"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:* 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 >>>> 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 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
