The Definition of Points
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From: Virgil on 18 Mar 2007 15:05 In article <45fd63b5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Bob Kolker wrote: > > Hero wrote: > > > >> Numbers are born in a huge family, the mother was time (the counting > >> of days into a moon cycle, displayed as the movement of the stars of > >> Nut ) and the father was space ( with features of Geb with calculi to > >> count the sheep and a container to measure the grain). > >> When Bob thinks, that numbers are grown up and do not need their > >> father any more, that they are not about spatial objects any more, so > >> why still call ,,geometry", why not call it ,,number theory"? > > > > Like Tevyeh in -Fiddler on the Roof- says: Tradition! > > > > Modern math has outgrown its parents and gone far beyond them, like any > > successful Son. > > > > Bob Kolker > > > > That is like saying your mind has outgrown your body, so you no longer > need to eat or breathe. The language of math is the more abstract > aspect, but the geometry of it is still the basis of its truth. The natural numbers, with their arithmetic, do not owe anything to geometry. Even Euclid separated them.
From: Lester Zick on 18 Mar 2007 15:08 On Sat, 17 Mar 2007 18:01:54 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Sat, 17 Mar 2007 11:27:48 -0500, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> Lester Zick wrote: >>>> On Fri, 16 Mar 2007 16:18:53 +0100, "SucMucPaProlij" >>>> <mrjohnpauldike2006(a)hotmail.com> wrote: >>>> >>>>> "Lester Zick" <dontbother(a)nowhere.net> wrote in message >>>>> news:1ukbv2hq1fo7ucv8971u9qo37b48bj6a5h(a)4ax.com... >>>>>> 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~~ >>>>> Can you prove that non-circular definition of existence exists? >>>> Well that depends on what you and others mean by "existence exists". >>>> On the face of it the phrase "existence exists" is itself circular and >>>> no more demonstrable than a phrase like "pointing points". It's just a >>>> phrase taken as a root axiomatic assumption of truth by Ayn Rand in my >>>> own personal experience whether others have used it or not I don't >>>> know. >>>> >>>> On the other hand if you're asking whether anything exists and is >>>> capable of being unambiguously defined the answer is yes. I've done >>>> exactly that on more than one occasion first in the root post to the >>>> thread "Epistemology 201: The Science of Science" of two years ago and >>>> more recently in the root post to the thread "Epistemology 401: >>>> Tautological Mechanics" from a month ago. >>>> >>>> The technique of unambiguous definition and the definition of truth is >>>> simply to show that all possible alternative are false. Empirics and >>>> mathematikers generally prefer to base their definitions on >>>> undemonstrable axiomatic assumptions of truth whereas I prefer to base >>>> definitions of truth on finite mechanical tautological reduction to >>>> self contradictory alternatives. The former technique is a practice in >>>> mystical insight while the latter entails exhaustive analysis and >>>> reduction in purely mechanical terms. >>>> >>>> ~v~~ >>> So, essentially, anything that's not self-contradictory exists, or is >>> "true"? In an infinite universe, perhaps.... >> >> Hey, Tony. Good to hear from you as always. The point is that any self >> contradictory predicate is perforce false.Therefore any alternative to >> self contradictory predicates must be perforce true. > >I think that's what I said. Pretty much, Tony. I just wanted to say it myself to make sure. >> However you need to be very careful here. It is certainly possible to >> combine predicates in various ways such that showing the combination >> is self contradictory and perforce false doesn't make it exactly clear >> what the tautological alternative may be that is true. This is why I >> invariably reduce consideration of such self contradictory predicates >> to"not not" or the "contradiction of contradiction" whose tautological >> alternatives are the clear and unambiguous "not" or "contradiction". > >If x is false, then not(x) is true, but not(x) doesn't define a true y. The problem here though is that for compound predicates we don't know what "not x" necessarily has to be. If x is self contradictory x is false. However in the case of single isolated predicates not x would be true. If x=red and red is self contradictory for whatever reason then not x would be true. The problem comes when you try to mix up combinations of predicates. And I think in your foregoing example you're implicitly imagining some combination of self contradictory predicates which are false and then trying to apply "not" to them somehow across the board which of course does not work because we have no way to say which predicate or predicate subcombination the "not" necessarily has to apply to. That's the reason most people think tautologies are useless because applying "not" to predicate combinations doesn't yield any necessary conclusion as to which predicate or subcombination should be negated. In other words if we know "not x" is true that still doesn't tell us what that "not x" is. "Not red" for example doesn't tell us whether the "not red" is "blue" "yellow" or whatever. But that's only because in the first instance we have some specific predicate such as "red" but in the latter instance of "not red" we have potentially unlimited predicate combinations to negate to arrive at any specific color. And the only time we can really use tautological regression abstractly and effectively to determine the truth of something specific is when the predicate we're looking at is "not" itself because tautological alternatives to "not" in the form of "not not" are self contradictory. >> On the other hand if we complain "blue ideas" are self contradictory >> it's not really clear at all just what the