Re: [agi] constructivist issues
Charles, It might be off-track here, but it would be perfectly on-track in the agi-philosophy list that Ben might eventually split off of this one. But, thanks, that clarifies what you were saying greatly. --Abram On Mon, Nov 3, 2008 at 10:50 PM, Charles Hixson [EMAIL PROTECTED] wrote: That's a lot stronger and more interesting that the theories that I was referring to. Also a lot more complex. **This is getting way off topic, so the rest should probably be ignored.** One of the theories that I was referring to contained only 0 and a rule that given a number allowed you to construct the successor to that number. It clearly couldn't prove it's own consistency, but given enough time and effort it could clearly [from external observation] generate all finite integers. If that were your theory, then just about all you could say about a number was what it's successor was and possibly what it's predecessors were...though you couldn't do that latter within the theory itself. My point was that numbers defined under that theory don't HAVE any other characteristics. They may be equivalent in some sense to numbers defined under Number Theory that were generated in an equivalent way, but they don't have the extra characteristics that the Number Theory numbers have. And this is because numbers aren't characteristics of the universe, but rather particular abstractions from various characteristics of the universe. And if I adopt a constructivist stance, then only a finite number of numbers exist, in fact precisely those numbers that have been generated. I don't happen to think that this is the best approach, because I think that it arises out of an attempt to reify numbers, but it has it's value in some cases. In a way this is like my argument against existentialism. I assert that a true existentialist couldn't walk across the room, because he couldn't be sure that the floor would exist when he took a step. Nobody is that complete an existentialist, and there is merit in the existentialist stance...but it's not the one that is usually described. The way this becomes important is that all simple representations of numbers within a computer are inherently finite. (I'm saying simple to avoid things like lazily evaluated functions which given a transfinite amount of time, RAM, and energy could generate transfinite numbers...and so by existing in an unevaluated state can be said to represent said transfinite numbers.) I.e., for computers constructivist theories describe what's actually possible, though typically they don't impose strict enough restrictions to serve that function, they COULD do so. But it's not clear to me that this is an appropriate approach, even though it models simply onto the physicality of the situation. In a way it reminds me of the arguments that used to be raised between the assembler programmers and the compiler language programmers. Compiler languages use a more abstract representation, but they require more system resources, and you can do anything in assembler that's actually possible. But there are good reasons why the more abstract and general choice is more commonly made. But for some purposes there's really no choice but to actually understand things at an assembler level. Similarly with the more abstract math and the constructivist approach. The constructivists are correct about what we can actually know and count and be certain of. But they aren't right when they claim that this is generally the appropriate stance to take towards math. Doing that is like deriving planetary motions from quantum equations. Sorry for going so off-track. Abram Demski wrote: Charles, defining formal system is just as difficult as defining number-- in fact, those problems are basically equivalent. Godel made use of that to make his proof work (or at least that's one way of looking at it). So, if you claim that the concept number is dependent on what formal system it is defined in, shouldn't you also say the same thing of formal system? So, for example, we might agree to interpret number as Peano-arithmatic number for the purpose of some discussion. But, we might still disagree on how to interpret Peano Arithmetic. I might say, statement X isn't derivable from PA, by transfinite induction, and you might reply, Hey, no, you're not allowed to use transfinite induction. Then we would need to settle which logic to interpret PA in: maybe you convince me that we've got to stick to robinson arithmetic. But then, I use some line of reasoning about the properties of Robinson arithmetic, and we've got to settle on an even higher system to resolve the argument... The point is, we've got to make some actual choice of default at some point. By arguing with anything I want to at every level, I'm using the classical-type default, which is essentially to use the strongest system available. Perhaps you would choose one of these
Re: [agi] constructivist issues
That's a lot stronger and more interesting that the theories that I was referring to. Also a lot more complex. **This is getting way off topic, so the rest should probably be ignored.** One of the theories that I was referring to contained only 0 and a rule that given a number allowed you to construct the successor to that number. It clearly couldn't prove it's own consistency, but given enough time and effort it could clearly [from external observation] generate all finite integers. If that were your theory, then just about all you could say about a number was what it's successor was and possibly what it's predecessors were...though you couldn't do that latter within the theory itself. My point was that numbers defined under that theory don't HAVE any other characteristics. They may be equivalent in some sense to numbers defined under Number Theory that were generated in an equivalent way, but they don't have the extra characteristics that the Number Theory numbers have. And this is because numbers aren't characteristics of the universe, but rather particular abstractions from various characteristics of the universe. And if I adopt a constructivist stance, then only a finite number of numbers exist, in fact precisely those numbers that have been generated. I don't happen to think that this is the best approach, because I think that it arises out of an attempt to reify numbers, but it has it's value in some cases. In a way this is like my argument against existentialism. I assert that a true existentialist couldn't walk across the room, because he couldn't be sure that the floor would exist when he took a step. Nobody is that complete an existentialist, and there is merit in the existentialist stance...but it's not the one that is usually described. The way this becomes important is that all simple representations of numbers within a computer are inherently finite. (I'm saying simple to avoid things like lazily evaluated functions which given a transfinite amount of time, RAM, and energy could generate transfinite numbers...and so by existing in an unevaluated state can be said to represent said transfinite numbers.) I.e., for computers constructivist theories describe what's actually possible, though typically they don't impose strict enough restrictions to serve that function, they COULD do so. But it's not clear to me that this is an appropriate approach, even though it models simply onto the physicality of the situation. In a way it reminds me of the arguments that used to be raised between the assembler programmers and the compiler language programmers. Compiler languages use a more abstract representation, but they require more system resources, and you can do anything in assembler that's actually possible. But there are good reasons why the more abstract and general choice is more commonly made. But for some purposes there's really no choice but to actually understand things at an assembler level. Similarly with the more abstract math and the constructivist approach. The constructivists are correct about what we can actually know and count and be certain of. But they aren't right when they claim that this is generally the appropriate stance to take towards math. Doing that is like deriving planetary motions from quantum equations. Sorry for going so off-track. Abram Demski wrote: Charles, defining formal system is just as difficult as defining number-- in fact, those problems are basically equivalent. Godel made use of that to make his proof work (or at least that's one way of looking at it). So, if you claim that the concept number is dependent on what formal system it is defined in, shouldn't you also say the same thing of formal system? So, for example, we might agree to interpret number as Peano-arithmatic number for the purpose of some discussion. But, we might still disagree on how to interpret Peano Arithmetic. I might say, statement X isn't derivable from PA, by transfinite induction, and you might reply, Hey, no, you're not allowed to use transfinite induction. Then we would need to settle which logic to interpret PA in: maybe you convince me that we've got to stick to robinson arithmetic. But then, I use some line of reasoning about the properties of Robinson arithmetic, and we've got to settle on an even higher system to resolve the argument... The point is, we've got to make some actual choice of default at some point. By arguing with anything I want to at every level, I'm using the classical-type default, which is essentially to use the strongest system available. Perhaps you would choose one of these logics that Godel's theorem fails for as your default. If we can figure out what default is normatively ideal, then in my opinion we've made important headway. I think I found the logics you're referring to? Looks *very* interesting. http://en.wikipedia.org/wiki/Self-verifying_theories --Abram On Fri, Oct
Re: [agi] constructivist issues
It all depends on what definition of number you are using. If it's constructive, then it must be a finite set of numbers. If it's based on full Number Theory, then it's either incomplete or inconsistent. If it's based on any of several subsets of Number Theory that don't allow incompleteness to be proven (or even described) then the numbers are precisely this which is included in that subset of the theory. Number Theory is the one with the largest (i.e., and infinite number) of unprovable theories about numbers of the variations that I have been considering. My point in the just prior post is that numbers are precisely that item which the theory you are using to describe them says they are, since they are artifacts created for computational convenience, as opposed to direct sensory experiences of the universe. As such, it doesn't make sense to say that a subset of number theory leaves more facts about numbers undefined. In the subsets those aren't facts about numbers. Abram Demski wrote: Charles, OK, but if you argue in that manner, then your original point is a little strange, doesn't it? Why worry about Godelian incompleteness if you think incompleteness is just fine? Therefore, I would assert that it isn't that it leaves *even more* about numbers left undefined, but that those characteristics aren't in such a case properties of numbers. Merely of the simplifications an abstractions made to ease computation. In this language, what I'm saying is that it is important to examine the simplifications and abstractions, and discover how they work, so that we can ease computation in our implementations. --Abram On Thu, Oct 30, 2008 at 7:58 PM, Charles Hixson [EMAIL PROTECTED] wrote: If you were talking about something actual, then you would have a valid point. Numbers, though, only exist in so far as they exist in the theory that you are using to define them. E.g., if I were to claim that no number larger than the power-set of energy states within the universe were valid, it would not be disprovable. That would immediately mean that only finite numbers were valid. P.S.: Just because you have a rule that could generate a particular number given a larger than possible number of steps doesn't mean that it is a valid number, as you can't actually ever generate it. I suspect that infinity is primarily a computational convenience. But one shouldn't mistake the fact that it's very convenient for meaning that it's true. Or, given Occam's Razor, should one? But Occam's Razor only detects provisional truths, not actual ones. If you're going to be constructive, then you must restrict yourself to finitely many steps, each composed of finitely complex reasoning. And this means that you must give up both infinite numbers and irrational numbers. To do otherwise means assuming that you can make infinitely precise measurements (which would, at any rate, allow irrational numbers back in). Therefore, I would assert that it isn't that it leaves *even more* about numbers left undefined, but that those characteristics aren't in such a case properties of numbers. Merely of the simplifications an abstractions made to ease computation. --- 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
If you were talking about something actual, then you would have a valid point. Numbers, though, only exist in so far as they exist in the theory that you are using to define them. E.g., if I were to claim that no number larger than the power-set of energy states within the universe were valid, it would not be disprovable. That would immediately mean that only finite numbers were valid. P.S.: Just because you have a rule that could generate a particular number given a larger than possible number of steps doesn't mean that it is a valid number, as you can't actually ever generate it. I suspect that infinity is primarily a computational convenience. But one shouldn't mistake the fact that it's very convenient for meaning that it's true. Or, given Occam's Razor, should one? But Occam's Razor only detects provisional truths, not actual ones. If you're going to be constructive, then you must restrict yourself to finitely many steps, each composed of finitely complex reasoning. And this means that you must give up both infinite numbers and irrational numbers. To do otherwise means assuming that you can make infinitely precise measurements (which would, at any rate, allow irrational numbers back in). Therefore, I would assert that it isn't that it leaves *even more* about numbers left undefined, but that those characteristics aren't in such a case properties of numbers. Merely of the simplifications an abstractions made to ease computation. Abram Demski wrote: 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
Re: [agi] constructivist issues
Charles, OK, but if you argue in that manner, then your original point is a little strange, doesn't it? Why worry about Godelian incompleteness if you think incompleteness is just fine? Therefore, I would assert that it isn't that it leaves *even more* about numbers left undefined, but that those characteristics aren't in such a case properties of numbers. Merely of the simplifications an abstractions made to ease computation. In this language, what I'm saying is that it is important to examine the simplifications and abstractions, and discover how they work, so that we can ease computation in our implementations. --Abram On Thu, Oct 30, 2008 at 7:58 PM, Charles Hixson [EMAIL PROTECTED] wrote: If you were talking about something actual, then you would have a valid point. Numbers, though, only exist in so far as they exist in the theory that you are using to define them. E.g., if I were to claim that no number larger than the power-set of energy states within the universe were valid, it would not be disprovable. That would immediately mean that only finite numbers were valid. P.S.: Just because you have a rule that could generate a particular number given a larger than possible number of steps doesn't mean that it is a valid number, as you can't actually ever generate it. I suspect that infinity is primarily a computational convenience. But one shouldn't mistake the fact that it's very convenient for meaning that it's true. Or, given Occam's Razor, should one? But Occam's Razor only detects provisional truths, not actual ones. If you're going to be constructive, then you must restrict yourself to finitely many steps, each composed of finitely complex reasoning. And this means that you must give up both infinite numbers and irrational numbers. To do otherwise means assuming that you can make infinitely precise measurements (which would, at any rate, allow irrational numbers back in). Therefore, I would assert that it isn't that it leaves *even more* about numbers left undefined, but that those characteristics aren't in such a case properties of numbers. Merely of the simplifications an abstractions made to ease computation. --- 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
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... Huh? Integers are a class (which I would argue is an entity) that is I would argue is well-defined and useful in science. What is meaning if not well-defined and useful? I need to go back to your paper because I didn't get that out of it at all. - Original Message - From: Ben Goertzel To: agi@v2.listbox.com Sent: Tuesday, October 28, 2008 6:41 PM Subject: 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_(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
Re: [agi] constructivist issues
but we never need arbitrarily large integers in any particular case, we only need integers going up to the size of the universe ;-) On Wed, Oct 29, 2008 at 7:24 AM, Mark Waser [EMAIL PROTECTED] wrote: 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... Huh? Integers are a class (which I would argue is an entity) that is I would argue is well-defined and useful in science. What is meaning if not well-defined and useful? I need to go back to your paper because I didn't get that out of it at all. - Original Message - *From:* Ben Goertzel [EMAIL PROTECTED] *To:* agi@v2.listbox.com *Sent:* Tuesday, October 28, 2008 6:41 PM *Subject:* 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
Re: [agi] constructivist issues
Ben, Thanks, that writeup did help me understand your viewpoint. However, I don't completely unserstand/agree with the argument (one of the two, not both!). My comments to that effect are posted on your blog. About the earlier question... (Mark) 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)? (Ben) well-defined is not well-defined in my view... To rephrase. Do you think there is a truth of the matter concerning formally undecidable statements about numbers? --Abram On Tue, Oct 28, 2008 at 5:26 PM, Ben Goertzel [EMAIL PROTECTED] wrote: 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 | 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
To rephrase. Do you think there is a truth of the matter concerning formally undecidable statements about numbers? --Abram That all depends on what the meaning of is, is ... ;-) --- 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
but we never need arbitrarily large integers in any particular case, we only need integers going up to the size of the universe ;-) But measured in which units? For any given integer, I can come up with (invent :-) a unit of measurement that requires a larger/greater number than that integer to describe the size of the universe. ;-) Nice try, but . . . . :-p - Original Message - From: Ben Goertzel To: agi@v2.listbox.com Sent: Wednesday, October 29, 2008 9:48 AM Subject: Re: [agi] constructivist issues but we never need arbitrarily large integers in any particular case, we only need integers going up to the size of the universe ;-) On Wed, Oct 29, 2008 at 7:24 AM, Mark Waser [EMAIL PROTECTED] wrote: 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... Huh? Integers are a class (which I would argue is an entity) that is I would argue is well-defined and useful in science. What is meaning if not well-defined and useful? I need to go back to your paper because I didn't get that out of it at all. - Original Message - From: Ben Goertzel To: agi@v2.listbox.com Sent: Tuesday, October 28, 2008 6:41 PM Subject: 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_(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
Re: [agi] constructivist issues
sorry, I should have been more precise. There is some K so that we never need integers with algorithmic information exceeding K. On Wed, Oct 29, 2008 at 10:32 AM, Mark Waser [EMAIL PROTECTED] wrote: but we never need arbitrarily large integers in any particular case, we only need integers going up to the size of the universe ;-) But measured in which units? For any given integer, I can come up with (invent :-) a unit of measurement that requires a larger/greater number than that integer to describe the size of the universe. ;-) Nice try, but . . . . :-p - Original Message - *From:* Ben Goertzel [EMAIL PROTECTED] *To:* agi@v2.listbox.com *Sent:* Wednesday, October 29, 2008 9:48 AM *Subject:* Re: [agi] constructivist issues but we never need arbitrarily large integers in any particular case, we only need integers going up to the size of the universe ;-) On Wed, Oct 29, 2008 at 7:24 AM, Mark Waser [EMAIL PROTECTED] wrote: 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... Huh? Integers are a class (which I would argue is an entity) that is I would argue is well-defined and useful in science. What is meaning if not well-defined and useful? I need to go back to your paper because I didn't get that out of it at all. - Original Message - *From:* Ben Goertzel [EMAIL PROTECTED] *To:* agi@v2.listbox.com *Sent:* Tuesday, October 28, 2008 6:41 PM *Subject:* 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
Re: [agi] constructivist issues
Here's another slant . . . . I really liked Pei's phrasing (which I consider to be the heart of Constructivism: The Epistemology :-) 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. Classicists (to me) seem to frequently want one and only one truth that must be accurate, complete, and not only provable but for proofs of all of it's implications to exist (which is obviously thwarted by Tarski and Gödel). So . . . . is true that light is a particle? is it true that light is a wave? That's why Ben and I are stuck answering many of your questions with requests for clarification -- Which question -- pi or cat? Which subset of what *might* be considered mathematics/arithmetic? Why are you asking the question? Certain statements appear obviously untrue (read inconsistent with the empirical world or our assumed extensions of it) in the vast majority of cases/contexts but many others are just/simply context-dependent. - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Wednesday, October 29, 2008 10:08 AM Subject: Re: [agi] constructivist issues Ben, Thanks, that writeup did help me understand your viewpoint. However, I don't completely unserstand/agree with the argument (one of the two, not both!). My comments to that effect are posted on your blog. About the earlier question... (Mark) 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)? (Ben) well-defined is not well-defined in my view... To rephrase. Do you think there is a truth of the matter concerning formally undecidable statements about numbers? --Abram On Tue, Oct 28, 2008 at 5:26 PM, Ben Goertzel [EMAIL PROTECTED] wrote: 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
Re: [agi] constructivist issues
