The Definition of Points
in [Math]
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From: Bob Kolker on 14 Mar 2007 10:07 SucMucPaProlij wrote:> > > One can assume that there are some objects other than points but I don't think Only if one makes this assumption explicit. This means introducing objects other than points and lines into the system and it means some axiom must somehow mention and characterize this additional object or kind of object. The idea of an axiom system such as Hilbert's is to -explicitly- mention those objects which are not defined and characterize them with the axioms. Thus, given two distinct points there is one and only one line containing the points. The containment relation expressed in a number of ways is also undefined. We we say a point is on a line. A line contains a point or a line passes through a point etc.. Look at hilbert's axiom system in wiki. Bob Kolker
From: Eckard Blumschein on 14 Mar 2007 10:51 On 3/13/2007 6:52 PM, Lester Zick wrote: > > In the swansong of modern math lines are composed of points. But then > we must ask how points are defined? I hate arbitrary definitions. I would rather like to pinpoint what makes the notion of a point different from the notion of a number: If a line is really continuous, then a mobile point can continuously glide on it. If the line just consists of points corresponding to rational numbers, then one can only jump from one discrete position to an other. A point has no parts, each part of continuum has parts, therefore continuum cannot be resolved into any finite amount of points. Real numbers must be understood like fictions. All this seems to be well-known. When will the battle between frogs and mices end with a return to Salviati?
From: PD on 14 Mar 2007 11:07 On Mar 14, 9:51 am, Eckard Blumschein <blumsch...(a)et.uni-magdeburg.de> wrote: > On 3/13/2007 6:52 PM, Lester Zick wrote: > > > > > In the swansong of modern math lines are composed of points. But then > > we must ask how points are defined? > > I hate arbitrary definitions. I would rather like to pinpoint what makes > the notion of a point different from the notion of a number: > > If a line is really continuous, then a mobile point can continuously > glide on it. If the line just consists of points corresponding to > rational numbers, then one can only jump from one discrete position to > an other. That's an interesting (but old) problem. How would one distinguish between continuous and discrete? As a proposal, I would suggest means that there is a finite, nonzero interval (where interval is measurable somehow) between successive positions, in which there is no intervening position. Unfortunately, the rational numbers do not satisfy this definition of discreteness, because between *any* two rational numbers, there is an intervening rational number. I'd be interested in your definition of discreteness that the rational numbers satisfy. PD > > A point has no parts, each part of continuum has parts, therefore > continuum cannot be resolved into any finite amount of points. > Real numbers must be understood like fictions. > > All this seems to be well-known. When will the battle between frogs and > mices end with a return to Salviati?
From: Bob Kolker on 14 Mar 2007 11:45 Eckard Blumschein wrote:> If a line is really continuous, then a mobile point can continuously > glide on it. If the line just consists of points corresponding to > rational numbers, then one can only jump from one discrete position to > an other. Points don't glide. In fact points don't move. You are still pushing discrete mathematics? All you will get is a means of totalling up your grocery bill. Bob Kolker
From: VK on 14 Mar 2007 13:10
On Mar 14, 1:28 am, Lester Zick <dontbot...(a)nowhere.net> wrote: > Are points and lines not still mathematical objects The point is Ïο Ïί ήν είναι ("to ti en einai") of the infinity. If you want a definition based on something fresher than Aristotle then: The point is nothing which is still something in potention to become everything. IMHO the Aristotle-based definition is better, but it's personal. Now after some thinking you may decide to stay with the crossing lines and hell on the cross-definition issues ;-) The speach is not a reflection of entities: it is a reflection - of different levels of quality - of the mind processes. This way a word doesn't have neither can decribe an entity. The purpose of the word - when read - to trig a "mentagram", state of mind, as close as possible to the original one - which caused the word to be written. This way it is not important how is the point defined: it is important that all people involved in the subject would think of appoximately the same entity so not say about triangles or squares. In this aspect crossing lines definition in math does the trick pretty well. From the other side some "sizeless thingy" as the definition would work in math as well - again as long as everyone involved would think the same entity when reading it. |