The Definition of Points
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From: Tony Orlow on 19 Mar 2007 11:51 Hero wrote: > Lester Zick wrote: > Hero 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. >> > > Your rac construction (of two distances in rational or algebraic > relation) is exact, a "mechanically exhaustive method of construction" > - in Plato's paradise. In Plato's hell a ruler is allowed to move with > constant speed. > Which gives us still another definition of point, that of a puncture. > The puncture of a compass-tip into a solid or through a surface. > > NB1: When You do a "mechanically exhaustive method of construction" of > a circle with a compass, the distance between the tips of the compass, > the radius of the circle is of course transcendental, when You regard > the length of the circle-perimeter as one [unit]. So now all Your rac- > constructions give trascendental length. Two different rooms in > Plato's space. bene notatus > NB2: Euclid was not totally a Platonist, he defines solids like cones > with a dynamic construction by means of rotation. > Plato discognoscerunt. > With friendly greetings > Hero > Platypus maximus Tony
From: Randy Poe on 19 Mar 2007 11:59 On Mar 19, 11:32 am, Tony Orlow <t...(a)lightlink.com> wrote: > Virgil wrote: > > In article <45fd9bf...(a)news2.lightlink.com>, > > Tony Orlow <t...(a)lightlink.com> wrote: > > >> Virgil wrote: > >>> In article <45fd6...(a)news2.lightlink.com>, > >>> Tony Orlow <t...(a)lightlink.com> wrote: > > >>>> Virgil wrote: > >>>>> In article <45fc6...(a)news2.lightlink.com>, > >>>>> Tony Orlow <t...(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. > > Geometrically incorrect. Unless there is a natural infinitely greater > than the origin, there is no infinite extent involved. The naturals don't have physical positions, since they are not defined geometrically. But at any rate, this is your own private misconception, and it stems from your insistence that there must be a member of N which is "at the end" of N. > > That the cardinality of the symmetric group on n elements is n!. > > Isn't that demonstrable geometrically? No. There is nothing geometrical in that statement. > > That a polynomial over the reals of degree n has no more than n real > > zeroes. > > Also geometrically demonstrable. Please describe such a demonstration. > > That a square matrix always 'satisfies' its characteristic polynomial. > > > Etc. > > Not sure what that means, but excuse me if I take "square" to be a > geometrical concept.... It's not, in this context. - Randy
From: Tony Orlow on 19 Mar 2007 12:03 SucMucPaProlij wrote: > "Lester Zick" <dontbother(a)nowhere.net> wrote in message > news:29erv29qotk1c65v9mruh7rdjl9biqmf0q(a)4ax.com... >> 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? >> > > no, "set" as "any set" > > Hi Suc - I've noted your comments.... As I see it, you are asking whether set theory itself can be described *in* set theory, and from what I've seen, you haven't yet received a good answer. Theory itself, with regards to sets, measures, evolution, or anything else, consists of a set of rules with which one can draw conclusions from premises, that is, statements assumed to be true. Before set theory is established, the rules of first order logic are established, which involve determining the truth values of statements which are derived according to logical rules from statements with known truth values. This system of logical analysis precedes set theory, or *any* theory. There is debate as to whether mathematics is a subset of logic, or vice versa. My position is that logic is a branch of mathematics, as it involves measure of its own ilk. Therefore, set theory, sitting on top of logical calculus, does not constitute the very foundation of math. Set theory depends on basic formal logical, and an additional primitive 'e', or "element of". Logic itself is the foundation. And logic is an exploration of the relationship between 0 and 1. :) TonLocJahPothole
From: Tony Orlow on 19 Mar 2007 12:06 Eckard Blumschein wrote: > On 3/15/2007 7:07 PM, Bob Kolker wrote: > >> 9/10 + 9/100 + etc converges to 1.0 >> >> Bob Kolker > > > Not to 1.0, > but why not to 1.00... as 1 + 0/10 + 0/100 + etc converges to 0.99... > Hi Eckard - Long time.... In the adics, what is successor to ...999? ...000. In an uncountable ring*, does successor mean measurable difference? Not really. Whether we make distinction between 0.999... and 1.000... is a matter of what we consider measurable. Smiles, Tony
From: Tony Orlow on 19 Mar 2007 12:12
Randy Poe wrote: > On Mar 19, 11:32 am, Tony Orlow <t...(a)lightlink.com> wrote: >> Virgil wrote: >>> In article <45fd9bf...(a)news2.lightlink.com>, >>> Tony Orlow <t...(a)lightlink.com> wrote: >>>> Virgil wrote: >>>>> In article <45fd6...(a)news2.lightlink.com>, >>>>> Tony Orlow <t...(a)lightlink.com> wrote: >>>>>> Virgil wrote: >>>>>>> In article <45fc6...(a)news2.lightlink.com>, >>>>>>> Tony Orlow <t...(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. >> Geometrically incorrect. Unless there is a natural infinitely greater >> than the origin, there is no infinite extent involved. > > The naturals don't have physical positions, since they are not > defined geometrically. But at any rate, this is your own private > misconception, and it stems from your insistence that there must > be a member of N which is "at the end" of N. You're within your own orb. Position the point of your compass at 0, and traverse the line - that's 1. Now position it at 1, and traverse the line in the same direction ... mark those points sequentially.... > >>> That the cardinality of the symmetric group on n elements is n!. >> Isn't that demonstrable geometrically? > > No. There is nothing geometrical in that statement. > Arrange them in a convex polygon as vertices, and draw lines between them.... >>> That a polynomial over the reals of degree n has no more than n real >>> zeroes. >> Also geometrically demonstrable. > > Please describe such a demonstration. > Let me think about that, among other things... In the meantime, let Virgule explain how I can't... >>> That a square matrix always 'satisfies' its characteristic polynomial. >>> Etc. >> Not sure what that means, but excuse me if I take "square" to be a >> geometrical concept.... > > It's not, in this context. > > - Randy > > Oh. Define the geometrical definition of square, that's not "analytical".... - Tony |