Re: [agi] constructivist issues
*That* is what I was asking about when I asked which side you fell on. Do you think such extensions are arbitrary, or do you think there is a fact of the matter? The extensions are clearly judged on whether or not they accurately reflect the empirical world *as currently known* -- so they aren't arbitrary in that sense. On the other hand, there may not be just a single set of extensions that accurately reflect the world so I guess that you could say that choosing among sets of extensions that both accurately reflect the world is (necessarily) an arbitrary process since there is no additional information to go on (though there are certainly heuristics like Occam's razor -- but they are more about getting a usable or more likely to hold up under future observations or more likely to be easily modified to match future observations theory . . . .). The world is real. Our explanations and theories are constructed. For any complete system, you can take the classical approach but incompleteness (of current information which then causes undecidability) ever forces you into constructivism to create an ever-expanding series of shells of stronger systems to explain those systems contained by them. - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Monday, October 27, 2008 5:43 PM Subject: Re: [agi] constructivist issues Mark, Sorry, I accidentally called you Mike in the previous email! Anyway, you said: Also, you seem to be ascribing arbitrariness to constructivism which is emphatically not the case. I didn't mean to ascribe arbitrariness to constructivism-- what I meant was that constructivists would (as I understand it) ascribe arbitrariness to extensions of arithmetic. A constructivist sees the fact of the matter as undefined for undecidable statements, so adding axioms that make them decidable is necessarily an arbitrary process. The classical view, on the other hand, sees it as an attempt to increase the amount of true information contained in the axioms-- so there is a right and wrong. *That* is what I was asking about when I asked which side you fell on. Do you think such extensions are arbitrary, or do you think there is a fact of the matter? --Abram On Mon, Oct 27, 2008 at 3:33 PM, Mark Waser [EMAIL PROTECTED] wrote: The number of possible descriptions is countable I disagree. if we were able to randomly pick a real number between 1 and 0, it would be indescribable with probability 1. If we were able to randomly pick a real number between 1 and 0, it would be indescribable with probability *approaching* 1. Which side do you fall on? I still say that the sides are parts of the same coin. In other words, we're proving arithmetic consistent only by adding to its definition, which hardly counts. The classical viewpoint, of course, is that the stronger system is actually correct. Its additional axioms are not arbitrary. So, the proof reflects the truth. What is the stronger system other than an addition? And the viewpoint that the stronger system is actually correct -- is that an assumption? a truth? what? (And how do you know?) Also, you seem to be ascribing arbitrariness to constructivism which is emphatically not the case. - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Monday, October 27, 2008 2:53 PM Subject: Re: [agi] constructivist issues Mark, The number of possible descriptions is countable, while the number of possible real numbers is uncountable. So, there are infinitely many more real numbers that are individually indescribable, then describable; so much so that if we were able to randomly pick a real number between 1 and 0, it would be indescribable with probability 1. I am getting this from Chaitin's book Meta Math!. I believe that arithmetic is a formal and complete system. I'm not a constructivist where formal and complete systems are concerned (since there is nothing more to construct). Oh, I believe there is some confusion here because of my use of the word arithmetic. I don't mean grade-school addition/subtraction/multiplication/division. What I mean is the axiomatic theory of numbers, which Godel showed to be incomplete if it is consistent. Godel also proved that one of the incompletenesses in arithmetic was that it could not prove its own consistency. Stronger logical systems can and have proven its consistency, but any particular logical system cannot prove its own consistency. It seems to me that the constructivist viewpoint says, The so-called stronger system merely defines truth in more cases; but, we could just as easily take the opposite definitions. In other words, we're proving arithmetic consistent only by adding to its definition, which hardly counts. The classical viewpoint, of course, is that the stronger system is actually correct. Its additional axioms are not arbitrary. So, the proof reflects the truth. Which side do you
Re: [agi] constructivist issues
Mark, You assert that the extensions are judged on how well they reflect the world. The extension currently under discussion is one that allows us to prove the consistency of Arithmetic. So, it seems, you count that as something observable in the world-- no mathematician has ever proved a contradiction from the axioms of arithmetic, so they seem consistent. If this is indeed what you are saying, then you are in line with the classical view in this respect (and with my opinion). But, if this is your view, I don't see how you can maintain the constructivist assertion that Godelian statements are undecidable because they are undefined by the axioms. It seems that, instead, you are agreeing with the classical notion that there is in fact a truth of the matter concerning Godelian statements, we're just unable to deduce that truth from the axioms. --Abram On Tue, Oct 28, 2008 at 7:21 AM, Mark Waser [EMAIL PROTECTED] wrote: *That* is what I was asking about when I asked which side you fell on. Do you think such extensions are arbitrary, or do you think there is a fact of the matter? The extensions are clearly judged on whether or not they accurately reflect the empirical world *as currently known* -- so they aren't arbitrary in that sense. On the other hand, there may not be just a single set of extensions that accurately reflect the world so I guess that you could say that choosing among sets of extensions that both accurately reflect the world is (necessarily) an arbitrary process since there is no additional information to go on (though there are certainly heuristics like Occam's razor -- but they are more about getting a usable or more likely to hold up under future observations or more likely to be easily modified to match future observations theory . . . .). The world is real. Our explanations and theories are constructed. For any complete system, you can take the classical approach but incompleteness (of current information which then causes undecidability) ever forces you into constructivism to create an ever-expanding series of shells of stronger systems to explain those systems contained by them. --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] constructivist issues
Abram, I could agree with the statement that there are uncountably many *potential* numbers but I'm going to argue that any number that actually exists is eminently describable. Take the set of all numbers that are defined far enough after the decimal point that they never accurately describe anything manifest in the physical universe and are never described or invoked by any entity in the physical universe (specifically including a method for the generation of that number). Pi is clearly not in the set since a) it describes all sorts of ratios in the physical universe and b) there is a clear formula for generating successive approximations of it. My question is -- do these numbers really exist? And, if so, by what definition of exist since my definition is meant to rule out any form of manifestation whether physical or as a concept. Clearly these numbers have the potential to exist -- but it should be equally clear that they do not actually exist (i.e. they are never individuated out of the class). Any number which truly exists has at least one description either of the type of a) the number which is manifest as or b) the number which is generated by. Classicists seem to want to insist that all of these potential numbers actually do exist -- so they can make statements like There are uncountably many real numbers that no one can ever describe in any manner. I ask of them (and you) -- Show me just one.:-) --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] constructivist issues
Hi, We keep going around and around because you keep dropping my distinction between two different cases . . . . The statement that The cat is red is undecidable by arithmetic because it can't even be defined in terms of the axioms of arithmetic (i.e. it has *meaning* outside of arithmetic). You need to construct additions/extensions to arithmetic to even start to deal with it. The statement that Pi is a normal number is decidable by arithmetic because each of the terms has meaning in arithmetic (so it certainly can be disproved by counter-example). It may not be deducible from the axioms but the meaning of the statement is contained within the axioms. The first example is what you call a constructivist view. The second example is what you call a classical view. Which one I take is eminently context-dependent and you keep dropping the context. If the meaning of the statement is contained within the system, it is decidable even if it is not deducible. If the meaning is beyond the system, then it is not decidable because you can't even express what you're deciding. Mark - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Tuesday, October 28, 2008 9:32 AM Subject: Re: [agi] constructivist issues Mark, You assert that the extensions are judged on how well they reflect the world. The extension currently under discussion is one that allows us to prove the consistency of Arithmetic. So, it seems, you count that as something observable in the world-- no mathematician has ever proved a contradiction from the axioms of arithmetic, so they seem consistent. If this is indeed what you are saying, then you are in line with the classical view in this respect (and with my opinion). But, if this is your view, I don't see how you can maintain the constructivist assertion that Godelian statements are undecidable because they are undefined by the axioms. It seems that, instead, you are agreeing with the classical notion that there is in fact a truth of the matter concerning Godelian statements, we're just unable to deduce that truth from the axioms. --Abram On Tue, Oct 28, 2008 at 7:21 AM, Mark Waser [EMAIL PROTECTED] wrote: *That* is what I was asking about when I asked which side you fell on. Do you think such extensions are arbitrary, or do you think there is a fact of the matter? The extensions are clearly judged on whether or not they accurately reflect the empirical world *as currently known* -- so they aren't arbitrary in that sense. On the other hand, there may not be just a single set of extensions that accurately reflect the world so I guess that you could say that choosing among sets of extensions that both accurately reflect the world is (necessarily) an arbitrary process since there is no additional information to go on (though there are certainly heuristics like Occam's razor -- but they are more about getting a usable or more likely to hold up under future observations or more likely to be easily modified to match future observations theory . . . .). The world is real. Our explanations and theories are constructed. For any complete system, you can take the classical approach but incompleteness (of current information which then causes undecidability) ever forces you into constructivism to create an ever-expanding series of shells of stronger systems to explain those systems contained by them. --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?; Powered by Listbox: http://www.listbox.com --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] constructivist issues
Mark, The question that is puzzling, though, is: how can it be that these uncomputable, inexpressible entities are so bloody useful ;-) ... for instance in differential calculus ... Also, to say that uncomputable entities don't exist because they can't be finitely described, is basically just to *define* existence as finite describability. So this is more a philosophical position on what exists means than an argument that could convince anyone. I have some more detailed thoughts on these issues that I'll write down sometime soon when I get the time. My position is fairly close to yours but I think that with these sorts of issues, the devil is in the details. ben On Tue, Oct 28, 2008 at 6:53 AM, Mark Waser [EMAIL PROTECTED] wrote: Abram, I could agree with the statement that there are uncountably many *potential* numbers but I'm going to argue that any number that actually exists is eminently describable. Take the set of all numbers that are defined far enough after the decimal point that they never accurately describe anything manifest in the physical universe and are never described or invoked by any entity in the physical universe (specifically including a method for the generation of that number). Pi is clearly not in the set since a) it describes all sorts of ratios in the physical universe and b) there is a clear formula for generating successive approximations of it. My question is -- do these numbers really exist? And, if so, by what definition of exist since my definition is meant to rule out any form of manifestation whether physical or as a concept. Clearly these numbers have the potential to exist -- but it should be equally clear that they do not actually exist (i.e. they are never individuated out of the class). Any number which truly exists has at least one description either of the type of a) the number which is manifest as or b) the number which is generated by. Classicists seem to want to insist that all of these potential numbers actually do exist -- so they can make statements like There are uncountably many real numbers that no one can ever describe in any manner. I ask of them (and you) -- Show me just one.:-) -- *agi* | Archives https://www.listbox.com/member/archive/303/=now https://www.listbox.com/member/archive/rss/303/ | Modifyhttps://www.listbox.com/member/?;Your Subscription http://www.listbox.com -- Ben Goertzel, PhD CEO, Novamente LLC and Biomind LLC Director of Research, SIAI [EMAIL PROTECTED] A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects. -- Robert Heinlein --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] constructivist issues
