The Definition of Points
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From: Virgil on 19 Mar 2007 16:48 In article <45FE95CD.8010609(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> 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, Oh! Since 1 = 1.0 = 1.00 = 1.000 = 1.000..., why not?
From: hagman on 19 Mar 2007 17:22 On 17 Mrz., 19:23, "Hero" <Hero.van.Jind...(a)gmx.de> wrote: > On 17 Mrz., 18:49, Bob Kolker <nowh...(a)nowhere.com> wrote:> SucMucPaProlij wrote: > > >>You can develop geometry based purely on real numbers and sets. You need not > > >>assume any geometrical notions to do the thing. One of the triumphs of > > >>mathematics in the modern era was to make geometry the child of analysis. > > > > And it means that lines, planes and points are defined in geometry. > > > Make up your mind, Bob! > > > Not true. One of the mathematical systems which satisfy Hilbert's Axioms > > for plane geometry is RxR , where R is the real number set. Points are > > ordered pairs of real numbers. Not a scintilla of geometry there. > > Left and right are geometrical concepts. > When You write down ( 3, 4 ) 3 is left in Your view and 4 is right. > > With friendly greetings > Hero No, (3,4) is {{3},{3,4}} and then 3 is the only element of the only singleton element and 4 is the other guy.
From: SucMucPaProlij on 19 Mar 2007 17:28 "hagman" <google(a)von-eitzen.de> wrote in message news:1174327593.061615.24560(a)y66g2000hsf.googlegroups.com... > On 16 Mrz., 15:35, "SucMucPaProlij" <mrjohnpauldike2...(a)hotmail.com> > wrote: >> "hagman" <goo...(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? >> >> Lester Zick has 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. >> >> I think it's time to call Determinator :)))) >> He is the only one who can help us! hahahahahahaha > > All I wanted to do is go along with Lester on his path of arguments as > far as bearable - which is more than "as far as correct". > Let me start again with a bit more sleep: > In his OP, Lester talked about lines as being composed of points by > definition and that the intersection of two lines determined a point. > This is his private theory although he states that it were somehow > usual math folklore. > > First imtermezzo: What is a definition? > I won't define that term here rogorously, but a definition should be > useful (that's just a pragmatic aspect) and make the defined term > eliminable. > Example: In the context of natural number we can define > DEF. k is a divisior of n if there exists some m such that k*m=n. > DEF. p is prime if p has eactly two divisors. > These are good definitions as we can eliminate the terms "prime" and > "divisor" from statements like > "There are infinitely many primes" = "There are infinitely many > numbers p such that there are exactly two numbers k such that there > exists some m such that p=k*m" > > So: What is the definition of "point"? And what is the definition of > "line"? > Mathematical theories where both the terms "points" and "lines" are > used usually go like this: > A tuple G=(P,L,E) is called a "geometry" (the precise term may depend > on the precise set of axioms used below and might as well be > "euklidean plane" or the like) if P and L are disjoint sets and E is a > relation among them, i.e. E subset PxL, such that > 1. For two distinct p,q in P, there is exactly one g in L such that > (p,g) in E and (q,g) in E > 2. For two distinct g,h in L, there is at most one p in P such that > (p,g) in E and (p,h) in E > 3. For each p in P there is at least one g in L such that (p,g) not in > E > 4. ... > > Only the context of such a geometry and the complete set of axioms > listed define point (and line and incidence). > > Thus, "Let p be a point ..." should be writen via elimination as > "Assume G=(P,L,E) is a geometry and p in P ..." > Note that lines are not "composed of" points. > However, two (non-parallel) lines determine a point in the sense that > it is a theorem (in fact an axiom) in this theory that for these lines > there is a unique point incident with both lines. > > As lines are not composed of points, the very first sentence of the OP > is nonsense. > However, this nonsense can be escaped from: > Given a geometry G(P,L,E), we ca define a mapping from > pointson: L -> 2^P, g |-> {p in P: (p,g) in E} > This mapping is injective and allows us to replace L by {pointson(): g > on L} and E by element containment to obtain an isomorphic geometry. > Hence one may assume wlog that lines are sets of points. > > However, we could alternatively have used a mapping > goesthru: P -> 2^L, p |-> {g in L: (p,g) in E} > and would then interprete lines as "atomic" and points as sets of > lines. > > The first method is more useful as it is more readily generalized to > other sets of points (circles etc.) > > In view of the above, "lines are composed of points" and "lines > determine points" is no more miraculous than "sets contain elements" > and "sometimes two sets have exactly one element in common" > General problem with this kind of discussions is that people don't want to accept "alternative truth" or the fact that you don't have to use the same words to describe same thing. I do try to understand what other people say and I do try to recognize my thoughts in other people's words but sometimes I just fool around :))))
From: Hero on 19 Mar 2007 17:55 hagman wrote: > Hero wrote: > > Bob Kolker wrote: > > > ...Points are ordered pairs of real numbers. Not a scintilla of geometry there. > > > Left and right are geometrical concepts. > > When You write down ( 3, 4 ) 3 is left in Your view and 4 is right. > > No, (3,4) is {{3},{3,4}} and then 3 is the only element of the only > singleton element and 4 is the other guy. Congratulations, You win. With friendly greetings Hero
From: Lester Zick on 19 Mar 2007 17:58