tautological alternative to >> "blue ideas" might be. We could just say "not blue ideas" but that >> doesn't tell us what exactly "not blue ideas" might mean. Obviously >> both "blue" and "ideas" are true in some ways but their combination is >> not for reasons which are not clear just from their combination. > >That seems to be an example, not of logical contradiction, but of >inappropriate context for the attribute. Well the difficulty here, Tony, is that people generally recognize there is a problem but just don't see the problem in terms of self contradiction alone in terms of predicate combinations. So they make up different names among which I've seen "ill formed" "category error" or your own "inappropriate context". The problem is that these are all just so many aliases for the same predicate combination structural problem where we try to apply one "not" to predicate combinations. That won't work. Whether we have a predicate combination such as "blue ideas" or "one sided triangle" it's all the same problem. And when we try to apply a tautological "not" to resolve the problem we are still left uncertain to which predicate or predicate combination the "not" should apply in any particular instance to get the truth. > Ideas are not generally >considered to have the attribute of color, but that doesn't contradict >the notion that color may be associated with some ideas. For instance, >"blue ideas" may be sad thoughts, or concepts regarding shades of blue, >or maybe even plans to go to the beach. So, I wouldn't say "blue ideas" >is self-contradictory, or that it has any particular negation. Well I agree if speaking metaphorically. However I was speaking literally not metaphorically. It's just tough to find examples which cannot be construed metaphorically if we're so disposed. >> The same would be true for "one sided triangles" except that here we >> can see the self contradiction in "one" versus "tri-" and recognize >> the alternative "three sided triangle". But most self contradictory >> predicate combinations do not have such clear cut tautological >> alternatives whose reduction to "truth" is so readily apparent. > >I think if they are truly self-contradictory, that could be determined >logically. In this case, since 1<3, and triangles are defined to have 3 >sides, the contradiction is clear. Yes but if we didn't understand that "triangles" are defined to have three sides or we didn't understand that "tri-" meant three we would be left unaware exactly where the self contradiction lay and we would be forced into the same kind of tautological reconstructive regression required to analyze the self contradiction in "blue ideas". >> It's a fascinating area of science, Tony, because it represents the >> way we actually think and mechanize ideas whether true or false. In >> other words conventional approaches to truth in empiricism and >> empirical mathematics emphasize truth by axiomatic assumption or >> reduction to such simple circumstances that the "truth" is readily >> apparent. But in point of fact "truth" has to be demonstrated in >> mechanical terms and cannot just be assumed regardless of how >> "intuitively obvious to the casual observer" an axiomatic reduction >> might appear. > >Yes, I think what you're advocating is the more inductive side of >discovering the rules, by analyzing evidence, rather than assuming >rules, and following them to their deductive conclusions. Both are >necessary for science, wouldn't you say? Sure. But the deductive branch is only a shortcut which relies on the correctness of the inductive branch. Tautological regression is the one actual mechanism involved in both and subjects can always be analyzed inductively whether or not they can analyzed deductively. >> I don't know if you caught my recent post "Epistemology 401: >> Tautological Mechanics" which illustrates the tautological reduction >> of conjunctions to compoundings of "contradiction" or "not" but that >> represented the last major hurdle in my efforts to reduce the origin >> of all things to finite tautologically true regressions in mechanical >> terms and its well worth checking out. >> >> ~v~~ > >I'll try to take a look. I've kinda been on a hiatus lately. Have a good >one. You too, Tony. My mathematical conclusions regarding the calculus, infintesimals, curves, straight lines, and so on are more speculative because I don't quite yet have the necessary insights to demonstrate them fully. But the E401 summary is rock solid and demonstrably so. ~v~~
From: Lester Zick on 18 Mar 2007 15:10 On Sat, 17 Mar 2007 22:38:30 GMT, mmeron(a)cars3.uchicago.edu wrote: >In article <1174170793.549628.45350(a)b75g2000hsg.googlegroups.com>, "VK" <schools_ring(a)yahoo.com> writes: >>On Mar 18, 12:59 am, "SucMucPaProlij" <mrjohnpauldike2...(a)hotmail.com> >>wrote: >>> Cabin is symetrical and you can't distinguish between left and right. >>> When you say "Left button is broken" question is "what left button?" >>> There is no left or right half of circle. >> >>Yep, this is what I mean. That was to argue with the Hero's statement >>that "left and right are geometrical concepts". Left and right are >>semantical concepts appeared grace to the particular human body >>symmetry. If octopuses got the intellect, I would die to see their >>geometry books. And I would sell my new car for any junior-high >>calculus book from a planet populated by creatures having three pods >>instead of ten fingers - so they are naturally using base-3 numeral >>system with base-10 system being a scientific domain obscurity. > >Why do you think that there is ***anything*** in calculus that depends >on whether you use base 3, 10, 42 or whatever? Good question. ~v~~
From: Lester Zick on 18 Mar 2007 15:12 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~~
From: Hero on 18 Mar 2007 15:22
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. With friendly greetings Hero |