The Definition of Points
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From: Mike Kelly on 21 Mar 2007 12:02 On 21 Mar, 15:47, Tony Orlow <t...(a)lightlink.com> wrote: > Mike Kelly wrote: > > On 21 Mar, 05:47, Tony Orlow <t...(a)lightlink.com> wrote: > >> step...(a)nomail.com wrote: > >>> In sci.math Tony Orlow <t...(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.... > > >>> 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 > > > Oh. And I suppose there *is* such a correlation in "real" geometry? > > > -- > > mike. > > In my geometry |{x in R: 0<x<=1}| = Big'Un (or oo). What does that have to do with geometry? -- mike.
From: Tony Orlow on 21 Mar 2007 12:02 Wolf wrote: > 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. Yes, I understand that. I have nothing against axiomatization and deductive proof methods. > >> 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. Postulate 1.7 is simply untrue in 4D or greater. :)] Lions and tigers and bears.... > >> 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. Sure. What does the axiomatization process say about points vs. lines? Anything? > > 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. If you say so. > > -- > > Wolf > > "Don't believe everything you think." (Maxine)
From: Tony Orlow on 21 Mar 2007 12:03 Wolf wrote: > Tony Orlow wrote: > [...]I'm not putting down axiomatization persay. > [...] > > I'm about to pick a nit: > > It's not "persay", it's "per se." Latin. Means "by means of itself." The > usual English rendering is "in and of itself." > > OK? > Yes, of course.
From: Hero on 21 Mar 2007 12:31 Tony Orlow wrote: > Bob Kolker wrote: > > Tony Orlow wrote: > > >> You know that's not what I mean. > > > I do? Then what do you 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. If some > > > But not in math. The only thing that matters is that the conclusions > > follow from the premises and the premises do not imply contradictions. > > Matters of empirical true, as such, have no place in mathematics. > > > 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. Hilbert didn't just pick statements out of a hat. > Rather, he didn't do so entirely, though they could have been > generalized better. 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 And Lester Zick also gave an answer. So, Bob, when Thales determined the height of the pyramids by measuring the shadow, at that time, when his own shadow equalled his own length -was he using a true theorem or did his theorem only became true, after Euclid derived it from his axioms? Hippocrates of Chios and Euclid did not do math, in deriving their axioms, and also Moritz Pasch, Mario Pieri, Oswald Veblen, Edward Vermilye Huntington, Gilbert Robinson, and Henry George Forder? May be You learned math from Your teacher in starting with premisses and in drawing conclusions from them - but there is more to math. Why most people think that it is necessary to add an axiom of Archimedes to other axioms, when talking about geometry? True statements about the world and logical reasoning showed, without Archimedes axiom, You only can do a handicapped geometry. With friendly greetings Hero
From: stephen on 21 Mar 2007 12:52
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: >>> 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. > Gee, I guess it's a novel idea, then. That might make it good, and not > necessarily bad. Hilbert also didn't bother to generalize his axioms to > include anything but Euclidean space, which is lazy. He's got too many > axioms, and they do too little. :) Then you should not be complaining about the "set of points" approach to geometry and instead should be complaining about all prior approaches to geometry. Apparently they are all "limited" to you. Of course you cannot identify any actual limitation, but that is par for the course. > There's no reason the circumference of the unit circle can't be > considered to have 2*pi*oo points. But what is the reason to consider that is does? All you are doing is multiplying the length by oo. You are not adding any new information. You are not learning anything. >> >> 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)? > It's got 2*pi*sqrt(1)*oo points, as a set, which is greater than the > number of points in the unit interval. :) And what good does that do? You are just giving a new name to "length". You have not added anything. Stephen |