The Definition of Points
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From: Lester Zick on 20 Mar 2007 20:17 On 20 Mar 2007 12:39:37 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> wrote: >> > This caused discussion to grind to a halt until >> >a common language can be re-established. >> >> Apparently there is no common language between us. At least there >> never has been. Anytime you prefer to evade an issue you just >> translate it into some other terms which you feel more comfortable >> with. > >No, this is the process by which I attain communication. Well it doesn't seem to work. >It is a concept you are unfamiliar with, but here's >how it works: Some language is not known to your >listener. They ask "did you mean X?" and attempt to >paraphrase their understanding of what you said. If the >answer is "no", then you provide a restatement in >alternate terminology, paraphrasing yourself. So are we to understand you don't know generic English? Or only that you just prefer to pretend you don't so you can revise the meaning of the question in neomathspeak for disingenuous rhetorical purposes? I can readily appreciate the latter as disingenuity would seem to be the lingua franca of modern math but I find it hard to credit the former. >> >Is my guess wrong? Fine. I assume you know what >> >you meant (oops, there I go making assumptions again). >> >If so, then explain it. >> >> "Associated with points". > >That would not be "alternate terminology". No it would be the question I asked in the terms I asked it. If you find the language unfathomable I suggest you abandon English altogether and adopt another tongue. > One >mark of whether you in fact understand a concept, >including one of your own, is whether you are >capable of paraphrasing. Oh I'm perfeclty capable of paraphrasing as you should know. I just see no reason to satisfy your disingenuous rhetorical ambitions when the question itself is perfectly plain. >I find it instructive that of all the many, many >examples of phrases you have thrown out on this >newsgroup, the ones that make people say "what the >hell is that supposed to mean?" you have never to >my recollection been able to explain a single >one. And how exactly would you know whether I've been able to or not, Randy. Certainly I've substantiated my claims in generic language many times whereas your neomathspeak claims have gone without demonstrations of truth whatsoever. ~v~~
From: Lester Zick on 20 Mar 2007 20:29 On 20 Mar 2007 16:15:06 -0700, "hagman" <google(a)von-eitzen.de> wrote: >On 20 Mrz., 20:24, Lester Zick <dontbot...(a)nowhere.net> wrote: >> On 20 Mar 2007 03:11:41 -0700, "hagman" <goo...(a)von-eitzen.de> wrote: >> >> >On 20 Mrz., 00:30, Lester Zick <dontbot...(a)nowhere.net> wrote: >> >> >> > I'm saying that a >> >> >model is just a model. The properties of the model do not cause >> >> >the thing it's modeling to have those properties. >> >> >> Oh great. So now the model of a thing has properties which don't model >> >> the properties of the thing it's modeling. So why model it? >> >> >> ~v~~ >> >> >You are trying to walk the path in the wrong direction. >> >E.g. 0:={}, 1:={{}}, 2:= {{},{{}}}, ... >> >is a model of the naturals: The Peano axioms hold. >> >> However we don't have a model of straight lines except by naive >> assumption. > >Wrong. >The field of reals provides a very good model of a straight line, >also in the form of an affine subspace of a higher dimensional real >vector space. Please can the neomathspeak for a moment to explain how application of the Peano and suc( ) axioms and a resultant succession of integers and straight line segments, even if we allow their assumption, defines straight lines. I mean are we just supposed to assume the sequence of line segments is necessarily colinear and lies on a straight line? >> >However, in this model we have "0 is a set", which does not follow >> >from Peano axioms. >> >Thus the model has some additional properties. What's wrong with that? >> >> It isn't a model of what we wish to model. > >You made a general statement against models (or rather grossly >misunderstood >someone elses general statement about models). >Thus I am allowed to pull out any model I like. Go right ahead. I'd just like to see you pull out straight lines first. >> >However, the model shows that the Peno axioms are consistent (provided >> >the set theory we used to construct the model is). >> >> Consistency is only a prerequisite not a final objective for a model. >> > >On the contrary. Producing a model for a theory is the common way to >show that >the theory is consistent (provided the theory used to construct the >model >is consistent) Well you know, hagman, this is curious. I complain that consistency is not the final objective of a model. And your reply is to assure me that some models are consistent. Do you consider that responsive? >A: Let's consider a theory where 'lines' consist of 'points' and >'lines' determine 'points'... >B: Hey, isn't that nonsense? Can such a theory exist? >A: Of course. Take pairs of real numbers for points and certain sets >of such pairs as lines... >B: I see. Except you aren't even bothering to demonstrate the existence of numbers, lines, straight lines, etc. You're just claiming they're there. How can you take something that isn't there? Then how can you demonstrate they are there? Let's get real for a change shall we. ~v~~
