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: <[email protected]> > 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* 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: <[email protected]> >> 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://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=8660244&id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
