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 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