On Mon, 19 Mar 2007 10:42:43 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On 18 Mar 2007 11:12:39 -0700, "VK" <schools_ring(a)yahoo.com> wrote: >> >>> On Mar 18, 8:33 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: >>>> Oh I don't actually disagree; I just can't tell exactly what all these >>>> qualifications amount to and mean. You've got "abstraction" and >>>> "perception" and "equivalence" and all sorts of terms mixed up in here >>>> that make me suspect none of us including you knows exactly what >>>> you're talking about in mechanically exhaustive terms. >>> If anyone of rivals (mathematics, philosophy, religion) would knew one >>> day "in mechanically exhaustive terms" what is a "thing without sides" >>> or say what is "infinity" - wow, the rest would come begging to clean >>> their shoos :-) >> >> Personally I don't agree. I see the issue of points with or without >> sides as almost frivolous as I've never known anyone who thought >> points had sides. >> > >You've met Ross, and I'm not adverse to the concept of points of various >dimensions with zero measure, depending on their spatial context. :) But zero is zero, Tony. I've speculated on the existence of "points" with infinitesimal dimensions. Just never on the existence of actual points with non zero dimensions. >>> <snip> >>>> Well maybe that would be true if your initial predicates had any >>>> specific and exhaustive value. But lots of things may be true of >>>> points without being essential to their definition. I don't understand >>>> what "ti en einai of infinity" is supposed to mean nor a "reversed >>>> infinity". >>> That was not a question which one of definition is correct, neither >>> "in mechanically exhaustive terms" nor even by some intuitive feeling; >>> well probably neither one. I was asking: do you believe that there is >>> one and only one correct definition of the point (a point on a line) >>> implied by the very nature of this entity? >> >> I think probably so. However my interest as I pointed out early on was >> more directed at whether lines are made up of points or not if points >> are in fact defined by the intersection of lines. >> > >Can't lines be defined mutually as the set of intersections with other >lines? Maybe, though that doesn't identify a root concept. More to the point though, Tony, it identifies a circular concept. The same problem with points defined as intersections of lines which in turn constitute lines. Just doesn't work in mechanically exhaustive terms. > I would say >that the point is undoubtedly the atom of space, indivisible while every >line or segment can be divided at any given internal point. In that >sense, the point is more elementary than the line. Elementary in a certain sense, yes. But constituents of lines, no. > The relevant >question, as I see it, is whether useful mathematics can be developed by >only looking at measureless points, or whether the concept of the line >is more central to what really flowers in science and math. A line >connects two points, and a point identifies the intersection between two >lines. Is one more "important" than the other? No. But lines occupy a similar station of relevance with respect to surfaces. Geometric figures are all boundaries. They aren't material artifacts or constituents of one another. What is the constituent of a shadow? >>> The fact that maybe no one can bring it in some mechanically >>> exhaustive terms right in this second does not change anything in the >>> question. >> >> No of course not. However the issue I'm really interested in doesn't >> require a mechanically exhaustive or any other kind of definition for >> points apart from what is mentioned directly above. >> > >Do you think ALL points are "intersections of lines"? I think so. But more to the point in this context is whether points are constituents of lines and the intersection of lines represent points. That's the specific circular issue I'm trying to address here. > I am not sure this >is so, especially when you consider a 1-D space like the number line, >where there are no other lines to intersect with. There are certainly >still some infinite number of points. But here, Tony, we just go back to the same old merry go round. Do geometric figures just represent boundaries or are they some kind of material substance with material constituents? >>> After all there is a number of unresolved problems not >>> because they don't have any solution but simply because they are not >>> solved yet due to different obstacles. >>> But as long as we arrived to such entities as "point", "line", >>> "infinite set", "natural number", "real number", "irrational number" >>> etc. - as long that: do you believe that each of them there is one and >>> only one proper mechanically exhaustive definition to find - coming >>>from the very nature of these entities? >> >> Once again probably so. I just haven't spent a lot of time on those >> issues as yet - if I ever do. >> > >Oh, do! Well, Tony, in the beginning I set myself three more or less physical objectives: mechanical elucidation of Michelson-Morley, analytical origin of Planck's constant and related quantum effects, and Hubble's constant. And so far the pickings have been mighty slim. I only got into mathematical issues because mathematikers turn out to be a bunch of silver tongued cloaked empirics on such issues as transcendental numbers and curves in addition to SOAP operas. So the outlook is a little grim at the moment. Apart from Bob's one time agreement on the non existence of a real number line I really haven't had any success. >>> So once found we may expect >>> them universally correct, so even for some civilization from another >>> star they will be necessary either the same or wrong (so the said >>> civilization did not find the proper definition yet)? >> >> Sure. The truth of what I'm after is universal in scope and not just a >> particular or local truth in the sense it is demonstrably so. >> >> ~v~~ > >I have some ideas concerning universal truth, but you might not find >them quite tautologically regressive enough. Or, you might. Well, Tony, the demonstration of universal truth is my idea. It's not really open to debate or alternatives. So unless you have something specific you'd like to ask about or suggest is wrong with my approach or defective in my arguments of what is necessarily universally true and why, tautological mechanics in general, or the mechanical basis for Boolean logic, I'd have to say the issue is pretty much just what it is and where it stands at the moment. This isn't a guessing game any more. ~v~~ |