MW:Pi is a normal number is decidable by arithmetic because each of the terms has meaning in arithmetic Can it be expressed in purely mathematical terms/signs without using language? --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] constructivist issues
yes On Tue, Oct 28, 2008 at 8:46 AM, Mike Tintner [EMAIL PROTECTED]wrote: MW:Pi is a normal number is decidable by arithmetic because each of the terms has meaning in arithmetic Can it be expressed in purely mathematical terms/signs without using language? --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?; Powered by Listbox: http://www.listbox.com -- Ben Goertzel, PhD CEO, Novamente LLC and Biomind LLC Director of Research, SIAI [EMAIL PROTECTED] A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects. -- Robert Heinlein --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
[agi] Occam's Razor and its abuse
Triggered by several recent discussions, I'd like to make the following position statement, though won't commit myself to long debate on it. ;-) Occam's Razor, in its original form, goes like entities must not be multiplied beyond necessity, and it is often stated as All other things being equal, the simplest solution is the best or when multiple competing theories are equal in other respects, the principle recommends selecting the theory that introduces the fewest assumptions and postulates the fewest entities --- all from http://en.wikipedia.org/wiki/Occam's_razor I fully agree with all of the above statements. However, to me, there are two common misunderstandings associated with it in the context of AGI and philosophy of science. (1) To take this statement as self-evident or a stand-alone postulate To me, it is derived or implied by the insufficiency of resources. If a system has sufficient resources, it has no good reason to prefer a simpler theory. (2) To take it to mean The simplest answer is usually the correct answer. This is a very different statement, which cannot be justified either analytically or empirically. When theory A is an approximation of theory B, usually the former is simpler than the latter, but less correct or accurate, in terms of its relation with all available evidence. When we are short in resources and have a low demand on accuracy, we often prefer A over B, but it does not mean that by doing so we judge A as more correct than B. In summary, in choosing among alternative theories or conclusions, the preference for simplicity comes from shortage of resources, though simplicity and correctness are logically independent of each other. Pei --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] constructivist issues
Mark, Yes, I do keep dropping the context. This is because I am concerned only with mathematical knowledge at the moment. I should have been more specific. So, if I understand you right, you are saying that you take the classical view when it comes to mathematics. In that case, shouldn't you agree with the classical perspective on Godelian incompleteness, since Godel's incompleteness theorem is about mathematical systems? --Abram On Tue, Oct 28, 2008 at 10:20 AM, Mark Waser [EMAIL PROTECTED] wrote: Hi, We keep going around and around because you keep dropping my distinction between two different cases . . . . The statement that The cat is red is undecidable by arithmetic because it can't even be defined in terms of the axioms of arithmetic (i.e. it has *meaning* outside of arithmetic). You need to construct additions/extensions to arithmetic to even start to deal with it. The statement that Pi is a normal number is decidable by arithmetic because each of the terms has meaning in arithmetic (so it certainly can be disproved by counter-example). It may not be deducible from the axioms but the meaning of the statement is contained within the axioms. The first example is what you call a constructivist view. The second example is what you call a classical view. Which one I take is eminently context-dependent and you keep dropping the context. If the meaning of the statement is contained within the system, it is decidable even if it is not deducible. If the meaning is beyond the system, then it is not decidable because you can't even express what you're deciding. Mark - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Tuesday, October 28, 2008 9:32 AM Subject: Re: [agi] constructivist issues --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] Occam's Razor and its abuse
Ben, Thanks. So the other people now see that I'm not attacking a straw man. My solution to Hume's problem, as embedded in the experience-grounded semantics, is to assume no predictability, but to justify induction as adaptation. However, it is a separate topic which I've explained in my other publications. Here I just want to point out that the original and basic meaning of Occam's Razor and those two common (mis)usages of it are not necessarily the same. I fully agree with the former, but not the latter, and I haven't seen any convincing justification of the latter. Instead, they are often taken as granted, under the name of Occam's Razor. Pei On Tue, Oct 28, 2008 at 12:37 PM, Ben Goertzel [EMAIL PROTECTED] wrote: Hi Pei, This is an interesting perspective; I just want to clarify for others on the list that it is a particular and controversial perspective, and contradicts the perspectives of many other well-informed research professionals and deep thinkers on relevant topics. Many serious thinkers in the area *do* consider Occam's Razor a standalone postulate. This fits in naturally with the Bayesian perspective, in which one needs to assume *some* prior distribution, so one often assumes some sort of Occam prior (e.g. the Solomonoff-Levin prior, the speed prior, etc.) as a standalone postulate. Hume pointed out that induction (in the old sense of extrapolating from the past into the future) is not solvable except by introducing some kind of a priori assumption. Occam's Razor, in one form or another, is a suitable a prior assumption to plug into this role. If you want to replace the Occam's Razor assumption with the assumption that the world is predictable by systems with limited resources, and we will prefer explanations that consume less resources, that seems unproblematic as it's basically equivalent to assuming an Occam prior. On the other hand, I just want to point out that to get around Hume's complaint you do need to make *some* kind of assumption about the regularity of the world. What kind of assumption of this nature underlies your work on NARS (if any)? ben On Tue, Oct 28, 2008 at 8:58 AM, Pei Wang [EMAIL PROTECTED] wrote: Triggered by several recent discussions, I'd like to make the following position statement, though won't commit myself to long debate on it. ;-) Occam's Razor, in its original form, goes like entities must not be multiplied beyond necessity, and it is often stated as All other things being equal, the simplest solution is the best or when multiple competing theories are equal in other respects, the principle recommends selecting the theory that introduces the fewest assumptions and postulates the fewest entities --- all from http://en.wikipedia.org/wiki/Occam's_razor I fully agree with all of the above statements. However, to me, there are two common misunderstandings associated with it in the context of AGI and philosophy of science. (1) To take this statement as self-evident or a stand-alone postulate To me, it is derived or implied by the insufficiency of resources. If a system has sufficient resources, it has no good reason to prefer a simpler theory. (2) To take it to mean The simplest answer is usually the correct answer. This is a very different statement, which cannot be justified either analytically or empirically. When theory A is an approximation of theory B, usually the former is simpler than the latter, but less correct or accurate, in terms of its relation with all available evidence. When we are short in resources and have a low demand on accuracy, we often prefer A over B, but it does not mean that by doing so we judge A as more correct than B. In summary, in choosing among alternative theories or conclusions, the preference for simplicity comes from shortage of resources, though simplicity and correctness are logically independent of each other. Pei --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?; Powered by Listbox: http://www.listbox.com -- Ben Goertzel, PhD CEO, Novamente LLC and Biomind LLC Director of Research, SIAI [EMAIL PROTECTED] A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects. -- Robert Heinlein agi | Archives | Modify Your Subscription --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed:
Re: [agi] Occam's Razor and its abuse
Ben, You assert that Pei is forced to make an assumption about the regulatiry of the world to justify adaptation. Pei could also take a different argument. He could try to show that *if* a strategy exists that can be implemented given the finite resources, NARS will eventually find it. Thus, adaptation is justified on a sort of we might as well try basis. (The proof would involve showing that NARS searches the state of finite-state-machines that can be implemented with the resources at hand, and is more probable to stay for longer periods of time in configurations that give more reward, such that NARS would eventually settle on a configuration if that configuration consistently gave the highest reward.) So, some form of learning can take place with no assumptions. The problem is that the search space is exponential in the resources available, so there is some maximum point where the system would perform best (because the amount of resources match the problem), but giving the system more resources would hurt performance (because the system searches the unnecessarily large search space). So, in this sense, the system's behavior seems counterintuitive-- it does not seem to be taking advantage of the increased resources. I'm not claiming NARS would have that problem, of course just that a theoretical no-assumption learner would. --Abram On Tue, Oct 28, 2008 at 2:12 PM, Ben Goertzel [EMAIL PROTECTED] wrote: On Tue, Oct 28, 2008 at 10:00 AM, Pei Wang [EMAIL PROTECTED] wrote: Ben, Thanks. So the other people now see that I'm not attacking a straw man. My solution to Hume's problem, as embedded in the experience-grounded semantics, is to assume no predictability, but to justify induction as adaptation. However, it is a separate topic which I've explained in my other publications. Right, but justifying induction as adaptation only works if the environment is assumed to have certain regularities which can be adapted to. In a random environment, adaptation won't work. So, still, to justify induction as adaptation you have to make *some* assumptions about the world. The Occam prior gives one such assumption: that (to give just one form) sets of observations in the world tend to be producible by short computer programs. For adaptation to successfully carry out induction, *some* vaguely comparable property to this must hold, and I'm not sure if you have articulated which one you assume, or if you leave this open. In effect, you implicitly assume something like an Occam prior, because you're saying that a system with finite resources can successfully adapt to the world ... which means that sets of observations in the world *must* be approximately summarizable via subprograms that can be executed within this system. So I argue that, even though it's not your preferred way to think about it, your own approach to AI theory and practice implicitly assumes some variant of the Occam prior holds in the real world. Here I just want to point out that the original and basic meaning of Occam's Razor and those two common (mis)usages of it are not necessarily the same. I fully agree with the former, but not the latter, and I haven't seen any convincing justification of the latter. Instead, they are often taken as granted, under the name of Occam's Razor. I agree that the notion of an Occam prior is a significant conceptual beyond the original Occam's Razor precept enounced long ago. Also, I note that, for those who posit the Occam prior as a **prior assumption**, there is not supposed to be any convincing justification for it. The idea is simply that: one must make *some* assumption (explicitly or implicitly) if one wants to do induction, and this is the assumption that some people choose to make. -- Ben G agi | Archives | Modify Your Subscription --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] Occam's Razor and its abuse