sorry, I should have been more precise. There is some K so that we never need integers with algorithmic information exceeding K. Ah . . . . but is K predictable? Or do we need all the integers above it as a safety margin? :-) (What is the meaning of need? :-) The inductive proof to show that all integers are necessary as a safety margin is pretty obvious . . . . - Original Message - From: Ben Goertzel To: agi@v2.listbox.com Sent: Wednesday, October 29, 2008 10:38 AM Subject: Re: [agi] constructivist issues sorry, I should have been more precise. There is some K so that we never need integers with algorithmic information exceeding K. On Wed, Oct 29, 2008 at 10:32 AM, Mark Waser [EMAIL PROTECTED] wrote: but we never need arbitrarily large integers in any particular case, we only need integers going up to the size of the universe ;-) But measured in which units? For any given integer, I can come up with (invent :-) a unit of measurement that requires a larger/greater number than that integer to describe the size of the universe. ;-) Nice try, but . . . . :-p - Original Message - From: Ben Goertzel To: agi@v2.listbox.com Sent: Wednesday, October 29, 2008 9:48 AM Subject: Re: [agi] constructivist issues but we never need arbitrarily large integers in any particular case, we only need integers going up to the size of the universe ;-) On Wed, Oct 29, 2008 at 7:24 AM, Mark Waser [EMAIL PROTECTED] wrote: 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... Huh? Integers are a class (which I would argue is an entity) that is I would argue is well-defined and useful in science. What is meaning if not well-defined and useful? I need to go back to your paper because I didn't get that out of it at all. - Original Message - From: Ben Goertzel To: agi@v2.listbox.com Sent: Tuesday, October 28, 2008 6:41 PM Subject: 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_(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
Re: [agi] constructivist issues
Ben, So, for example, if I describe a Turing machine whose halting I prove formally undecidable by the axioms of peano arithmetic (translating the Turing machine's operation into numerical terms, of course), and then I ask you, is this Turing machine non-halting, then would you answer, That depends on what the meaning of is, is? Or does the context provide enough additional information to provide a more full answer? --Abram On Wed, Oct 29, 2008 at 10:21 AM, Ben Goertzel [EMAIL PROTECTED] wrote: To rephrase. Do you think there is a truth of the matter concerning formally undecidable statements about numbers? --Abram That all depends on what the meaning of is, is ... ;-) 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
On Wed, Oct 29, 2008 at 11:19 AM, Abram Demski [EMAIL PROTECTED]wrote: Ben, So, for example, if I describe a Turing machine whose halting I prove formally undecidable by the axioms of peano arithmetic (translating the Turing machine's operation into numerical terms, of course), and then I ask you, is this Turing machine non-halting, then would you answer, That depends on what the meaning of is, is? Or does the context provide enough additional information to provide a more full answer? --Abram hmmm... you're saying the halting is provable in some more powerful axiom system but not in Peano arithmetic? The thing is, a Turing machine is not a real machine: it's a mathematical abstraction. A mathematical abstraction only has meaning inside a certain formal system. So, the Turing machine inside the Peano arithmetic system would neither provably halt nor not-halt ... the Turing machine inside some other formal system might potentially provably halt... But the question is what does this mean about any actual computer, or any actual physical object -- which we can only communicate about clearly insofar as it can be boiled down to a finite dataset. The use of the same term machine for an observable object and a mathematical abstraction seems to confuse the issue. -- Ben --- 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, OK, that is a pretty good answer. I don't think I have any questions left about your philosophy :). Some comments, though. hmmm... you're saying the halting is provable in some more powerful axiom system but not in Peano arithmetic? Yea, it would be provable in whatever formal system I used to prove the undecidability in the first place. (Probably PA plus an axiom asserting PA is consistent.) The thing is, a Turing machine is not a real machine: it's a mathematical abstraction. A mathematical abstraction only has meaning inside a certain formal system. So, the Turing machine inside the Peano arithmetic system would neither provably halt nor not-halt ... the Turing machine inside some other formal system might potentially provably halt... Basically, I see this this as a no to my original Do you think there is a truth of the matter question. After all, if we need more definitions to determine the truth of a statement, then surely the statement's truth without the additional context is undefined. Take-home message for me: Yes, Ben really is a constructivist. But the question is what does this mean about any actual computer, or any actual physical object -- which we can only communicate about clearly insofar as it can be boiled down to a finite dataset. What it means to me is that Any actual computer will not halt (with a correct output) for this program. An actual computer will keep crunching away until some event happens that breaks the metaphor between it and the abstract machine-- memory overload, power failure, et cetera. This does not seem to me to depend on the formal system that we choose. Argument: very basic axioms fill in all the positive facts, and will tell us that a Turing machine halts when such is the case. Any additional axioms are attempts to fill in the negative space, so that we can prove some Turing machines non-halting. It seems perfectly reasonable to think hypothetically about the formal system that has *all* the negative cases filled in properly, even though this is impossible to actually do. This system is the truth of the matter. So, when we choose a formal system to reason about Turing machines with, we are justified in choosing the strongest one available to us (more specifically, the strongest one we suspect to be consistent). The use of the same term machine for an observable object and a mathematical abstraction seems to confuse the issue. Sure. Do you have a preferred term? I can't think of any... -- Ben 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
But the question is what does this mean about any actual computer, or any actual physical object -- which we can only communicate about clearly insofar as it can be boiled down to a finite dataset. What it means to me is that Any actual computer will not halt (with a correct output) for this program. An actual computer will keep crunching away until some event happens that breaks the metaphor between it and the abstract machine-- memory overload, power failure, et cetera. Yes ... this can be concluded **if** you can convince yourself that the formal model corresponds to the physical machine. And to do *this*, you need to use a finite set of finite data points ;-) ben --- 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, The difference can I think be best illustrated with two hypothetical AGIs. Both are supposed to be learning that computers are approximately Turing machines. The first, made by you, interprets this constructively (let's say relative to PA). The second, made by me, interprets this classically (so it will always take the strongest set of axioms that it suspects to be consistent). The first AGI will be checking to see how well the computer's halting matches with the positive cases it can prove in PA, and the non-halting with the negative cases it can prove in PA. It will be ignoring the halting/nonhalting behavior when it can prove nothing. The second AGI will be checking to see how well the computer's halting matches with the positive cases it can prove in the axiom system of its choice, and the non-halting with the negative cases it can prove in PA, *plus* it will look to see if it is non-halting in the cases where it can prove nothing (after significant effort). Of course, both will conclude nearly the same thing: the computer is similar to the formal entity within specific restrictions. The second AGI will have slightly more data (extra axioms plus information in cases when it can't prove anything), but it will be learning a formally different statement too, so a direct comparison isn't quite fair. Anyway, I think this clarifies the difference. --Abram On Wed, Oct 29, 2008 at 1:13 PM, Ben Goertzel [EMAIL PROTECTED] wrote: But the question is what does this mean about any actual computer, or any actual physical object -- which we can only communicate about clearly insofar as it can be boiled down to a finite dataset. What it means to me is that Any actual computer will not halt (with a correct output) for this program. An actual computer will keep crunching away until some event happens that breaks the metaphor between it and the abstract machine-- memory overload, power failure, et cetera. Yes ... this can be concluded **if** you can convince yourself that the formal model corresponds to the physical machine. And to do *this*, you need to use a finite set of finite data points ;-) ben 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
Ben, No, I wasn't intending any weird chips. For me, the most important way in which you are a constructivist is that you think AIXI is the ideal that finite intelligence should approach. --Abram On Wed, Oct 29, 2008 at 2:33 PM, Ben Goertzel [EMAIL PROTECTED] wrote: OK ... but are both of these hypothetical computer programs on standard contemporary chips, or do any of them use weird supposedly-uncomputability-supporting chips? ;-) Of course, a computer program can use any axiom set it wants to analyze its data ... just as we can now use automated theorem-provers to prove stuff about uncomputable entities, in a formal sense... By the way, I'm not sure the sense in which I'm a constructivist. I'm not willing to commit to the statement that the universe is finite, or that only finite math has meaning. But, it seems to me that, within the scope of *science* and *language*, as currently conceived, there is no *need* to posit anything non-finite. Science and language are not necessarily comprehensive of the universe Potentially (though I doubt it) mind is uncomputable in a way that makes it impossible for science and math to grasp it well enough to guide us in building an AGI ;-) ... and, interestingly, in this case we could still potentially build an AGI via copying a human brain ... and then randomly tinkering with it!! ben On Wed, Oct 29, 2008 at 1:45 PM, Abram Demski [EMAIL PROTECTED] wrote: Ben, The difference can I think be best illustrated with two hypothetical AGIs. Both are supposed to be learning that computers are approximately Turing machines. The first, made by you, interprets this constructively (let's say relative to PA). The second, made by me, interprets this classically (so it will always take the strongest set of axioms that it suspects to be consistent). The first AGI will be checking to see how well the computer's halting matches with the positive cases it can prove in PA, and the non-halting with the negative cases it can prove in PA. It will be ignoring the halting/nonhalting behavior when it can prove nothing. The second AGI will be checking to see how well the computer's halting matches with the positive cases it can prove in the axiom system of its choice, and the non-halting with the negative cases it can prove in PA, *plus* it will look to see if it is non-halting in the cases where it can prove nothing (after significant effort). Of course, both will conclude nearly the same thing: the computer is similar to the formal entity within specific restrictions. The second AGI will have slightly more data (extra axioms plus information in cases when it can't prove anything), but it will be learning a formally different statement too, so a direct comparison isn't quite fair. Anyway, I think this clarifies the difference. --Abram On Wed, Oct 29, 2008 at 1:13 PM, Ben Goertzel [EMAIL PROTECTED] wrote: But the question is what does this mean about any actual computer, or any actual physical object -- which we can only communicate about clearly insofar as it can be boiled down to a finite dataset. What it means to me is that Any actual computer will not halt (with a correct output) for this program. An actual computer will keep crunching away until some event happens that breaks the metaphor between it and the abstract machine-- memory overload, power failure, et cetera. Yes ... this can be concluded **if** you can convince yourself that the formal model corresponds to the physical machine. And to do *this*, you need to use a finite set of finite data points ;-) ben 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 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
On Wed, Oct 29, 2008 at 4:47 PM, Abram Demski [EMAIL PROTECTED] wrote: Ben, No, I wasn't intending any weird chips. For me, the most important way in which you are a constructivist is that you think AIXI is the ideal that finite intelligence should approach. Hmmm... I'm not sure I think that. AIXI is ideal in terms of a certain formal definition of intelligence, which I don't necessarily accept as the end-all of intelligence... It may be that future science identifies conceptual shortcomings in the theoretical framework within which AIXI lives. But, I do think that AIXI is interesting as a source of inspiration for some aspects of the process of creating practical AGI systems. -- Ben G --- 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
*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
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] 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] 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] 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] 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] 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
Re: [agi] constructivist issues
Hmmm. I think that some of our miscommunication might have been due to the fact that you seem to be talking about two things while I think that I'm talking about third . . . . I believe that *meaning* is constructed. I believe that truth is absolute (within a given context) and is a proper subset of meaning. I believe that proof is constructed and is a proper subset of truth (and therefore a proper subset of meaning as well). So, fundamentally, I *am* a constructivist as far as meaning is concerned and take Gödel's theorem to say that meaning is not completely defined or definable. Since I'm being a constructionist about meaning, it would seem that your statement that A constructivist would be justified in asserting the equivalence of Gödel's incompleteness theorem and Tarski's undefinability theorem, would mean that I was correct (or, at least, not wrong) in using Gödel's theorem but probably not as clear as I could have been if I'd used Tarski since an additional condition/assumption (constructivism) was required. So, interchanging the two theorems is fully justifiable in some intellectual circles! Just don't do it when non-constructivists are around :). I guess the question is . . . . How many people *aren't* constructivists when it comes to meaning? Actually, I get the impression that this mailing list is seriously split . . . . Where do you fall on the constructivism of meaning? - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Sunday, October 26, 2008 10:00 PM Subject: Re: [agi] constructivist issues Mark, After some thought... A constructivist would be justified in asserting the equivalence of Godel's incompleteness theorem and Tarski's undefinability theorem, based on the idea that truth is constructable truth. Where classical logicians take Godels theorem to prove that provability cannot equal truth, constructivists can take it to show that provability is not completely defined or definable (and neither is truth, since they are the same). So, interchanging the two theorems is fully justifiable in some intellectual circles! Just don't do it when non-constructivists are around :). --Abram On Sat, Oct 25, 2008 at 6:18 PM, Mark Waser [EMAIL PROTECTED] wrote: OK. A good explanation and I stand corrected and more educated. Thank you. - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Saturday, October 25, 2008 6:06 PM Subject: Re: [agi] constructivist issues Mark, Yes. I wouldn't normally be so picky, but Godel's theorem *really* gets misused. Using Godel's theorem to say made it sound (to me) as if you have a very fundamental confusion. You were using a theorem about the incompleteness of proof to talk about the incompleteness of truth, so it sounded like you thought logically true and logically provable were equivalent, which is of course the *opposite* of what Godel proved. Intuitively, Godel's theorem says If a logic can talk about number theory, it can't have a complete system of proof. Tarski's says, If a logic can talk about number theory, it can't talk about its own notion of truth. Both theorems rely on the Diagonal Lemma, which states If a logic can talk about number theory, it can talk about its own proof method. So, Tarski's theorem immediately implies Godel's theorem: if a logic can talk about its own notion of proof, but not its own notion of truth, then the two can't be equivalent! So, since Godel's theorem follows so closely from Tarski's (even though Tarski's came later), it is better to invoke Tarski's by default if you aren't sure which one applies. --Abram On Sat, Oct 25, 2008 at 4:22 PM, Mark Waser [EMAIL PROTECTED] wrote: So you're saying that if I switch to using Tarski's theory (which I believe is fundamentally just a very slightly different aspect of the same critical concept -- but unfortunately much less well-known and therefore less powerful as an explanation) that you'll agree with me? That seems akin to picayune arguments over phrasing when trying to simply reach general broad agreement . . . . (or am I misinterpreting?) - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Friday, October 24, 2008 5:29 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/?; Powered by Listbox: http://www.listbox.com --- agi Archives: https://www.listbox.com/member/archive/303
Re: [agi] constructivist issues
Mark, I'm a classicalist in the sense that I think classical mathematics needs to be accounted for in a theory of meaning. (Ben seems to think that a constructivist can do this by equating classical mathematics with axiom-systems-of-classical-mathematics, but I am unconvinced.) I am also a classicalist in the sense that I think that the mathematically true is a proper subset of the mathematically provable, so that Godelian truths are not undefined, just unprovable. I might be called a constructivist in the sense that I think there needs to be a tight, well-defined connection between syntax and semantics... The semantics of an AGI's internal logic needs to follow from its manipulation rules. But, partly because I accept the implementability of super-recursive algorithms, I think there is a chance to allow at least *some* classical mathematics into the picture. And, since I believe in the computational nature of the mind, I think that and classical mathematics that *can't* fit into the picture is literally nonsense! So, since I don't feel like much of math is nonsense, I won't be satisfied until I've fit most of it in. I'm not sure what you mean when you say that meaning is constructed, yet truth is absolute. Could you clarify? --Abram On Mon, Oct 27, 2008 at 10:27 AM, Mark Waser [EMAIL PROTECTED] wrote: Hmmm. I think that some of our miscommunication might have been due to the fact that you seem to be talking about two things while I think that I'm talking about third . . . . I believe that *meaning* is constructed. I believe that truth is absolute (within a given context) and is a proper subset of meaning. I believe that proof is constructed and is a proper subset of truth (and therefore a proper subset of meaning as well). So, fundamentally, I *am* a constructivist as far as meaning is concerned and take Gödel's theorem to say that meaning is not completely defined or definable. Since I'm being a constructionist about meaning, it would seem that your statement that A constructivist would be justified in asserting the equivalence of Gödel's incompleteness theorem and Tarski's undefinability theorem, would mean that I was correct (or, at least, not wrong) in using Gödel's theorem but probably not as clear as I could have been if I'd used Tarski since an additional condition/assumption (constructivism) was required. So, interchanging the two theorems is fully justifiable in some intellectual circles! Just don't do it when non-constructivists are around :). I guess the question is . . . . How many people *aren't* constructivists when it comes to meaning? Actually, I get the impression that this mailing list is seriously split . . . . Where do you fall on the constructivism of meaning? - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Sunday, October 26, 2008 10:00 PM Subject: Re: [agi] constructivist issues Mark, After some thought... A constructivist would be justified in asserting the equivalence of Godel's incompleteness theorem and Tarski's undefinability theorem, based on the idea that truth is constructable truth. Where classical logicians take Godels theorem to prove that provability cannot equal truth, constructivists can take it to show that provability is not completely defined or definable (and neither is truth, since they are the same). So, interchanging the two theorems is fully justifiable in some intellectual circles! Just don't do it when non-constructivists are around :). --Abram On Sat, Oct 25, 2008 at 6:18 PM, Mark Waser [EMAIL PROTECTED] wrote: OK. A good explanation and I stand corrected and more educated. Thank you. - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Saturday, October 25, 2008 6:06 PM Subject: Re: [agi] constructivist issues Mark, Yes. I wouldn't normally be so picky, but Godel's theorem *really* gets misused. Using Godel's theorem to say made it sound (to me) as if you have a very fundamental confusion. You were using a theorem about the incompleteness of proof to talk about the incompleteness of truth, so it sounded like you thought logically true and logically provable were equivalent, which is of course the *opposite* of what Godel proved. Intuitively, Godel's theorem says If a logic can talk about number theory, it can't have a complete system of proof. Tarski's says, If a logic can talk about number theory, it can't talk about its own notion of truth. Both theorems rely on the Diagonal Lemma, which states If a logic can talk about number theory, it can talk about its own proof method. So, Tarski's theorem immediately implies Godel's theorem: if a logic can talk about its own notion of proof, but not its own notion of truth, then the two can't be equivalent! So, since Godel's theorem follows so closely from Tarski's (even though Tarski's came later), it is better to invoke
Re: [agi] constructivist issues
