The Definition of Points
in [Math]
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From: Brian Chandler on 16 Mar 2007 03:46 Lester Zick wrote: > On Wed, 14 Mar 2007 22:30:21 +0100, "SucMucPaProlij" > <mrjohnpauldike2006(a)hotmail.com> wrote: > > >> If the point is defined by the intersection what happens to the point > >> and what defines the point when the lines don't intersect? > >> On the other hand if the point is not defined by the intersection of lines > >> how can one assume the line is made up of things which aren't defined? > >> > > > >hahahahaha you are poor philosopher. > > Obviously. That's why I became a mathematician. You did? Gosh, congratulations! Brian Chandler http://imaginatorium.org (just wanting to be part of this golden thread, this irridescent braid, this)
From: Eckard Blumschein on 16 Mar 2007 04:52 On 3/16/2007 12:05 AM, PD wrote: > You did not answer my question about your definition of discreteness > that rational numbers satisfy. Is being countable your definition of > discreteness? What we are calling definition is usually the attempt to pinpoint something by means of a non-circular description. Why should I define discreteness of something by saying it is uncountable? If so, wouldn't the relationship be valid the other way round too? I would rather like to state that there is a fundamental property which utters itself in that ... and in that... . What about countability, we have to anticipate the mistake that it is so far merely understood like a property of a set. Incidentally, Cantor himself used the word countable in the sense 'there is a bijection to the naturals' while he used "counted" (abgezaehlt) in the common sense meaning. I say, already the decimal representation of the unresolvable task called pi is uncountable. In my understanding even 0.99... is an uncountable representation. Any number is uncountable if only existing or just embedded in IR. Embedded rationals are uncountable as long as they do not belong to Q but to IR. Is this too strange to you? Eckard Blumschein
From: SucMucPaProlij on 16 Mar 2007 07:40 >>I gave a book suggestion [Sibley's geometry] and a Wikipedia link that >>mirrors what is said in Sibley, plus I already explained that there >>are undefined terms in geometry - and that 'point' is one of them. > > But a line made up of points is not one of them. > and will you share with us your secret definition for points and lines or not?
From: hagman on 16 Mar 2007 10:00 On 13 Mrz., 18:52, Lester Zick <dontbot...(a)nowhere.net> wrote: > The Definition of Points > ~v~~ > > In the swansong of modern math lines are composed of points. But then > we must ask how points are defined? However I seem to recollect > intersections of lines determine points. But if so then we are left to > consider the rather peculiar proposition that lines are composed of > the intersection of lines. Now I don't claim the foregoing definitions > are circular. Only that the ratio of definitional logic to conclusions > is a transcendental somewhere in the neighborhood of 3.14159 . . . > > ~v~~ Please look up the difference between "define" and "determine". In a theory that deals with "points" and "lines" (these are typically theories about geometry), it is usual to leave these terms themselves undefined and to investigate an incidence relation "P on L" (for points P and lines L) with certain properties Then the intersection of two lines /determines/ a point in the sense that IF we have two lines L1 and L2 AND there exists a point P such that both P on L1 and P on L2 THEN this point is unique. This is usually stated as an axiom. And it does not define points nor lines.
From: hagman on 16 Mar 2007 10:26
On 15 Mrz., 23:54, Lester Zick <dontbot...(a)nowhere.net> wrote: > On Thu, 15 Mar 2007 11:38:50 -0400, Bob Kolker <nowh...(a)nowhere.com> > wrote: > > >Sam Wormley wrote: > >The fact that RxR with a metric satisfies the Hilbert Axioms for plane > >geometry implies that points can be taken to be pairs of real numbers. > > As a guess not bad. As a mathematical assumption pretty awful. There's no assumption in here. "RxR satisfies Hilbert axioms for plane geometry" is provable. "Foo satisfies the axioms of a Bar object" means that all theroems of Bar theory are true when interpreted as statements about Foo. |