From: Lester Zick on 20 Mar 2007 20:31 On Tue, 20 Mar 2007 14:02:17 -0400, Bob Kolker <nowhere(a)nowhere.com> wrote: >Tony Orlow wrote: > >> >> You know that's not what I mean. > >I do? Then what do you mean. > > >How do you measure the accuracy of the >> premises you use for your arguments? You check the results. That's the >> way it works in science, and that's the way t works in geometry. If some > >But not in math. The only thing that matters is that the conclusions >follow from the premises and the premises do not imply contradictions. >Matters of empirical true, as such, have no place in mathematics. Then how do you explain axioms, Bob? Are they demonstrated or are they empirically assumed? >Math is about what follows from assumptions, not true statements about >the world. Well certainly your math isn't about truth one way or the other, Bob. ~v~~
From: Lester Zick on 20 Mar 2007 20:37 On Tue, 20 Mar 2007 16:31:02 -0500, Wolf <ElLoboViejo(a)ruddy.moss> wrote: >Tony Orlow wrote: >> Virgil wrote: >[...] >>> There have even been blind geometers, such as Lev Pontryagin. >> >> They are geometric, even if not visual. If they are purely auditory, >> then those symbols are not likely to lead to much progress in spatial >> analysis. > > >Even sighted geometers have to leave their visual intuitions behind to >do geometry. Why? Because our visual system is prone to error and >illusion, is why. > >That's why the first thing we were taught in math class was not to trust >our eyes. You want to show that two triangles are congruent? Your >drawing will not prove it - only logic. And what's logic? Yeah, Wolf, what's logic indeed? > The >manipulation of symbols, is what. So if I just manipulate symbols I've done logic? Well that was certainly easy enough. And here I had no idea. > You don't have to be able to see those >symbols, nor what they refer to (if anything.) So if symbols refer to color on the one hand and ideas on the other one might come up with perfectly logical colorful ideas? Curiouser and curiouser I must say. Now I think I'm beginning to understand what's behind the looking glass. >Actually, being able to see the symbols may be a hindrance. I was taught >geometry both in Austria and in England. In England, they were still >using Euclid's notation: "the line AB", for example, instead of "the >line a". I found the use of capital letters to refer to both vertices >and lines less than transparent, shall we say. Then get a looking glass and look through it instead of at it. ~v~~
From: Lester Zick on 20 Mar 2007 20:50
On 20 Mar 2007 14:09:33 -0700, "Hero" <Hero.van.Jindelt(a)gmx.de> wrote: > Lester Zick wrote: > >> >> >And another question: is the trace, left by a movement, not part of >> >static geometry? It is an invariant of dynamic geometry. >> >> You know, Hero, there are some extraordinarily subtle considerations >> involved here which need to be considered for any exact analysis of >> static rac versus dynamic non rac construction methods. However I'd >> rather not get into them just at present because they really aren't >> germane to the basic topics we're considering here at the moment. >> > >It's a pity. Well as I mentioned above there are some extraordinarily subtle considerations here which I prefer not to get into at the moment mainly because I'm not really confident I fully understand them myself. So anything I said would be more speculative in nature than I'd feel comfortable speculating on with such a hostile audience. > (just one more comment, when You can get to exact >algebraic lengths on the real line and because of an open window the >temperature did change a bit and with it Your compass, so You have a >chance of an exact transcendental). Well you always have chance encounters with points which are on a straight line but that doesn't mean there is any point on a straight line which corresponds with a transcendental, the proof being that every point on a straight line corresponds with some rac construction and no straight line rac construction corresponds to transcendentals. >Okay, back to points and lines. There's topology, just the simple >beginning: > A space (mathematical) is a set with structure. >A point is a geometrical space without geometrical structure, but it >can give structure to geometry. >Think of a vertex or a center and so forth. >A line is made up of points and sets of points ( the open intervalls >between each two points),which obey three topological rules. Hero, exactly what makes you think the foregoing observations are true? >What i learned recently: >With adding a point to an open (open in standard topology) flexible >surface one can enclose a solid, with adding a point to an open line >one can enclose a figure, and two points are the boundary of an >intervall on a line. But there is no point at or beyond infinity. All very interesting but I still have no idea why any of this is supposed to be true. ~v~~ |