Hi, It's interesting (and useful) that you didn't use the word meaning until your last paragraph. I'm not sure what you mean when you say that meaning is constructed, yet truth is absolute. Could you clarify? Hmmm. What if I say that meaning is your domain model and that truth is whether that domain model (or rather, a given preposition phrased in the semantics of the domain model) accurately represents the empirical world? = = = = = = = = I'm a classicalist in the sense that I think classical mathematics needs to be accounted for in a theory of meaning. Would *anyone* argue with this? Is there anyone (with a clue ;-) who isn't a classicist in this sense? I am also a classicalist in the sense that I think that the mathematically true is a proper subset of the mathematically provable, so that Gödelian truths are not undefined, just unprovable. OK. But that is talking about a formal (and complete -- though still infinite) system. I might be called a constructivist in the sense that I think there needs to be a tight, well-defined connection between syntax and semantics... Agreed but you seem to be overlooking the question of Syntax and semantics of what? The semantics of an AGI's internal logic needs to follow from its manipulation rules. Absolutely. But, partly because I accept the implementability of super-recursive algorithms, I think there is a chance to allow at least *some* classical mathematics into the picture. And, since I believe in the computational nature of the mind, I think that and classical mathematics that *can't* fit into the picture is literally nonsense! So, since I don't feel like much of math is nonsense, I won't be satisfied until I've fit most of it in. OK. But I'm not sure where this is going . . . . I agree with all that you're saying but can't see where/how it's supposed to address/go back into my domain model ;-) - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Monday, October 27, 2008 11:05 AM Subject: Re: [agi] constructivist issues Mark, I'm a classicalist in the sense that I think classical mathematics needs to be accounted for in a theory of meaning. (Ben seems to think that a constructivist can do this by equating classical mathematics with axiom-systems-of-classical-mathematics, but I am unconvinced.) I am also a classicalist in the sense that I think that the mathematically true is a proper subset of the mathematically provable, so that Godelian truths are not undefined, just unprovable. I might be called a constructivist in the sense that I think there needs to be a tight, well-defined connection between syntax and semantics... The semantics of an AGI's internal logic needs to follow from its manipulation rules. But, partly because I accept the implementability of super-recursive algorithms, I think there is a chance to allow at least *some* classical mathematics into the picture. And, since I believe in the computational nature of the mind, I think that and classical mathematics that *can't* fit into the picture is literally nonsense! So, since I don't feel like much of math is nonsense, I won't be satisfied until I've fit most of it in. I'm not sure what you mean when you say that meaning is constructed, yet truth is absolute. Could you clarify? --Abram On Mon, Oct 27, 2008 at 10:27 AM, Mark Waser [EMAIL PROTECTED] wrote: Hmmm. I think that some of our miscommunication might have been due to the fact that you seem to be talking about two things while I think that I'm talking about third . . . . I believe that *meaning* is constructed. I believe that truth is absolute (within a given context) and is a proper subset of meaning. I believe that proof is constructed and is a proper subset of truth (and therefore a proper subset of meaning as well). So, fundamentally, I *am* a constructivist as far as meaning is concerned and take Gödel's theorem to say that meaning is not completely defined or definable. Since I'm being a constructionist about meaning, it would seem that your statement that A constructivist would be justified in asserting the equivalence of Gödel's incompleteness theorem and Tarski's undefinability theorem, would mean that I was correct (or, at least, not wrong) in using Gödel's theorem but probably not as clear as I could have been if I'd used Tarski since an additional condition/assumption (constructivism) was required. So, interchanging the two theorems is fully justifiable in some intellectual circles! Just don't do it when non-constructivists are around :). I guess the question is . . . . How many people *aren't* constructivists when it comes to meaning? Actually, I get the impression that this mailing list is seriously split . . . . Where do you fall on the constructivism of meaning? - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Sunday, October 26, 2008 10:00 PM Subject: Re
Re: [agi] constructivist issues
Mark, An example of people who would argue with the meaningfulness of classical mathematics: there are some people who contest the concept of real numbers. The cite things like that the vast majority of real numbers cannot even be named or referenced in any way as individuals, since the infinity of real numbers is larger than the infinity of possible names/descriptions. OK. But I'm not sure where this is going . . . . I agree with all that you're saying but can't see where/how it's supposed to address/go back into my domain model ;-) Well, you already agreed that classical mathematics is meaningful. But, you also asserted that you are a constructivist where meaning is concerned, and therefore collapse Godel's and Tarski's theorems. I do not think you can consistently assert both! If the Godelian truths are unreachable because they are undefined, then there is something *wrong* with the classical insistence that they are true or false but we just don't know which. To take a concrete example: One of these truths that suffers from Godelian incompleteness is the consistency of arithmetic. I, being of the classical persuasion, believe that arithmetic is either consistent or inconsistent. You, to the extent that you are a constructivist, should say that the matter is undecidable and therefore undefined. --Abram On Mon, Oct 27, 2008 at 12:04 PM, Mark Waser [EMAIL PROTECTED] wrote: Hi, It's interesting (and useful) that you didn't use the word meaning until your last paragraph. I'm not sure what you mean when you say that meaning is constructed, yet truth is absolute. Could you clarify? Hmmm. What if I say that meaning is your domain model and that truth is whether that domain model (or rather, a given preposition phrased in the semantics of the domain model) accurately represents the empirical world? = = = = = = = = I'm a classicalist in the sense that I think classical mathematics needs to be accounted for in a theory of meaning. Would *anyone* argue with this? Is there anyone (with a clue ;-) who isn't a classicist in this sense? I am also a classicalist in the sense that I think that the mathematically true is a proper subset of the mathematically provable, so that Gödelian truths are not undefined, just unprovable. OK. But that is talking about a formal (and complete -- though still infinite) system. I might be called a constructivist in the sense that I think there needs to be a tight, well-defined connection between syntax and semantics... Agreed but you seem to be overlooking the question of Syntax and semantics of what? The semantics of an AGI's internal logic needs to follow from its manipulation rules. Absolutely. But, partly because I accept the implementability of super-recursive algorithms, I think there is a chance to allow at least *some* classical mathematics into the picture. And, since I believe in the computational nature of the mind, I think that and classical mathematics that *can't* fit into the picture is literally nonsense! So, since I don't feel like much of math is nonsense, I won't be satisfied until I've fit most of it in. OK. But I'm not sure where this is going . . . . I agree with all that you're saying but can't see where/how it's supposed to address/go back into my domain model ;-) - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Monday, October 27, 2008 11:05 AM Subject: Re: [agi] constructivist issues Mark, I'm a classicalist in the sense that I think classical mathematics needs to be accounted for in a theory of meaning. (Ben seems to think that a constructivist can do this by equating classical mathematics with axiom-systems-of-classical-mathematics, but I am unconvinced.) I am also a classicalist in the sense that I think that the mathematically true is a proper subset of the mathematically provable, so that Godelian truths are not undefined, just unprovable. I might be called a constructivist in the sense that I think there needs to be a tight, well-defined connection between syntax and semantics... The semantics of an AGI's internal logic needs to follow from its manipulation rules. But, partly because I accept the implementability of super-recursive algorithms, I think there is a chance to allow at least *some* classical mathematics into the picture. And, since I believe in the computational nature of the mind, I think that and classical mathematics that *can't* fit into the picture is literally nonsense! So, since I don't feel like much of math is nonsense, I won't be satisfied until I've fit most of it in. I'm not sure what you mean when you say that meaning is constructed, yet truth is absolute. Could you clarify? --Abram On Mon, Oct 27, 2008 at 10:27 AM, Mark Waser [EMAIL PROTECTED] wrote: Hmmm. I think that some of our miscommunication might have been due to the fact that you seem to be talking about two things
Re: [agi] constructivist issues
I, being of the classical persuasion, believe that arithmetic is either consistent or inconsistent. You, to the extent that you are a constructivist, should say that the matter is undecidable and therefore undefined. 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). On the other hand, if you want to try to get into the meaning of arithmetic . . . . = = = = = = = since the infinity of real numbers is larger than the infinity of possible names/descriptions. Huh? The constructivist in me points out that via compound constructions the infinity of possible names/descriptions is exponentially larger than the infinity of real numbers. You can reference *any* real number to the extent that you can define it. And yes, that is both a trick statement AND also the crux of the matter at the same time -- you can't name pi as a sequence of numbers but you certainly can define it by a description of what it is and what it does and any description can also be said to be a name (or a true name if you will :-). If the Gödelian truths are unreachable because they are undefined, then there is something *wrong* with the classical insistence that they are true or false but we just don't know which. They are undefined unless they are part of a formal and complete system. If they are part of a formal and complete system, then they are defined but may be indeterminable. There is nothing *wrong* with the classical insistence as long as it is applied to a limited domain (i.e. that of closed systems) which is what you are doing. - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Monday, October 27, 2008 12:29 PM Subject: Re: [agi] constructivist issues Mark, An example of people who would argue with the meaningfulness of classical mathematics: there are some people who contest the concept of real numbers. The cite things like that the vast majority of real numbers cannot even be named or referenced in any way as individuals, since the infinity of real numbers is larger than the infinity of possible names/descriptions. OK. But I'm not sure where this is going . . . . I agree with all that you're saying but can't see where/how it's supposed to address/go back into my domain model ;-) Well, you already agreed that classical mathematics is meaningful. But, you also asserted that you are a constructivist where meaning is concerned, and therefore collapse Godel's and Tarski's theorems. I do not think you can consistently assert both! If the Godelian truths are unreachable because they are undefined, then there is something *wrong* with the classical insistence that they are true or false but we just don't know which. To take a concrete example: One of these truths that suffers from Godelian incompleteness is the consistency of arithmetic. I, being of the classical persuasion, believe that arithmetic is either consistent or inconsistent. You, to the extent that you are a constructivist, should say that the matter is undecidable and therefore undefined. --Abram On Mon, Oct 27, 2008 at 12:04 PM, Mark Waser [EMAIL PROTECTED] wrote: Hi, It's interesting (and useful) that you didn't use the word meaning until your last paragraph. I'm not sure what you mean when you say that meaning is constructed, yet truth is absolute. Could you clarify? Hmmm. What if I say that meaning is your domain model and that truth is whether that domain model (or rather, a given preposition phrased in the semantics of the domain model) accurately represents the empirical world? = = = = = = = = I'm a classicalist in the sense that I think classical mathematics needs to be accounted for in a theory of meaning. Would *anyone* argue with this? Is there anyone (with a clue ;-) who isn't a classicist in this sense? I am also a classicalist in the sense that I think that the mathematically true is a proper subset of the mathematically provable, so that Gödelian truths are not undefined, just unprovable. OK. But that is talking about a formal (and complete -- though still infinite) system. I might be called a constructivist in the sense that I think there needs to be a tight, well-defined connection between syntax and semantics... Agreed but you seem to be overlooking the question of Syntax and semantics of what? The semantics of an AGI's internal logic needs to follow from its manipulation rules. Absolutely. But, partly because I accept the implementability of super-recursive algorithms, I think there is a chance to allow at least *some* classical mathematics into the picture. And, since I believe in the computational nature of the mind, I think that and classical mathematics that *can't* fit into the picture is literally nonsense! So, since I don't feel like much of math is nonsense, I won't be satisfied until I've
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 fall on? --Abram On Mon, Oct 27, 2008 at 1:03 PM, Mark Waser [EMAIL PROTECTED] wrote: I, being of the classical persuasion, believe that arithmetic is either consistent or inconsistent. You, to the extent that you are a constructivist, should say that the matter is undecidable and therefore undefined. 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). On the other hand, if you want to try to get into the meaning of arithmetic . . . . = = = = = = = since the infinity of real numbers is larger than the infinity of possible names/descriptions. Huh? The constructivist in me points out that via compound constructions the infinity of possible names/descriptions is exponentially larger than the infinity of real numbers. You can reference *any* real number to the extent that you can define it. And yes, that is both a trick statement AND also the crux of the matter at the same time -- you can't name pi as a sequence of numbers but you certainly can define it by a description of what it is and what it does and any description can also be said to be a name (or a true name if you will :-). If the Gödelian truths are unreachable because they are undefined, then there is something *wrong* with the classical insistence that they are true or false but we just don't know which. They are undefined unless they are part of a formal and complete system. If they are part of a formal and complete system, then they are defined but may be indeterminable. There is nothing *wrong* with the classical insistence as long as it is applied to a limited domain (i.e. that of closed systems) which is what you are doing. - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Monday, October 27, 2008 12:29 PM Subject: Re: [agi] constructivist issues Mark, An example of people who would argue with the meaningfulness of classical mathematics: there are some people who contest the concept of real numbers. The cite things like that the vast majority of real numbers cannot even be named or referenced in any way as individuals, since the infinity of real numbers is larger than the infinity of possible names/descriptions. OK. But I'm not sure where this is going . . . . I agree with all that you're saying but can't see where/how it's supposed to address/go back into my domain model ;-) Well, you already agreed that classical mathematics is meaningful. But, you also asserted that you are a constructivist where meaning is concerned, and therefore collapse Godel's and Tarski's theorems. I do not think you can consistently assert both! If the Godelian truths are unreachable because they are undefined, then there is something *wrong* with the classical insistence that they are true or false but we just don't know which. To take a concrete example: One of these truths that suffers from Godelian incompleteness is the consistency of arithmetic. I, being of the classical persuasion, believe that arithmetic is either consistent or inconsistent. You, to the extent that you are a constructivist, should say that the matter is undecidable and therefore undefined. --Abram --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https
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 fall on? --Abram On Mon, Oct 27, 2008 at 1:03 PM, Mark Waser [EMAIL PROTECTED] wrote: I, being of the classical persuasion, believe that arithmetic is either consistent or inconsistent. You, to the extent that you are a constructivist, should say that the matter is undecidable and therefore undefined. 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). On the other hand, if you want to try to get into the meaning of arithmetic . . . . = = = = = = = since the infinity of real numbers is larger than the infinity of possible names/descriptions. Huh? The constructivist in me points out that via compound constructions the infinity of possible names/descriptions is exponentially larger than the infinity of real numbers. You can reference *any* real number to the extent that you can define it. And yes, that is both a trick statement AND also the crux of the matter at the same time -- you can't name pi as a sequence of numbers but you certainly can define it by a description of what it is and what it does and any description can also be said to be a name (or a true name if you will :-). If the Gödelian truths are unreachable because they are undefined, then there is something *wrong
Re: [agi] constructivist issues
I don't think this is reasonable. For the experiment, we would isolate you with various shielding. It is a question of the design of an experiment, like any other physics experiment. At some point, Occam's Razor tells you that the best theory is a non-computational system. And, I hate to be defending people who make this kind of claim, because their claims are wrong-- since what they are claiming to have observed the mind do could easily be done by a computer. And the kind of stuff I am saying you would use to test it I don't believe people could do. But the point is only that one could perform experiments that would test the hypothesis. The claim that such experiments would have to be infinitely long to be convincing is not valid, I don't believe. Ben Eric, According to your argument, there are some cases in which Ben you could demonstrate that I was producing outputs that could not Ben be generated by the specific computer that is **my brain** Ben according to our current understanding of my brain. Ben However, this would not demonstrate that the source is Ben noncomputational. There are other possible explanations, such as Ben the explanation that there is some more powerful computer Ben somewhere generating the outputs, in a way that we don't Ben currently understand. Ben So the question then becomes how would you distinguish between Ben the hypothesis of a hidden noncomputational source, and a hidden Ben more-powerful-computer source? Again, you need to make this Ben distinction using a finite set of finite-precision Ben observations And so my argument then applies again to this Ben additional set of observations Ben So I don't see that you have really provided a counterexample. Ben However, I can see the value of formalizing my argument Ben mathematically so as to avoid the appearance of such loopholes... Ben ben g Ben On Fri, Oct 24, 2008 at 7:01 PM, Eric Baum [EMAIL PROTECTED] Ben wrote: You have not convinced me that you can do anything a computer can't do. And, using language or math, you never will -- because any finite set of symbols you can utter, could also be uttered by some computational system. -- Ben G I have the sense that this argument is not air tight, because I can imagine a zero-knowledge proof that you can do something a computer can't do. Any finite set of symbols you utter *could*, of course, be utterable by some computational system, but if they are generated in response to queries that are not known in advance, it might be arbitrarily unlikely that they *would* be uttered by any particular computational system. For example, to make this concrete and airtight, I can add a time element. Say I compute offline the answers to a large number of problems that, if one were to solve them with a computation, provably could only be solved by extremely long sequential computations, each longer than any sequential computation that a computer that could possibly be built out of the matter in your brain could compute in an hour, and I present you these problems and you answer 1 of them in half an hour. At this point, I am going, I think, to be pursuaded that you are doing something that can not be captured by a Turing machine. Not that I believe, of course, that you can do anything a computer can't do. I'm just saying, the above argument is not a proof that, if you could, it could not be demonstrated. --- 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 -- Ben Goertzel, PhD CEO, Novamente LLC and Biomind LLC Director Ben of Research, SIAI [EMAIL PROTECTED] Ben A human being should be able to change a diaper, plan an Ben invasion, butcher a hog, conn a ship, design a building, write a Ben sonnet, balance accounts, build a wall, set a bone, comfort the Ben dying, take orders, give orders, cooperate, act alone, solve Ben equations, analyze a new problem, pitch manure, program a Ben computer, cook a tasty meal, fight efficiently, die gallantly. Ben Specialization is for insects. -- Robert Heinlein Ben --- agi Archives: Ben https://www.listbox.com/member/archive/303/=now RSS Feed: Ben https://www.listbox.com/member/archive/rss/303/ Modify Your Ben Subscription: Ben https://www.listbox.com/member/?; Ben Powered by Listbox: http://www.listbox.com Ben brEric,brbrAccording to your argument, there are some cases Ben in which you could demonstrate that I was producing outputs that Ben could not be generated by the specific computer that is **my Ben brain** according to our current understanding of my brain.br Ben brHowever, this would not demonstrate that the source is Ben noncomputational.nbsp; There are other possible explanations, Ben such as the explanation
Re: [agi] constructivist issues
