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:* [email protected] > *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:* [email protected] >> *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 things> Argh! >>> >>> = = = = = = = = >>> >>> 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<http://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:* [email protected] >>> *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: <[email protected]> >>>> 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> >>> <https://www.listbox.com/member/archive/rss/303/> | >>> Modify<https://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/> | >>> Modify<https://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> >> <https://www.listbox.com/member/archive/rss/303/> | >> Modify<https://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/> | >> Modify<https://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> > <https://www.listbox.com/member/archive/rss/303/> | > Modify<https://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/> | > Modify<https://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=8660244&id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
