The Definition of Points
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From: Lester Zick on 20 Mar 2007 15:24 On 20 Mar 2007 03:11:41 -0700, "hagman" <google(a)von-eitzen.de> wrote: >On 20 Mrz., 00:30, Lester Zick <dontbot...(a)nowhere.net> wrote: >> >> > I'm saying that a >> >model is just a model. The properties of the model do not cause >> >the thing it's modeling to have those properties. >> >> Oh great. So now the model of a thing has properties which don't model >> the properties of the thing it's modeling. So why model it? >> >> ~v~~ > >You are trying to walk the path in the wrong direction. >E.g. 0:={}, 1:={{}}, 2:= {{},{{}}}, ... >is a model of the naturals: The Peano axioms hold. However we don't have a model of straight lines except by naive assumption. >However, in this model we have "0 is a set", which does not follow >from Peano axioms. >Thus the model has some additional properties. What's wrong with that? It isn't a model of what we wish to model. >However, the model shows that the Peno axioms are consistent (provided >the set theory we used to construct the model is). Consistency is only a prerequisite not a final objective for a model. ~v~~
From: Lester Zick on 20 Mar 2007 15:30 On 20 Mar 2007 08:58:26 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> wrote: >On Mar 19, 7:30 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: >> On 19 Mar 2007 11:51:47 -0700, "Randy Poe" <poespam-t...(a)yahoo.com> >> wrote: >> >> >> >> >On Mar 19, 2:44 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: >> >> On 19 Mar 2007 08:59:24 -0700, "Randy Poe" <poespam-t...(a)yahoo.com> >> >> wrote: >> >> >> >> > That the set of naturals is infinite. >> >> >> >> Geometrically incorrect. Unless there is a natural infinitely greater >> >> >> than the origin, there is no infinite extent involved. >> >> >> >The naturals don't have physical positions, since they are not >> >> >defined geometrically. >> >> >> They are if they're associated with points and points define line >> >> segments. >> >> >By "associated with points" I assume you mean something >> >like using points to model the naturals. >> >> Why do you assume that, Randy? > >Because I can't think of another meaning for this >phrase. As usual, you throw out language which is all >but meaningless, and it is up to people to guess by >successive approximations what it is you mean. You just mean "meaningless" in your own personal lingo. >> I mean if I raise an issue and you >> willy-nilly recast it in terms amenable to you > >You didn't "throw out an issue", you threw out an undefined >phrase. I didn't "throw out an issue" at all. I "raised" an issue in terms which you consider "undefined" because you prefer to consider the issue in terms of your own private language instead of the terms in which I raised it. > This caused discussion to grind to a halt until >a common language can be re-established. Apparently there is no common language between us. At least there never has been. Anytime you prefer to evade an issue you just translate it into some other terms which you feel more comfortable with. >Is my guess wrong? Fine. I assume you know what >you meant (oops, there I go making assumptions again). >If so, then explain it. "Associated with points". ~v~~
From: Lester Zick on 20 Mar 2007 15:33 On 19 Mar 2007 11:51:47 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> wrote: >On Mar 19, 2:44 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: >> On 19 Mar 2007 08:59:24 -0700, "Randy Poe" <poespam-t...(a)yahoo.com> >> wrote: >> >> >> > That the set of naturals is infinite. >> >> >> Geometrically incorrect. Unless there is a natural infinitely greater >> >> than the origin, there is no infinite extent involved. >> >> >The naturals don't have physical positions, since they are not >> >defined geometrically. >> >> They are if they're associated with points and points define line >> segments. > >By "associated with points" I assume you mean something >like using points to model the naturals. In that case the points >in your model have positions, but nevertheless the naturals >themselves don't have physical positions or exist as geometric >entities. > >Do you have any idea what I'm saying? I'm a physicist, Randy, not a psychologist. > I'm saying that a >model is just a model. The properties of the model do not cause >the thing it's modeling to have those properties. ~v~~