It's not solved by shielding, because the hypothetical computable source whose algorithmic information is too high for me to grok it could be within the molecules of the brain, just where the hypothetical uncomputable source is hypothesized to be by Penrose and Hammeroff and so forth. You can never do any experiment to distinguish directly between A = X is uncomputable and B = X is a computable but has an algorithmic information far higher than my brain. You can distinguish between them indirectly via inference according to some theory, but, then the extension of theory to deal with A and B is going to be speculative and unsupported, etc. -- Ben G On Sun, Oct 26, 2008 at 9:19 AM, Eric Baum [EMAIL PROTECTED] wrote: I don't think this is reasonable. For the experiment, we would isolate you with various shielding. It is a question of the design of an experiment, like any other physics experiment. At some point, Occam's Razor tells you that the best theory is a non-computational system. And, I hate to be defending people who make this kind of claim, because their claims are wrong-- since what they are claiming to have observed the mind do could easily be done by a computer. And the kind of stuff I am saying you would use to test it I don't believe people could do. But the point is only that one could perform experiments that would test the hypothesis. The claim that such experiments would have to be infinitely long to be convincing is not valid, I don't believe. Ben Eric, According to your argument, there are some cases in which Ben you could demonstrate that I was producing outputs that could not Ben be generated by the specific computer that is **my brain** Ben according to our current understanding of my brain. Ben However, this would not demonstrate that the source is Ben noncomputational. There are other possible explanations, such as Ben the explanation that there is some more powerful computer Ben somewhere generating the outputs, in a way that we don't Ben currently understand. Ben So the question then becomes how would you distinguish between Ben the hypothesis of a hidden noncomputational source, and a hidden Ben more-powerful-computer source? Again, you need to make this Ben distinction using a finite set of finite-precision Ben observations And so my argument then applies again to this Ben additional set of observations Ben So I don't see that you have really provided a counterexample. Ben However, I can see the value of formalizing my argument Ben mathematically so as to avoid the appearance of such loopholes... Ben ben g Ben On Fri, Oct 24, 2008 at 7:01 PM, Eric Baum [EMAIL PROTECTED] Ben wrote: You have not convinced me that you can do anything a computer can't do. And, using language or math, you never will -- because any finite set of symbols you can utter, could also be uttered by some computational system. -- Ben G I have the sense that this argument is not air tight, because I can imagine a zero-knowledge proof that you can do something a computer can't do. Any finite set of symbols you utter *could*, of course, be utterable by some computational system, but if they are generated in response to queries that are not known in advance, it might be arbitrarily unlikely that they *would* be uttered by any particular computational system. For example, to make this concrete and airtight, I can add a time element. Say I compute offline the answers to a large number of problems that, if one were to solve them with a computation, provably could only be solved by extremely long sequential computations, each longer than any sequential computation that a computer that could possibly be built out of the matter in your brain could compute in an hour, and I present you these problems and you answer 1 of them in half an hour. At this point, I am going, I think, to be pursuaded that you are doing something that can not be captured by a Turing machine. Not that I believe, of course, that you can do anything a computer can't do. I'm just saying, the above argument is not a proof that, if you could, it could not be demonstrated. --- 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 -- Ben Goertzel, PhD CEO, Novamente LLC and Biomind LLC Director Ben of Research, SIAI [EMAIL PROTECTED] Ben A human being should be able to change a diaper, plan an Ben invasion, butcher a hog, conn a ship, design a building, write a Ben sonnet, balance accounts, build a wall, set a bone, comfort the Ben dying, take orders, give orders, cooperate, act alone, solve Ben equations, analyze a new problem, pitch manure, program a Ben computer, cook a tasty meal,
Re: [agi] constructivist issues
Algorithmic information has nothing to do with my argument. I'm talking about time complexity. There are limits to how fast a computer can run its clock, for example because delta E times Delta T must be greater than hbar, so if you try to make delta T too small you explode. Ben It's not solved by shielding, because the hypothetical Ben computable source whose algorithmic information is too high for Ben me to grok it could be within the molecules of the brain, just Ben where the hypothetical uncomputable source is hypothesized to Ben be by Penrose and Hammeroff and so forth. Ben You can never do any experiment to distinguish directly between Ben A = X is uncomputable Ben and Ben B = X is a computable but has an algorithmic information far Ben higher than my brain. Ben You can distinguish between them indirectly via inference Ben according to some theory, but, then the extension of theory to Ben deal with A and B is going to be speculative and unsupported, Ben etc. Ben -- Ben G Ben On Sun, Oct 26, 2008 at 9:19 AM, Eric Baum [EMAIL PROTECTED] Ben wrote: I don't think this is reasonable. For the experiment, we would isolate you with various shielding. It is a question of the design of an experiment, like any other physics experiment. At some point, Occam's Razor tells you that the best theory is a non-computational system. And, I hate to be defending people who make this kind of claim, because their claims are wrong-- since what they are claiming to have observed the mind do could easily be done by a computer. And the kind of stuff I am saying you would use to test it I don't believe people could do. But the point is only that one could perform experiments that would test the hypothesis. The claim that such experiments would have to be infinitely long to be convincing is not valid, I don't believe. Ben Eric, According to your argument, there are some cases in which Ben you could demonstrate that I was producing outputs that could not Ben be generated by the specific computer that is **my brain** Ben according to our current understanding of my brain. Ben However, this would not demonstrate that the source is Ben noncomputational. There are other possible explanations, such as Ben the explanation that there is some more powerful computer Ben somewhere generating the outputs, in a way that we don't Ben currently understand. Ben So the question then becomes how would you distinguish between Ben the hypothesis of a hidden noncomputational source, and a hidden Ben more-powerful-computer source? Again, you need to make this Ben distinction using a finite set of finite-precision Ben observations And so my argument then applies again to this Ben additional set of observations Ben So I don't see that you have really provided a counterexample. Ben However, I can see the value of formalizing my argument Ben mathematically so as to avoid the appearance of such loopholes... Ben ben g Ben On Fri, Oct 24, 2008 at 7:01 PM, Eric Baum [EMAIL PROTECTED] Ben wrote: You have not convinced me that you can do anything a computer can't do. And, using language or math, you never will -- because any finite set of symbols you can utter, could also be uttered by some computational system. -- Ben G I have the sense that this argument is not air tight, because I can imagine a zero-knowledge proof that you can do something a computer can't do. Any finite set of symbols you utter *could*, of course, be utterable by some computational system, but if they are generated in response to queries that are not known in advance, it might be arbitrarily unlikely that they *would* be uttered by any particular computational system. For example, to make this concrete and airtight, I can add a time element. Say I compute offline the answers to a large number of problems that, if one were to solve them with a computation, provably could only be solved by extremely long sequential computations, each longer than any sequential computation that a computer that could possibly be built out of the matter in your brain could compute in an hour, and I present you these problems and you answer 1 of them in half an hour. At this point, I am going, I think, to be pursuaded that you are doing something that can not be captured by a Turing machine. Not that I believe, of course, that you can do anything a computer can't do. I'm just saying, the above argument is not a proof that, if you could, it could not be demonstrated. --- 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 -- Ben Goertzel, PhD CEO, Novamente LLC and Biomind LLC Director Ben of Research, SIAI [EMAIL PROTECTED] Ben A human being should be
Re: [agi] constructivist issues
Mark, After some thought... A constructivist would be justified in asserting the equivalence of Godel's incompleteness theorem and Tarski's undefinability theorem, based on the idea that truth is constructable truth. Where classical logicians take Godels theorem to prove that provability cannot equal truth, constructivists can take it to show that provability is not completely defined or definable (and neither is truth, since they are the same). So, interchanging the two theorems is fully justifiable in some intellectual circles! Just don't do it when non-constructivists are around :). --Abram On Sat, Oct 25, 2008 at 6:18 PM, Mark Waser [EMAIL PROTECTED] wrote: OK. A good explanation and I stand corrected and more educated. Thank you. - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Saturday, October 25, 2008 6:06 PM Subject: Re: [agi] constructivist issues Mark, Yes. I wouldn't normally be so picky, but Godel's theorem *really* gets misused. Using Godel's theorem to say made it sound (to me) as if you have a very fundamental confusion. You were using a theorem about the incompleteness of proof to talk about the incompleteness of truth, so it sounded like you thought logically true and logically provable were equivalent, which is of course the *opposite* of what Godel proved. Intuitively, Godel's theorem says If a logic can talk about number theory, it can't have a complete system of proof. Tarski's says, If a logic can talk about number theory, it can't talk about its own notion of truth. Both theorems rely on the Diagonal Lemma, which states If a logic can talk about number theory, it can talk about its own proof method. So, Tarski's theorem immediately implies Godel's theorem: if a logic can talk about its own notion of proof, but not its own notion of truth, then the two can't be equivalent! So, since Godel's theorem follows so closely from Tarski's (even though Tarski's came later), it is better to invoke Tarski's by default if you aren't sure which one applies. --Abram On Sat, Oct 25, 2008 at 4:22 PM, Mark Waser [EMAIL PROTECTED] wrote: So you're saying that if I switch to using Tarski's theory (which I believe is fundamentally just a very slightly different aspect of the same critical concept -- but unfortunately much less well-known and therefore less powerful as an explanation) that you'll agree with me? That seems akin to picayune arguments over phrasing when trying to simply reach general broad agreement . . . . (or am I misinterpreting?) - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Friday, October 24, 2008 5:29 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/?; 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
Eric, Nobody here is actually arguing that the brain is non-computational, though. (The quote you refer to was a misunderstanding). I was arguing that we have an understanding of noncomputational entities, and Ben was arguing (approximately) that any actual behavior could be explained equally well by an understanding of a computational entity (namely an axiomatic system describing the noncomp. entity). But, your argument *does* apply roughly to my claim that we could usefully learn that the world contained noncomputable entities... any behavior can be explained computationally, but to me this is only a little better than saying that any behavior can be explained by a hidden markov model. It is technically true, but sometimes a more sophisticated model will provide a *better* explanation. You can never know for sure that something is noncomputational as opposed to large-resource-computable, but you can similarly never know whether something is generated by a recursive definition (fractals, context-free grammers) or simply a complicated state-transition definition (regular grammars, hidden markov models). --Abram On Fri, Oct 24, 2008 at 7:01 PM, Eric Baum [EMAIL PROTECTED] wrote: You have not convinced me that you can do anything a computer can't do. And, using language or math, you never will -- because any finite set of symbols you can utter, could also be uttered by some computational system. -- Ben G I have the sense that this argument is not air tight, because I can imagine a zero-knowledge proof that you can do something a computer can't do. Any finite set of symbols you utter *could*, of course, be utterable by some computational system, but if they are generated in response to queries that are not known in advance, it might be arbitrarily unlikely that they *would* be uttered by any particular computational system. For example, to make this concrete and airtight, I can add a time element. Say I compute offline the answers to a large number of problems that, if one were to solve them with a computation, provably could only be solved by extremely long sequential computations, each longer than any sequential computation that a computer that could possibly be built out of the matter in your brain could compute in an hour, and I present you these problems and you answer 1 of them in half an hour. At this point, I am going, I think, to be pursuaded that you are doing something that can not be captured by a Turing machine. Not that I believe, of course, that you can do anything a computer can't do. I'm just saying, the above argument is not a proof that, if you could, it could not be demonstrated. --- 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
The limitations of Godelian completeness/incompleteness are a subset of the much stronger limitations of finite automata. Can we get a listing of what you believe these limitations are and whether or not you believe that they apply to humans? I believe that humans are constrained by *all* the limits of finite automata yet are general intelligences so I'm not sure of your point. - Original Message - From: Dr. Matthias Heger [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Friday, October 24, 2008 4:09 AM Subject: AW: [agi] constructivist issues The limitations of Godelian completeness/incompleteness are a subset of the much stronger limitations of finite automata. If you want to build a spaceship to go to mars it is of no practical relevance to think whether it is theoretically possible to move through wormholes in the universe. I think, this comparison is adequate to evaluate the role of Gödel's theorem for AGI. - Matthias Abram Demski [mailto:[EMAIL PROTECTED] wrote I agree with your point in this context, but I think you also mean to imply that Godel's incompleteness theorem isn't of any importance for artificial intelligence, which (probably pretty obviously) I wouldn't agree with. Godel's incompleteness theorem tells us important limitations of the logical approach to AI (and, indeed, any approach that can be implemented on normal computers). It *has* however been overused and abused throughout the years... which is one reason I jumped on Mark... --Abram --- 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
Forget consensus! I don't even see a discussion forming. This is all quite long and impenetrable. What have we learned here? If possible I want to catch up. Eric B --- 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
Matthias, OK, that seems fair. Perhaps you will let me get away with a weaker statement: Since it is convenient to *pretend* that computers are Turing machines rather than finite-state machines when doing theoretical work, it is *also* convenient to pretend that Godelian limitations are all that apply to AI designs. To actually implement the thing, we need to keep the finite-state limitations in mind. As hardware improves, and *particular* finite-state inability will melt away (providing some justification for pretending that Godelian limitations are the important ones). But, of course, an infinite number of such restrictions will remain. --Abram On Fri, Oct 24, 2008 at 4:09 AM, Dr. Matthias Heger [EMAIL PROTECTED] wrote: The limitations of Godelian completeness/incompleteness are a subset of the much stronger limitations of finite automata. If you want to build a spaceship to go to mars it is of no practical relevance to think whether it is theoretically possible to move through wormholes in the universe. I think, this comparison is adequate to evaluate the role of Gödel's theorem for AGI. - Matthias --- 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
I know I've expressed frustration with this thread in the past. But I don't want to discourage its development. If someone wants to hit me with a summary off-list maybe I can contribute something _ --- 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, It makes sense but I'm arguing that you're making my point for me . . . . I'm making the point natural language is incompletely defined for you, but *not* the point natural language suffers from Godelian incompleteness, unless you specify what concept of proof applies to natural language. It emphatically does *not* tell us anything about any approach that can be implemented on normal computers and this is where all the people who insist that because computers operate algorithmically, they will never achieve true general intelligence are going wrong. It tells us that any approach that is implementable on a normal computer will not always be able to come up with correct answers to all halting-problem questions (along with other problems that suffer from incompleteness). You are correct in saying that Godel's theory has been improperly overused and abused over the years but my point was merely that AGI is Godellian Incomplete, natural language is Godellian Incomplete, Specify truth and proof in these domains before applying the theorem, please. For agi I am OK, since X is provable would mean the AGI will come to believe X, and X is true would mean something close to what it intuitively means. But for natural language? Natural language will come to believe X makes no sense, so it can't be our definition of proof... Really, it is a small objection, and I'm only making it because I don't want the theorem abused. You could fix your statement just by saying any proof system we might want to provide will be incomplete for any well-defined subset of natural language semantics that is large enough to talk fully about numbers. Doing this just seems pointless, because the real point you are trying to make is that the semantics is ill-defined in general, *not* that some hypothetical proof system is incomplete. and effectively AGI-Complete most probably pretty much exactly means Godellian-Incomplete. (Yes, that is a radically new phrasing and not necessarily quite what I mean/meant but . . . . ). I used to agree that Godelian incompleteness was enough to show that the semantics of a knowledge representation was strong enough for AGI. But, that alone doesn't seem to guarantee that a knowledge representation can faithfully reflect concepts like continuous differentiable function (which gets back to the whole discussion with Ben). Have you heard of Tarski's undefinability theorem? It is relevant to this discussion. http://en.wikipedia.org/wiki/Indefinability_theory_of_truth --Abram On Fri, Oct 24, 2008 at 9:19 AM, Mark Waser [EMAIL PROTECTED] wrote: I'm saying Godelian completeness/incompleteness can't be easily defined in the context of natural language, so it shouldn't be applied there without providing justification for that application (specifically, unambiguous definitions of provably true and semantically true for natural language). Does that make sense, or am I still confusing? It makes sense but I'm arguing that you're making my point for me . . . . agree with. Godel's incompleteness theorem tells us important limitations of the logical approach to AI (and, indeed, any approach that can be implemented on normal computers). It *has* however been overused and abused throughout the years... which is one reason I jumped on Mark... Godel's incompleteness theorem tells us important limitations of all formal *and complete* approaches and systems (like logic). It clearly means that any approach to AI is going to have to be open-ended (Godellian-incomplete? ;-) It emphatically does *not* tell us anything about any approach that can be implemented on normal computers and this is where all the people who insist that because computers operate algorithmically, they will never achieve true general intelligence are going wrong. The later argument is similar to saying that because an inductive mathematical proof always operates only on just the next number, it will never successfully prove anything about infinity. I'm a firm believe in inductive proofs and the fact that general intelligences can be implemented on the computers that we have today. You are correct in saying that Godel's theory has been improperly overused and abused over the years but my point was merely that AGI is Godellian Incomplete, natural language is Godellian Incomplete, and effectively AGI-Complete most probably pretty much exactly means Godellian-Incomplete. (Yes, that is a radically new phrasing and not necessarily quite what I mean/meant but . . . . ). --- 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
But I do not agree that most humans can be scientists. If this is necessary for general intelligence then most humans are not general intelligences. Soften be scientists to generally use the scientific method. Does this change your opinion? - Original Message - From: Dr. Matthias Heger [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Friday, October 24, 2008 10:27 AM Subject: AW: [agi] constructivist issues Mark Waser wrote: Can we get a listing of what you believe these limitations are and whether or not you believe that they apply to humans? I believe that humans are constrained by *all* the limits of finite automata yet are general intelligences so I'm not sure of your point. It is also my opinion that humans are constrained by *all* the limits of finite automata. But I do not agree that most humans can be scientists. If this is necessary for general intelligence then most humans are not general intelligences. It depends on your definition of general intelligence. Surely there are rules (=algorithms) to be a scientist. If not, AGI would not be possible and there would not be any scientist at all. But you cannot separate the rules (algorithm) from the evaluation whether a human or a machine is intelligent. Intelligence comes essentially from these rules and from a lot of data. The mere ability to use arbitrary rules does not imply general intelligence. Your computer has this ability but without the rules it is not intelligent at all. - Matthias --- 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