Ben, It seems that you agree the issue I pointed out really exists, but just take it as a necessary evil. Furthermore, you think I also assumed the same thing, though I failed to see it. I won't argue against the necessary evil part --- as far as you agree that those postulates (such as the universe is computable) are not convincingly justified. I won't try to disprove them. As for the latter part, I don't think you can convince me that you know me better than I know myself. ;-) The following is from http://nars.wang.googlepages.com/wang.semantics.pdf , page 28: If the answers provided by NARS are fallible, in what sense these answers are better than arbitrary guesses? This leads us to the concept of rationality. When infallible predictions cannot be obtained (due to insufficient knowledge and resources), answers based on past experience are better than arbitrary guesses, if the environment is relatively stable. To say an answer is only a summary of past experience (thus no future confirmation guaranteed) does not make it equal to an arbitrary conclusion — it is what adaptation means. Adaptation is the process in which a system changes its behaviors as if the future is similar to the past. It is a rational process, even though individual conclusions it produces are often wrong. For this reason, valid inference rules (deduction, induction, abduction, and so on) are the ones whose conclusions correctly (according to the semantics) summarize the evidence in the premises. They are truth-preserving in this sense, not in the model-theoretic sense that they always generate conclusions which are immune from future revision. --- so you see, I don't assume adaptation will always be successful, even successful to a certain probability. You can dislike this conclusion, though you cannot say it is the same as what is assumed by Novamente and AIXI. Pei On Tue, Oct 28, 2008 at 2:12 PM, Ben Goertzel [EMAIL PROTECTED] wrote: On Tue, Oct 28, 2008 at 10:00 AM, Pei Wang [EMAIL PROTECTED] wrote: Ben, Thanks. So the other people now see that I'm not attacking a straw man. My solution to Hume's problem, as embedded in the experience-grounded semantics, is to assume no predictability, but to justify induction as adaptation. However, it is a separate topic which I've explained in my other publications. Right, but justifying induction as adaptation only works if the environment is assumed to have certain regularities which can be adapted to. In a random environment, adaptation won't work. So, still, to justify induction as adaptation you have to make *some* assumptions about the world. The Occam prior gives one such assumption: that (to give just one form) sets of observations in the world tend to be producible by short computer programs. For adaptation to successfully carry out induction, *some* vaguely comparable property to this must hold, and I'm not sure if you have articulated which one you assume, or if you leave this open. In effect, you implicitly assume something like an Occam prior, because you're saying that a system with finite resources can successfully adapt to the world ... which means that sets of observations in the world *must* be approximately summarizable via subprograms that can be executed within this system. So I argue that, even though it's not your preferred way to think about it, your own approach to AI theory and practice implicitly assumes some variant of the Occam prior holds in the real world. Here I just want to point out that the original and basic meaning of Occam's Razor and those two common (mis)usages of it are not necessarily the same. I fully agree with the former, but not the latter, and I haven't seen any convincing justification of the latter. Instead, they are often taken as granted, under the name of Occam's Razor. I agree that the notion of an Occam prior is a significant conceptual beyond the original Occam's Razor precept enounced long ago. Also, I note that, for those who posit the Occam prior as a **prior assumption**, there is not supposed to be any convincing justification for it. The idea is simply that: one must make *some* assumption (explicitly or implicitly) if one wants to do induction, and this is the assumption that some people choose to make. -- Ben G agi | Archives | Modify Your Subscription --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] Occam's Razor and its abuse
Most certainly ... and the human mind seems to make a lot of other, more specialized assumptions about the environment also ... so that unless the environment satisfies a bunch of these other more specialized assumptions, its adaptation will be very slow and resource-inefficient... ben g On Tue, Oct 28, 2008 at 12:05 PM, Pei Wang [EMAIL PROTECTED] wrote: We can say the same thing for the human mind, right? Pei On Tue, Oct 28, 2008 at 2:54 PM, Ben Goertzel [EMAIL PROTECTED] wrote: Sure ... but my point is that unless the environment satisfies a certain Occam-prior-like property, NARS will be useless... ben On Tue, Oct 28, 2008 at 11:52 AM, Abram Demski [EMAIL PROTECTED] wrote: Ben, You assert that Pei is forced to make an assumption about the regulatiry of the world to justify adaptation. Pei could also take a different argument. He could try to show that *if* a strategy exists that can be implemented given the finite resources, NARS will eventually find it. Thus, adaptation is justified on a sort of we might as well try basis. (The proof would involve showing that NARS searches the state of finite-state-machines that can be implemented with the resources at hand, and is more probable to stay for longer periods of time in configurations that give more reward, such that NARS would eventually settle on a configuration if that configuration consistently gave the highest reward.) So, some form of learning can take place with no assumptions. The problem is that the search space is exponential in the resources available, so there is some maximum point where the system would perform best (because the amount of resources match the problem), but giving the system more resources would hurt performance (because the system searches the unnecessarily large search space). So, in this sense, the system's behavior seems counterintuitive-- it does not seem to be taking advantage of the increased resources. I'm not claiming NARS would have that problem, of course just that a theoretical no-assumption learner would. --Abram On Tue, Oct 28, 2008 at 2:12 PM, Ben Goertzel [EMAIL PROTECTED] wrote: On Tue, Oct 28, 2008 at 10:00 AM, Pei Wang [EMAIL PROTECTED] wrote: Ben, Thanks. So the other people now see that I'm not attacking a straw man. My solution to Hume's problem, as embedded in the experience-grounded semantics, is to assume no predictability, but to justify induction as adaptation. However, it is a separate topic which I've explained in my other publications. Right, but justifying induction as adaptation only works if the environment is assumed to have certain regularities which can be adapted to. In a random environment, adaptation won't work. So, still, to justify induction as adaptation you have to make *some* assumptions about the world. The Occam prior gives one such assumption: that (to give just one form) sets of observations in the world tend to be producible by short computer programs. For adaptation to successfully carry out induction, *some* vaguely comparable property to this must hold, and I'm not sure if you have articulated which one you assume, or if you leave this open. In effect, you implicitly assume something like an Occam prior, because you're saying that a system with finite resources can successfully adapt to the world ... which means that sets of observations in the world *must* be approximately summarizable via subprograms that can be executed within this system. So I argue that, even though it's not your preferred way to think about it, your own approach to AI theory and practice implicitly assumes some variant of the Occam prior holds in the real world. Here I just want to point out that the original and basic meaning of Occam's Razor and those two common (mis)usages of it are not necessarily the same. I fully agree with the former, but not the latter, and I haven't seen any convincing justification of the latter. Instead, they are often taken as granted, under the name of Occam's Razor. I agree that the notion of an Occam prior is a significant conceptual beyond the original Occam's Razor precept enounced long ago. Also, I note that, for those who posit the Occam prior as a **prior assumption**, there is not supposed to be any convincing justification for it. The idea is simply that: one must make *some* assumption (explicitly or implicitly) if one wants to do induction, and this is the assumption that some people choose to make. -- Ben G agi | Archives | Modify Your Subscription --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed:
Re: [agi] Occam's Razor and its abuse
We can say the same thing for the human mind, right? Pei On Tue, Oct 28, 2008 at 2:54 PM, Ben Goertzel [EMAIL PROTECTED] wrote: Sure ... but my point is that unless the environment satisfies a certain Occam-prior-like property, NARS will be useless... ben On Tue, Oct 28, 2008 at 11:52 AM, Abram Demski [EMAIL PROTECTED] wrote: Ben, You assert that Pei is forced to make an assumption about the regulatiry of the world to justify adaptation. Pei could also take a different argument. He could try to show that *if* a strategy exists that can be implemented given the finite resources, NARS will eventually find it. Thus, adaptation is justified on a sort of we might as well try basis. (The proof would involve showing that NARS searches the state of finite-state-machines that can be implemented with the resources at hand, and is more probable to stay for longer periods of time in configurations that give more reward, such that NARS would eventually settle on a configuration if that configuration consistently gave the highest reward.) So, some form of learning can take place with no assumptions. The problem is that the search space is exponential in the resources available, so there is some maximum point where the system would perform best (because the amount of resources match the problem), but giving the system more resources would hurt performance (because the system searches the unnecessarily large search space). So, in this sense, the system's behavior seems counterintuitive-- it does not seem to be taking advantage of the increased resources. I'm not claiming NARS would have that problem, of course just that a theoretical no-assumption learner would. --Abram On Tue, Oct 28, 2008 at 2:12 PM, Ben Goertzel [EMAIL PROTECTED] wrote: On Tue, Oct 28, 2008 at 10:00 AM, Pei Wang [EMAIL PROTECTED] wrote: Ben, Thanks. So the other people now see that I'm not attacking a straw man. My solution to Hume's problem, as embedded in the experience-grounded semantics, is to assume no predictability, but to justify induction as adaptation. However, it is a separate topic which I've explained in my other publications. Right, but justifying induction as adaptation only works if the environment is assumed to have certain regularities which can be adapted to. In a random environment, adaptation won't work. So, still, to justify induction as adaptation you have to make *some* assumptions about the world. The Occam prior gives one such assumption: that (to give just one form) sets of observations in the world tend to be producible by short computer programs. For adaptation to successfully carry out induction, *some* vaguely comparable property to this must hold, and I'm not sure if you have articulated which one you assume, or if you leave this open. In effect, you implicitly assume something like an Occam prior, because you're saying that a system with finite resources can successfully adapt to the world ... which means that sets of observations in the world *must* be approximately summarizable via subprograms that can be executed within this system. So I argue that, even though it's not your preferred way to think about it, your own approach to AI theory and practice implicitly assumes some variant of the Occam prior holds in the real world. Here I just want to point out that the original and basic meaning of Occam's Razor and those two common (mis)usages of it are not necessarily the same. I fully agree with the former, but not the latter, and I haven't seen any convincing justification of the latter. Instead, they are often taken as granted, under the name of Occam's Razor. I agree that the notion of an Occam prior is a significant conceptual beyond the original Occam's Razor precept enounced long ago. Also, I note that, for those who posit the Occam prior as a **prior assumption**, there is not supposed to be any convincing justification for it. The idea is simply that: one must make *some* assumption (explicitly or implicitly) if one wants to do induction, and this is the assumption that some people choose to make. -- Ben G agi | Archives | Modify Your Subscription --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?; Powered by Listbox: http://www.listbox.com -- Ben Goertzel, PhD CEO, Novamente LLC and Biomind LLC Director of Research, SIAI [EMAIL PROTECTED] A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze
Re: [agi] constructivist issues
The question that is puzzling, though, is: how can it be that these uncomputable, inexpressible entities are so bloody useful ;-) ... for instance in differential calculus ... Differential calculus doesn't use those individual entities . . . . Also, to say that uncomputable entities don't exist because they can't be finitely described, is basically just to *define* existence as finite describability. I never said any such thing. I referenced a class of numbers that I defined as never physically manifesting and never being conceptually distinct and then asked if they existed. Clearly some portion of your liver that I can't define finitely still exists because it is physically manifest. So this is more a philosophical position on what exists means than an argument that could convince anyone. Yes, in that I basically defined my version of exists as physically manifest and/or described or invoked and then asked if that matched Abram's definition. No, in that you're now coming in with half (or less) of my definition and arguing that I'm unconvincing. :-) - Original Message - From: Ben Goertzel To: agi@v2.listbox.com Sent: Tuesday, October 28, 2008 11:44 AM Subject: Re: [agi] constructivist issues Mark, The question that is puzzling, though, is: how can it be that these uncomputable, inexpressible entities are so bloody useful ;-) ... for instance in differential calculus ... Also, to say that uncomputable entities don't exist because they can't be finitely described, is basically just to *define* existence as finite describability. So this is more a philosophical position on what exists means than an argument that could convince anyone. I have some more detailed thoughts on these issues that I'll write down sometime soon when I get the time. My position is fairly close to yours but I think that with these sorts of issues, the devil is in the details. ben On Tue, Oct 28, 2008 at 6:53 AM, Mark Waser [EMAIL PROTECTED] wrote: Abram, I could agree with the statement that there are uncountably many *potential* numbers but I'm going to argue that any number that actually exists is eminently describable. Take the set of all numbers that are defined far enough after the decimal point that they never accurately describe anything manifest in the physical universe and are never described or invoked by any entity in the physical universe (specifically including a method for the generation of that number). Pi is clearly not in the set since a) it describes all sorts of ratios in the physical universe and b) there is a clear formula for generating successive approximations of it. My question is -- do these numbers really exist? And, if so, by what definition of exist since my definition is meant to rule out any form of manifestation whether physical or as a concept. Clearly these numbers have the potential to exist -- but it should be equally clear that they do not actually exist (i.e. they are never individuated out of the class). Any number which truly exists has at least one description either of the type of a) the number which is manifest as or b) the number which is generated by. Classicists seem to want to insist that all of these potential numbers actually do exist -- so they can make statements like There are uncountably many real numbers that no one can ever describe in any manner. I ask of them (and you) -- Show me just one.:-) agi | Archives | Modify Your Subscription -- Ben Goertzel, PhD CEO, Novamente LLC and Biomind LLC Director of Research, SIAI [EMAIL PROTECTED] A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects. -- Robert Heinlein -- agi | Archives | Modify Your Subscription --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] constructivist issues