From: Randy Poe on 20 Mar 2007 15:39 On Mar 20, 3:30 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: > On 20 Mar 2007 08:58:26 -0700, "Randy Poe" <poespam-t...(a)yahoo.com> > wrote: > > > > >On Mar 19, 7:30 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: > >> On 19 Mar 2007 11:51:47 -0700, "Randy Poe" <poespam-t...(a)yahoo.com> > >> wrote: > > >> >On Mar 19, 2:44 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: > >> >> On 19 Mar 2007 08:59:24 -0700, "Randy Poe" <poespam-t...(a)yahoo.com> > >> >> wrote: > > >> >> >> > That the set of naturals is infinite. > > >> >> >> Geometrically incorrect. Unless there is a natural infinitely greater > >> >> >> than the origin, there is no infinite extent involved. > > >> >> >The naturals don't have physical positions, since they are not > >> >> >defined geometrically. > > >> >> They are if they're associated with points and points define line > >> >> segments. > > >> >By "associated with points" I assume you mean something > >> >like using points to model the naturals. > > >> Why do you assume that, Randy? > > >Because I can't think of another meaning for this > >phrase. As usual, you throw out language which is all > >but meaningless, and it is up to people to guess by > >successive approximations what it is you mean. > > You just mean "meaningless" in your own personal lingo. > > >> I mean if I raise an issue and you > >> willy-nilly recast it in terms amenable to you > > >You didn't "throw out an issue", you threw out an undefined > >phrase. > > I didn't "throw out an issue" at all. I "raised" an issue in terms > which you consider "undefined" because you prefer to consider the > issue in terms of your own private language instead of the terms in > which I raised it. > > > This caused discussion to grind to a halt until > >a common language can be re-established. > > Apparently there is no common language between us. At least there > never has been. Anytime you prefer to evade an issue you just > translate it into some other terms which you feel more comfortable > with. No, this is the process by which I attain communication. It is a concept you are unfamiliar with, but here's how it works: Some language is not known to your listener. They ask "did you mean X?" and attempt to paraphrase their understanding of what you said. If the answer is "no", then you provide a restatement in alternate terminology, paraphrasing yourself. > >Is my guess wrong? Fine. I assume you know what > >you meant (oops, there I go making assumptions again). > >If so, then explain it. > > "Associated with points". That would not be "alternate terminology". One mark of whether you in fact understand a concept, including one of your own, is whether you are capable of paraphrasing. I find it instructive that of all the many, many examples of phrases you have thrown out on this newsgroup, the ones that make people say "what the hell is that supposed to mean?" you have never to my recollection been able to explain a single one. - Randy
From: Virgil on 20 Mar 2007 16:06
In article <46001b67(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Bob Kolker wrote: > > Tony Orlow wrote:> > >> > >> How do you know the conclusions are correct, > > > > You mean how do you know conclusions correctly follow from the axioms. > > You look. Proofs are precisely the evidence that the conclusion follows > > from the premises. > > You know that's not what I mean. How do you measure the accuracy of the > premises you use for your arguments? You check the results. That's the > way it works in science, and that's the way t works in geometry. The only way to "check results" in an axiomatic system, whether algebraic or geometrical, is to assure that all conclusions follow from their explicitely stated premises, and that no hidden assumptions are involved. Since axiomatic systems in mathematics say nothing about the physical world, there is no physical way of checking any of ones "results". If a particular axiom system is supposed to reflect some physical situation, one can also test how well the theory matches the facts, but that has nothing to do with the derivations from the axioms, but only with how well the axioms model the "reality". > If some > set of rules you define leads you to conclude that the volume of a > sphere is equal to the volume of two spheres of the same radius as the > first, well, you probably want to go back and reexamine your premises > and make sure you didn't err somewhere in the derivation of your > conclusions. In this case, one can easily show that the mathematical model and any physical interpretation are inconsistent, but that does not establish any inconsistency in the mathematics at all. > > > > > Look any any standard treatise on first order logic for a definiton of > > proof. Checking to see that a proof indeed shows the desired conclusion > > follows from the axioms is a purely mechanical algorithmeic proceedure. > > It does not involve intelligence. -Finding- a proof does. Checking a > > proof can be done by a trained gorilla or Lester Zick on one of his > > better days. > > > > Bob Kolker > > > > Bob - wake up. How do we know relativity is correct? Because it follows > from e=mc^2? Relativity is not a mathematical theory, it is a physical theory. The mathematical model is consistent, but its consistency is not a physics question, and whether it matches the physics is not at all a mathemtical question. > > Oy! Indeed! |