The Definition of Points
in [Math]
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From: Lester Zick on 16 Mar 2007 14:56 On Fri, 16 Mar 2007 15:35:47 +0100, "SucMucPaProlij" <mrjohnpauldike2006(a)hotmail.com> wrote: > >"hagman" <google(a)von-eitzen.de> wrote in message >news:1174053602.723585.89690(a)l77g2000hsb.googlegroups.com... >> 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. >> > >This is interesting observation :)))) > >But how do you define difference between "define" and "determine"? >Can "definition" determine and can "determination" define? Well as noted in my collateral reply to hagman I more or less agree with you here. At least I couldn't come up with any significant distinction between the two because as far as I can tell definitions determine a thing and conversely determinations of a thing define it. >Lester Zick has problem with "circular definitions" Lester Zick doesn't have a problem with circular definitions. Science has a problem with circular definitions. > and you used term "point" in >your "determination" to determine it. Maybe you want to say that in definition >you can't use term you define to define it and in termination you can use it to >determine it. And I think you're making a little too much out of nominal circular regressions. All you really have to do to define definition is prove it satisfies it own definition. >I think it's time to call Determinator :)))) >He is the only one who can help us! hahahahahahaha > ~v~~
From: Lester Zick on 16 Mar 2007 15:14 On 15 Mar 2007 16:03:14 -0700, "PD" <TheDraperFamily(a)gmail.com> wrote: >On Mar 14, 6:24 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: >> On 14 Mar 2007 08:07:33 -0700, "PD" <TheDraperFam...(a)gmail.com> wrote: >> >> >> >> >> >> >On Mar 14, 9:51 am, Eckard Blumschein <blumsch...(a)et.uni-magdeburg.de> >> >wrote: >> >> On 3/13/2007 6:52 PM, Lester Zick wrote: >> >> >> > In the swansong of modern math lines are composed of points. But then >> >> > we must ask how points are defined? >> >> >> I hate arbitrary definitions. I would rather like to pinpoint what makes >> >> the notion of a point different from the notion of a number: >> >> >> If a line is really continuous, then a mobile point can continuously >> >> glide on it. If the line just consists of points corresponding to >> >> rational numbers, then one can only jump from one discrete position to >> >> an other. >> >> >That's an interesting (but old) problem. How would one distinguish >> >between continuous and discrete? As a proposal, I would suggest means >> >that there is a finite, nonzero interval (where interval is measurable >> >somehow) between successive positions, in which there is no >> >intervening position. Unfortunately, the rational numbers do not >> >satisfy this definition of discreteness, because between *any* two >> >rational numbers, there is an intervening rational number. I'd be >> >interested in your definition of discreteness that the rational >> >numbers satisfy. >> >> That there is a straight line segment between rational numbers? > >Well, that's of course true, provided that you're associating numbers >with points on the line. I suppose it's true for line segments defined by points regardless of whether you're associating numbers with them or not. It's modern mathematikers who insist on associating points with numbers, not me. > Oh, wait, you don't believe that a line >consists of points. Of course not. Perhaps you can show how this has any bearing on the discreteness of line segments defined between points regardless? > Perhaps you want to associate points with numbers, >regardless of a line, and then say that there is a line segment >between those two points. That is probably correct. Probably??? That's mighty swell of you. > The question, >though is whether there is a line segment between two points >corresponding to rational numbers, such that no other rational number >corresponding to a point on the line segment between those two >endpoints. Funny I don't see that as the question at all. The question I see is whether there are line segments defined between points at all. If mathematikers wish to correlate numbers with those line segments. don't put it off on me. I don't much care what you do with them. > Are you saying that there is such a thing? I think I'm saying only what I'm saying. Of course that may not be quite what you say I'm saying. > If so, then I >invite you to come up with two rational numbers that satisfy that >criterion. First I would invite you to show that there are two rational numbers which necessarily lie on any common line segment at all. For the life of me I can really find nothing in the Peano and suc( ) axioms which shows any necessity for this. You just have various rational numbers associated with various line segments which have no connection to one another and could presumably each run off in any various direction. >> >> A point has no parts, each part of continuum has parts, therefore >> >> continuum cannot be resolved into any finite amount of points. >> >> Real numbers must be understood like fictions. >> >> >> All this seems to be well-known. When will the battle between frogs and >> >> mices end with a return to Salviati? >> >> ~v~~- Hide quoted text - >> >> - Show quoted text - > ~v~~
From: Lester Zick on 16 Mar 2007 15:23 On Fri, 16 Mar 2007 12:40:55 +0100, "SucMucPaProlij" <mrjohnpauldike2006(a)hotmail.com> wrote: >>>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? The specific problem I raised in connection with the definition for lines and points was whether ANY definition could satify the circular implication for lines constituted of points and points defined by the intersection of lines.That was the substance and direction of my post. However for what it's worth I'll be happy to share my take on the definition of geometric figures in general. Geometric figures are boundaries. That and nothing more. They're related to one another through the calculus and processes of derivation and integration but are not constituted or composed of one another. (Hmm although I'm beginning to wonder here whether that would include cubes composed of cubes etc.Oh well.Grist for another windmill to tilt at I daresay.) ~v~~
