The Definition of Points
in [Math]
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From: Hero on 21 Mar 2007 11:56 Lester Zick wrote: > Hero wrote: > > Lester Zick wrote: > > >.... There's topology, just the simple > > beginning: > > A space (mathematical) is a set with structure. > > A point is a geometrical space without geometrical structure, but it > > can give structure to geometry. > > Think of a vertex or a center and so forth. > > A line is made up of points and sets of points ( the open intervalls > > between each two points),which obey three topological rules. > > Hero, exactly what makes you think the foregoing observations are > true? > > >What i learned recently: > >With adding a point to an open (open in standard topology) flexible > >surface one can enclose a solid, with adding a point to an open line > >one can enclose a figure, and two points are the boundary of an > >intervall on a line. But there is no point at or beyond infinity. > > All very interesting but I still have no idea why any of this is > supposed to be true. > Geometry doesn't start in school. You do a lot of observations and more over You practise geometry - just one example: a football-match. This needs lots of practical knowledge about differences of directions (angles) and different coordinate systems (while moving You shoot the ball to another person, which is moving an a different way), not to speak of simpler things like lines and points. Later on, and in school, one learns analyzing and synthesizing too, also with logical reasoning. And talking about sets ( not only from me) - just in the beginning of it is the definition of a set by Georg Cantor: "By a set we understand any collection M of definite, distinct objects m of our perception ("Anschauung") or of our thought (which will be called the elements of M) into a whole." That a line is made up of points is not sufficient, this is shown here and else. The topological property, that a line is made up of points and sets of points, was never questioned. May be one needs more, but not less for a line. And this is not a circular definition. With friendly greetings Hero
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 |