In that case, shouldn't you agree with the classical perspective on Godelian incompleteness, since Godel's incompleteness theorem is about mathematical systems? It depends. Are you asking me a fully defined question within the current axioms of what you call mathematical systems (i.e. a pi question) or a cat question (which could *eventually* be defined by some massive extensions to your mathematical systems but which isn't currently defined in what you're calling mathematical systems)? Saying that Gödel is about mathematical systems is not saying that it's not about cat-including systems. - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Tuesday, October 28, 2008 12:06 PM Subject: Re: [agi] constructivist issues Mark, Yes, I do keep dropping the context. This is because I am concerned only with mathematical knowledge at the moment. I should have been more specific. So, if I understand you right, you are saying that you take the classical view when it comes to mathematics. In that case, shouldn't you agree with the classical perspective on Godelian incompleteness, since Godel's incompleteness theorem is about mathematical systems? --Abram On Tue, Oct 28, 2008 at 10:20 AM, Mark Waser [EMAIL PROTECTED] wrote: Hi, We keep going around and around because you keep dropping my distinction between two different cases . . . . The statement that The cat is red is undecidable by arithmetic because it can't even be defined in terms of the axioms of arithmetic (i.e. it has *meaning* outside of arithmetic). You need to construct additions/extensions to arithmetic to even start to deal with it. The statement that Pi is a normal number is decidable by arithmetic because each of the terms has meaning in arithmetic (so it certainly can be disproved by counter-example). It may not be deducible from the axioms but the meaning of the statement is contained within the axioms. The first example is what you call a constructivist view. The second example is what you call a classical view. Which one I take is eminently context-dependent and you keep dropping the context. If the meaning of the statement is contained within the system, it is decidable even if it is not deducible. If the meaning is beyond the system, then it is not decidable because you can't even express what you're deciding. Mark - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Tuesday, October 28, 2008 9:32 AM Subject: Re: [agi] constructivist issues --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?; Powered by Listbox: http://www.listbox.com --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] Occam's Razor and its abuse
Abram, I agree with your basic idea in the following, though I usually put it in different form. Pei On Tue, Oct 28, 2008 at 2:52 PM, Abram Demski [EMAIL PROTECTED] wrote: Ben, You assert that Pei is forced to make an assumption about the regulatiry of the world to justify adaptation. Pei could also take a different argument. He could try to show that *if* a strategy exists that can be implemented given the finite resources, NARS will eventually find it. Thus, adaptation is justified on a sort of we might as well try basis. (The proof would involve showing that NARS searches the state of finite-state-machines that can be implemented with the resources at hand, and is more probable to stay for longer periods of time in configurations that give more reward, such that NARS would eventually settle on a configuration if that configuration consistently gave the highest reward.) So, some form of learning can take place with no assumptions. The problem is that the search space is exponential in the resources available, so there is some maximum point where the system would perform best (because the amount of resources match the problem), but giving the system more resources would hurt performance (because the system searches the unnecessarily large search space). So, in this sense, the system's behavior seems counterintuitive-- it does not seem to be taking advantage of the increased resources. I'm not claiming NARS would have that problem, of course just that a theoretical no-assumption learner would. --Abram On Tue, Oct 28, 2008 at 2:12 PM, Ben Goertzel [EMAIL PROTECTED] wrote: On Tue, Oct 28, 2008 at 10:00 AM, Pei Wang [EMAIL PROTECTED] wrote: Ben, Thanks. So the other people now see that I'm not attacking a straw man. My solution to Hume's problem, as embedded in the experience-grounded semantics, is to assume no predictability, but to justify induction as adaptation. However, it is a separate topic which I've explained in my other publications. Right, but justifying induction as adaptation only works if the environment is assumed to have certain regularities which can be adapted to. In a random environment, adaptation won't work. So, still, to justify induction as adaptation you have to make *some* assumptions about the world. The Occam prior gives one such assumption: that (to give just one form) sets of observations in the world tend to be producible by short computer programs. For adaptation to successfully carry out induction, *some* vaguely comparable property to this must hold, and I'm not sure if you have articulated which one you assume, or if you leave this open. In effect, you implicitly assume something like an Occam prior, because you're saying that a system with finite resources can successfully adapt to the world ... which means that sets of observations in the world *must* be approximately summarizable via subprograms that can be executed within this system. So I argue that, even though it's not your preferred way to think about it, your own approach to AI theory and practice implicitly assumes some variant of the Occam prior holds in the real world. Here I just want to point out that the original and basic meaning of Occam's Razor and those two common (mis)usages of it are not necessarily the same. I fully agree with the former, but not the latter, and I haven't seen any convincing justification of the latter. Instead, they are often taken as granted, under the name of Occam's Razor. I agree that the notion of an Occam prior is a significant conceptual beyond the original Occam's Razor precept enounced long ago. Also, I note that, for those who posit the Occam prior as a **prior assumption**, there is not supposed to be any convincing justification for it. The idea is simply that: one must make *some* assumption (explicitly or implicitly) if one wants to do induction, and this is the assumption that some people choose to make. -- Ben G agi | Archives | Modify Your Subscription --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?; Powered by Listbox: http://www.listbox.com --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] constructivist issues
Mark, Thank you, that clarifies somewhat. But, *my* answer to *your* question would seem to depend on what you mean when you say fully defined. Under the classical interpretation, yes: the question is fully defined, so it is a pi question. Under the constructivist interpretation, no: the question is not fully defined, so it is a cat question. Numbers can be fully defined in the classical sense, but not in the constructivist sense. So, when you say fully defined question, do you mean a question for which all answers are stipulated by logical necessity (classical), or logical deduction (constructivist)? --Abram Demski On Tue, Oct 28, 2008 at 3:28 PM, Mark Waser [EMAIL PROTECTED] wrote: In that case, shouldn't you agree with the classical perspective on Godelian incompleteness, since Godel's incompleteness theorem is about mathematical systems? It depends. Are you asking me a fully defined question within the current axioms of what you call mathematical systems (i.e. a pi question) or a cat question (which could *eventually* be defined by some massive extensions to your mathematical systems but which isn't currently defined in what you're calling mathematical systems)? Saying that Gödel is about mathematical systems is not saying that it's not about cat-including systems. - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Tuesday, October 28, 2008 12:06 PM Subject: Re: [agi] constructivist issues --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] constructivist issues
Numbers can be fully defined in the classical sense, but not in the constructivist sense. So, when you say fully defined question, do you mean a question for which all answers are stipulated by logical necessity (classical), or logical deduction (constructivist)? How (or why) are numbers not fully defined in a constructionist sense? (I was about to ask you whether or not you had answered your own question until that caught my eye on the second or third read-through). - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Tuesday, October 28, 2008 3:47 PM Subject: Re: [agi] constructivist issues Mark, Thank you, that clarifies somewhat. But, *my* answer to *your* question would seem to depend on what you mean when you say fully defined. Under the classical interpretation, yes: the question is fully defined, so it is a pi question. Under the constructivist interpretation, no: the question is not fully defined, so it is a cat question. Numbers can be fully defined in the classical sense, but not in the constructivist sense. So, when you say fully defined question, do you mean a question for which all answers are stipulated by logical necessity (classical), or logical deduction (constructivist)? --Abram Demski On Tue, Oct 28, 2008 at 3:28 PM, Mark Waser [EMAIL PROTECTED] wrote: In that case, shouldn't you agree with the classical perspective on Godelian incompleteness, since Godel's incompleteness theorem is about mathematical systems? It depends. Are you asking me a fully defined question within the current axioms of what you call mathematical systems (i.e. a pi question) or a cat question (which could *eventually* be defined by some massive extensions to your mathematical systems but which isn't currently defined in what you're calling mathematical systems)? Saying that Gödel is about mathematical systems is not saying that it's not about cat-including systems. - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Tuesday, October 28, 2008 12:06 PM Subject: Re: [agi] constructivist issues --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?; Powered by Listbox: http://www.listbox.com --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] Occam's Razor and its abuse
2008/10/28 Ben Goertzel [EMAIL PROTECTED]: On the other hand, I just want to point out that to get around Hume's complaint you do need to make *some* kind of assumption about the regularity of the world. What kind of assumption of this nature underlies your work on NARS (if any)? Not directed to me, but my take on this interesting question. The initial architecture would have limited assumptions about the world. Then the programming in the architecture would for the bias. Initially the system would divide up the world into the simple (inanimate) and highly complex (animate). Why should the system expect animate things to be complex? Because it applies the intentional stance and thinks that they are optimal problem solvers. Optimal problems solvers in a social environment tend to high complexity, as there is an arms race as to who can predict the others, but not be predicted and exploited by the others. Thinking, there are other things like me out here, when you are a complex entity entails thinking things are complex, even when there might be simpler explanations. E.g. what causes weather. Will Pearson --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] constructivist issues
Mark, That is thanks to Godel's incompleteness theorem. Any formal system that describes numbers is doomed to be incomplete, meaning there will be statements that can be constructed purely by reference to numbers (no red cats!) that the system will fail to prove either true or false. So my question is, do you interpret this as meaning Numbers are not well-defined and can never be (constructivist), or do you interpret this as It is impossible to pack all true information about numbers into an axiom system (classical)? Hmm By the way, I might not be using the term constructivist in a way that all constructivists would agree with. I think intuitionist (a specific type of constructivist) would be a better term for the view I'm referring to. --Abram Demski On Tue, Oct 28, 2008 at 4:13 PM, Mark Waser [EMAIL PROTECTED] wrote: Numbers can be fully defined in the classical sense, but not in the constructivist sense. So, when you say fully defined question, do you mean a question for which all answers are stipulated by logical necessity (classical), or logical deduction (constructivist)? How (or why) are numbers not fully defined in a constructionist sense? (I was about to ask you whether or not you had answered your own question until that caught my eye on the second or third read-through). --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] constructivist issues
--- On Tue, 10/28/08, Mike Tintner [EMAIL PROTECTED] wrote: MW:Pi is a normal number is decidable by arithmetic because each of the terms has meaning in arithmetic Can it be expressed in purely mathematical terms/signs without using language? No, because mathematics is a language. -- Matt Mahoney, [EMAIL PROTECTED] --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] constructivist issues