From: Bob Cain on 16 Mar 2007 16:41 Lester Zick wrote: > 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. Can you provide a useful physical example of the latter? Hell, even a useless one would be better than what you've provided so far. What won't suffice is any example that defies consensus understanding. If you provide a symbol string that only you can "understand" it won't satisfy the challenge because it has no practical value. Bob -- "Things should be described as simply as possible, but no simpler." A. Einstein
From: PD on 16 Mar 2007 16:54
On Mar 16, 2:14 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: > On 15 Mar 2007 16:03:14 -0700, "PD" <TheDraperFam...(a)gmail.com> wrote: > > > >> >On Mar 14, 9:51 am, Eckard Blumschein <blumsch...(a)et.uni-magdeburg.de> > >> >wrote: > >> >> On 3/13/2007 6:52 PM, Lester Zick wrote: > > >> >> > In the swansong of modern math lines are composed of points. But then > >> >> > we must ask how points are defined? > > >> >> I hate arbitrary definitions. I would rather like to pinpoint what makes > >> >> the notion of a point different from the notion of a number: > > >> >> If a line is really continuous, then a mobile point can continuously > >> >> glide on it. If the line just consists of points corresponding to > >> >> rational numbers, then one can only jump from one discrete position to > >> >> an other. > > >> >That's an interesting (but old) problem. How would one distinguish > >> >between continuous and discrete? As a proposal, I would suggest means > >> >that there is a finite, nonzero interval (where interval is measurable > >> >somehow) between successive positions, in which there is no > >> >intervening position. Unfortunately, the rational numbers do not > >> >satisfy this definition of discreteness, because between *any* two > >> >rational numbers, there is an intervening rational number. I'd be > >> >interested in your definition of discreteness that the rational > >> >numbers satisfy. > > >> That there is a straight line segment between rational numbers? > > >Well, that's of course true, provided that you're associating numbers > >with points on the line. > > I suppose it's true for line segments defined by points regardless of > whether you're associating numbers with them or not. It's modern > mathematikers who insist on associating points with numbers, not me. Ah, good. Then please sketch for me the line segment between 3/16 and 4/7, without any assocation between any points on the line segment and those rational numbers. > > > Oh, wait, you don't believe that a line > >consists of points. > > Of course not. Perhaps you can show how this has any bearing on the > discreteness of line segments defined between points regardless? > > > Perhaps you want to associate points with numbers, > >regardless of a line, and then say that there is a line segment > >between those two points. That is probably correct. > > Probably??? That's mighty swell of you. Gee, and just above you said it was true whether you associated the numbers with points defining the line segment or not. Now you say you want to associate points with the numbers. Clever strategy, Lester. Say one thing. Mock the reader for not seeing it. Say the converse. Mock the reader for not seeing that, too. > > > The question, > >though is whether there is a line segment between two points > >corresponding to rational numbers, such that no other rational number > >corresponding to a point on the line segment between those two > >endpoints. > > Funny I don't see that as the question at all. The question I see is > whether there are line segments defined between points at all. Gee, and just a few posts ago you said a line segment was defined by its two endpoints. Now you say the question is whether there are line segments defined between points at all. There's your lovely strategy again. > If > mathematikers wish to correlate numbers with those line segments. > don't put it off on me. I don't much care what you do with them. > > > Are you saying that there is such a thing? > > I think I'm saying only what I'm saying. Of course that may not be > quite what you say I'm saying. And then there's the strategy of declining to answer any questions, and then mocking those that ask the question. > > > If so, then I > >invite you to come up with two rational numbers that satisfy that > >criterion. > > First I would invite you to show that there are two rational numbers > which necessarily lie on any common line segment at all. For the life > of me I can really find nothing in the Peano and suc( ) axioms which > shows any necessity for this. You just have various rational numbers > associated with various line segments which have no connection to one > another and could presumably each run off in any various direction. > I didn't say that rational numbers necessarily existed on a line at all. The first to mention a line segment was you. Apparently you lose track of what you say hour to hour. See above or below, your choice: Me: "I'd be interested in your definition of discreteness that the rational numbers satisfy." You: "That there is a straight line segment between rational numbers?" Me: "Well, that's of course true, provided that you're associating numbers with points on the line." Blather away, Lester. Constructing a pretty sentence does not make you look appear any brighter than a charcoal briquet. If you find this entertainment, Lester, you have a tad too much time on your hands and should be letting the occupational therapist show you how to make yarn potholders. I've grown bored because of your transparent tactics and hour-to-hour self-contradictory incoherence. A schizoid that babbles in Shakespearean prose is still a schizoid. PD |