for this reason infinite. I fear I must be missing your point. Emphasizing "a limit is NEVER reached" makes it no less opaque. Knowing Zeno's paradox, you must realize that the limit is reached every time you overtake a slower moving object. Since this seems to be the crux of your argument, please clarify what you mean by asserting that "a limit is NEVER reached". Bob Kohout
rad@genco.bungi.com (Bob Daniel) (04/16/91)
In article <32913@mimsy.umd.edu> kohout@drinkme.cs.umd.edu (Robert Kohout) writes: >In article <383@zeus.enst-bretagne.fr> beugnard@zeus.enst-bretagne.fr (Antoine Beugnard) writes: >What are you talking about? What do you mean **a limit is NEVER reached**. >Is that supposed to be some property of limits? When we say, "the limit >of f(x), as x approaches infinity = Z", we don't mean "if only we could >ever get there". for example > >0.1 + 0.01 + 0.001 + 0.0001 + 0.00001 .... = 1/9 . > >The sum EQUALS 1/9. '=' does NOT mean "would be 1/9 if we could ever add >all of these things up." IF the hare travels 10 times as fast as the tortoise, But if you use the half life series, f(x) = 1/(2x) where x approaches infinity, f(x) will continue to get smaller and never reach zero or any finite result. You all are talking about velocity however. What limit are you approaching? Time? Distance? Velocity? Bob Daniel rad@genco.bungi.com
sarima@tdatirv.UUCP (Stanley Friesen) (04/16/91)
In article <19628@ogicse.ogi.edu> maxwebb@moe.cse.ogi.edu (Max G. Webb) writes: >Possibly, but if these systems are so computationally nonrobust >that rounding and discretization errors in a computer simulation >obliterate their value, then how could they have evolved? Keep >in mind that the neurons and the architecture of the nets have >been changing and evolving at the same time? Quite right. I suspect that one of the most important factors controlling neural evolution is stability in the face of noisy input. Ineed in some sense that could be said to be the primary dunction of the nervous system. >Actually, no. The tremendous range of light values detectable >(1 photon up to a sunbeam) is compressed to a narrower range >before the nervous system ever sees it, by the photoreceptor >(first of all). Secondly, there is plenty of evidence that it >is _edge_ information that is passed back, possibly other compressions >of the data. An article in the *latest* issue of Scientific Amerian (April, 1991) is relevant here. The primary encoding coming out of the eye seems to be local relative intensity on a logarithmic scale. (This does indeed amplify edges since they inovle abrupt changes in light levels). That is V = log(X/X-bar). (Where X-bar is the local weighted average of the light intensity). > While i also like floating point, >you would have a very hard time convincing a biologist that >a change of 2^-24 in the operating levels of a neuron would >destabilize the system! In fact one of the features of the eye circuitry is its stability over a wide range of inputs levels, and its ability to extract detail from limited, noisy data. -- --------------- uunet!tdatirv!sarima (Stanley Friesen)
sinkkone@kruuna.Helsinki.FI (Janne Sinkkonen) (04/18/91)
In article <381@sein.enst-bretagne.fr> beugnard@sein.enst-bretagne.fr (Antoine Beugnard) writes: > The paradox... Achilles and a turtle are about to run a race. > Obviously as Achilles can run faster than the turtle, he decides > to let her start N meters ahead. They start running at the same > time, but Achilles never reaches the turtle... > When Achilles reaches the previous Turtle location, she has gone > forward. d(n) denotes the distance she has covered while Achilles > was reaching her previous location. So, > > A(0) = 0 and A(n) = T(n-1) defines A(n). > T(0) = N and T(n) = T(n-1) + d(n) defines T(n). > According to Zenon, d(n) is positive then Achilles never reaches > the Turtle. "Never" here means "not for any value of N", and this is quite true. However, with these formulas you can only describe the system during the timepoints defined by some event A(n) or T(n) happening. (Timepoint is here referring to the time we are experiencing.) If you compute the timepoints when A(n) or T(n) happens, you find that all these times are _before_ the time (t0) Achilles really reaches the Turtle. So, "not for any value of N" really means "not before t0". Zenon's use of the word "never", or whatever he used, is misleading. It is only referring to the timepoints defined by A(n) or T(n), and misses the future after all these. > But in every day life, we can notice that Achilles overtakes her. > We do not call into question the mathematical modelling. So where is > the problem? Most people would say our model is wrong since it > does not describe the real world. We do ask why it is wrong!! It matches the real world perfectly, but describes only a part of it. ;) > Therefore the world is discontinuous !! No ?? Yes, I believe so! janne
beugnard@molene.enst-bretagne.fr (Antoine Beugnard) (04/19/91)
The talk is about continuous vs discrete but despite some remarks, we don't consider to waste the bandwidth. We have proposed an interpretation of an old paradox (refered as Zeno's) to show that the world could be discrete. Zeno's mind experiment leads to a mismatch between reality and its model. That is: Achilles that runs faster than the Turtle never reaches her. We have sent news to request advices or ideas...and we get 5 kinds of answer: 1 - "ok, it seems to prove that either time is not infinitely divisible, or space is not infinitively divisible" 2 - "this model is nonsense, you should have used another one, such as, Xa = N + Va*t and Xt = Vt*t" 3 - "An inifinite sum is not necessary infinite" 4 - "a limit is eventually reached" (Achilles never reaches the turtle until he does) We, now, would like to discus the four last answers. 