The Definition of Points
in [Logic]
|
Prev: On Ultrafinitism
Next: Modal logic example
From: hagman on 21 Mar 2007 07:38 On 21 Mrz., 01:29, Lester Zick <dontbot...(a)nowhere.net> wrote: > On 20 Mar 2007 16:15:06 -0700, "hagman" <goo...(a)von-eitzen.de> wrote: > > > > >On 20 Mrz., 20:24, Lester Zick <dontbot...(a)nowhere.net> wrote: > >> On 20 Mar 2007 03:11:41 -0700, "hagman" <goo...(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. > > >Wrong. > >The field of reals provides a very good model of a straight line, > >also in the form of an affine subspace of a higher dimensional real > >vector space. > > Please can the neomathspeak for a moment to explain how application of > the Peano and suc( ) axioms and a resultant succession of integers and > straight line segments, even if we allow their assumption, defines > straight lines. I mean are we just supposed to assume the sequence of > line segments is necessarily colinear and lies on a straight line? > > >> >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. > > >You made a general statement against models (or rather grossly > >misunderstood > >someone elses general statement about models). > >Thus I am allowed to pull out any model I like. > > Go right ahead. I'd just like to see you pull out straight lines > first. I did that below: R is as straight as can get. To get back to the problem of excess properties: R identified with Rx{0} as a subset of RxR "is" a straight line in a plane. Again, the model shares all properties of the abstract line in a plane, but it also has additional proerties, e.g. there is a special point (0,0). Btw, {42}xR is another line of that plane model and it happens to be the case that - lines are sets of points in this model by construction - two non-parallel lines determine a unique point of intersection (e.g. (42,0)) > > >> >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. > > >On the contrary. Producing a model for a theory is the common way to > >show that > >the theory is consistent (provided the theory used to construct the > >model > >is consistent) > > Well you know, hagman, this is curious. I complain that consistency is > not the final objective of a model. And your reply is to assure me > that some models are consistent. Do you consider that responsive? Am I fooled by some subtle notion because I'm not a native speaker of English? For me "final objective" means something like "purpose". So your claim is: purpose of model =/= consistency My reply: purpose of model == demonstration of consistency of modeled theory Your paraphrasing of my reply: Some models are consistent > > >A: Let's consider a theory where 'lines' consist of 'points' and > >'lines' determine 'points'... > >B: Hey, isn't that nonsense? Can such a theory exist? > >A: Of course. Take pairs of real numbers for points and certain sets > >of such pairs as lines... > >B: I see. > > Except you aren't even bothering to demonstrate the existence of > numbers, lines, straight lines, etc. You're just claiming they're > there. How can you take something that isn't there? Then how can you > demonstrate they are there? Let's get real for a change shall we. > Of course we have to start *somewhere*. It depends on the point where B really says "I see". The usual accepted startiong point today is ZF(C) set theory, which allows to construct a model of the natural numbers N. Given N, one can construct a model for the field Q of rationals. Given Q, one can construct a model for the complete archimedean field R. Given R, well, we had that already... So you have to redirect any attack to ZFC and, yes, that's about the point where I stop bothering. And maybe I don't even claim "they're there" but only "it makes sense talking about them".
From: Bob Kolker on 21 Mar 2007 07:40 Tony Orlow wrote: > > There is no correlation between length and number of points, because > there is no workable infinite or infinitesimal units. Allow oo points > per unit length, oo^2 per square unit area, etc, in line with the > calculus. Nuthin' big. Jes' give points a size. :) Points (taken individually or in countable bunches) have measure zero. Bob Kolekr
From: stephen on 21 Mar 2007 07:44 In sci.math Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: >> In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >>> PD wrote: >>>> No one says a set of points IS in fact the constitution of physical >>>> object. >>>> Whether it is rightly the constitution of a mentally formed object >>>> (such as a geometric object), that seems to be an issue of arbitration >>>> and convention, not of truth. Is the concept of "blue" a correct one? >>>> >>>> PD >>>> >> >>> The truth of the "convention" of considering higher geometric objects to >>> be "sets" of points is ascertained by the conclusions one can draw from >>> that consideration, which are rather limited. >> >> How is it limited Tony? Consider points in a plane, where each >> point is identified by a pair of real numbers. The set of >> points { (x,y) | (x-3)^2+(y+4)^2=10 } describes a geometric object. > That's a very nice circle, Stephen, very nice.... Yes, it is a circle. You knew exactly what this supposedly "limited" description was supposed to be. >> In what way is this description "limited"? Can you provide a >> better description, and explain how it overcomes those limitations? > There is no correlation between length and number of points, because > there is no workable infinite or infinitesimal units. Allow oo points > per unit length, oo^2 per square unit area, etc, in line with the > calculus. Nuthin' big. Jes' give points a size. :) How is that a limitation? You knew exactly what shape the set of points described. There is no feature of the circle that cannot be determined by the above description. There is no need to correlate length and number of points. Neither Euclid or Hilbert ever did that. So where is your better description, and where is the explanation as to why it is better? What more can you say about a circle centered at (3,-4) with a radius of sqrt(10)? Stephen
From: stephen on 21 Mar 2007 07:55 In sci.math hagman <google(a)von-eitzen.de> wrote: > On 21 Mrz., 01:29, Lester Zick <dontbot...(a)nowhere.net> wrote: >> >> Well you know, hagman, this is curious. I complain that consistency is >> not the final objective of a model. And your reply is to assure me >> that some models are consistent. Do you consider that responsive? > Am I fooled by some subtle notion because I'm not a native speaker of > English? No, your English is not the problem. The problem is Lester's. Lester speaks his own language where words mean whatever he wants them to mean, so are in fact meaningless. Often he is just parroting back phrases he does not understand in a vain attempt to appear knowledgable. Stephen
From: Wolf on 21 Mar 2007 10:37
Tony Orlow wrote: > Bob Kolker wrote: [...] >> Math is about what follows from assumptions, not true statements about >> the world. >> >> Bob Kolker > > If the algebraic portions of your mathematics that describe the > geometric entities therein do not produce the same conclusions as would > be derived geometrically, then the algebraic representation of the > geometry fails. Actually, the algebra enables you to draw conclusions that would be difficult or impossible to do "geometrically" (by which I presume you mean by geometric construction.) They are still true "geometrically", ie, if interpreted as applying to geometric entities, including ones that can't be drawn with ruler and compasses. > Hilbert didn't just pick statements out of a hat. > Rather, he didn't do so entirely, though they could have been > generalized better. Oh my, another genius who understands math better than Hilbert, et al. > In any case, they represent facts that are > justifiable, not within the language of axiomatic description, but > within the spatial context of that which is described. > > Tony Orlow Algebraisation frees geometry from mere 3D physical space. You can apply it any set of objects, for the elements of a set can distinguished from each other along at least one dimension. IOW, you can use some axiomatised system S to construct a model M of some phenomena {P}. The validity of M as a description of {P} is tested by the predictions it makes about {P}. But the success or failure of M as a description of {P} has no bearing on the mathematical truth of S. -- Wolf "Don't believe everything you think." (Maxine) |