The Definition of Points
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From: Lester Zick on 15 Mar 2007 13:11 On 14 Mar 2007 18:57:28 -0700, "Eric Gisse" <jowr.pi(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, "Eric Gisse" <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, "Eric Gisse" <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, "Eric Gisse" <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. ~v~~
From: Eckard Blumschein on 15 Mar 2007 13:46 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
From: Bob Kolker on 15 Mar 2007 14:07 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 Bob Kolker
From: Lester Zick on 15 Mar 2007 14:24 On Thu, 15 Mar 2007 02:40:11 GMT, Sam Wormley <swormley1(a)mchsi.com> wrote: >Lester Zick wrote: > >> Straight lines are derivatives of curves. At least according to Newton >> and his method of drawing tangents. Tell Euler et al. they can stop >> rolling. Euler couldn't even get the definition of angular mechanics >> right. >> >> > > > Hey Lester > Line > http://mathworld.wolfram.com/Line.html > > "A line is uniquely determined by two points, and the line passing > through points A and B". Well technically, Sam, I should think two points determine a straight line segment not a straight line.The writer above seems to think there are just mystery assumptions called lines and points somewhere out there and points determine a particular line. In other words he just seems to consider straight lines and points givens without derivation. My idea for straight lines depends on their derivation from curves. Perhaps you can appreciate the problem from the perspective of the Peano axioms. There we have a series of integers derived through the suc( ) axiom and a succession of points associated with them. And the points define a succession of straight line segments. However I see no reason to assume those straight line segments are colinear and form a single straight line as is commonly assumed. > "A line is a straight one-dimensional figure having no thickness and > extending infinitely in both directions. A line is sometimes called > a straight line or, more archaically, a right line (Casey 1893), to > emphasize that it has no "wiggles" anywhere along its length. While > lines are intrinsically one-dimensional objects, they may be embedded > in higher dimensional spaces". I don't agree with the notion that lines and straight lines mean the same thing, Sam, mainly because we're then at a loss to account for curves. In informal terms I suppose there's no harm done referring to straight lines as just lines. But in formal terms we have to consider curves in addition to straight lines and to consider the properties of each in relation to the other. As I've mentioned to Bob Kolker in the past given curves we can derive straight lines through tangency but given just straight lines we can't go the other way and determine curves from tangents alone without factors pertinent to the calculus, derivation, and integration. That's what makes the whole problem intractable if we just proceed with straight lines and segments by assumption as neomathematikers do. Further if we then define points by the intersection of lines we must also ask to which line a particular point belongs.Obviously it belongs to both intersecting lines and is a property of their intersection and is not a constituent of either line in itself. At least that's my general take on the subject of lines and points. But I appreciate your contribution nonetheless. I just don't consider the problem quite as trivial and frivolous as neomathematikers appear willing to assume. ~v~~
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~~ |