I'm making the point natural language is incompletely defined for you, but *not* the point natural language suffers from Godelian incompleteness, unless you specify what concept of proof applies to natural language. I'm back to being lost I think. You agree that natural language is incompletely defined. Cool. My saying that natural language suffers from Godelian incompleteness merely adds that it *can't* be defined. Do you mean to say that natural languages *can* be completely defined? Or are you arguing that I can't *prove* that they can't be defined? If it is the last, then that's like saying that Godel's theorem can't prove itself -- which is exactly the point to what Godel's theorem says . . . . Have you heard of Tarski's undefinability theorem? It is relevant to this discussion. http://en.wikipedia.org/wiki/Indefinability_theory_of_truth Yes. In fact, the restatement of Tarski's theory as No sufficiently powerful language is strongly-semantically-self-representational also fundamentally says that I can't prove in natural language what you're asking me to prove about natural language. Personally, I always have trouble separating out Godel and Tarski as they are obviously both facets of the same underlying principles. I'm still not sure of what you're getting at. If it's a proof, then Godel says I can't give it to you. If it's something else, then I'm not getting it. - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Friday, October 24, 2008 11:31 AM Subject: Re: [agi] constructivist issues Mark, It makes sense but I'm arguing that you're making my point for me . . . . I'm making the point natural language is incompletely defined for you, but *not* the point natural language suffers from Godelian incompleteness, unless you specify what concept of proof applies to natural language. It emphatically does *not* tell us anything about any approach that can be implemented on normal computers and this is where all the people who insist that because computers operate algorithmically, they will never achieve true general intelligence are going wrong. It tells us that any approach that is implementable on a normal computer will not always be able to come up with correct answers to all halting-problem questions (along with other problems that suffer from incompleteness). You are correct in saying that Godel's theory has been improperly overused and abused over the years but my point was merely that AGI is Godellian Incomplete, natural language is Godellian Incomplete, Specify truth and proof in these domains before applying the theorem, please. For agi I am OK, since X is provable would mean the AGI will come to believe X, and X is true would mean something close to what it intuitively means. But for natural language? Natural language will come to believe X makes no sense, so it can't be our definition of proof... Really, it is a small objection, and I'm only making it because I don't want the theorem abused. You could fix your statement just by saying any proof system we might want to provide will be incomplete for any well-defined subset of natural language semantics that is large enough to talk fully about numbers. Doing this just seems pointless, because the real point you are trying to make is that the semantics is ill-defined in general, *not* that some hypothetical proof system is incomplete. and effectively AGI-Complete most probably pretty much exactly means Godellian-Incomplete. (Yes, that is a radically new phrasing and not necessarily quite what I mean/meant but . . . . ). I used to agree that Godelian incompleteness was enough to show that the semantics of a knowledge representation was strong enough for AGI. But, that alone doesn't seem to guarantee that a knowledge representation can faithfully reflect concepts like continuous differentiable function (which gets back to the whole discussion with Ben). Have you heard of Tarski's undefinability theorem? It is relevant to this discussion. http://en.wikipedia.org/wiki/Indefinability_theory_of_truth --Abram On Fri, Oct 24, 2008 at 9:19 AM, Mark Waser [EMAIL PROTECTED] wrote: I'm saying Godelian completeness/incompleteness can't be easily defined in the context of natural language, so it shouldn't be applied there without providing justification for that application (specifically, unambiguous definitions of provably true and semantically true for natural language). Does that make sense, or am I still confusing? It makes sense but I'm arguing that you're making my point for me . . . . agree with. Godel's incompleteness theorem tells us important limitations of the logical approach to AI (and, indeed, any approach that can be implemented on normal computers). It *has* however been overused and abused throughout the years... which is one reason I jumped on Mark... Godel's incompleteness theorem tells us important limitations of all formal
Re: [agi] constructivist issues
Mike, Personally, I always have trouble separating out Godel and Tarski as they are obviously both facets of the same underlying principles. This is essentially what I'm complaining about. If you had used Tarski's theorem to begin with, I wouldn't be bugging you :). --Abram On Fri, Oct 24, 2008 at 12:58 PM, Mark Waser [EMAIL PROTECTED] wrote: I'm making the point natural language is incompletely defined for you, but *not* the point natural language suffers from Godelian incompleteness, unless you specify what concept of proof applies to natural language. I'm back to being lost I think. You agree that natural language is incompletely defined. Cool. My saying that natural language suffers from Godelian incompleteness merely adds that it *can't* be defined. Do you mean to say that natural languages *can* be completely defined? Or are you arguing that I can't *prove* that they can't be defined? If it is the last, then that's like saying that Godel's theorem can't prove itself -- which is exactly the point to what Godel's theorem says . . . . Have you heard of Tarski's undefinability theorem? It is relevant to this discussion. http://en.wikipedia.org/wiki/Indefinability_theory_of_truth Yes. In fact, the restatement of Tarski's theory as No sufficiently powerful language is strongly-semantically-self-representational also fundamentally says that I can't prove in natural language what you're asking me to prove about natural language. Personally, I always have trouble separating out Godel and Tarski as they are obviously both facets of the same underlying principles. I'm still not sure of what you're getting at. If it's a proof, then Godel says I can't give it to you. If it's something else, then I'm not getting it. - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Friday, October 24, 2008 11:31 AM Subject: Re: [agi] constructivist issues Mark, It makes sense but I'm arguing that you're making my point for me . . . . I'm making the point natural language is incompletely defined for you, but *not* the point natural language suffers from Godelian incompleteness, unless you specify what concept of proof applies to natural language. It emphatically does *not* tell us anything about any approach that can be implemented on normal computers and this is where all the people who insist that because computers operate algorithmically, they will never achieve true general intelligence are going wrong. It tells us that any approach that is implementable on a normal computer will not always be able to come up with correct answers to all halting-problem questions (along with other problems that suffer from incompleteness). You are correct in saying that Godel's theory has been improperly overused and abused over the years but my point was merely that AGI is Godellian Incomplete, natural language is Godellian Incomplete, Specify truth and proof in these domains before applying the theorem, please. For agi I am OK, since X is provable would mean the AGI will come to believe X, and X is true would mean something close to what it intuitively means. But for natural language? Natural language will come to believe X makes no sense, so it can't be our definition of proof... Really, it is a small objection, and I'm only making it because I don't want the theorem abused. You could fix your statement just by saying any proof system we might want to provide will be incomplete for any well-defined subset of natural language semantics that is large enough to talk fully about numbers. Doing this just seems pointless, because the real point you are trying to make is that the semantics is ill-defined in general, *not* that some hypothetical proof system is incomplete. and effectively AGI-Complete most probably pretty much exactly means Godellian-Incomplete. (Yes, that is a radically new phrasing and not necessarily quite what I mean/meant but . . . . ). I used to agree that Godelian incompleteness was enough to show that the semantics of a knowledge representation was strong enough for AGI. But, that alone doesn't seem to guarantee that a knowledge representation can faithfully reflect concepts like continuous differentiable function (which gets back to the whole discussion with Ben). Have you heard of Tarski's undefinability theorem? It is relevant to this discussion. http://en.wikipedia.org/wiki/Indefinability_theory_of_truth --Abram --- 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 Sat, Oct 25, 2008 at 3:01 AM, Eric Baum [EMAIL PROTECTED] wrote: For example, to make this concrete and airtight, I can add a time element. Say I compute offline the answers to a large number of problems that, if one were to solve them with a computation, provably could only be solved by extremely long sequential computations, each longer than any sequential computation that a computer that could possibly be built out of the matter in your brain could compute in an hour, and I present you these problems and you answer 1 of them in half an hour. At this point, I am going, I think, to be pursuaded that you are doing something that can not be captured by a Turing machine. Maybe your brain patches into a huge ultrafast machine concealed in an extra dimension. We'd just need to find a way to hack in there and exploit its computational potential on industrial scale. ;-) -- Vladimir Nesov [EMAIL PROTECTED] http://causalityrelay.wordpress.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
Eric, According to your argument, there are some cases in which you could demonstrate that I was producing outputs that could not be generated by the specific computer that is **my brain** according to our current understanding of my brain. However, this would not demonstrate that the source is noncomputational. There are other possible explanations, such as the explanation that there is some more powerful computer somewhere generating the outputs, in a way that we don't currently understand. So the question then becomes how would you distinguish between the hypothesis of a hidden noncomputational source, and a hidden more-powerful-computer source? Again, you need to make this distinction using a finite set of finite-precision observations And so my argument then applies again to this additional set of observations So I don't see that you have really provided a counterexample. However, I can see the value of formalizing my argument mathematically so as to avoid the appearance of such loopholes... ben g On Fri, Oct 24, 2008 at 7:01 PM, Eric Baum [EMAIL PROTECTED] wrote: You have not convinced me that you can do anything a computer can't do. And, using language or math, you never will -- because any finite set of symbols you can utter, could also be uttered by some computational system. -- Ben G I have the sense that this argument is not air tight, because I can imagine a zero-knowledge proof that you can do something a computer can't do. Any finite set of symbols you utter *could*, of course, be utterable by some computational system, but if they are generated in response to queries that are not known in advance, it might be arbitrarily unlikely that they *would* be uttered by any particular computational system. For example, to make this concrete and airtight, I can add a time element. Say I compute offline the answers to a large number of problems that, if one were to solve them with a computation, provably could only be solved by extremely long sequential computations, each longer than any sequential computation that a computer that could possibly be built out of the matter in your brain could compute in an hour, and I present you these problems and you answer 1 of them in half an hour. At this point, I am going, I think, to be pursuaded that you are doing something that can not be captured by a Turing machine. Not that I believe, of course, that you can do anything a computer can't do. I'm just saying, the above argument is not a proof that, if you could, it could not be demonstrated. --- 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
But, I still do not agree with the way you are using the incompleteness theorem. Um. OK. Could you point to a specific example where you disagree? I'm a little at a loss here . . . . It is important to distinguish between two different types of incompleteness. 1. Normal Incompleteness-- a logical theory fails to completely specify something. 2. Godelian Incompleteness-- a logical theory fails to completely specify something, even though we want it to. I'm also not getting this. If I read the words, it looks like the difference between Normal and Godelian incompleteness is based upon our desires. I think I'm having a complete disconnect with your intended meaning. However, it seems like all you need is type 1 completeness for what you are saying. So, Godel's theorem is way overkill here in my opinion. Um. OK. So I used a bazooka on a fly? If it was a really pesky fly and I didn't destroy anything else, is that wrong? :-) It seems as if you're not arguing with my conclusion but saying that my arguments were way better than they needed to be (like I'm being over-efficient?) . . . . = = = = = Seriously though, I having a complete disconnect here. Maybe I'm just having a bad morning but . . . huh? :-) If I read the words, all I'm getting is that you disagree with the way that I am using the theory because the theory is overkill for what is necessary. - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Wednesday, October 22, 2008 9:05 PM Subject: Re: [agi] constructivist issues Mark, I own and have read the book-- but my first introduction to Godel's Theorem was Douglas Hofstadter's earlier work, Godel Escher Bach. Since I had already been guided through the details of the proof (and grappled with the consequences), to be honest chapter 10 you refer to was a little boring :). But, I still do not agree with the way you are using the incompleteness theorem. It is important to distinguish between two different types of incompleteness. 1. Normal Incompleteness-- a logical theory fails to completely specify something. 2. Godelian Incompleteness-- a logical theory fails to completely specify something, even though we want it to. Logicians always mean type 2 incompleteness when they use the term. To formalize the difference between the two, the measuring stick of semantics is used. If a logic's provably-true statements don't match up to its semantically-true statements, it is incomplete. However, it seems like all you need is type 1 completeness for what you are saying. Nobody claims that there is a complete, well-defined semantics for natural language against which we could measure the provably-true (whatever THAT would mean). So, Godel's theorem is way overkill here in my opinion. --Abram On Wed, Oct 22, 2008 at 7:48 PM, Mark Waser [EMAIL PROTECTED] wrote: Most of what I was thinking of and referring to is in Chapter 10. Gödel's Quintessential Strange Loop (pages 125-145 in my version) but I would suggest that you really need to read the shorter Chapter 9. Pattern and Provability (pages 113-122) first. I actually had them conflated into a single chapter in my memory. I think that you'll enjoy them tremendously. - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Wednesday, October 22, 2008 4:19 PM Subject: Re: [agi] constructivist issues Mark, Chapter number please? --Abram On Wed, Oct 22, 2008 at 1:16 PM, Mark Waser [EMAIL PROTECTED] wrote: Douglas Hofstadter's newest book I Am A Strange Loop (currently available from Amazon for $7.99 - http://www.amazon.com/Am-Strange-Loop-Douglas-Hofstadter/dp/B001FA23HM) has an excellent chapter showing Godel in syntax and semantics. I highly recommend it. The upshot is that while it is easily possible to define a complete formal system of syntax, that formal system can always be used to convey something (some semantics) that is (are) outside/beyond the system -- OR, to paraphrase -- meaning is always incomplete because it can always be added to even inside a formal system of syntax. This is why I contend that language translation ends up being AGI-complete (although bounded subsets clearly don't need to be -- the question is whether you get a usable/useful subset more easily with or without first creating a seed AGI). - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Wednesday, October 22, 2008 12:38 PM Subject: Re: [agi] constructivist issues Mark, The way you invoke Godel's Theorem is strange to me... perhaps you have explained your argument more fully elsewhere, but as it stands I do not see your reasoning. --Abram On Wed, Oct 22, 2008 at 12:20 PM, Mark Waser [EMAIL PROTECTED] wrote: It looks like all this disambiguation by moving to a more formal language is about sweeping the problem under the rug, removing the need for uncertain reasoning from surface
Re: Lojban (was Re: [agi] constructivist issues)
Hi. I don't understand the following statements. Could you explain it some more? - Natural language can be learned from examples. Formal language can not. I think that you're basing this upon the methods that *you* would apply to each of the types of language. It makes sense to me that because of the regularities of a formal language that you would be able to use more effective methods -- but it doesn't mean that the methods used on natural language wouldn't work (just that they would be as inefficient as they are on natural languages. I would also argue that the same argument applies to the first statement of following the following two. - Formal language must be parsed before it can be understood. Natural language must be understood before it can be parsed. - Original Message - From: Matt Mahoney To: agi@v2.listbox.com Sent: Wednesday, October 22, 2008 9:23 PM Subject: Lojban (was Re: [agi] constructivist issues) Why would anyone use a simplified or formalized English (with regular grammar and no ambiguities) as a path to natural language understanding? Formal language processing has nothing to do with natural language processing other than sharing a common lexicon that make them appear superficially similar. - Natural language can be learned from examples. Formal language can not. - Formal language has an exact grammar and semantics. Natural language does not. - Formal language must be parsed before it can be understood. Natural language must be understood before it can be parsed. - Formal language is designed to be processed efficiently on a fast, reliable, sequential computer that neither makes nor tolerates errors, between systems that have identical, fixed language models. Natural language evolved to be processed efficiently by a slow, unreliable, massively parallel computer with enormous memory in a noisy environment between systems that have different but adaptive language models. So how does yet another formal language processing system help us understand natural language? This route has been a dead end for 50 years, in spite of the ability to always make some initial progress before getting stuck. -- Matt Mahoney, [EMAIL PROTECTED] --- On Wed, 10/22/08, Ben Goertzel [EMAIL PROTECTED] wrote: From: Ben Goertzel [EMAIL PROTECTED] Subject: Re: [agi] constructivist issues To: agi@v2.listbox.com Cc: [EMAIL PROTECTED] Date: Wednesday, October 22, 2008, 12:27 PM This is the standard Lojban dictionary http://jbovlaste.lojban.org/ I am not so worried about word meanings, they can always be handled via reference to WordNet via usages like run_1, run_2, etc. ... or as you say by using rarer, less ambiguous words Prepositions are more worrisome, however, I suppose they can be handled in a similar way, e.g. by defining an ontology of preposition meanings like with_1, with_2, with_3, etc. In fact we had someone spend a couple months integrating existing resources into a preposition-meaning ontology like this a while back ... the so-called PrepositionWordNet ... or as it eventually came to be called the LARDict or LogicalArgumentRelationshipDictionary ... I think it would be feasible to tweak RelEx to recognize these sorts of subscripts, and in this way to recognize a highly controlled English that would be unproblematic to map semantically... We would then say e.g. I ate dinner with_2 my fork I live in_2 Maryland I have lived_6 for_3 41 years (where I suppress all _1's, so that e.g. ate means ate_1) Because, RelEx already happily parses the syntax of all simple sentences, so the only real hassle to deal with is disambiguation. We could use similar hacking for reference resolution, temporal sequencing, etc. The terrorists_v1 robbed_v2 my house. After that_v2, the jerks_v1 urinated in_3 my yard. I think this would be a relatively pain-free way to communicate with an AI that lacks the common sense to carry out disambiguation and reference resolution reliably. Also, the log of communication would provide a nice training DB for it to use in studying disambiguation. -- Ben G On Wed, Oct 22, 2008 at 12:00 PM, Mark Waser [EMAIL PROTECTED] wrote: IMHO that is an almost hopeless approach, ambiguity is too integral to English or any natural language ... e.g preposition ambiguity Actually, I've been making pretty good progress. You just always use big words and never use small words and/or you use a specific phrase as a word. Ambiguous prepositions just disambiguate to one of three/four/five/more possible unambiguous words/phrases. The problem is that most previous subsets (Simplified English, Basic
Re: Lojban (was Re: [agi] constructivist issues)
--- On Thu, 10/23/08, Mark Waser [EMAIL PROTECTED] wrote: Hi. I don't understand the following statements. Could you explain it some more? - Natural language can be learned from examples. Formal language can not. I really mean that formal languages like C++ and HTML are not designed to be learned by the machines that implement them. We write a formal specification of their syntax and semantics. Obviously they are learnable by humans in the same way that humans learn natural languages -- by generalizing from lots of examples. Formal languages serve as a bridge between humans and machines. As such, a language is designed as a compromise between ease of machine specification and ease of human learnability. - Formal language must be parsed before it can be understood. Natural language must be understood before it can be parsed. In formal languages, the meaning of sentence depends heavily on its parse, for example: a = b - c; // a comment b = c - a; // a comment // a - b = c; a comment In natural language, a parse depends greatly on the meanings of the words. For example: - I ate pizza with chopsticks. - I ate pizza with pepperoni. - I ate pizza with Bob. But word order has only a small effect on meaning: - With Bob I ate pizza. - I with Bob ate pizza. - Pizza Bob I ate with. This is my objection to using formal languages to train AGI in a childhood development model like OpenCog (artificial toddler, child, adult, scientist). A child would be trained on single words with semantic content like pizza. Then an adult would learn increasingly complex grammatical structures. Only at the scientist level would an AGI be capable of learning formal languages. There really isn't any stage where a clean language like Lojban or Esperanto seems to help much with knowledge acquisition. If it did, then we would be teaching it in our schools. -- 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
