The Definition of Points
in [Math]
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From: Bob Kolker on 17 Mar 2007 13:49 SucMucPaProlij wrote: >>You can develop geometry based purely on real numbers and sets. You need not >>assume any geometrical notions to do the thing. One of the triumphs of >>mathematics in the modern era was to make geometry the child of analysis. >> > > > And it means that lines, planes and points are defined in geometry. > Make up your mind, Bob! Not true. One of the mathematical systems which satisfy Hilbert's Axioms for plane geometry is RxR , where R is the real number set. Points are ordered pairs of real numbers. Not a scintilla of geometry there. Bob Kolker > >
From: Bob Kolker on 17 Mar 2007 13:50 alanmc95210(a)yahoo.com wrote:> > Euclid established the foundation for our mathematical deduction > system. As he realized from his Axioms and Postulates, you can't > prove everything. You've got to start with some given Axioms. Lines > and points are among those basic assumptions- A. McIntire The lines and points are undefined objects. It is the axioms concerning lines and points that are the basic assumptions. Bob Kolker >
From: Bob Kolker on 17 Mar 2007 13:52 nonsense(a)unsettled.com wrote: > > Sometimes even a troll asks a good question. > > A point and an apple are self defining. We only > get to report about them. Apples are defined by ostention. One points to an apple and says "apple". That is how babies learn what basic words mean. Many of the basic worlds we use are defined by pointing to objects and attaching the word to the object. Logical definitions occur at a higher level of abstraction. Bob Kolker
From: Bob Kolker on 17 Mar 2007 13:54 SucMucPaProlij wrote: > > > You can't define points and lines with numbers and sets? > Try it. It is not hard. Points (in n-dimesnsional space) are ordered n-tuples of real numbers. Bob Kolker
From: Hero on 17 Mar 2007 14:09
Bob Kolker wrote: > SucMucPaProlij wrote: > > > And I agree but can you tell me does point exist? > > How do you explain it? > > Point is an idea or a notion. It has no physical existence. Neither does > the integer 1. > > Point is a place holder for an intuition about space. Nothing more. > Along with line, plane and a few other place holders they constitute the > undefined terms of geometry. Intuitive notions are useful guides for > finding logical proofs, but they have not probatory or logical standing. > Referring to .." they have not probatory...standing". This associates: If You want to put down a glass onto a table and You are holding it's base a bit skew it might get a standing, but being pushed by someone at this moment it might tumble - and this has to do with it's point of gravity. With friendly greetings Hero PS. I just wonder, if a point relates to the word "pointing"? |