Torkel Franzen on truth
in [Logic]
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From: MoeBlee on 18 Dec 2007 18:50 On Dec 18, 2:59 pm, "Nam D. Nguyen" <namducngu...(a)shaw.ca> wrote: > A model > *always already* includes a *chosen* interpretation: hence a belief has been > "believed" already. A model is a mathematical object. I don't know how you would argue that a model requires "a belief has been "believed" already". MoeBlee
From: george on 18 Dec 2007 20:16 > >Almost nobody more than 10 years younger than you ever did any > >such thing. The rest of us STARTED with PA. It embodies the basic > >things we know, at first-order, about this realm. On Dec 18, 2:38 pm, tc...(a)lsa.umich.edu wrote: > George, you've *got* to be kidding here. Do you not have any clue how > idiosyncratic your viewpoint is? Name one person other than yourself > who "STARTED with PA." > > I've seen lots of "new math" movements, but never have I heard of anyone > teaching arithmetic to five-year-olds by starting with PA. At some point at an age not MUCH later than that one, YOU (yes, YOU, Tim Chow) were taught that adding 0 to something does not change it. You were taught that 5+0=5. You were also taught that 5 was not special in this regard, and that EVERY natural number was like this. The fact that you were not told that "This is an axiom of PA" did NOT STOP you from HAVING, FACTUALLY, ACTUALLY been taught ONE axiom of PA, namely Ax[x+0=x]. The fact that it wasn't phrased specifically that way and that PA as an entity was not mentioned IS NOT relevant. You were also taught that Ax[x*1=1] from the FIRST day you were taught any multiplication tables. I repeat, style of spelling this stuff IS NOT relevant. The content of the axioms is what is relevant.
From: george on 18 Dec 2007 20:19 On Dec 18, 2:38 pm, tc...(a)lsa.umich.edu wrote: > George, you've *got* to be kidding here. Do you not have any clue how > idiosyncratic your viewpoint is? Name one person other than yourself > who "STARTED with PA." Well, people in general, regardless of age, certainly don't start thinking about the totality of models of some axioms. The intended model that most of us started with thought that "the numbers" were finite digit-strings, with digit=={0..9}. Well, actually, we were maybe taught Roman numerals so that we would know that digit-strings were numerals rather than numbers, but that is actually a distinction that does NOT even matter; numerals are enough LIKE numbers to serve all the same purposes; the important point was just to remind people that numeration systems were possible as OPPOSED to necessary. Axiomatizations too, for that matter.
From: george on 18 Dec 2007 20:23 On Dec 18, 2:38 pm, tc...(a)lsa.umich.edu wrote: > George, you've *got* to be kidding here. Do you not have any clue how > idiosyncratic your viewpoint is? Name one person other than yourself > who "STARTED with PA." Started DOING WHAT? We *start* by CALCULATING. That is *0th*-order. I am trying to remember what the FIRST first-order results I ever learned or PROVED were. I would say that the ones I learned first were identities. Those ARE axioms of PA *even* when they are not presented as such. In defense of your point I would say that I was next taught commutativity and associativity as axioms; I was not taught how to prove them using induction. But there is a pedagogical question regarding what sort of "easier" axiom-systems would be better than PA *to* start with.
From: MoeBlee on 18 Dec 2007 20:35
On Dec 18, 5:16 pm, george <gree...(a)cs.unc.edu> wrote: > > >Almost nobody more than 10 years younger than you ever did any > > >such thing. The rest of us STARTED with PA. It embodies the basic > > >things we know, at first-order, about this realm. > > On Dec 18, 2:38 pm, tc...(a)lsa.umich.edu wrote: > > > George, you've *got* to be kidding here. Do you not have any clue how > > idiosyncratic your viewpoint is? Name one person other than yourself > > who "STARTED with PA." > > > I've seen lots of "new math" movements, but never have I heard of anyone > > teaching arithmetic to five-year-olds by starting with PA. > > At some point at an age not MUCH later than that one, > YOU (yes, YOU, Tim Chow) were taught that adding 0 to something > does not change it. You were taught that 5+0=5. You were also taught > that 5 was not special in this regard, and that EVERY natural number > was like this. > The fact that you were not told that "This is an axiom of PA" did NOT > STOP you > from HAVING, FACTUALLY, ACTUALLY been taught ONE axiom of PA, namely > Ax[x+0=x]. The fact that it wasn't phrased specifically that way and > that PA as > an entity was not mentioned IS NOT relevant. > > You were also taught that Ax[x*1=1] from the FIRST day you were taught > any multiplication tables. I repeat, style of spelling this stuff IS > NOT relevant. > The content of the axioms is what is relevant. They teach the induction schema to little kids? MoeBlee |