The Definition of Points
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From: PD on 15 Mar 2007 18:58 On Mar 14, 9:24 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: > On 14 Mar 2007 13:02:00 -0700, "PD" <TheDraperFam...(a)gmail.com> wrote: > > > >> >On Mar 14, 1:28 am, Lester Zick <dontbot...(a)nowhere.net> wrote: > >> >> Are points and lines not still mathematical objects > > >> > The point is ?? ?? ?? ????? ("to ti en einai") of the infinity. > >> >If you want a definition based on something fresher than Aristotle > >> >then: > >> > The point is nothing which is still something in potention to > >> >become everything. > >> >IMHO the Aristotle-based definition is better, but it's personal. > > >> I don't want a definition for points fresher or not than Aristotle. > >> I'm trying to ascertain whether lines are made up of points. > > >Let's see if I can help. > > Oh that'll be refreshing for a change. > > >I believe Lester is asking whether a line is a composite object or an > >atomic primitive. > > Actually I'm interested in whether vectors exist and have > constituents. Well, since that's not particularly on topic, I don't care that you're interested in that. > > >One way of asking the question is whether a point sits ON a line or > >whether the point is part OF the line. > > Like I said before you're not very good at philosophy but you're much > worse at science. > > >Of course, since both the point and the line are idealizations, > >conceptual constructions out of the human mind that don't have any > >independent reality, then one could rightly ask why the hell it > >matters, since there is no way to verify either statement through an > >external discriminator. > > An external what-inator? Why don't you just call it magic and be done > with it? No need to dress it up like a dog's dinner with all the > philosophical badinage. You're a mystic. So what? > > > Lester doesn't believe in external > >discriminators anyway, because that is the work of evil empirics, and > >he'd rather spend his day mentally diddling away at issues like this. > > Whereas obviously you don't. Not really, no. I don't like the smell of stink bait, either, but it does come in handy when hooking a nice fish. > > >But to provide him with some prurient prose by which to diddle > > You know, sport, if you were even half as witty as I am that might > indeed make you a half wit. However in this instance you're trying too > hard and you wind up appearing more trying than witty. > > >further, let's toss him the idea that we can clearly cleave a line in > >two by picking a point (either on the line or part of the line, take > >your pick) and assigning one direction to one semi-infinite segment > >and the other direction to the other semi-infinite segment -- > >sometimes called rays. One can then take one of those rays and cleave > >it again, and one of the results will be a line segment, which is > >distinguished by having two end *points*. Now the interesting question > >is whether those end points are ON the line segment or part OF the > >line segment. > > Neither. The end points contain the line segment. That's how the line > segment is defined. And where did those points come from? Did we have to bring them in from Points Depot or PointsMart? Or were they already there when we cleaved the line? Or did they just suddenly appear, created in the act of cleaving? Or did they fall of the line they were resting on? > > > One way to answer this is to take the geometric limit of > >one end point approaching the other end point, > > Of course another way to answer this is to ask what defines the line > segment to begin with. Well, that would be a question, not an answer. Perhaps there is an answer to the question. Oh yes, those two points at the end. Where did they come from again? > > > and ask what the limit > >of the line segment is. > > When it gets to zero do be sure to let us know. Gee, and I was thinking of a geometric limit, not a numerical limit. I don't recall any measure being introduced so far. > > > That should either settle it or send Lester > >into an orgasmic frenzy. > > Gee with another swell foop you might actually get to the calculus. Of > course Newton and Leibniz and probably a thousand other wannabe's are > waiting in the wings ahead of you and the other neomathematikers. > Nicely done, there, Lester. Spend a good chunk of your reply talking about anything (mostly your evaluation of me, which I don't find relevant to anything) other than the subject matter of your original post. PD
From: Eric Gisse on 15 Mar 2007 19:01 On Mar 15, 2:54 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: [...] What is your background in mathematics, Lester?