Mark, I'm saying Godelian completeness/incompleteness can't be easily defined in the context of natural language, so it shouldn't be applied there without providing justification for that application (specifically, unambiguous definitions of provably true and semantically true for natural language). Does that make sense, or am I still confusing? Matthias, I agree with your point in this context, but I think you also mean to imply that Godel's incompleteness theorem isn't of any importance for artificial intelligence, which (probably pretty obviously) I wouldn't agree with. Godel's incompleteness theorem tells us important limitations of the logical approach to AI (and, indeed, any approach that can be implemented on normal computers). It *has* however been overused and abused throughout the years... which is one reason I jumped on Mark... --Abram On Thu, Oct 23, 2008 at 4:07 PM, Mark Waser [EMAIL PROTECTED] wrote: So to sum up, while you think linguistic vagueness comes from Godelian incompleteness, I think Godelian incompleteness can't even be defined in this context, due to linguistic vagueness. OK. Personally, I think that you did a good job of defining Godelian Incompleteness this time but arguably you did it by reference and by building a new semantic structure as you went along. On the other hand, you now seem to be arguing that my thinking that linguistic vagueness comes from Godelian incompleteness is wrong because Godelian incompleteness can't be defined . . . . I'm sort of at a loss as to how to proceed from here. If Godelian Incompleteness can't be defined, then by definition I can't prove anything but you can't disprove anything. This is nicely Escheresque and very Hofstadterian but . . . . - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Thursday, October 23, 2008 11:54 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] constructivist issues
You may not like Therefore, we cannot understand the math needed to define our own intelligence., but I'm rather convinced that it's correct. Do you mean to say that there are parts that we can't understand or that the totality is too large to fit and that it can't be cleanly and completely decomposed into pieces (i.e. it's a complex system ;-). Personally, I believe that the foundational pieces necessary to construct/boot-strap an intelligence are eminently understandable (if not even fairly simple) but that the resulting intelligence that a) organically grows from it's interaction with an environment that it can only extract partial, dirty, and ambiguous data and b) does not have the time, computational capability, or data to make itself even remotely consistent past a certain level IS large and complex enough that you will never truly understand it (which is where I have sympathy with Richard Loosemore's arguments -- but don't buy that the interaction of the pieces is necessarily so complex that we can't make broad predictions that are accurate enough to be able to engineer intelligence). If you say parts we can't understand, how do you reconcile that with your statements of yesterday about what general intelligences can learn? --- 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
It doesn't, because **I see no evidence that humans can understand the semantics of formal system in X in any sense that a digital computer program cannot** I just argued that humans can't understand the totality of any formal system X due to Godel's Incompleteness Theorem but the rest of this is worth addressing . . . . Whatever this mysterious understanding is that you believe you possess, **it cannot be communicated to me in language or mathematics**. Because any series of symbols you give me, could equally well be produced by some being without this mysterious understanding. Excellent! Except for the fact that the probability of the being *continuing* to emit those symbols without this mysterious understanding rapidly approaches zero. So I'm going to argue that understanding *can* effectively be communicated/determined. Arguing otherwise is effectively arguing for vanishingly small probabilities in infinities (and why I hate most arguments involving AIXI as proving *anything* except absolute limits c.f. Matt Mahoney and compression = intelligence). I'm going to continue arguing that understanding exactly equates to having a competent domain model and being able to communicate about it (i.e. that there is no mystery about understanding -- other than not understanding it ;-). Can you describe any possible finite set of finite-precision observations that could provide evidence in favor of the hypothesis that you possess this posited understanding, and against the hypothesis that you are something equivalent to a digital computer? I think you cannot. But I would argue that this is because a digital computer can have understanding (and must and will in order to be an AGI). So, your belief in this posited understanding has nothing to do with science, it's basically a kind of religious faith, it seems to me... '-) If you're assuming that humans have it and computers can't, then I have to strenuously agree. There is no data (that I am aware of) to support this conclusion so it's pure faith, not science. --- 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
I don't want to diss the personal value of logically inconsistent thoughts. But I doubt their scientific and engineering value. I doesn't seem to make sense that something would have personal value and then not have scientific or engineering value. I can sort of understand science if you're interpreting science looking for the final correct/optimal value but engineering generally goes for either good enough or the best of the currently known available options and anything that really/truly has personal value would seem to have engineering value. --- 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
(1) We humans understand the semantics of formal system X. No. This is the root of your problem. For example, replace formal system X with XML. Saying that We humans understand the semantics of XML certainly doesn't work and why I would argue that natural language understanding is AGI-complete (i.e. by the time all the RDF, OWL, and other ontology work is completed -- you'll have an AGI). Any formal system can always be extended *within it's defined syntax* to have any meaning. That is the essence of Godel's Incompleteness Theorem. It's also sort of the basis for my argument with Dr. Matthias Heger. Semantics are never finished except when your model of the world is finished (including all possibilities and infinitudes) so language understanding can't be simple and complete. Personally, rather than starting with NLP, I think that we're going to need to start with a formal language that is a disambiguated subset of English and figure out how to use our world model/knowledge to translate English to this disambiguated subset -- and then we can build from there. (or maybe this makes Heger's argument for him . . . . ;-) --- 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 Wed, Oct 22, 2008 at 10:51 AM, Mark Waser [EMAIL PROTECTED] wrote: I don't want to diss the personal value of logically inconsistent thoughts. But I doubt their scientific and engineering value. I doesn't seem to make sense that something would have personal value and then not have scientific or engineering value. Come by the house, we'll drop some acid together and you'll be convinced ;-) --- 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
Well, if you are a computable system, and if by think you mean represent accurately and internally then you can only think that odd thought via being logically inconsistent... ;-) True -- but why are we assuming *internally*? Drop that assumption as Charles clearly did and there is no problem. It's like infrastructure . . . . I don't have to know all the details of something to use it under normal circumstances though I frequently need to know the details is I'm doing something odd with it or looking for extreme performance and I definitely need to know the details if I'm diagnosing/fixing/debugging it -- but I can always learn them as I go . . . . - Original Message - From: Ben Goertzel To: agi@v2.listbox.com Sent: Tuesday, October 21, 2008 11:26 PM Subject: Re: [agi] constructivist issues Well, if you are a computable system, and if by think you mean represent accurately and internally then you can only think that odd thought via being logically inconsistent... ;-) On Tue, Oct 21, 2008 at 11:23 PM, charles griffiths [EMAIL PROTECTED] wrote: I disagree, and believe that I can think X: This is a thought (T) that is way too complex for me to ever have. Obviously, I can't think T and then think X, but I might represent T as a combination of myself plus a notebook or some other external media. Even if I only observe part of T at once, I might appreciate that it is one thought and believe (perhaps in error) that I could never think it. I might even observe T in action, if T is the result of billions of measurements, comparisons and calculations in a computer program. Isn't it just like thinking This is an image that is way too detailed for me to ever see? Charles Griffiths --- On Tue, 10/21/08, Ben Goertzel [EMAIL PROTECTED] wrote: From: Ben Goertzel [EMAIL PROTECTED] Subject: Re: [agi] constructivist issues To: agi@v2.listbox.com Date: Tuesday, October 21, 2008, 7:56 PM I am a Peircean pragmatist ... I have no objection to using infinities in mathematics ... they can certainly be quite useful. I'd rather use differential calculus to do calculations, than do everything using finite differences. It's just that, from a science perspective, these mathematical infinities have to be considered finite formal constructs ... they don't existP except in this way ... I'm not going to claim the pragmatist perspective is the only subjectively meaningful one. But so far as I can tell it's the only useful one for science and engineering... To take a totally different angle, consider the thought X = This is a thought that is way too complex for me to ever have Can I actually think X? Well, I can understand the *idea* of X. I can manipulate it symbolically and formally. I can reason about it and empathize with it by analogy to A thought that is way too complex for my three-year-old past-self to have ever had , and so forth. But it seems I can't ever really think X, except by being logically inconsistent within that same thought ... this is the Godel limitation applied to my own mind... I don't want to diss the personal value of logically inconsistent thoughts. But I doubt their scientific and engineering value. -- Ben G On Tue, Oct 21, 2008 at 10:43 PM, Abram Demski [EMAIL PROTECTED] wrote: Ben, How accurate would it be to describe you as a finitist or ultrafinitist? I ask because your view about restricting quantifiers seems to reject even the infinities normally allowed by constructivists. --Abram --- 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] Nothing will ever be attempted if all possible objections must be first overcome - Dr Samuel Johnson agi | Archives | Modify Your Subscription agi | Archives | Modify Your Subscription -- Ben Goertzel, PhD CEO, Novamente LLC and Biomind LLC Director of Research, SIAI [EMAIL PROTECTED] Nothing will ever be attempted if all possible objections must be first overcome - Dr
Re: [agi] constructivist issues
I disagree, and believe that I can think X: This is a thought (T) that is way too complex for me to ever have. Obviously, I can't think T and then think X, but I might represent T as a combination of myself plus a notebook or some other external media. Even if I only observe part of T at once, I might appreciate that it is one thought and believe (perhaps in error) that I could never think it. I might even observe T in action, if T is the result of billions of measurements, comparisons and calculations in a computer program. Isn't it just like thinking This is an image that is way too detailed for me to ever see? Excellent! This is precisely how I feel about intelligence . . . . (and why we *can* understand it even if we can't hold the totality of it -- or fully predict it -- sort of like the weather :-) --- 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
You have not convinced me that you can do anything a computer can't do. And, using language or math, you never will -- because any finite set of symbols you can utter, could also be uttered by some computational system. -- Ben G Can we pin this somewhere? (Maybe on Penrose? ;-) --- 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
The problem is to gradually improve overall causal model of environment (and its application for control), including language and dynamics of the world. Better model allows more detailed experience, and so through having a better inbuilt model of an aspect of environment, such as language, it's possible to communicate richer description of other aspects of environment. But it's not obvious that bandwidth of experience is the bottleneck here. No, but nor is it obvious that this *isn't* one of the major bottlenecks... It's probably just limitations of the cognitive algorithm that simply can't efficiently improve its model, and so feeding it more experience through tricks like this is like trying to get a hundredfold speedup in the O(log(log(n))) algorithm by feeding it more hardware. Hard to say... Remember, we humans have a load of evolved inductive bias for understanding human language ... AGI's don't ... so using Lojban to talk to an AGI could be a way to partly make up for this deficit in inductive bias... It should be possible to get a proof-of-concept level results about efficiency without resorting to Cycs and Lojbans, and after that they'll turn out to be irrelevant. Cyc and Lojban are not comparable, one is a knowledge-base, the other is a language Cyc-L and Lojban are more closely comparable, though still very different because Lojban allows for more ambiguity (as well as Cyc-L level precision, depending on speaker's choice) ... and of course Lojban is intended for interactive conversation rather than knowledge entry ben g --- 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
IMHO that is an almost hopeless approach, ambiguity is too integral to English or any natural language ... e.g preposition ambiguity Actually, I've been making pretty good progress. You just always use big words and never use small words and/or you use a specific phrase as a word. Ambiguous prepositions just disambiguate to one of three/four/five/more possible unambiguous words/phrases. The problem is that most previous subsets (Simplified English, Basic English) actually *favored* the small tremendously over-used/ambiguous words (because you got so much more bang for the buck with them). Try only using big unambiguous words and see if you still have the same opinion. If you want to take this sort of approach, you'd better start with Lojban instead Learning Lojban is a pain but far less pain than you'll have trying to make a disambiguated subset of English. My first reaction is . . . . Take a Lojban dictionary and see if you can come up with an unambiguous English word or very short phrase for each Lojban word. If you can do it, my approach will work and will have the advantage that the output can be read by anyone (i.e. it's the equivalent of me having done it in Lojban and then added a Lojban - English translation on the end) though the input is still *very* problematical (thus the need for a semantically-driven English-subset translator). If you can't do it, then my approach won't work. Can you do it? Why or why not? If you can, do you still believe that my approach won't work? Oh, wait . . . . a Lojban-to-English dictionary *does* attempt to come up with an unambiguous English word or very short phrase for each Lojban word. :-) Actually, h . . . . a Lojban dictionary would probably help me focus my efforts a bit better and highlight things that I may have missed . . . . do you have a preferred dictionary or resource? (Google has too many for me to do a decent perusal quickly) - Original Message - From: Ben Goertzel To: agi@v2.listbox.com Sent: Wednesday, October 22, 2008 11:11 AM Subject: Re: [agi] constructivist issues Personally, rather than starting with NLP, I think that we're going to need to start with a formal language that is a disambiguated subset of English IMHO that is an almost hopeless approach, ambiguity is too integral to English or any natural language ... e.g preposition ambiguity If you want to take this sort of approach, you'd better start with Lojban instead Learning Lojban is a pain but far less pain than you'll have trying to make a disambiguated subset of English. 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] constructivist issues
(joke) What? You don't love me any more? /thread - Original Message - From: Ben Goertzel To: agi@v2.listbox.com Sent: Wednesday, October 22, 2008 11:11 AM Subject: Re: [agi] constructivist issues (joke) On Wed, Oct 22, 2008 at 11:11 AM, Ben Goertzel [EMAIL PROTECTED] wrote: On Wed, Oct 22, 2008 at 10:51 AM, Mark Waser [EMAIL PROTECTED] wrote: I don't want to diss the personal value of logically inconsistent thoughts. But I doubt their scientific and engineering value. I doesn't seem to make sense that something would have personal value and then not have scientific or engineering value. Come by the house, we'll drop some acid together and you'll be convinced ;-) -- 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
Come by the house, we'll drop some acid together and you'll be convinced ;-) Been there, done that. Just because some logically inconsistent thoughts have no value doesn't mean that all logically inconsistent thoughts have no value. Not to mention the fact that hallucinogens, if not the subsequently warped thoughts, do have the serious value of raising your mental Boltzmann temperature. - Original Message - From: Ben Goertzel To: agi@v2.listbox.com Sent: Wednesday, October 22, 2008 11:11 AM Subject: Re: [agi] constructivist issues On Wed, Oct 22, 2008 at 10:51 AM, Mark Waser [EMAIL PROTECTED] wrote: I don't want to diss the personal value of logically inconsistent thoughts. But I doubt their scientific and engineering value. I doesn't seem to make sense that something would have personal value and then not have scientific or engineering value. Come by the house, we'll drop some acid together and you'll be convinced ;-) -- 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
On Wed, Oct 22, 2008 at 7:47 PM, Ben Goertzel [EMAIL PROTECTED] wrote: The problem is to gradually improve overall causal model of environment (and its application for control), including language and dynamics of the world. Better model allows more detailed experience, and so through having a better inbuilt model of an aspect of environment, such as language, it's possible to communicate richer description of other aspects of environment. But it's not obvious that bandwidth of experience is the bottleneck here. No, but nor is it obvious that this *isn't* one of the major bottlenecks... My intuition is that it's very easy to steadily increase bandwidth of experience, the more you know the more you understand. If you start from simple sensors/actuators (or even chess or Go), progress is gradual and open-ended. It's probably just limitations of the cognitive algorithm that simply can't efficiently improve its model, and so feeding it more experience through tricks like this is like trying to get a hundredfold speedup in the O(log(log(n))) algorithm by feeding it more hardware. Hard to say... Remember, we humans have a load of evolved inductive bias for understanding human language ... AGI's don't ... so using Lojban to talk to an AGI could be a way to partly make up for this deficit in inductive bias... Any language at all is a way of increasing experiential bandwidth about environment. If bandwidth isn't essential, bootstrapping this process through a language is equally irrelevant. At some point, however inefficiently, language can be learned if system allows open-ended learning. This is a question of not doing premature optimization of a program that is not even designed yet, not talking about being implemented and profiled. It should be possible to get a proof-of-concept level results about efficiency without resorting to Cycs and Lojbans, and after that they'll turn out to be irrelevant. Cyc and Lojban are not comparable, one is a knowledge-base, the other is a language Cyc-L and Lojban are more closely comparable, though still very different because Lojban allows for more ambiguity (as well as Cyc-L level precision, depending on speaker's choice) ... and of course Lojban is intended for interactive conversation rather than knowledge entry (as tools towards improving bandwidth of experience, they do the same thing) -- Vladimir Nesov [EMAIL PROTECTED] http://causalityrelay.wordpress.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
Too many responses for me to comment on everything! So, sorry to those I don't address... Ben, When I claim a mathematical entity exists, I'm saying loosely that meaningful statements can be made using it. So, I think meaning is more basic. I mentioned already what my current definition of meaning is: a statement is meaningful if it is associated with a computable rule of deduction that it can use to operate on other (meaningful) statements. This is in contrast to positivist-style definitions of meaning, that would instead require a computable test of truth and/or falsehood. So, a statement is meaningful if it has procedural deductive meaning. We *understand* a statement if we are capable of carrying out the corresponding deductive procedure. A statement is *true* if carrying out that deductive procedure only produces more true statements. We *believe* a statement if we not only understand it, but proceed to apply its deductive procedure. There is of course some basic level of meaningful statements, such as sensory observations, so that this is a working recursive definition of meaning and truth. By this definition of meaning, any statement in the arithmetical hierarchy is meaningful (because each statement can be represented by computable consequences on other statements in the arithmetical hierarchy). I think some hyperarithmetical truths are captured as well. I am more doubtful about it capturing anything beyond the first level of the analytic hierarchy, and general set-theoretic discourse seems far beyond its reach. Regardless, the definition of meaning makes a very large number of uncomputable truths nonetheless meaningful. Russel, I think both Ben and I would approximately agree with everything you said, but that doesn't change our disagreeing with each other :). Mark, Good call... I shouldn't be talking like I think it is terrifically unlikely that some more-intelligent alien species would find humans mathematically crude. What I meant was, it seems like humans are logically complete in some sense. In practice we are greatly limited by memory and processing speed and so on; but I *don't* think we're limited by lacking some important logical construct. It would be like us discovering some alien species whose mathematicians were able to understand each individual case of mathematical induction, but were unable to comprehend the argument for accepting it as a general principle, because they lack the abstraction. Something like that is what I find implausible. --Abram --- 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