2 -> The problem was not to find a model that gives "wider" results, the problem was to explain why Zeno's modeling is not correct. Well, it is correct, because when you calculate limits, you obtain that Achilles *could* reach the turtle at time t = N / (Va-Vt), and at position ... which are the results obtained by more classical models. To decide that this model is partial (truncated), you have to assume that time and space are not limited and then you can complete your modeling. But you have to be *external* to your model. From the inside, your model is definitivelly, for ever, correct, and Achilles cannot reach the Turtle. 3 -> We agree that an old Greek could not have thougth to that...but we are aware of that. The model is truncated. But we know it is truncated because we know time and space are not limited, the model doesn't. The calculus in the model never terminates, so, why would it be necessary to change the model... 4 -> take f(x) = 1 / x when x -> oo never reaches 0. So, we reiterate the question: why Zeno's model is false? We were advised to talk on a net about physics, sorry not to be connected to. We would like to add a physicist remark. In our physics, we know 3 fundamental constants: G : gravitation attraction (Newton) h\: energy quantum (Planck) c : ligth velocity (Einstein) These 3 constants may be expressed with 3 basic "units" T (in seconds), L (in meters) and M (in kilograms). As a system of 3 equations these units may be expressed with the 3 previous values (gravitation (G), velocity (v), energy (E)); we obtain: t = (EGv^-5)^(1/2) And since c >= v and h\ < E then t >= 5.4E-44 s Some physicist propose to call that quantity of time a Chronon. This physicist deduction lies on our modern physics, that may be incorrect... but Zeno's paradox is only a mind experiment...that is based only on our own experience...and is, therefore, much more general... Mind experiments lead Einstein to the theory of relativity... ; ) And they both (do they really?) conclude that time could be discontinuous. Are you so troubled thinking time could be discrete? Antoine Beugnard and Didier Guy ENST de Bretagne, LIBr, Brest, France beugnard@enstb.enst-bretagne.fr guy@enstb.enst-bretagne.fr
sinkkone@klaava.Helsinki.FI (Janne Sinkkonen) (04/22/91)
In article <384@molene.enst-bretagne.fr> beugnard@molene.enst-bretagne.fr (Antoine Beugnard and Didier Guy) write: [...] > 4 - "a limit is eventually reached" (Achilles never reaches the > turtle until he does) And 5 - saying "Achilles never reaches the turtle" makes sense only under some models, if you want to use all the words under the model with the same meanings they have in our everyday life. Within your model this is not possible. > Well, it is correct, because when you calculate limits, you obtain > that Achilles *could* reach the turtle at time t = N / (Va-Vt), > and at position ... which are the results obtained by more > classical models. To decide that this model is partial > (truncated), you have to assume that time and space are not > limited and then you can complete your modeling. But you have to > be *external* to your model. From the inside, your model is > definitivelly, for ever, correct, and Achilles cannot reach the > Turtle. You could model every phenomenon in the nature by tranforming the time axis somehow and by reformulating all the laws. As an example, let's set n=-t, and say "we will never die". This model is not even partial (truncated). Even better, let's set n=t0 for odd values of n, and n=t1 for even values. This is a little truncated model, but calculation is never terminating, as you would say (it is defined for all values of n). The problem is that "ever" and "never" are still external (in the same sense as you say). You should do _the_same_transformation_ for these two words, and to all others, as you do to the time axis. But then, "inside" the model, _everything_ is as here outside. This should be no surprising, if the model is correct. Thus, for a correct model, there is no such thing as inside-model-world being different from the outside world. If there was, then the model would be incorrect, or incomplete. Your model is incomplete, because inside it there does not exist all the outside things: some timepoints are missed. As a result, some concepts we use in this world are referring to something nonexistential if translated inside to your model. "Never" and "ever" belong to these words. So, inside the Zenon model, it makes no sense saying: "Achilles never reaches the turtle". You could define a new word inside your model world, which means "not for any value of n". But taking up all the context (remember that your thoughts are also running together along Achilles and the turtle), what it really means to "them" is by no means our "never". At the final gates of singularity, the model world is just the same as ours, and then suddenly, it will be undefined. There is no eternity. After these considerations, it should be clear why Zenon model is not as good as our conventional. It is not false, but it is incomplete. Incompleteness does not become apparent when compared to a hypothetical real world, but when compared to our conventional model. > Are you so troubled thinking time could be discrete? Still repeating, I'm not. But this has nothing to do with Zenon. I think continuum is just a convenient mathematical abstraction, a nice tool to work with. It is a form of infinity, and in the world there is no infinities. (a subjective belief) Janne
ziane@nuri.inria.fr (ziane mikal @) (04/25/91)
In the article cited above A. Beugnard proposes a demonstration that the world is not continuous based on the famous Zeno's paradox: "Achilles and the tortoise". I think the problem is that the description of Achilles's trajectory is biased. That description obviously only considers a series of observation points that, by construction, are always behind the turtle. It does not mean that this description describes Achilles' trajectory completely, that is precisely after he has passed the tortoise. The description of Achilles's trajectory has an infinite number of observation points, but by no means this implies an infinite time, and thus the complete trajectory of Achilles ! Whether or not, assuming an infinite number of observation points before Achilles passes the tortoise is a proper model of the world, is another question. Sorry if somebody has already pointed it out. Mikal Ziane.