From: PD on 15 Mar 2007 19:03 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. Oh, wait, you don't believe that a line consists of points. 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. 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. Are you saying that there is such a thing? If so, then I invite you to come up with two rational numbers that satisfy that criterion. > > >> 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 -
From: PD on 15 Mar 2007 19:05 On Mar 15, 12:46 pm, Eckard Blumschein <blumsch...(a)et.uni- magdeburg.de> wrote: > On 3/14/2007 4:07 PM, PD wrote: > > > 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. > > Rational numbers are countable because all of them are different from > each other. > The two real numbers 0.9... and 1.0... with actually indefinite length > merely hypothetically exhibit a difference of value zero that tells us > the left one is nonetheless smaller than the right one. > > In other words: Real numbers must differ from rational ones by the > unreasonable claim of providing infinite acuity. IR just constitutes the > hypothetical border of the rationals. The continuum IS the tertium. > > Do not destroy this fortunate insight into how the border between number > and continuum works by stupid definitions. We need this heresy in order > to resolve several practical problems. > > Eckard Blumschein You did not answer my question about your definition of discreteness that rational numbers satisfy. Is being countable your definition of discreteness? PD
From: Lester Zick on 15 Mar 2007 19:20
On 15 Mar 2007 15:21:44 -0700, "Eric Gisse" <jowr.pi(a)gmail.com> wrote: >On Mar 15, 9:11 am, Lester Zick <dontbot...(a)nowhere.net> wrote: >> On 14 Mar 2007 18:57:28 -0700, "EricGisse" <jowr...(a)gmail.com> wrote: >> >> >> >> >On Mar 14, 5:23 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: >> >> On 14 Mar 2007 14:54:55 -0700, "EricGisse" <jowr...(a)gmail.com> wrote: >> >> >> >On Mar 14, 11:15 am, Lester Zick <dontbot...(a)nowhere.net> wrote: >> >> >> On 13 Mar 2007 23:21:54 -0700, "EricGisse" <jowr...(a)gmail.com> wrote: >> >> >> >> >On Mar 13, 9:54 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: >> >> >> >> On 13 Mar 2007 17:18:03 -0700, "EricGisse" <jowr...(a)gmail.com> wrote: >> >> >> >> >> >On Mar 13, 9:52 am, 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~~ >> >> >> >> >> >Points, lines, etc aren't defined. Only their relations to eachother. >> >> >> >> >> So is the relation between points and lines is that lines are made up >> >> >> >> of points and is the relation between lines and points that the >> >> >> >> intersection of lines defines a point? >> >> >> >> >No, it is more complicated than that. >> >> >> >> Well that's certainly a relief. I thought you said "only their >> >> >> relations to each other". It's certainly good to know that what lines >> >> >> are made up of is not "only a relation" between points and lines. >> >> >> >> ~v~~ >> >> >> >No, I said "it is more complicated than that." >> >> >> No what you said is "Points, lines, etc aren't defined. Only their >> >> relations to eachother". Your comment that "No, it is more complicated >> >> than that" was simply a naive extraneous appeal to circumvent my >> >> observation that relations between points and lines satisfy your >> >> original observation. Your trivial ideas on complexity are irrelevant. >> >> >> ~v~~ >> >> >*sigh* >> >> >It isn't my fault you cannot read for comprehension. >> >> But it is your fault you cannot argue for comprehension by others. >> >> >Points and lines are undefined - it is as simple as that. >> >> Problem is that when you want to endorse an idea you say "it is as >> simple as that" and when you want to oppose an idea you say "it is >> more complicated than that" such that we have a pretty good idea what >> your opinions might be but no idea at all why your opinions matter or >> are what they are or should be considered true by others. >> >> > Every >> >question you ask that is of the form "So <idiotic idea> defines >> >[point,line]" will have "no" as an answer. >> >> So we should just accept your opinions as true without justification? >> Excuse moi but this is still a science forum and not merely a polemics >> forum. > >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. >Why don't you just stop posting and leave science to those who are at >least marginally capable, unlike yourself? As you said, this is not a >polemics forum. Well I agree that unlike me you're only marginably capable of science. That's why I post to and for science while you're reduced to polemics. ~v~~ |