This is the standard Lojban dictionary http://jbovlaste.lojban.org/ I am not so worried about word meanings, they can always be handled via reference to WordNet via usages like run_1, run_2, etc. ... or as you say by using rarer, less ambiguous words Prepositions are more worrisome, however, I suppose they can be handled in a similar way, e.g. by defining an ontology of preposition meanings like with_1, with_2, with_3, etc. In fact we had someone spend a couple months integrating existing resources into a preposition-meaning ontology like this a while back ... the so-called PrepositionWordNet ... or as it eventually came to be called the LARDict or LogicalArgumentRelationshipDictionary ... I think it would be feasible to tweak RelEx to recognize these sorts of subscripts, and in this way to recognize a highly controlled English that would be unproblematic to map semantically... We would then say e.g. I ate dinner with_2 my fork I live in_2 Maryland I have lived_6 for_3 41 years (where I suppress all _1's, so that e.g. ate means ate_1) Because, RelEx already happily parses the syntax of all simple sentences, so the only real hassle to deal with is disambiguation. We could use similar hacking for reference resolution, temporal sequencing, etc. The terrorists_v1 robbed_v2 my house. After that_v2, the jerks_v1 urinated in_3 my yard. I think this would be a relatively pain-free way to communicate with an AI that lacks the common sense to carry out disambiguation and reference resolution reliably. Also, the log of communication would provide a nice training DB for it to use in studying disambiguation. -- Ben G On Wed, Oct 22, 2008 at 12:00 PM, Mark Waser [EMAIL PROTECTED] wrote: IMHO that is an almost hopeless approach, ambiguity is too integral to English or any natural language ... e.g preposition ambiguity Actually, I've been making pretty good progress. You just always use big words and never use small words and/or you use a specific phrase as a word. Ambiguous prepositions just disambiguate to one of three/four/five/more possible unambiguous words/phrases. The problem is that most previous subsets (Simplified English, Basic English) actually *favored* the small tremendously over-used/ambiguous words (because you got so much more bang for the buck with them). Try only using big unambiguous words and see if you still have the same opinion. If you want to take this sort of approach, you'd better start with Lojban instead Learning Lojban is a pain but far less pain than you'll have trying to make a disambiguated subset of English. My first reaction is . . . . Take a Lojban dictionary and see if you can come up with an unambiguous English word or very short phrase for each Lojban word. If you can do it, my approach will work and will have the advantage that the output can be read by anyone (i.e. it's the equivalent of me having done it in Lojban and then added a Lojban - English translation on the end) though the input is still *very* problematical (thus the need for a semantically-driven English-subset translator). If you can't do it, then my approach won't work. Can you do it? Why or why not? If you can, do you still believe that my approach won't work? Oh, wait . . . . a Lojban-to-English dictionary *does* attempt to come up with an unambiguous English word or very short phrase for each Lojban word. :-) Actually, h . . . . a Lojban dictionary would probably help me focus my efforts a bit better and highlight things that I may have missed . . . . do you have a preferred dictionary or resource? (Google has too many for me to do a decent perusal quickly) - Original Message - *From:* Ben Goertzel [EMAIL PROTECTED] *To:* agi@v2.listbox.com *Sent:* Wednesday, October 22, 2008 11:11 AM *Subject:* Re: [agi] constructivist issues Personally, rather than starting with NLP, I think that we're going to need to start with a formal language that is a disambiguated subset of English IMHO that is an almost hopeless approach, ambiguity is too integral to English or any natural language ... e.g preposition ambiguity If you want to take this sort of approach, you'd better start with Lojban instead Learning Lojban is a pain but far less pain than you'll have trying to make a disambiguated subset of English. ben g -- *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 -- *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
Re: [agi] constructivist issues
So, a statement is meaningful if it has procedural deductive meaning. We *understand* a statement if we are capable of carrying out the corresponding deductive procedure. A statement is *true* if carrying out that deductive procedure only produces more true statements. We *believe* a statement if we not only understand it, but proceed to apply its deductive procedure. OK, then according to your definition, Godel's Theorem says that if humans are computable there are some things that we cannot understand ... just as, for any computer program, there are some things it can't understand. It just happens that according to your definition, a computer system can understand some fabulously uncomputable entities. But there's no contradiction there. Just like a human can, a digital theorem prover can understand some uncomputable entities in the sense you specify... ben g --- 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 way you invoke Godel's Theorem is strange to me... perhaps you have explained your argument more fully elsewhere, but as it stands I do not see your reasoning. --Abram On Wed, Oct 22, 2008 at 12:20 PM, Mark Waser [EMAIL PROTECTED] wrote: It looks like all this disambiguation by moving to a more formal language is about sweeping the problem under the rug, removing the need for uncertain reasoning from surface levels of syntax and semantics, to remember about it 10 years later, retouch the most annoying holes with simple statistical techniques, and continue as before. That's an excellent criticism but not the intent. Godel's Incompleteness Theorem means that you will be forever building . . . . All that disambiguation does is provides a solid, commonly-agreed upon foundation to build from. English and all natural languages are *HARD*. They are not optimal for simple understanding particularly given the realms we are currently in and ambiguity makes things even worse. Languages have so many ambiguities because of the way that they (and concepts) develop. You see something new, you grab the nearest analogy and word/label and then modify it to fit. That's why you then later need the much longer words and very specific scientific terms and names. Simple language is what you need to build the more specific complex language. Having an unambiguous constructed language is simply a template or mold that you can use as scaffolding while you develop NLU. Children start out very unambiguous and concrete and so should we. (And I don't believe in statistical techniques unless you have the resources of Google or AIXI) --- 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
What I meant was, it seems like humans are logically complete in some sense. In practice we are greatly limited by memory and processing speed and so on; but I *don't* think we're limited by lacking some important logical construct. It would be like us discovering some alien species whose mathematicians were able to understand each individual case of mathematical induction, but were unable to comprehend the argument for accepting it as a general principle, because they lack the abstraction. Something like that is what I find implausible. I like the phrase logically complete. The way that I like to think about it is that we have the necessary seed of whatever intelligence/competence is that can be logically extended to cover all circumstances. We may not have the personal time or resources to do so but given infinite time and resources there is no block on the path from what we have to getting there. Note, however, that it is my understanding that a number of people on this list do not agree with this statement (feel free to chime in with you reasons why folks). - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Wednesday, October 22, 2008 12:20 PM Subject: Re: [agi] constructivist issues Too many responses for me to comment on everything! So, sorry to those I don't address... Ben, When I claim a mathematical entity exists, I'm saying loosely that meaningful statements can be made using it. So, I think meaning is more basic. I mentioned already what my current definition of meaning is: a statement is meaningful if it is associated with a computable rule of deduction that it can use to operate on other (meaningful) statements. This is in contrast to positivist-style definitions of meaning, that would instead require a computable test of truth and/or falsehood. So, a statement is meaningful if it has procedural deductive meaning. We *understand* a statement if we are capable of carrying out the corresponding deductive procedure. A statement is *true* if carrying out that deductive procedure only produces more true statements. We *believe* a statement if we not only understand it, but proceed to apply its deductive procedure. There is of course some basic level of meaningful statements, such as sensory observations, so that this is a working recursive definition of meaning and truth. By this definition of meaning, any statement in the arithmetical hierarchy is meaningful (because each statement can be represented by computable consequences on other statements in the arithmetical hierarchy). I think some hyperarithmetical truths are captured as well. I am more doubtful about it capturing anything beyond the first level of the analytic hierarchy, and general set-theoretic discourse seems far beyond its reach. Regardless, the definition of meaning makes a very large number of uncomputable truths nonetheless meaningful. Russel, I think both Ben and I would approximately agree with everything you said, but that doesn't change our disagreeing with each other :). Mark, Good call... I shouldn't be talking like I think it is terrifically unlikely that some more-intelligent alien species would find humans mathematically crude. What I meant was, it seems like humans are logically complete in some sense. In practice we are greatly limited by memory and processing speed and so on; but I *don't* think we're limited by lacking some important logical construct. It would be like us discovering some alien species whose mathematicians were able to understand each individual case of mathematical induction, but were unable to comprehend the argument for accepting it as a general principle, because they lack the abstraction. Something like that is what I find implausible. --Abram --- 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: [OpenCog] Re: [agi] constructivist issues
I think this would be a relatively pain-free way to communicate with an AI that lacks the common sense to carry out disambiguation and reference resolution reliably. Also, the log of communication would provide a nice training DB for it to use in studying disambiguation. Awesome. Like I said, it's a piece of something that I'm trying currently. If I get positive results, I'm certainly not going to hide the fact. ;-) (or, it could turn into a learning experience like my attempts with Simplified English and Basic English :-) - Original Message - From: Ben Goertzel To: agi@v2.listbox.com Cc: [EMAIL PROTECTED] Sent: Wednesday, October 22, 2008 12:27 PM Subject: [OpenCog] Re: [agi] constructivist issues This is the standard Lojban dictionary http://jbovlaste.lojban.org/ I am not so worried about word meanings, they can always be handled via reference to WordNet via usages like run_1, run_2, etc. ... or as you say by using rarer, less ambiguous words Prepositions are more worrisome, however, I suppose they can be handled in a similar way, e.g. by defining an ontology of preposition meanings like with_1, with_2, with_3, etc. In fact we had someone spend a couple months integrating existing resources into a preposition-meaning ontology like this a while back ... the so-called PrepositionWordNet ... or as it eventually came to be called the LARDict or LogicalArgumentRelationshipDictionary ... I think it would be feasible to tweak RelEx to recognize these sorts of subscripts, and in this way to recognize a highly controlled English that would be unproblematic to map semantically... We would then say e.g. I ate dinner with_2 my fork I live in_2 Maryland I have lived_6 for_3 41 years (where I suppress all _1's, so that e.g. ate means ate_1) Because, RelEx already happily parses the syntax of all simple sentences, so the only real hassle to deal with is disambiguation. We could use similar hacking for reference resolution, temporal sequencing, etc. The terrorists_v1 robbed_v2 my house. After that_v2, the jerks_v1 urinated in_3 my yard. I think this would be a relatively pain-free way to communicate with an AI that lacks the common sense to carry out disambiguation and reference resolution reliably. Also, the log of communication would provide a nice training DB for it to use in studying disambiguation. -- Ben G On Wed, Oct 22, 2008 at 12:00 PM, Mark Waser [EMAIL PROTECTED] wrote: IMHO that is an almost hopeless approach, ambiguity is too integral to English or any natural language ... e.g preposition ambiguity Actually, I've been making pretty good progress. You just always use big words and never use small words and/or you use a specific phrase as a word. Ambiguous prepositions just disambiguate to one of three/four/five/more possible unambiguous words/phrases. The problem is that most previous subsets (Simplified English, Basic English) actually *favored* the small tremendously over-used/ambiguous words (because you got so much more bang for the buck with them). Try only using big unambiguous words and see if you still have the same opinion. If you want to take this sort of approach, you'd better start with Lojban instead Learning Lojban is a pain but far less pain than you'll have trying to make a disambiguated subset of English. My first reaction is . . . . Take a Lojban dictionary and see if you can come up with an unambiguous English word or very short phrase for each Lojban word. If you can do it, my approach will work and will have the advantage that the output can be read by anyone (i.e. it's the equivalent of me having done it in Lojban and then added a Lojban - English translation on the end) though the input is still *very* problematical (thus the need for a semantically-driven English-subset translator). If you can't do it, then my approach won't work. Can you do it? Why or why not? If you can, do you still believe that my approach won't work? Oh, wait . . . . a Lojban-to-English dictionary *does* attempt to come up with an unambiguous English word or very short phrase for each Lojban word. :-) Actually, h . . . . a Lojban dictionary would probably help me focus my efforts a bit better and highlight things that I may have missed . . . . do you have a preferred dictionary or resource? (Google has too many for me to do a decent perusal quickly) - Original Message - From: Ben Goertzel To: agi@v2.listbox.com Sent: Wednesday, October 22, 2008 11:11 AM Subject: Re: [agi] constructivist issues Personally, rather than starting with NLP, I think that we're going to need to start with a formal language that is a disambiguated subset of English IMHO that is an almost hopeless approach, ambiguity is too integral to English or any natural language ... e.g
Re: [agi] constructivist issues
All theorems in the same formal system are equivalent anyways ;-) On Wed, Oct 22, 2008 at 1:03 PM, Abram Demski [EMAIL PROTECTED] wrote: Ben, What, then, do you make of my definition? Do you think deductive consequence is insufficient for meaningfulness? I am not sure exactly where you draw the line as to what is really meaningful (as in finite collections of finite statements about finite-precision measurements) and what is only indirectly meaningful by its usefulness (as in differential calculus). Perhaps any universal statements are only meaningful by usefulness? Also, it seems like when you say Godel's Incompleteness, you mean Tarski's Undefinability? (Can't let the theorems be misused!) About the theorem prover; yes, absolutely, so long as the mathematical entity is understandable by the definition I gave. Unfortunately, I still have some work to do, because as far as I can tell that definition does not explain how uncountable sets are meaningful... (maybe it does and I am just missing something...) --Abram On Wed, Oct 22, 2008 at 12:30 PM, Ben Goertzel [EMAIL PROTECTED] wrote: So, a statement is meaningful if it has procedural deductive meaning. We *understand* a statement if we are capable of carrying out the corresponding deductive procedure. A statement is *true* if carrying out that deductive procedure only produces more true statements. We *believe* a statement if we not only understand it, but proceed to apply its deductive procedure. OK, then according to your definition, Godel's Theorem says that if humans are computable there are some things that we cannot understand ... just as, for any computer program, there are some things it can't understand. It just happens that according to your definition, a computer system can understand some fabulously uncomputable entities. But there's no contradiction there. Just like a human can, a digital theorem prover can understand some uncomputable entities in the sense you specify... 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 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
Also, I don't prefer to define meaning the way you do ... so clarifying issues with your definition is your problem, not mine!! On Wed, Oct 22, 2008 at 1:03 PM, Abram Demski [EMAIL PROTECTED] wrote: Ben, What, then, do you make of my definition? Do you think deductive consequence is insufficient for meaningfulness? I am not sure exactly where you draw the line as to what is really meaningful (as in finite collections of finite statements about finite-precision measurements) and what is only indirectly meaningful by its usefulness (as in differential calculus). Perhaps any universal statements are only meaningful by usefulness? Also, it seems like when you say Godel's Incompleteness, you mean Tarski's Undefinability? (Can't let the theorems be misused!) About the theorem prover; yes, absolutely, so long as the mathematical entity is understandable by the definition I gave. Unfortunately, I still have some work to do, because as far as I can tell that definition does not explain how uncountable sets are meaningful... (maybe it does and I am just missing something...) --Abram On Wed, Oct 22, 2008 at 12:30 PM, Ben Goertzel [EMAIL PROTECTED] wrote: So, a statement is meaningful if it has procedural deductive meaning. We *understand* a statement if we are capable of carrying out the corresponding deductive procedure. A statement is *true* if carrying out that deductive procedure only produces more true statements. We *believe* a statement if we not only understand it, but proceed to apply its deductive procedure. OK, then according to your definition, Godel's Theorem says that if humans are computable there are some things that we cannot understand ... just as, for any computer program, there are some things it can't understand. It just happens that according to your definition, a computer system can understand some fabulously uncomputable entities. But there's no contradiction there. Just like a human can, a digital theorem prover can understand some uncomputable entities in the sense you specify... 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 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
Douglas Hofstadter's newest book I Am A Strange Loop (currently available from Amazon for $7.99 - http://www.amazon.com/Am-Strange-Loop-Douglas-Hofstadter/dp/B001FA23HM) has an excellent chapter showing Godel in syntax and semantics. I highly recommend it. The upshot is that while it is easily possible to define a complete formal system of syntax, that formal system can always be used to convey something (some semantics) that is (are) outside/beyond the system -- OR, to paraphrase -- meaning is always incomplete because it can always be added to even inside a formal system of syntax. This is why I contend that language translation ends up being AGI-complete (although bounded subsets clearly don't need to be -- the question is whether you get a usable/useful subset more easily with or without first creating a seed AGI). - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Wednesday, October 22, 2008 12:38 PM Subject: Re: [agi] constructivist issues Mark, The way you invoke Godel's Theorem is strange to me... perhaps you have explained your argument more fully elsewhere, but as it stands I do not see your reasoning. --Abram On Wed, Oct 22, 2008 at 12:20 PM, Mark Waser [EMAIL PROTECTED] wrote: It looks like all this disambiguation by moving to a more formal language is about sweeping the problem under the rug, removing the need for uncertain reasoning from surface levels of syntax and semantics, to remember about it 10 years later, retouch the most annoying holes with simple statistical techniques, and continue as before. That's an excellent criticism but not the intent. Godel's Incompleteness Theorem means that you will be forever building . . . . All that disambiguation does is provides a solid, commonly-agreed upon foundation to build from. English and all natural languages are *HARD*. They are not optimal for simple understanding particularly given the realms we are currently in and ambiguity makes things even worse. Languages have so many ambiguities because of the way that they (and concepts) develop. You see something new, you grab the nearest analogy and word/label and then modify it to fit. That's why you then later need the much longer words and very specific scientific terms and names. Simple language is what you need to build the more specific complex language. Having an unambiguous constructed language is simply a template or mold that you can use as scaffolding while you develop NLU. Children start out very unambiguous and concrete and so should we. (And I don't believe in statistical techniques unless you have the resources of Google or AIXI) --- 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 --- 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: [OpenCog] Re: [agi] constructivist issues
