The Definition of Points
in [Math]
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From: Lester Zick on 15 Mar 2007 15:07 On Thu, 15 Mar 2007 08:09:37 -0400, Bob Kolker <nowhere(a)nowhere.com> wrote: >Lester Zick wrote: > >> >> Actually I'm interested in whether vectors exist and have >> constituents. > >Yes they do, in the mathematical sense. They lead to a successful >description of forces for one thing. The constituents of a vector are >length and direction. Actually, Bob, my comment here was completely facetious because when I explained my solution to the correction of Michelson-Morley, Draper more or less just lapsed into an intellectual coma and maintained there was no vector analysis pertinent to Michelson-Morley. >> Neither. The end points contain the line segment. That's how the line >> segment is defined. > >That is admirably correct. And given the end points of a segment one can >readily define the set of points that make up the line determined by the >end points of the segment. Learn some analytic geometry to see how. Well, Bob, this is only partly correct. Defining end points contain the line and not vice versa. So there's no reason to suggest lines are made up of points nor SOAP's. Consider the problem this way. Let's say you have a SOAP. Does that define a straight line? We can infer that straight line segments are defined between points in the SOAP as is commonly assumed. But we cannot then just infer that the series of straight line segments defined by any SOAP lie together on any one common straight line. It's the same problem we face with the Peano and suc( ) axioms and a putative real number line. Those axioms may or may not generate series of integers and points associated with those integers but nothing can guarantee the series of successive straight line segments are colinear and align together on any common straight line real or otherwise. >> Of course another way to answer this is to ask what defines the line >> segment to begin with. > >A pair of points. Obviously. This does not however justify the inference that the line itself is made up of constituent points. >>> and ask what the limit >>>of the line segment is. >> >> >> When it gets to zero do be sure to let us know. > >I see you are channeling Bishop Berkeley again. All of hist objections >have been answered by the theory of hyperreal numbers on which >non-standard analysis is based. Berkeley raised cogent objections to >Newton and Leibniz which were finally and complete answered in the late >1950's by Abraham Robinson. I think you mistake my intent here, Bob. My complaint is not that there is no limit or that the limit is never reached by infinitesimal approximation. It's merely what happens when the limit is reached. When the limit is reached the line segment becomes zero, disappears, and the points themselves disappear as distinct points in addition. >By the way, if lines (or other curves) do not consist of the points on >them, what do they consist of? Well I don't quite agree that straight lines are just a garden variety curve because all straight lines and segments are a uniquely common shape whereas curves are not. Speaking historically as you did above I'm inclined to think the whole problem is an atomistic anachronism commonly associated I believe with Democritus. I'm not sure it's a question that can even properly be posed without necessarily assuming there are constituents to begin with. I see straight lines as derived through tangency to curves and points as the intersection of curves and tangents to those curves or as the intersection of lines of any kind. However I don't believe it makes sense to get into hypothetical constituents for any of these figures because we have no indication there is any constituency involved. Those figures just share intersections with one another but the intersections themselves have certain properties not evident in the figures themselves. Of course this suggests we need to revisit the axiomatic foundations of modern math but I suppose that's grist for another windmill to tilt at. ~v~~
From: Ralf Bader on 15 Mar 2007 17:02 Bob Kolker wrote: > Eckard Blumschein wrote: >> >> Rational numbers are countable because all of them are different from >> each other. > > All real numbers are pairwise distinct but they constitute an > uncountable set. > >> 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. > > This is nonsense. Have you ever heard of a convergent series? > > 9/10 + 9/100 + etc converges to 1.0 Be warned - if you try to explain anything to Mr. Blumschein it will drive you crazy. Ralf
From: Lester Zick on 15 Mar 2007 18:15 On 14 Mar 2007 18:08:47 -0700, "Aaron" <anodide(a)hotmail.com> wrote: >On Mar 13, 3:13 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: >> On Tue, 13 Mar 2007 18:43:09 GMT, Sam Wormley <sworml...(a)mchsi.com> >> wrote: >> >> >> >> >> >> >Lester Zick 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~~ >> >> > Point >> > http://mathworld.wolfram.com/Point.html >> >> > A point 0-dimensional mathematical object, which can be specified in >> > n-dimensional space using n coordinates. Although the notion of a point >> > is intuitively rather clear, the mathematical machinery used to deal >> > with points and point-like objects can be surprisingly slippery. This >> > difficulty was encountered by none other than Euclid himself who, in >> > his Elements, gave the vague definition of a point as "that which has >> > no part." >> >> Sure, Sam. I understand that there are things we call points which >> have no exhaustive definition. However my point is the contention of >> mathematikers that lines are made up of points is untenable if lines >> are required to define points through their intersection.It's vacuous. > >I'm like not getting it here. Are we just talking about graphing of >functions? > >Isn't this splitting hairs or am I missing something? > >A point is like a spot and has the same number of information elements >as there are dimensions in the space it models, right? > >A line is then all the spots from one spot to another. > >If two lines share a spot, big deal. They ahare a spot. > >It's just numbers in a co-ordinate system, which in tern is an >abstract device to count numbers and model things we see using math >functions. > >It it really more complicated than that? It's really more complicated than that when you begin to speculate on the composition of lines and other geometric figures. As far as I can tell geomtric figures are not composed of other more elementary figures. They just are what they are.It's relatively easy to visualize geometric figures as composed of other more elementrary constituent figures but I think badly misrepresents what we are dealing with. Then you wind up trying to deal with all sorts of nonsense from real number lines to curves composed of straight line segments defined by points. ~v~~
From: Eric Gisse on 15 Mar 2007 18:21 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. 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. > > ~v~~
From: Lester Zick on 15 Mar 2007 18:22
On Wed, 14 Mar 2007 21:51:01 -0400, Bob Kolker <nowhere(a)nowhere.com> wrote: >Lester Zick wrote: >> >> >> Obviously. That's why I became a mathematician. >You are not now, nor were you ever a mathematician. Nor will you ever be >one unless you get a brain transplant. Come, come, Bob. I went trolling and suckered you into just such a statement. What I'm definitely not is a SOAP speculator. I do though know a little something about demonstrations of truth and such which seem to escape yourself and most self styled modern mathematikers. >Your postings indicate not only a profound ignorance of things >mathetmicatical but a definite lack of talent for and competence in >mathematics. Well you seem to confuse SOAP speculations with mathematics, Bob. Two entirely different things. When mathematikers want to score points they do it with SOAP speculation and terminology and insist others do likewise. Then they turn right around and pretend what they're doing is mathematics instead of rank speculation and assumptions of truth. ~v~~ |