Ben, What are the mathematical or logical signs for normal number/ rational number? My assumption would be that neither logic nor maths can be done without some language attached - such as the term rational number - but I'm asking from extensive ignorance. Ben:yes MT:MW:Pi is a normal number is decidable by arithmetic because each of the terms has meaning in arithmetic Can it be expressed in purely mathematical terms/signs without using language? --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] constructivist issues
All of math can be done without any words ... it just gets annoying to read for instance, all math can be formalized in this sort of manner http://www.cs.miami.edu/~tptp/MizarTPTP/TPTPProofs/arithm/arithm__t1_arithm and the words in there like v1_ordinal1(B) could be replaced with v1_1234(B) or whatever, and it wouldn't make any difference... ben On Tue, Oct 28, 2008 at 2:10 PM, Mike Tintner [EMAIL PROTECTED]wrote: Ben, What are the mathematical or logical signs for normal number/ rational number? My assumption would be that neither logic nor maths can be done without some language attached - such as the term rational number - but I'm asking from extensive ignorance. Ben:yes MT:MW:Pi is a normal number is decidable by arithmetic because each of the terms has meaning in arithmetic Can it be expressed in purely mathematical terms/signs without using language? -- *agi* | Archives https://www.listbox.com/member/archive/303/=now https://www.listbox.com/member/archive/rss/303/ | Modifyhttps://www.listbox.com/member/?;Your Subscription http://www.listbox.com -- Ben Goertzel, PhD CEO, Novamente LLC and Biomind LLC Director of Research, SIAI [EMAIL PROTECTED] A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects. -- Robert Heinlein --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] constructivist issues
Hi guys, I took a couple hours on a red-eye flight last night to write up in more detail my argument as to why uncomputable entities are useless for science: http://multiverseaccordingtoben.blogspot.com/2008/10/are-uncomputable-entities-useless-for.html Of course, I had to assume a specific formal model of science which may be controversial. But at any rate, I think I did succeed in writing down my argument in a more clear way than I'd been able to do in scattershot emails. The only real AGI relevance here is some comments on Penrose's nasty AI theories, e.g. in the last paragraph and near the intro... -- Ben G On Tue, Oct 28, 2008 at 2:02 PM, Abram Demski [EMAIL PROTECTED] wrote: Mark, That is thanks to Godel's incompleteness theorem. Any formal system that describes numbers is doomed to be incomplete, meaning there will be statements that can be constructed purely by reference to numbers (no red cats!) that the system will fail to prove either true or false. So my question is, do you interpret this as meaning Numbers are not well-defined and can never be (constructivist), or do you interpret this as It is impossible to pack all true information about numbers into an axiom system (classical)? Hmm By the way, I might not be using the term constructivist in a way that all constructivists would agree with. I think intuitionist (a specific type of constructivist) would be a better term for the view I'm referring to. --Abram Demski On Tue, Oct 28, 2008 at 4:13 PM, Mark Waser [EMAIL PROTECTED] wrote: Numbers can be fully defined in the classical sense, but not in the constructivist sense. So, when you say fully defined question, do you mean a question for which all answers are stipulated by logical necessity (classical), or logical deduction (constructivist)? How (or why) are numbers not fully defined in a constructionist sense? (I was about to ask you whether or not you had answered your own question until that caught my eye on the second or third read-through). --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?; Powered by Listbox: http://www.listbox.com -- Ben Goertzel, PhD CEO, Novamente LLC and Biomind LLC Director of Research, SIAI [EMAIL PROTECTED] A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects. -- Robert Heinlein --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] Occam's Razor and its abuse
What Hutter proved is (very roughly) that given massive computational resources, following Occam's Razor will be -- within some possibly quite large constant -- the best way to achieve goals in a computable environment... That's not exactly proving Occam's Razor, though it is a proof related to Occam's Razor... One could easily argue it is totally irrelevant to AI due to its assumption of massive computational resources ben g On Tue, Oct 28, 2008 at 2:23 PM, Matt Mahoney [EMAIL PROTECTED] wrote: Hutter proved Occam's Razor (AIXI) for the case of any environment with a computable probability distribution. It applies to us because the observable universe is Turing computable according to currently known laws of physics. Specifically, the observable universe has a finite description length (approximately 2.91 x 10^122 bits, the Bekenstein bound of the Hubble radius). AIXI has nothing to do with insufficiency of resources. Given unlimited resources we would still prefer the (algorithmically) simplest explanation because it is the most likely under a Solomonoff distribution of possible environments. Also, AIXI does not state the simplest answer is the best answer. It says that the simplest answer consistent with observation so far is the best answer. When we are short on resources (and we always are because AIXI is not computable), then we may choose a different explanation than the simplest one. However this does not make the alternative correct. -- Matt Mahoney, [EMAIL PROTECTED] --- On Tue, 10/28/08, Pei Wang [EMAIL PROTECTED] wrote: From: Pei Wang [EMAIL PROTECTED] Subject: [agi] Occam's Razor and its abuse To: agi@v2.listbox.com Date: Tuesday, October 28, 2008, 11:58 AM Triggered by several recent discussions, I'd like to make the following position statement, though won't commit myself to long debate on it. ;-) Occam's Razor, in its original form, goes like entities must not be multiplied beyond necessity, and it is often stated as All other things being equal, the simplest solution is the best or when multiple competing theories are equal in other respects, the principle recommends selecting the theory that introduces the fewest assumptions and postulates the fewest entities --- all from http://en.wikipedia.org/wiki/Occam's_razorhttp://en.wikipedia.org/wiki/Occam%27s_razor I fully agree with all of the above statements. However, to me, there are two common misunderstandings associated with it in the context of AGI and philosophy of science. (1) To take this statement as self-evident or a stand-alone postulate To me, it is derived or implied by the insufficiency of resources. If a system has sufficient resources, it has no good reason to prefer a simpler theory. (2) To take it to mean The simplest answer is usually the correct answer. This is a very different statement, which cannot be justified either analytically or empirically. When theory A is an approximation of theory B, usually the former is simpler than the latter, but less correct or accurate, in terms of its relation with all available evidence. When we are short in resources and have a low demand on accuracy, we often prefer A over B, but it does not mean that by doing so we judge A as more correct than B. In summary, in choosing among alternative theories or conclusions, the preference for simplicity comes from shortage of resources, though simplicity and correctness are logically independent of each other. Pei --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?; Powered by Listbox: http://www.listbox.com -- Ben Goertzel, PhD CEO, Novamente LLC and Biomind LLC Director of Research, SIAI [EMAIL PROTECTED] A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects. -- Robert Heinlein --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] Occam's Razor and its abuse
Au contraire, I suspect that the fact that biological organisms grow via the same sorts of processes as the biological environment in which the live, causes the organisms' minds to be built with **a lot** of implicit bias that is useful for surviving in the environment... Some have argued that this kind of bias is **all you need** for evolution... see Evolution without Selection by A. Lima de Faria. I think that is wrong, but it's interesting that there's enough evidence to even try to make the argument... ben g On Tue, Oct 28, 2008 at 2:37 PM, Ed Porter [EMAIL PROTECTED] wrote: It appears to me that the assumptions about initial priors used by a self learning AGI or an evolutionary line of AGI's could be quite minimal. My understanding is that once a probability distribution starts receiving random samples from its distribution the effect of the original prior becomes rapidly lost, unless it is a rather rare one. Such rare problem priors would get selected against quickly by evolution. Evolution would tend to tune for the most appropriate priors for the success of subsequent generations (either or computing in the same system if it is capable of enough change or of descendant systems). Probably the best priors would generally be ones that could be trained moderately rapidly by data. So it seems an evolutionary system or line could initially learn priors without any assumptions for priors other than a random picking of priors. Over time and multiple generations it might develop hereditary priors, an perhaps even different hereditary priors for parts of its network connected to different inputs, outputs or internal controls. The use of priors in an AGI could be greatly improved by having a gen/comp hiearachy in which models for a given concept could be inherited from the priors of sets of models for similar concepts, and that the set of priors appropriate could change contextually. It would also seem that the notion of a prior could be improve by blending information from episodic and probabilistic models. It would appear than in almost any generally intelligent system, being able to approximate reality in a manner sufficient for evolutionary success with the most efficient representations would be a characteristic that would be greatly preferred by evolution, because it would allow systems to better model more of their environement sufficiently well for evolutionary success with whatever current modeling capacity they have. So, although a completely accurate description of virtually anything may not find much use for Occam's Razor, as a practically useful representation it often will. It seems to me that Occam's Razor is more oriented to deriving meaningful generalizations that it is exact descriptions of anything. Furthermore, it would seem to me that a more simple set of preconditions, is generally more probable than a more complex one, because it requires less coincidence. It would seem to me this would be true under most random sets of priors for the probabilities of the possible sets of components involved and Occam's Razor type selection. The are the musings of an untrained mind, since I have not spent much time studying philosophy, because such a high percent of it was so obviously stupid (such as what was commonly said when I was young, that you can't have intelligence without language) and my understanding of math is much less than that of many on this list. But none the less I think much of what I have said above is true. I think its gist is not totally dissimilar to what Abram has said. Ed Porter -Original Message- From: Pei Wang [mailto:[EMAIL PROTECTED] Sent: Tuesday, October 28, 2008 3:05 PM To: agi@v2.listbox.com Subject: Re: [agi] Occam's Razor and its abuse Abram, I agree with your basic idea in the following, though I usually put it in different form. Pei On Tue, Oct 28, 2008 at 2:52 PM, Abram Demski [EMAIL PROTECTED] wrote: Ben, You assert that Pei is forced to make an assumption about the regulatiry of the world to justify adaptation. Pei could also take a different argument. He could try to show that *if* a strategy exists that can be implemented given the finite resources, NARS will eventually find it. Thus, adaptation is justified on a sort of we might as well try basis. (The proof would involve showing that NARS searches the state of finite-state-machines that can be implemented with the resources at hand, and is more probable to stay for longer periods of time in configurations that give more reward, such that NARS would eventually settle on a configuration if that configuration consistently gave the highest reward.) So, some form of learning can take place with no assumptions. The problem is that the search space is exponential in the resources available, so there is some maximum point where the system would perform best (because the amount of
Re: [agi] constructivist issues
Any formal system that contains some basic arithmetic apparatus equivalent to http://en.wikipedia.org/wiki/Peano_axioms is doomed to be incomplete with respect to statements about numbers... that is what Godel originally showed... On Tue, Oct 28, 2008 at 2:50 PM, Mark Waser [EMAIL PROTECTED] wrote: That is thanks to Godel's incompleteness theorem. Any formal system that describes numbers is doomed to be incomplete Yes, any formal system is doomed to be incomplete. Emphatically, NO! It is not true that any formal system is doomed to be incomplete WITH RESPECT TO NUMBERS. It is entirely possible (nay, almost certain) that there is a larger system where the information about numbers is complete but that the other things that the system describes are incomplete. So my question is, do you interpret this as meaning Numbers are not well-defined and can never be (constructivist), or do you interpret this as It is impossible to pack all true information about numbers into an axiom system (classical)? Hmmm. From a larger reference framework, the former claimed-to-be-constructivist view isn't true/correct because it clearly *is* possible that numbers may be well-defined within a larger system (i.e. the can never be is incorrect). Does that mean that I'm a classicist or that you are mis-interpreting constructivism (because you're attributing a provably false statement to constructivists)? I'm leaning towards the latter currently. ;-) - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Tuesday, October 28, 2008 5:02 PM Subject: Re: [agi] constructivist issues Mark, That is thanks to Godel's incompleteness theorem. Any formal system that describes numbers is doomed to be incomplete, meaning there will be statements that can be constructed purely by reference to numbers (no red cats!) that the system will fail to prove either true or false. So my question is, do you interpret this as meaning Numbers are not well-defined and can never be (constructivist), or do you interpret this as It is impossible to pack all true information about numbers into an axiom system (classical)? Hmm By the way, I might not be using the term constructivist in a way that all constructivists would agree with. I think intuitionist (a specific type of constructivist) would be a better term for the view I'm referring to. --Abram Demski On Tue, Oct 28, 2008 at 4:13 PM, Mark Waser [EMAIL PROTECTED] wrote: Numbers can be fully defined in the classical sense, but not in the constructivist sense. So, when you say fully defined question, do you mean a question for which all answers are stipulated by logical necessity (classical), or logical deduction (constructivist)? How (or why) are numbers not fully defined in a constructionist sense? (I was about to ask you whether or not you had answered your own question until that caught my eye on the second or third read-through). --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?; Powered by Listbox: http://www.listbox.com --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?; Powered by Listbox: http://www.listbox.com -- Ben Goertzel, PhD CEO, Novamente LLC and Biomind LLC Director of Research, SIAI [EMAIL PROTECTED] A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects. -- Robert Heinlein --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] constructivist issues
