The Definition of Points
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From: Lester Zick on 15 Mar 2007 18:54 On Thu, 15 Mar 2007 11:38:50 -0400, Bob Kolker <nowhere(a)nowhere.com> wrote: >Sam Wormley wrote: > > >> Fair enough--However, for conceptualizing "defining" a point >> with coordinate systems suffices. > >Yes indeed. Point is a tuple of elements from a ring. But even these >have be grounded upon undefined terms. As is all of your logic, Bob. >The fact that RxR with a metric satisfies the Hilbert Axioms for plane >geometry implies that points can be taken to be pairs of real numbers. As a guess not bad. As a mathematical assumption pretty awful. >The fact that the Hilbert Axioms for the plane is a categorical system >makes me feel warm and fuzzy about identifying a line with a set of >points (number pairs) that satisfy a first degree equation in the >co-ordinate variables. Categorical system? What categorical system? A system with nothing more than empirical assumptions of truth to guide it? And that makes you feel warm and fuzzy, Bob? Fuzzy I can understand. Most everything you say is fuzzy. But warm? >This is a point (sic!) that Lester Zick is genetically incapable of >grasping. In your position, Bob, I might be a little more circumspect when talking genetics instead of mathematics. You're hardly qualified. ~v~~
From: Lester Zick on 15 Mar 2007 18:57 On Thu, 15 Mar 2007 12:11:36 +0100, "SucMucPaProlij" <mrjohnpauldike2006(a)hotmail.com> wrote: >> >> Look. If you have something to say responsive to my modest little >> essay I would hope you could abbreviate it with some kind of non >> circular philosophical extract running to oh maybe twenty lines or >> less. Obviously you think lines are made up of points. Big deal. So do >> most other neoplatonic mathematikers. >> > >I think that you think that mathematikers are stupid Lazy and/or stupid. Six of one, half dozen of the other. > and it has nothing to do with lines and point. Well thanks. I certainly appreciate the liners and pointer. >I only know that they are convergent because they are limited and monotone but >this is subject for another topic :)))) Let's hope so. ~v~~
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 - |