The Definition of Points
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From: Lester Zick on 18 Mar 2007 18:12 On Sun, 18 Mar 2007 18:07:13 +0100, "�u�Mu�PaProlij" <mrjohnpauldike2006(a)hotmail.com> wrote: >I have one question regarding sets but I can't find the answer. Maybe someone >can help me. > > > >I wonder if sets theory is self describing. > >Can you describe sets theory as a set? Are you talking about a set of all points or what? ~v~~
From: Virgil on 18 Mar 2007 18:23 In article <45fd9bfe$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <45fd6045(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <45fc6fd6(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>> > >>>> Except that linear order (trichotomy) and continuity are inherent in R. > >>>> Those may be considered geometric properties. > >>> If one defines them algebraically, as one often does, are they still > >>> purely geometric? > >>>> Tony Orlow > >> One may express them algebraically, but their truth is derived and > >> justified geometrically. > > > > How does one prove geometrically what is only defined algebraically? > example? That the set of naturals is infinite. That the cardinality of the symmetric group on n elements is n!. That a polynomial over the reals of degree n has no more than n real zeroes. That a square matrix always 'satisfies' its characteristic polynomial. Etc.
From: Lester Zick on 18 Mar 2007 18:40 On Sun, 18 Mar 2007 19:25:30 GMT, Sam Wormley <swormley1(a)mchsi.com> wrote: >Lester Zick wrote: >> On Sat, 17 Mar 2007 20:06:01 -0400, Bob Kolker <nowhere(a)nowhere.com> >> wrote: >> >>> Tony Orlow wrote:> >>>> Except that linear order (trichotomy) and continuity are inherent in R. >>>> Those may be considered geometric properties. >>> They can be defined in a purely analytic and algebraic manner starting >>> with the Peano axioms. While linear order is suggestive of geometric >>> notions, one can define such order without any geometric content >>> whatsoever. The order of the postive integers is more temporal than >>> spatial. Fist comes an integer -then- comes its successor. Etc. Etc. >> >> While much of this is true and pregnant with implications the Peano >> and suc( )axioms even if true cannot be used to define straight lines. >> >> ~v~~ > > > Lester, why are you posting on sci.physics? What purpose? Why am I posting this particular thread to sci.physics or why am I posting to sci.physics at all, Sam? I suppose the easy answer is that I'm interested in cross disciplinary feedback. These are all issues in science and with the exception of c.a.p. the groups are all prefaced by sci.But even in c.a.p. they're at least interested in the mechanics of sentience which is a directly related issue. I get the impression most all of the conventional thinkers on sci. physics and math are only interested in established approaches to science, math, and physics. I'm more interested in what is true in science and demonstrably so. I'm not interested in twentieth century interpretations by empirics of empirical experimental results. That's just a sacred cow which badly needs to be put out of our misery. At a glance I gather you find my approaches to truth and science or specific subjects offensive. I don't understand why. My educational credentials are only marginally relevant. I'm a serious thinker and not a gadfly looking to extort attention from the academic scholastic community. If you're trying to understand why I approach various subjects the way I do it is for only one reason: demonstrable truth. If you want to understand why I'm happy to discuss it with you or anyone else. But I'm not going to shunted off to one side with airs of patronizing condescension. Especially by those with no scientific credits of their own except the willingness to accept the status quo. ~v~~
From: Lester Zick on 18 Mar 2007 18:50 On 18 Mar 2007 11:12:39 -0700, "VK" <schools_ring(a)yahoo.com> wrote: >On Mar 18, 8:33 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: >> Oh I don't actually disagree; I just can't tell exactly what all these >> qualifications amount to and mean. You've got "abstraction" and >> "perception" and "equivalence" and all sorts of terms mixed up in here >> that make me suspect none of us including you knows exactly what >> you're talking about in mechanically exhaustive terms. > >If anyone of rivals (mathematics, philosophy, religion) would knew one >day "in mechanically exhaustive terms" what is a "thing without sides" >or say what is "infinity" - wow, the rest