Any formal system that contains some basic arithmetic apparatus equivalent to http://en.wikipedia.org/wiki/Peano_axioms is doomed to be incomplete with respect to statements about numbers... that is what Godel originally showed... Oh. Ick! My bad phrasing. WITH RESPECT TO NUMBERS should have been WITH RESPECT TO THE DEFINITION OF NUMBERS since I was responding to Numbers are not well-defined and can never be. Further, I should not have said information about numbers when I meant definition of numbers. two radically different thingsArgh! = = = = = = = = So Ben, how would you answer Abram's question So my question is, do you interpret this as meaning Numbers are not well-defined and can never be (constructivist), or do you interpret this as It is impossible to pack all true information about numbers into an axiom system (classical)? Does the statement that a formal system is incomplete with respect to statements about numbers mean that Numbers are not well-defined and can never be. = = = = = = = (Semi-)Retraction - maybe? (mostly for Abram). Ick again! I was assuming that we were talking about constructivism as in Constructivist epistemology (http://en.wikipedia.org/wiki/Constructivist_epistemology). I have just had Constructivism (mathematics) pointed out to me (http://en.wikipedia.org/wiki/Constructivism_(mathematics)) All I can say is Ick! I emphatically do not believe When one assumes that an object does not exist and derives a contradiction from that assumption, one still has not found the object and therefore not proved its existence. = = = = = = = = I'm quitting and going home now to avoid digging myself a deeper hole :-) Mark PS. Ben, I read and, at first glance, liked and agreed with your argument as to why uncomputable entities are useless for science. I'm going to need to go back over it a few more times though.:-) - Original Message - From: Ben Goertzel To: agi@v2.listbox.com Sent: Tuesday, October 28, 2008 5:55 PM Subject: Re: [agi] constructivist issues Any formal system that contains some basic arithmetic apparatus equivalent to http://en.wikipedia.org/wiki/Peano_axioms is doomed to be incomplete with respect to statements about numbers... that is what Godel originally showed... On Tue, Oct 28, 2008 at 2:50 PM, Mark Waser [EMAIL PROTECTED] wrote: That is thanks to Godel's incompleteness theorem. Any formal system that describes numbers is doomed to be incomplete Yes, any formal system is doomed to be incomplete. Emphatically, NO! It is not true that any formal system is doomed to be incomplete WITH RESPECT TO NUMBERS. It is entirely possible (nay, almost certain) that there is a larger system where the information about numbers is complete but that the other things that the system describes are incomplete. So my question is, do you interpret this as meaning Numbers are not well-defined and can never be (constructivist), or do you interpret this as It is impossible to pack all true information about numbers into an axiom system (classical)? Hmmm. From a larger reference framework, the former claimed-to-be-constructivist view isn't true/correct because it clearly *is* possible that numbers may be well-defined within a larger system (i.e. the can never be is incorrect). Does that mean that I'm a classicist or that you are mis-interpreting constructivism (because you're attributing a provably false statement to constructivists)? I'm leaning towards the latter currently. ;-) - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Tuesday, October 28, 2008 5:02 PM Subject: Re: [agi] constructivist issues Mark, That is thanks to Godel's incompleteness theorem. Any formal system that describes numbers is doomed to be incomplete, meaning there will be statements that can be constructed purely by reference to numbers (no red cats!) that the system will fail to prove either true or false. So my question is, do you interpret this as meaning Numbers are not well-defined and can never be (constructivist), or do you interpret this as It is impossible to pack all true information about numbers into an axiom system (classical)? Hmm By the way, I might not be using the term constructivist in a way that all constructivists would agree with. I think intuitionist (a specific type of constructivist) would be a better term for the view I'm referring to. --Abram Demski On Tue, Oct 28, 2008 at 4:13 PM, Mark Waser [EMAIL PROTECTED] wrote: Numbers can be fully defined in the classical sense, but not in the constructivist sense. So, when you say fully defined question, do you mean a question for which all answers are stipulated by logical
Re: [agi] constructivist issues
well-defined is not well-defined in my view... However, it does seem clear that the integers (for instance) is not an entity with *scientific* meaning, if you accept my formalization of science in the blog entry I recently posted... On Tue, Oct 28, 2008 at 3:34 PM, Mark Waser [EMAIL PROTECTED] wrote: Any formal system that contains some basic arithmetic apparatus equivalent to http://en.wikipedia.org/wiki/Peano_axioms is doomed to be incomplete with respect to statements about numbers... that is what Godel originally showed... Oh. Ick! My bad phrasing. WITH RESPECT TO NUMBERS should have been WITH RESPECT TO THE DEFINITION OF NUMBERS since I was responding to Numbers are not well-defined and can never be. Further, I should not have said information about numbers when I meant definition of numbers. two radically different thingsArgh! = = = = = = = = So Ben, how would you answer Abram's question So my question is, do you interpret this as meaning Numbers are not well-defined and can never be (constructivist), or do you interpret this as It is impossible to pack all true information about numbers into an axiom system (classical)? Does the statement that a formal system is incomplete with respect to statements about numbers mean that Numbers are not well-defined and can never be. = = = = = = = (Semi-)Retraction - maybe? (mostly for Abram). Ick again! I was assuming that we were talking about constructivism as in Constructivist epistemology ( http://en.wikipedia.org/wiki/Constructivist_epistemology). I have just had Constructivism (mathematics) pointed out to me ( http://en.wikipedia.org/wiki/Constructivism_(mathematicshttp://en.wikipedia.org/wiki/Constructivism_%28mathematics)) All I can say is Ick! I emphatically do not believe When one assumes that an object does not exist and derives a contradiction from that assumption http://en.wikipedia.org/wiki/Reductio_ad_absurdum, one still has not found the object and therefore not proved its existence. = = = = = = = = I'm quitting and going home now to avoid digging myself a deeper hole :-) Mark PS. Ben, I read and, at first glance, liked and agreed with your argument as to why uncomputable entities are useless for science. I'm going to need to go back over it a few more times though.:-) - Original Message - *From:* Ben Goertzel [EMAIL PROTECTED] *To:* agi@v2.listbox.com *Sent:* Tuesday, October 28, 2008 5:55 PM *Subject:* Re: [agi] constructivist issues Any formal system that contains some basic arithmetic apparatus equivalent to http://en.wikipedia.org/wiki/Peano_axioms is doomed to be incomplete with respect to statements about numbers... that is what Godel originally showed... On Tue, Oct 28, 2008 at 2:50 PM, Mark Waser [EMAIL PROTECTED] wrote: That is thanks to Godel's incompleteness theorem. Any formal system that describes numbers is doomed to be incomplete Yes, any formal system is doomed to be incomplete. Emphatically, NO! It is not true that any formal system is doomed to be incomplete WITH RESPECT TO NUMBERS. It is entirely possible (nay, almost certain) that there is a larger system where the information about numbers is complete but that the other things that the system describes are incomplete. So my question is, do you interpret this as meaning Numbers are not well-defined and can never be (constructivist), or do you interpret this as It is impossible to pack all true information about numbers into an axiom system (classical)? Hmmm. From a larger reference framework, the former claimed-to-be-constructivist view isn't true/correct because it clearly *is* possible that numbers may be well-defined within a larger system (i.e. the can never be is incorrect). Does that mean that I'm a classicist or that you are mis-interpreting constructivism (because you're attributing a provably false statement to constructivists)? I'm leaning towards the latter currently. ;-) - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Tuesday, October 28, 2008 5:02 PM Subject: Re: [agi] constructivist issues Mark, That is thanks to Godel's incompleteness theorem. Any formal system that describes numbers is doomed to be incomplete, meaning there will be statements that can be constructed purely by reference to numbers (no red cats!) that the system will fail to prove either true or false. So my question is, do you interpret this as meaning Numbers are not well-defined and can never be (constructivist), or do you interpret this as It is impossible to pack all true information about numbers into an axiom system (classical)? Hmm By the way, I might not be using the term constructivist in a way that all constructivists would agree with. I think intuitionist (a specific type of constructivist) would be a better term for the view I'm referring to. --Abram Demski On Tue, Oct
Re: [agi] constructivist issues
Matt, Interesting question re the differences between mathematics - i.e. arithmetic, algebra - and logic vs language. I haven't really thought about this, but I wouldn't call maths a language. Maths consists of symbolic systems of quantification and schematic patterns (geometry) which can only be applied to distinct entities - and is very limited in its capacity to describe the world. Language is vastly more general and abstract and actually not normally meant to be reduced to distinct quantities, patterns or entities, or pinned down, period, as maths is e,g. LIFE TAKES LOTS OF FORMS [life is a supra-entity, lots a supra-quantity, form a supra-pattern ] ditto: MATT MAHONEY IS A PERSONALITY IN PROGRESS Verbal statements like these aren't meant to be pinned down or definitively defined - and beyond the reach of maths. Language consists of open-ended classes; maths consists of closed-ended classes. Only language has the capacity to comprehensively describe the world. Maths is more of a sub-language than a true, full language. Matt: MW:Pi is a normal number is decidable by arithmetic because each of the terms has meaning in arithmetic Can it be expressed in purely mathematical terms/signs without using language? No, because mathematics is a language. --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] Occam's Razor and its abuse
Matt, The currently known laws of physics is a *description* of the universe at a certain level, which is fundamentally different from the universe itself. Also, All human knowledge can be reduced into physics is not a view point accepted by everyone. Furthermore, computable is a property of a mathematical function. It takes a bunch of assumptions to be applied to a statement, and some additional ones to be applied to an object --- Is the Earth computable? Does the previous question ever make sense? Whenever someone prove something outside mathematics, it is always based on certain assumptions. If the assumptions are not well justified, there is no strong reason for people to accept the conclusion, even though the proof process is correct. Pei On Tue, Oct 28, 2008 at 5:23 PM, Matt Mahoney [EMAIL PROTECTED] wrote: Hutter proved Occam's Razor (AIXI) for the case of any environment with a computable probability distribution. It applies to us because the observable universe is Turing computable according to currently known laws of physics. Specifically, the observable universe has a finite description length (approximately 2.91 x 10^122 bits, the Bekenstein bound of the Hubble radius). AIXI has nothing to do with insufficiency of resources. Given unlimited resources we would still prefer the (algorithmically) simplest explanation because it is the most likely under a Solomonoff distribution of possible environments. Also, AIXI does not state the simplest answer is the best answer. It says that the simplest answer consistent with observation so far is the best answer. When we are short on resources (and we always are because AIXI is not computable), then we may choose a different explanation than the simplest one. However this does not make the alternative correct. -- Matt Mahoney, [EMAIL PROTECTED] --- On Tue, 10/28/08, Pei Wang [EMAIL PROTECTED] wrote: From: Pei Wang [EMAIL PROTECTED] Subject: [agi] Occam's Razor and its abuse To: agi@v2.listbox.com Date: Tuesday, October 28, 2008, 11:58 AM Triggered by several recent discussions, I'd like to make the following position statement, though won't commit myself to long debate on it. ;-) Occam's Razor, in its original form, goes like entities must not be multiplied beyond necessity, and it is often stated as All other things being equal, the simplest solution is the best or when multiple competing theories are equal in other respects, the principle recommends selecting the theory that introduces the fewest assumptions and postulates the fewest entities --- all from http://en.wikipedia.org/wiki/Occam's_razor I fully agree with all of the above statements. However, to me, there are two common misunderstandings associated with it in the context of AGI and philosophy of science. (1) To take this statement as self-evident or a stand-alone postulate To me, it is derived or implied by the insufficiency of resources. If a system has sufficient resources, it has no good reason to prefer a simpler theory. (2) To take it to mean The simplest answer is usually the correct answer. This is a very different statement, which cannot be justified either analytically or empirically. When theory A is an approximation of theory B, usually the former is simpler than the latter, but less correct or accurate, in terms of its relation with all available evidence. When we are short in resources and have a low demand on accuracy, we often prefer A over B, but it does not mean that by doing so we judge A as more correct than B. In summary, in choosing among alternative theories or conclusions, the preference for simplicity comes from shortage of resources, though simplicity and correctness are logically independent of each other. Pei --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?; Powered by Listbox: http://www.listbox.com --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] Occam's Razor and its abuse