Well, I am confident my approach with subscripts to handle disambiguation and reference resolution would work, in conjunction with the existing link-parser/RelEx framework... If anyone wants to implement it, it seems like just some hacking with the open-source Java RelEx code... Like what I called a semantically-driven English-subset translator?. Oh, I'm pretty confidant that it will work as well . . . . after the LaBrea tar pit of implementations . . . . (exactly how little semantic-related coding do you think will be necessary? ;-) - Original Message - From: Ben Goertzel To: agi@v2.listbox.com Cc: [EMAIL PROTECTED] Sent: Wednesday, October 22, 2008 1:06 PM Subject: Re: [OpenCog] Re: [agi] constructivist issues Well, I am confident my approach with subscripts to handle disambiguation and reference resolution would work, in conjunction with the existing link-parser/RelEx framework... If anyone wants to implement it, it seems like just some hacking with the open-source Java RelEx code... ben g On Wed, Oct 22, 2008 at 12:59 PM, Mark Waser [EMAIL PROTECTED] wrote: I think this would be a relatively pain-free way to communicate with an AI that lacks the common sense to carry out disambiguation and reference resolution reliably. Also, the log of communication would provide a nice training DB for it to use in studying disambiguation. Awesome. Like I said, it's a piece of something that I'm trying currently. If I get positive results, I'm certainly not going to hide the fact. ;-) (or, it could turn into a learning experience like my attempts with Simplified English and Basic English :-) - Original Message - From: Ben Goertzel To: agi@v2.listbox.com Cc: [EMAIL PROTECTED] Sent: Wednesday, October 22, 2008 12:27 PM Subject: [OpenCog] Re: [agi] constructivist issues This is the standard Lojban dictionary http://jbovlaste.lojban.org/ I am not so worried about word meanings, they can always be handled via reference to WordNet via usages like run_1, run_2, etc. ... or as you say by using rarer, less ambiguous words Prepositions are more worrisome, however, I suppose they can be handled in a similar way, e.g. by defining an ontology of preposition meanings like with_1, with_2, with_3, etc. In fact we had someone spend a couple months integrating existing resources into a preposition-meaning ontology like this a while back ... the so-called PrepositionWordNet ... or as it eventually came to be called the LARDict or LogicalArgumentRelationshipDictionary ... I think it would be feasible to tweak RelEx to recognize these sorts of subscripts, and in this way to recognize a highly controlled English that would be unproblematic to map semantically... We would then say e.g. I ate dinner with_2 my fork I live in_2 Maryland I have lived_6 for_3 41 years (where I suppress all _1's, so that e.g. ate means ate_1) Because, RelEx already happily parses the syntax of all simple sentences, so the only real hassle to deal with is disambiguation. We could use similar hacking for reference resolution, temporal sequencing, etc. The terrorists_v1 robbed_v2 my house. After that_v2, the jerks_v1 urinated in_3 my yard. I think this would be a relatively pain-free way to communicate with an AI that lacks the common sense to carry out disambiguation and reference resolution reliably. Also, the log of communication would provide a nice training DB for it to use in studying disambiguation. -- Ben G On Wed, Oct 22, 2008 at 12:00 PM, Mark Waser [EMAIL PROTECTED] wrote: IMHO that is an almost hopeless approach, ambiguity is too integral to English or any natural language ... e.g preposition ambiguity Actually, I've been making pretty good progress. You just always use big words and never use small words and/or you use a specific phrase as a word. Ambiguous prepositions just disambiguate to one of three/four/five/more possible unambiguous words/phrases. The problem is that most previous subsets (Simplified English, Basic English) actually *favored* the small tremendously over-used/ambiguous words (because you got so much more bang for the buck with them). Try only using big unambiguous words and see if you still have the same opinion. If you want to take this sort of approach, you'd better start with Lojban instead Learning Lojban is a pain but far less pain than you'll have trying to make a disambiguated subset of English. My first reaction is . . . . Take a Lojban dictionary and see if you can come up with an unambiguous English word or very short phrase for each Lojban word. If you can do it, my approach will work and will have the advantage that the output can be read by anyone (i.e. it's the equivalent
Re: [agi] constructivist issues
Mark, I own and have read the book-- but my first introduction to Godel's Theorem was Douglas Hofstadter's earlier work, Godel Escher Bach. Since I had already been guided through the details of the proof (and grappled with the consequences), to be honest chapter 10 you refer to was a little boring :). But, I still do not agree with the way you are using the incompleteness theorem. It is important to distinguish between two different types of incompleteness. 1. Normal Incompleteness-- a logical theory fails to completely specify something. 2. Godelian Incompleteness-- a logical theory fails to completely specify something, even though we want it to. Logicians always mean type 2 incompleteness when they use the term. To formalize the difference between the two, the measuring stick of semantics is used. If a logic's provably-true statements don't match up to its semantically-true statements, it is incomplete. However, it seems like all you need is type 1 completeness for what you are saying. Nobody claims that there is a complete, well-defined semantics for natural language against which we could measure the provably-true (whatever THAT would mean). So, Godel's theorem is way overkill here in my opinion. --Abram On Wed, Oct 22, 2008 at 7:48 PM, Mark Waser [EMAIL PROTECTED] wrote: Most of what I was thinking of and referring to is in Chapter 10. Gödel's Quintessential Strange Loop (pages 125-145 in my version) but I would suggest that you really need to read the shorter Chapter 9. Pattern and Provability (pages 113-122) first. I actually had them conflated into a single chapter in my memory. I think that you'll enjoy them tremendously. - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Wednesday, October 22, 2008 4:19 PM Subject: Re: [agi] constructivist issues Mark, Chapter number please? --Abram On Wed, Oct 22, 2008 at 1:16 PM, Mark Waser [EMAIL PROTECTED] wrote: Douglas Hofstadter's newest book I Am A Strange Loop (currently available from Amazon for $7.99 - http://www.amazon.com/Am-Strange-Loop-Douglas-Hofstadter/dp/B001FA23HM) has an excellent chapter showing Godel in syntax and semantics. I highly recommend it. The upshot is that while it is easily possible to define a complete formal system of syntax, that formal system can always be used to convey something (some semantics) that is (are) outside/beyond the system -- OR, to paraphrase -- meaning is always incomplete because it can always be added to even inside a formal system of syntax. This is why I contend that language translation ends up being AGI-complete (although bounded subsets clearly don't need to be -- the question is whether you get a usable/useful subset more easily with or without first creating a seed AGI). - Original Message - From: Abram Demski [EMAIL PROTECTED] To: agi@v2.listbox.com Sent: Wednesday, October 22, 2008 12:38 PM Subject: Re: [agi] constructivist issues Mark, The way you invoke Godel's Theorem is strange to me... perhaps you have explained your argument more fully elsewhere, but as it stands I do not see your reasoning. --Abram On Wed, Oct 22, 2008 at 12:20 PM, Mark Waser [EMAIL PROTECTED] wrote: It looks like all this disambiguation by moving to a more formal language is about sweeping the problem under the rug, removing the need for uncertain reasoning from surface levels of syntax and semantics, to remember about it 10 years later, retouch the most annoying holes with simple statistical techniques, and continue as before. That's an excellent criticism but not the intent. Godel's Incompleteness Theorem means that you will be forever building . . . . All that disambiguation does is provides a solid, commonly-agreed upon foundation to build from. English and all natural languages are *HARD*. They are not optimal for simple understanding particularly given the realms we are currently in and ambiguity makes things even worse. Languages have so many ambiguities because of the way that they (and concepts) develop. You see something new, you grab the nearest analogy and word/label and then modify it to fit. That's why you then later need the much longer words and very specific scientific terms and names. Simple language is what you need to build the more specific complex language. Having an unambiguous constructed language is simply a template or mold that you can use as scaffolding while you develop NLU. Children start out very unambiguous and concrete and so should we. (And I don't believe in statistical techniques unless you have the resources of Google or AIXI) --- 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
Re: Lojban (was Re: [agi] constructivist issues)
[Usual disclaimer: this is not the approach I'm taking, but I don't find it stupid] The idea is that by teaching an AI in a minimally-ambiguous language, one can build up its commonsense understanding such that it can then deal with the ambiguities of natural language better, using this understanding... Just because Cyc failed doesn't mean teaching a system using Lojban would necessarily fail. Lojban is a lot more interesting than Cyc-L because it can tractably be used by people to informally chat with AI's, just as can a natural language... For instance, one could chat in Lojban with an embodied AI system, and it would then get strong symbol groundings for its Lojban ;-) ben g On Wed, Oct 22, 2008 at 9:23 PM, Matt Mahoney [EMAIL PROTECTED] wrote: Why would anyone use a simplified or formalized English (with regular grammar and no ambiguities) as a path to natural language understanding? Formal language processing has nothing to do with natural language processing other than sharing a common lexicon that make them appear superficially similar. - Natural language can be learned from examples. Formal language can not. - Formal language has an exact grammar and semantics. Natural language does not. - Formal language must be parsed before it can be understood. Natural language must be understood before it can be parsed. - Formal language is designed to be processed efficiently on a fast, reliable, sequential computer that neither makes nor tolerates errors, between systems that have identical, fixed language models. Natural language evolved to be processed efficiently by a slow, unreliable, massively parallel computer with enormous memory in a noisy environment between systems that have different but adaptive language models. So how does yet another formal language processing system help us understand natural language? This route has been a dead end for 50 years, in spite of the ability to always make some initial progress before getting stuck. -- Matt Mahoney, [EMAIL PROTECTED] --- On *Wed, 10/22/08, Ben Goertzel [EMAIL PROTECTED]* wrote: From: Ben Goertzel [EMAIL PROTECTED] Subject: Re: [agi] constructivist issues To: agi@v2.listbox.com Cc: [EMAIL PROTECTED] Date: Wednesday, October 22, 2008, 12:27 PM This is the standard Lojban dictionary http://jbovlaste.lojban.org/ I am not so worried about word meanings, they can always be handled via reference to WordNet via usages like run_1, run_2, etc. ... or as you say by using rarer, less ambiguous words Prepositions are more worrisome, however, I suppose they can be handled in a similar way, e.g. by defining an ontology of preposition meanings like with_1, with_2, with_3, etc. In fact we had someone spend a couple months integrating existing resources into a preposition-meaning ontology like this a while back ... the so-called PrepositionWordNet ... or as it eventually came to be called the LARDict or LogicalArgumentRelationshipDictionary ... I think it would be feasible to tweak RelEx to recognize these sorts of subscripts, and in this way to recognize a highly controlled English that would be unproblematic to map semantically... We would then say e.g. I ate dinner with_2 my fork I live in_2 Maryland I have lived_6 for_3 41 years (where I suppress all _1's, so that e.g. ate means ate_1) Because, RelEx already happily parses the syntax of all simple sentences, so the only real hassle to deal with is disambiguation. We could use similar hacking for reference resolution, temporal sequencing, etc. The terrorists_v1 robbed_v2 my house. After that_v2, the jerks_v1 urinated in_3 my yard. I think this would be a relatively pain-free way to communicate with an AI that lacks the common sense to carry out disambiguation and reference resolution reliably. Also, the log of communication would provide a nice training DB for it to use in studying disambiguation. -- Ben G On Wed, Oct 22, 2008 at 12:00 PM, Mark Waser [EMAIL PROTECTED] wrote: IMHO that is an almost hopeless approach, ambiguity is too integral to English or any natural language ... e.g preposition ambiguity Actually, I've been making pretty good progress. You just always use big words and never use small words and/or you use a specific phrase as a word. Ambiguous prepositions just disambiguate to one of three/four/five/more possible unambiguous words/phrases. The problem is that most previous subsets (Simplified English, Basic English) actually *favored* the small tremendously over-used/ambiguous words (because you got so much more bang for the buck with them). Try only using big unambiguous words and see if you still have the same opinion. If you want to take this sort of approach, you'd better start with Lojban instead Learning Lojban is a pain but far less pain than you'll have trying to make a disambiguated subset of English. My first reaction is . . . . Take a Lojban dictionary and see
Re: Lojban (was Re: [agi] constructivist issues)
On Thu, Oct 23, 2008 at 11:23 AM, Matt Mahoney [EMAIL PROTECTED] wrote: So how does yet another formal language processing system help us understand natural language? This route has been a dead end for 50 years, in spite of the ability to always make some initial progress before getting stuck. Although I mostly agree with you, I do often think that humans understand formal languages very differently to, say, compilers (if they can be said to understand them at all) and I think it is interesting to study how one might build an AGI system that understands formal languages the way humans do. I have no idea whether it is easier to do this with formal languages than it is to do this with natural languages. Trent --- 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, Unfortunately, this response is going to be (somewhat) long, because I have several points that I want to make. If I understand what you are saying, you're claiming that if I pointed to the black box and said That's a halting oracle, I'm not describing the box directly, but instead describing it in terms of a (semi)formal system in my head that defines halting oracle. This system is computable. This seems to fit back with the comment I made about William Pearson's system: we don't assume that the universe is computable, instead we just assume that our mental substrate is. But, we need to be careful about what computable means here. Things like a mandelbrot set rendering are computably enumerable, which is formally separated from computable, but still easily implemented on a computer. The same is true of first-order theories that describe halting oracle and related notions. Technically these are not computable, because there is no halting criteria (or, in the case of the mandelbrot renderer, no halting criteria *yet*, although many mathematicians expect that one can be formulated.) We can list positive cases (provably halting/nonhalting programs, provably escaping points) but we have no way of deciding when to give up on the stubborn points. A third type of computability is computably co-enumerable, which is what the halting problem is. I imagine you know the definition of this term already. So, halting-related things such as halting oracles have no computable description, but they do have a description-implementable-on-a-computer. Unfortunately, AIXI does not use models of this variety, since it only considers models that are computable in the strict technical sense. But, worse, there are mathematically well-defined entities that are not even enumerable or co-enumerable, and in no sense seem computable. Of course, any axiomatic theory of these objects *is* enumerable and therefore intuitively computable (but technically only computably enumerable). Schmidhuber's super-omegas are one example. Concerning your statement, It is not clear what you really mean by the description length of something uncomputable, since the essence of uncomputability is the property of **not being finitely describable**. That statement basically agrees with the following definition of meaning: A statement is meaningful if we have a (finite) rule that tells us whether it is true or false. The idea of finite rule here is a program that takes finite input (the facts we currently know) and halts in finite time with an output. This agrees with the formal definition of computable, so that meaningful facts and computable facts are one and the same. Here is a slightly broader definition: A statement is meaningful if we have a (finite) rule that tells us whether it is true. This agrees instead with the definition of enumerable. Or, the scientific testability version: A statement is meaningful if we have a (finite) rule that tells us whether it is false. This of course agrees with the definition of co-enumerable. Now here is a rather broad one: A statement is meaningful if we have a (finite) rule that tells us how we can reason if it is true. So, each statement corresponds to a program that operates on known statements to produce more statements; applying the rule corresponds to using the fact in our reasoning. So the direct consequences of a statement given some other statements are computable, but the truth or falsehood is not necessarily. As it happens, this definition of meaning admits horribly-terribly-uncomputable-things to be described! (Far worse than the above-mentioned super-omegas.) So, the truth or falsehood is very much not computable. I'm hesitant to provide the mathematical proof in this email, since it is already long enough... let's just say it is available upon request. Anyway, you'll probably have some more basic objection. --Abram On Mon, Oct 20, 2008 at 10:38 PM, Ben Goertzel [EMAIL PROTECTED] wrote: On Mon, Oct 20, 2008 at 5:29 PM, Abram Demski [EMAIL PROTECTED] wrote: Ben, [my statement] seems to incorporate the assumption of a finite period of time because a finite set of sentences or observations must occur during a finite period of time. A finite set of observations, sure, but a finite set of statements can include universal statements. Ok ... let me clarify what I meant re sentences I'll define what I mean by a **descriptive sentence** What I mean by a sentence is a finite string of symbols drawn from a finite alphabet. What I mean by a *descriptive sentence* is a sentence that is agreed by a certain community to denote some subset of the total set of observations (where all observations have finite precision and are drawn from a certain finite set). So, whether or not a descriptive sentence contains universal quantifiers or quantum-gravity quantifiers or psychospirituometaphysical quantifiers, or whatever, in the end there are some observation-sets it
Re: [agi] constructivist issues
But, worse, there are mathematically well-defined entities that are not even enumerable or co-enumerable, and in no sense seem computable. Of course, any axiomatic theory of these objects *is* enumerable and therefore intuitively computable (but technically only computably enumerable). Schmidhuber's super-omegas are one example. My contention is that the first use of the word are in the first sentence of the above is deceptive. The whole problem with the question of whether there are uncomputable entities is the ambiguity of the natural language term is / are, IMO ... If by A exists you mean communicable-existence, i.e. It is possible to communicate A using a language composed of discrete symbols, in a finite time then uncomputable numbers do not exist If by A exists you mean I can take some other formal property F(X) that applies to communicatively-existent things X, and apply it to A then this will often be true ... depending on the property F ... My question to you is: how do you interpret are in your statement that uncomputable entities are? ben --- 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, My discussion of meaning was supposed to clarify that. The final definition is the broadest I currently endorse, and it admits meaningful uncomputable facts about numbers. It does not appear to get into the realm of set theory, though. --Abram On Tue, Oct 21, 2008 at 12:07 PM, Ben Goertzel [EMAIL PROTECTED] wrote: But, worse, there are mathematically well-defined entities that are not even enumerable or co-enumerable, and in no sense seem computable. Of course, any axiomatic theory of these objects *is* enumerable and therefore intuitively computable (but technically only computably enumerable). Schmidhuber's super-omegas are one example. My contention is that the first use of the word are in the first sentence of the above is deceptive. The whole problem with the question of whether there are uncomputable entities is the ambiguity of the natural language term is / are, IMO ... If by A exists you mean communicable-existence, i.e. It is possible to communicate A using a language composed of discrete symbols, in a finite time then uncomputable numbers do not exist If by A exists you mean I can take some other formal property F(X) that applies to communicatively-existent things X, and apply it to A then this will often be true ... depending on the property F ... My question to you is: how do you interpret are in your statement that uncomputable entities are? ben 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
On Tue, Oct 21, 2008 at 4:53 PM, Abram Demski [EMAIL PROTECTED] wrote: As it happens, this definition of meaning admits horribly-terribly-uncomputable-things to be described! (Far worse than the above-mentioned super-omegas.) So, the truth or falsehood is very much not computable. I'm hesitant to provide the mathematical proof in this email, since it is already long enough... let's just say it is available upon request. Now I'm curious -- can these horribly-terribly-uncomputable-things be described to a non-mathematician? If so, consider this a request. --- 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
Try Rudy Rucker's book Infinity and the Mind for a good nontechnical treatment of related ideas http://www.amazon.com/Infinity-Mind-Rudy-Rucker/dp/0691001723 The related wikipedia pages are a bit technical ;-p , e.g. http://en.wikipedia.org/wiki/Inaccessible_cardinal On Tue, Oct 21, 2008 at 2:27 PM, Russell Wallace [EMAIL PROTECTED]wrote: On Tue, Oct 21, 2008 at 4:53 PM, Abram Demski [EMAIL PROTECTED] wrote: As it happens, this definition of meaning admits horribly-terribly-uncomputable-things to be described! (Far worse than the above-mentioned super-omegas.) So, the truth or falsehood is very much not computable. I'm hesitant to provide the mathematical proof in this email, since it is already long enough... let's just say it is available upon request. Now I'm curious -- can these horribly-terribly-uncomputable-things be described to a non-mathematician? If so, consider this a request. --- 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] Nothing will ever be attempted if all possible objections must be first overcome - Dr Samuel Johnson --- 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