would come begging to clean >their shoos :-) Personally I don't agree. I see the issue of points with or without sides as almost frivolous as I've never known anyone who thought points had sides. ><snip> >> Well maybe that would be true if your initial predicates had any >> specific and exhaustive value. But lots of things may be true of >> points without being essential to their definition. I don't understand >> what "ti en einai of infinity" is supposed to mean nor a "reversed >> infinity". > >That was not a question which one of definition is correct, neither >"in mechanically exhaustive terms" nor even by some intuitive feeling; >well probably neither one. I was asking: do you believe that there is >one and only one correct definition of the point (a point on a line) >implied by the very nature of this entity? I think probably so. However my interest as I pointed out early on was more directed at whether lines are made up of points or not if points are in fact defined by the intersection of lines. >The fact that maybe no one can bring it in some mechanically >exhaustive terms right in this second does not change anything in the >question. No of course not. However the issue I'm really interested in doesn't require a mechanically exhaustive or any other kind of definition for points apart from what is mentioned directly above. > After all there is a number of unresolved problems not >because they don't have any solution but simply because they are not >solved yet due to different obstacles. >But as long as we arrived to such entities as "point", "line", >"infinite set", "natural number", "real number", "irrational number" >etc. - as long that: do you believe that each of them there is one and >only one proper mechanically exhaustive definition to find - coming >from the very nature of these entities? Once again probably so. I just haven't spent a lot of time on those issues as yet - if I ever do. > So once found we may expect >them universally correct, so even for some civilization from another >star they will be necessary either the same or wrong (so the said >civilization did not find the proper definition yet)? Sure. The truth of what I'm after is universal in scope and not just a particular or local truth in the sense it is demonstrably so. ~v~~
From: Lester Zick on 18 Mar 2007 18:54
On 18 Mar 2007 12:22:27 -0700, "Hero" <Hero.van.Jindelt(a)gmx.de> wrote: >Lester Zick wrote: >> Hero wrote: >> > Lester Zick wrote: >> >> Hero wrote: >> >> >> >PS. I just wonder, if a point relates to the word "pointing"? >> >> >> I'm convinced the phrase "pointing out" is definitely related to >> >> "point". You can easily enough "point out" an irrational on a straight >> >> line using rac construction but you can't "point out" a transcendental >> >> on a straight line at all. >> >> >Using only rac construction ( ruler and compass) results in a >> >geometric handicap. Already before Euclid Hippias of Elis did his >> >quadratrix with other tools. >> >> Well to the best of my knowledge rac construction is the only >> mechanically exhaustive method of construction that actually specifies >> or defines some point. >> >> >Actually a transcendental, as well as an rational, is a mutual >> >relation to a one, a measure. A point can live an egocentric life, a >> >real number ( not natural number) arises out of a minimum of three >> >points. >> >> Not sure what this comment is in aid of. Transcendentals are defined >> on curves not straight lines. > >The quadratrix is defined with two moving straight lines, one with >constant velocity, the other with constant change of angle, look here: >http://de.wikipedia.org/wiki/Quadratrix > >And having just a line, one can not point at a point and tell, this >point is transcendental. Mark one point as a zero and another one as >One, so You have a measure. Now a wheel with radius 1, that is this >measure, placed with a contact point onto the zero and rolled along >the line exact one revolution will end up with a contact point on the >line and measure out a distance, which is in relation to the distance >between zero and one transcendental. Well sure, Hero, this is pretty much what I imagined. The difficulty is one of dynamic measures. Rac construction is static not dynamic. It requires motion to set up but none to measure. Your wheel of diameter one will roll out to an approximation of pi but since the measure is dynamic it will be affected by dynamic factors such as friction, temperature fluctuation, stretching, contraction, and so on. ~v~~ |