--- On Tue, 10/28/08, Ben Goertzel [EMAIL PROTECTED] wrote: What Hutter proved is (very roughly) that given massive computational resources, following Occam's Razor will be -- within some possibly quite large constant -- the best way to achieve goals in a computable environment... That's not exactly proving Occam's Razor, though it is a proof related to Occam's Razor... No, that's AIXI^tl. I was talking about AIXI. Hutter proved both. One could easily argue it is totally irrelevant to AI due to its assumption of massive computational resources If you mean AIXI^tl, I agree. However, it is AIXI that proves Occam's Razor. AIXI is useful to AGI exactly because it proves noncomputability. We can stop looking for a neat solution. -- Matt Mahoney, [EMAIL PROTECTED] --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
[agi] Occam's Razor and its abuse
Pei Triggered by several recent discussions, I'd like to make the Pei following position statement, though won't commit myself to long Pei debate on it. ;-) Pei Occam's Razor, in its original form, goes like entities must not Pei be multiplied beyond necessity, and it is often stated as All Pei other things being equal, the simplest solution is the best or Pei when multiple competing theories are equal in other respects, Pei the principle recommends selecting the theory that introduces the Pei fewest assumptions and postulates the fewest entities --- all Pei from http://en.wikipedia.org/wiki/Occam's_razor Pei I fully agree with all of the above statements. Pei However, to me, there are two common misunderstandings associated Pei with it in the context of AGI and philosophy of science. Pei (1) To take this statement as self-evident or a stand-alone Pei postulate Pei To me, it is derived or implied by the insufficiency of Pei resources. If a system has sufficient resources, it has no good Pei reason to prefer a simpler theory. With all due respect, this is mistaken. Occam's Razor, in some form, is the heart of Generalization, which is the essence (and G) of GI. For example, if you study concept learning from examples, say in the PAC learning context (related theorems hold in some other contexts as well), there are theorems to the effect that if you find a hypothesis from a simple enough class of a hypotheses it will with very high probability accurately classify new examples chosen from the same distribution, and conversely theorems that state (roughly speaking) that any method that chooses a hypothesis from too expressive a class of hypotheses will have a probability that can be bounded below by some reasonable number like 1/7, of having large error in its predictions on new examples-- in other words it is impossible to PAC learn without respecting Occam's Razor. For discussion of the above paragraphs, I'd refer you to Chapter 4 of What is Thought? (MIT Press, 2004). In other words, if you are building some system that learns about the world, it had better respect Occam's razor if you want whatever it learns to apply to new experience. (I use the term Occam's razor loosely; using hypotheses that are highly constrained in ways other than just being concise may work, but you'd better respect simplicity broadly defined. See Chap 6 of WIT? for more discussion of this point.) The core problem of GI is generalization: you want to be able to figure out new problems as they come along that you haven't seen before. In order to do that, you basically must implicitly or explicitly employ some version of Occam's Razor, independent of how much resources you have. In my view, the first and most important question to ask about any proposal for AGI is, in what way is it going to produce Occam hypotheses. If you can't answer that, don't bother implementing a huge system in hopes of capturing your many insights, because the bigger your implementation gets, the less likely it is to get where you want in the end. --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
RE: [agi] Occam's Razor and its abuse
===Below Ben wrote=== I suspect that the fact that biological organisms grow via the same sorts of processes as the biological environment in which the live, causes the organisms' minds to be built with **a lot** of implicit bias that is useful for surviving in the environment... ===My Response== Au Similaire. That was one of the points I was trying to make! And that arguably supports at least part of what Pei was arguing. I am not arguing it is all you need. You at least need some mechanism for exploring at least some subspace space of possible priors, but you don't need any specific pre-selected set of priors. Ed Porter -Original Message- From: Ben Goertzel [mailto:[EMAIL PROTECTED] Sent: Tuesday, October 28, 2008 5:50 PM To: agi@v2.listbox.com Subject: Re: [agi] Occam's Razor and its abuse Au contraire, I suspect that the fact that biological organisms grow via the same sorts of processes as the biological environment in which the live, causes the organisms' minds to be built with **a lot** of implicit bias that is useful for surviving in the environment... Some have argued that this kind of bias is **all you need** for evolution... see Evolution without Selection by A. Lima de Faria. I think that is wrong, but it's interesting that there's enough evidence to even try to make the argument... ben g On Tue, Oct 28, 2008 at 2:37 PM, Ed Porter [EMAIL PROTECTED] wrote: It appears to me that the assumptions about initial priors used by a self learning AGI or an evolutionary line of AGI's could be quite minimal. My understanding is that once a probability distribution starts receiving random samples from its distribution the effect of the original prior becomes rapidly lost, unless it is a rather rare one. Such rare problem priors would get selected against quickly by evolution. Evolution would tend to tune for the most appropriate priors for the success of subsequent generations (either or computing in the same system if it is capable of enough change or of descendant systems). Probably the best priors would generally be ones that could be trained moderately rapidly by data. So it seems an evolutionary system or line could initially learn priors without any assumptions for priors other than a random picking of priors. Over time and multiple generations it might develop hereditary priors, an perhaps even different hereditary priors for parts of its network connected to different inputs, outputs or internal controls. The use of priors in an AGI could be greatly improved by having a gen/comp hiearachy in which models for a given concept could be inherited from the priors of sets of models for similar concepts, and that the set of priors appropriate could change contextually. It would also seem that the notion of a prior could be improve by blending information from episodic and probabilistic models. It would appear than in almost any generally intelligent system, being able to approximate reality in a manner sufficient for evolutionary success with the most efficient representations would be a characteristic that would be greatly preferred by evolution, because it would allow systems to better model more of their environement sufficiently well for evolutionary success with whatever current modeling capacity they have. So, although a completely accurate description of virtually anything may not find much use for Occam's Razor, as a practically useful representation it often will. It seems to me that Occam's Razor is more oriented to deriving meaningful generalizations that it is exact descriptions of anything. Furthermore, it would seem to me that a more simple set of preconditions, is generally more probable than a more complex one, because it requires less coincidence. It would seem to me this would be true under most random sets of priors for the probabilities of the possible sets of components involved and Occam's Razor type selection. The are the musings of an untrained mind, since I have not spent much time studying philosophy, because such a high percent of it was so obviously stupid (such as what was commonly said when I was young, that you can't have intelligence without language) and my understanding of math is much less than that of many on this list. But none the less I think much of what I have said above is true. I think its gist is not totally dissimilar to what Abram has said. Ed Porter -Original Message- From: Pei Wang [mailto:[EMAIL PROTECTED] Sent: Tuesday, October 28, 2008 3:05 PM To: agi@v2.listbox.com Subject: Re: [agi] Occam's Razor and its abuse Abram, I agree with your basic idea in the following, though I usually put it in different form. Pei On Tue, Oct 28, 2008 at 2:52 PM, Abram Demski [EMAIL PROTECTED] wrote: Ben, You assert that Pei is forced to make an assumption about the regulatiry of the world to justify adaptation. Pei could also take a different argument. He could try to show that
Re: [agi] Occam's Razor and its abuse
Eric:The core problem of GI is generalization: you want to be able to figure out new problems as they come along that you haven't seen before. In order to do that, you basically must implicitly or explicitly employ some version of Occam's Razor It all depends on the subject matter of the generalization. It's a fairly good principle, but there is such a thing as simple-mindedness. For example, what is the cluster of associations evoked in the human brain by any given idea, and what is the principle [or principles] that determines how many associations in how many domains and how many brain areas? The answers to these questions are unlikely to be simple. IOW if the subject matter is complex, the generalization may also have to be complex. --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] Occam's Razor and its abuse
Ed, Since NARS doesn't follow the Bayesian approach, there is no initial priors to be assumed. If we use a more general term, such as initial knowledge or innate beliefs, then yes, you can add them into the system, will will improve the system's performance. However, they are optional. In NARS, all object-level (i.e., not meta-level) innate beliefs can be learned by the system afterward. Pei On Tue, Oct 28, 2008 at 5:37 PM, Ed Porter [EMAIL PROTECTED] wrote: It appears to me that the assumptions about initial priors used by a self learning AGI or an evolutionary line of AGI's could be quite minimal. My understanding is that once a probability distribution starts receiving random samples from its distribution the effect of the original prior becomes rapidly lost, unless it is a rather rare one. Such rare problem priors would get selected against quickly by evolution. Evolution would tend to tune for the most appropriate priors for the success of subsequent generations (either or computing in the same system if it is capable of enough change or of descendant systems). Probably the best priors would generally be ones that could be trained moderately rapidly by data. So it seems an evolutionary system or line could initially learn priors without any assumptions for priors other than a random picking of priors. Over time and multiple generations it might develop hereditary priors, an perhaps even different hereditary priors for parts of its network connected to different inputs, outputs or internal controls. The use of priors in an AGI could be greatly improved by having a gen/comp hiearachy in which models for a given concept could be inherited from the priors of sets of models for similar concepts, and that the set of priors appropriate could change contextually. It would also seem that the notion of a prior could be improve by blending information from episodic and probabilistic models. It would appear than in almost any generally intelligent system, being able to approximate reality in a manner sufficient for evolutionary success with the most efficient representations would be a characteristic that would be greatly preferred by evolution, because it would allow systems to better model more of their environement sufficiently well for evolutionary success with whatever current modeling capacity they have. So, although a completely accurate description of virtually anything may not find much use for Occam's Razor, as a practically useful representation it often will. It seems to me that Occam's Razor is more oriented to deriving meaningful generalizations that it is exact descriptions of anything. Furthermore, it would seem to me that a more simple set of preconditions, is generally more probable than a more complex one, because it requires less coincidence. It would seem to me this would be true under most random sets of priors for the probabilities of the possible sets of components involved and Occam's Razor type selection. The are the musings of an untrained mind, since I have not spent much time studying philosophy, because such a high percent of it was so obviously stupid (such as what was commonly said when I was young, that you can't have intelligence without language) and my understanding of math is much less than that of many on this list. But none the less I think much of what I have said above is true. I think its gist is not totally dissimilar to what Abram has said. Ed Porter -Original Message- From: Pei Wang [mailto:[EMAIL PROTECTED] Sent: Tuesday, October 28, 2008 3:05 PM To: agi@v2.listbox.com Subject: Re: [agi] Occam's Razor and its abuse Abram, I agree with your basic idea in the following, though I usually put it in different form. Pei On Tue, Oct 28, 2008 at 2:52 PM, Abram Demski [EMAIL PROTECTED] wrote: Ben, You assert that Pei is forced to make an assumption about the regulatiry of the world to justify adaptation. Pei could also take a different argument. He could try to show that *if* a strategy exists that can be implemented given the finite resources, NARS will eventually find it. Thus, adaptation is justified on a sort of we might as well try basis. (The proof would involve showing that NARS searches the state of finite-state-machines that can be implemented with the resources at hand, and is more probable to stay for longer periods of time in configurations that give more reward, such that NARS would eventually settle on a configuration if that configuration consistently gave the highest reward.) So, some form of learning can take place with no assumptions. The problem is that the search space is exponential in the resources available, so there is some maximum point where the system would perform best (because the amount of resources match the problem), but giving the system more resources would hurt performance (because the system searches the unnecessarily large search
