The Definition of Points
in [Math]
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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)
From: Wolf on 21 Mar 2007 10:44 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? -- Wolf "Don't believe everything you think." (Maxine)
From: Tony Orlow on 21 Mar 2007 11:47 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).
From: Tony Orlow on 21 Mar 2007 11:50 Bob Kolker wrote: > 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 > They certainly have measure less than any finite measure, which in standard mathematics means 0, however, extensions which allow an infinite unit may define the size of the point as the infinitesimal reciprocal of that unit. Given that, the sum of this infinite number of infinitesimal atoms gives the actual measure of the object. :) Tony Orlow
From: Tony Orlow on 21 Mar 2007 11:55
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. :) There's no reason the circumference of the unit circle can't be considered to have 2*pi*oo points. > > 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. :) > > Stephen > |