Re: [agi] Occam's Razor and its abuse
Eric, I highly respect your work, though we clearly have different opinions on what intelligence is, as well as on how to achieve it. For example, though learning and generalization play central roles in my theory about intelligence, I don't think PAC learning (or the other learning algorithms proposed so far) provides a proper conceptual framework for the typical situation of this process. Generally speaking, I'm not building some system that learns about the world, in the sense that there is a correct way to describe the world waiting to be discovered, which can be captured by some algorithm. Instead, learning to me is a non-algorithmic open-ended process by which the system summarizes its own experience, and uses it to predict the future. I fully understand that most people in this field probably consider this opinion wrong, though I haven't been convinced yet by the arguments I've seen so far. Instead of addressing all of the relevant issues, in this discussion I have a very limited goal. To rephrase what I said initially, I see that under the term Occam's Razor, currently there are three different statements: (1) Simplicity (in conclusions, hypothesis, theories, etc.) is preferred. (2) The preference to simplicity does not need a reason or justification. (3) Simplicity is preferred because it is correlated with correctness. I agree with (1), but not (2) and (3). I know many people have different opinions, and I don't attempt to argue with them here --- these problems are too complicated to be settled by email exchanges. However, I do hope to convince people in this discussion that the three statements are not logically equivalent, and (2) and (3) are not implied by (1), so to use Occam's Razor to refer to all of them is not a good idea, because it is going to mix different issues. Therefore, I suggest people to use Occam's Razor in its original and basic sense, that is (1), and to use other terms to refer to (2) and (3). Otherwise, when people talk about Occam's Razor, I just don't know what to say. Pei On Tue, Oct 28, 2008 at 8:09 PM, Eric Baum [EMAIL PROTECTED] wrote: Pei Triggered by several recent discussions, I'd like to make the Pei following position statement, though won't commit myself to long Pei debate on it. ;-) Pei Occam's Razor, in its original form, goes like entities must not Pei be multiplied beyond necessity, and it is often stated as All Pei other things being equal, the simplest solution is the best or Pei when multiple competing theories are equal in other respects, Pei the principle recommends selecting the theory that introduces the Pei fewest assumptions and postulates the fewest entities --- all Pei from http://en.wikipedia.org/wiki/Occam's_razor Pei I fully agree with all of the above statements. Pei However, to me, there are two common misunderstandings associated Pei with it in the context of AGI and philosophy of science. Pei (1) To take this statement as self-evident or a stand-alone Pei postulate Pei To me, it is derived or implied by the insufficiency of Pei resources. If a system has sufficient resources, it has no good Pei reason to prefer a simpler theory. With all due respect, this is mistaken. Occam's Razor, in some form, is the heart of Generalization, which is the essence (and G) of GI. For example, if you study concept learning from examples, say in the PAC learning context (related theorems hold in some other contexts as well), there are theorems to the effect that if you find a hypothesis from a simple enough class of a hypotheses it will with very high probability accurately classify new examples chosen from the same distribution, and conversely theorems that state (roughly speaking) that any method that chooses a hypothesis from too expressive a class of hypotheses will have a probability that can be bounded below by some reasonable number like 1/7, of having large error in its predictions on new examples-- in other words it is impossible to PAC learn without respecting Occam's Razor. For discussion of the above paragraphs, I'd refer you to Chapter 4 of What is Thought? (MIT Press, 2004). In other words, if you are building some system that learns about the world, it had better respect Occam's razor if you want whatever it learns to apply to new experience. (I use the term Occam's razor loosely; using hypotheses that are highly constrained in ways other than just being concise may work, but you'd better respect simplicity broadly defined. See Chap 6 of WIT? for more discussion of this point.) The core problem of GI is generalization: you want to be able to figure out new problems as they come along that you haven't seen before. In order to do that, you basically must implicitly or explicitly employ some version of Occam's Razor, independent of how much resources you have. In my view, the first and most important question to ask about any proposal for AGI is, in what way is it going to
Re: [agi] Occam's Razor and its abuse
If not verify, what about falsify? To me Occam's Razor has always been seen as a tool for selecting the first argument to attempt to falsify. If you can't, or haven't, falsified it, then it's usually the best assumption to go on (presuming that the costs of failing are evenly distributed). OTOH, Occam's Razor clearly isn't quantitative, and it doesn't always pick the right answer, just one that's good enough based on what we know at the moment. (Again presuming evenly distributed costs of failure.) (And actually that's an oversimplification. I've been considering the costs of applying the presumption of the theory chosen by Occam's Razor to be equal to or lower then the costs of the alternatives. Whoops! The simplest workable approach isn't always the cheapest, and given that all non-falsified-as-of-now approaches have closely equal plausibility...perhaps one should instead choose the cheapest to presume of all theories that have been vetted against current knowledge.) Occam's Razor is fine for it's original purposes, but when you try to apply it to practical rather than logical problems then you start needing to evaluate relative costs. Both costs of presuming and costs of failure. And actually often it turns out that a solution based on a theory known to be incorrect (e.g. Newton's laws) is good enough, so you don't need to decide about the correct answer. NASA uses Newton, not Einstein, even though Einstein might be correct and Newton is known to be wrong. Pei Wang wrote: Ben, It seems that you agree the issue I pointed out really exists, but just take it as a necessary evil. Furthermore, you think I also assumed the same thing, though I failed to see it. I won't argue against the necessary evil part --- as far as you agree that those postulates (such as the universe is computable) are not convincingly justified. I won't try to disprove them. As for the latter part, I don't think you can convince me that you know me better than I know myself. ;-) The following is from http://nars.wang.googlepages.com/wang.semantics.pdf , page 28: If the answers provided by NARS are fallible, in what sense these answers are better than arbitrary guesses? This leads us to the concept of rationality. When infallible predictions cannot be obtained (due to insufficient knowledge and resources), answers based on past experience are better than arbitrary guesses, if the environment is relatively stable. To say an answer is only a summary of past experience (thus no future confirmation guaranteed) does not make it equal to an arbitrary conclusion — it is what adaptation means. Adaptation is the process in which a system changes its behaviors as if the future is similar to the past. It is a rational process, even though individual conclusions it produces are often wrong. For this reason, valid inference rules (deduction, induction, abduction, and so on) are the ones whose conclusions correctly (according to the semantics) summarize the evidence in the premises. They are truth-preserving in this sense, not in the model-theoretic sense that they always generate conclusions which are immune from future revision. --- so you see, I don't assume adaptation will always be successful, even successful to a certain probability. You can dislike this conclusion, though you cannot say it is the same as what is assumed by Novamente and AIXI. Pei On Tue, Oct 28, 2008 at 2:12 PM, Ben Goertzel [EMAIL PROTECTED] wrote: On Tue, Oct 28, 2008 at 10:00 AM, Pei Wang [EMAIL PROTECTED] wrote: Ben, Thanks. So the other people now see that I'm not attacking a straw man. My solution to Hume's problem, as embedded in the experience-grounded semantics, is to assume no predictability, but to justify induction as adaptation. However, it is a separate topic which I've explained in my other publications. Right, but justifying induction as adaptation only works if the environment is assumed to have certain regularities which can be adapted to. In a random environment, adaptation won't work. So, still, to justify induction as adaptation you have to make *some* assumptions about the world. The Occam prior gives one such assumption: that (to give just one form) sets of observations in the world tend to be producible by short computer programs. For adaptation to successfully carry out induction, *some* vaguely comparable property to this must hold, and I'm not sure if you have articulated which one you assume, or if you leave this open. In effect, you implicitly assume something like an Occam prior, because you're saying that a system with finite resources can successfully adapt to the world ... which means that sets of observations in the world *must* be approximately summarizable via subprograms that can be executed within this system. So I argue that, even though it's not your preferred way to think about it, your own approach to AI theory and practice implicitly assumes some variant of the
Re: [agi] constructivist issues
Excuse me, but I thought there were subsets of Number theory which were strong enough to contain all the integers, and perhaps all the rational, but which weren't strong enough to prove Gödel's incompleteness theorem in. I seem to remember, though, that you can't get more than a finite number of irrationals in such a theory. And I think that there are limitations on what operators can be defined. Still, depending on what you mean my Number, that would seem to mean that Number was well-defined. Just not in Number Theory, but that's because Number Theory itself wasn't well-defined. Abram Demski wrote: Mark, That is thanks to Godel's incompleteness theorem. Any formal system that describes numbers is doomed to be incomplete, meaning there will be statements that can be constructed purely by reference to numbers (no red cats!) that the system will fail to prove either true or false. So my question is, do you interpret this as meaning Numbers are not well-defined and can never be (constructivist), or do you interpret this as It is impossible to pack all true information about numbers into an axiom system (classical)? Hmm By the way, I might not be using the term constructivist in a way that all constructivists would agree with. I think intuitionist (a specific type of constructivist) would be a better term for the view I'm referring to. --Abram Demski On Tue, Oct 28, 2008 at 4:13 PM, Mark Waser [EMAIL PROTECTED] wrote: Numbers can be fully defined in the classical sense, but not in the constructivist sense. So, when you say fully defined question, do you mean a question for which all answers are stipulated by logical necessity (classical), or logical deduction (constructivist)? How (or why) are numbers not fully defined in a constructionist sense? (I was about to ask you whether or not you had answered your own question until that caught my eye on the second or third read-through). --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
Re: [agi] constructivist issues
Charles, Interesting point-- but, all of these theories would be weaker then the standard axioms, and so there would be *even more* about numbers left undefined in them. --Abram On Tue, Oct 28, 2008 at 10:46 PM, Charles Hixson [EMAIL PROTECTED] wrote: Excuse me, but I thought there were subsets of Number theory which were strong enough to contain all the integers, and perhaps all the rational, but which weren't strong enough to prove Gödel's incompleteness theorem in. I seem to remember, though, that you can't get more than a finite number of irrationals in such a theory. And I think that there are limitations on what operators can be defined. Still, depending on what you mean my Number, that would seem to mean that Number was well-defined. Just not in Number Theory, but that's because Number Theory itself wasn't well-defined. --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
