The Definition of Points
in [Math]
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From: Lester Zick on 17 Apr 2007 18:18 On Tue, 17 Apr 2007 12:20:59 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Fri, 13 Apr 2007 14:33:20 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> Lester Zick wrote: >>>> On Thu, 12 Apr 2007 14:35:36 -0400, Tony Orlow <tony(a)lightlink.com> >>>> wrote: >>>> >>>>> Lester Zick wrote: >>>>>> On Sat, 31 Mar 2007 20:58:31 -0500, Tony Orlow <tony(a)lightlink.com> >>>>>> wrote: >>>>>> >>>>>>> How many arguments do true() and false() take? Zero? (sigh) >>>>>>> Well, there they are. Zero-place operators for your dining pleasure. >>>>>> Or negative place operators, or imaginary place operators, or maybe >>>>>> even infinite and infinitesimal operators. I'd say the field's pretty >>>>>> wide open when all you're doing is guessing and making assumptions of >>>>>> truth. Pretty much whatever you'd want I expect.Don't let me stop you. >>>>>> >>>>>> ~v~~ >>>>> Okay, so if there are no parameters to the function, you would like to >>>>> say there's an imaginary, or real, or natural, or whatever kind of >>>>> parameter, that doesn't matter? Oy! It doesn't matter. true() and >>>>> false() take no parameters at all, and return a logical truth value. >>>>> They are logical functions, like not(x), or or(x,y) and and(x,y). Not >>>>> like not(). That requires a logical parameter to the function. >>>> Tony, you might just as well be making all this up as you go along >>>> according to what seems reasonable to you. My point was that you have >>>> no demonstration any of these characteristics in terms of one another >>>> which proves or disproves any of these properties in mechanical terms >>>> starting right at the beginning with the ideas of true and false. >>>> >>>> ~v~~ >>> Sorry, Lester, but that's an outright lie. I clearly laid it out for >>> you, starting with only true and false, demonstrating how not(x) is the >>> only 1-place operator besides x, true and false, and how the 2-place >>> operators follow. For someone who claims to want mechanical ground-up >>> derivations of truth, you certainly seem unappreciative. >> >> Only because you're not doing a ground up mechanical derivation of >> true or false. You're just telling me how you employ the terms true >> and false in particular contexts whereas what I'm interested in is how >> true and false are defined in mechanically reduced exhaustive terms. >> What you clearly laid out are the uses of true and false with respect >> to one another once established. But you haven't done anything to >> establish true and false themselves in mechanically exhaustive terms. >> >> ~v~~ > >Again, define "mechanics". Tony, time for you to do a little work for yourself. I've already gone through this. You describe for me the mechanics of using binary truth values and I explain to you I'm interested in truth not binary truth values and how to ascertain truth in mechanical terms initially and not how to work with truth values mechanically once ascertained. By the way what is the truth value of "square triangles" and how does that differ from the truth value of "blue squares" and how do you know the difference? ~v~~
From: Lester Zick on 17 Apr 2007 18:19 On Tue, 17 Apr 2007 12:21:59 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Fri, 13 Apr 2007 16:52:21 +0000 (UTC), stephen(a)nomail.com wrote: >> >>> In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >>>> Lester Zick wrote: >>>>> On Mon, 2 Apr 2007 16:12:46 +0000 (UTC), stephen(a)nomail.com wrote: >>>>> >>>>>>> It is not true that the set of consecutive naturals starting at 1 with >>>>>>> cardinality x has largest element x. A set of consecutive naturals >>>>>>> starting at 1 need not have a largest element at all. >>>>>> To be fair to Tony, he said "size", not "cardinality". If Tony wishes to define >>>>>> "size" such that set of consecutive naturals starting at 1 with size x has a >>>>>> largest element x, he can, but an immediate consequence of that definition >>>>>> is that N does not have a size. >>>>> Is that true? >>>>> >>>>> ~v~~ >>>> Yes, Lester, Stephen is exactly right. I am very happy to see this >>>> response. It follows from the assumptions. Axioms have merit, but >>>> deserve periodic review. >>>> 01oo >>> Everything follows from the assumptions and definitions. >> >> And since definitions are considered neither true nor false everything >> follows from raw assumptions which are considered neither true nor >> false. >> >> ~v~~ > >Oh come on. Assumptions are considered true for the sake of the argument >at hand. That's what an assumption IS. So are "square triangles" or "blue squares" considered true for the sake of the argument at hand? Strange argument I must say. ~v~~
From: Lester Zick on 17 Apr 2007 18:20 On Tue, 17 Apr 2007 12:51:28 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Fri, 13 Apr 2007 13:42:06 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> Lester Zick wrote: >>>> On Thu, 12 Apr 2007 14:23:04 -0400, Tony Orlow <tony(a)lightlink.com> >>>> wrote: >>>> >>>>> Lester Zick wrote: >>>>>> On Sat, 31 Mar 2007 16:18:16 -0400, Bob Kolker <nowhere(a)nowhere.com> >>>>>> wrote: >>>>>> >>>>>>> Lester Zick wrote: >>>>>>> >>>>>>>> Mathematikers still can't say what an infinity is, Bob, and when they >>>>>>>> try to they're just guessing anyway. So I suppose if we were to take >>>>>>>> your claim literally we would just have to conclude that what made >>>>>>>> physics possible was guessing and not mathematics at all. >>>>>>> Not true. Transfite cardinality is well defined. >>>>>> I didn't say it wasn't, Bob. You can do all the transfinite zen you >>>>>> like. I said "infinity". >>>>>> >>>>>>> In projective geometry points at infinity are well defined (use >>>>>>> homogeneous coordinates). >>>>>> That's nice, Bob. >>>>>> >>>>>>> You are batting 0 for n, as usual. >>>>>> Considerably higher than second guessers. >>>>>> >>>>>> ~v~~ >>>>> That's okay. 0 for 0 is 100%!!! :) >>>> Not exactly, Tony. 0/0 would have to be evaluated under L'Hospital's >>>> rule. >>>> >>>> ~v~~ >>> Well, you put something together that one can take a derivative of, and >>> let's see what happens with that. >> >> Or let's see you put something together that you can't take the >> deriviative of and let's see how you managed to do it. >> >> ~v~~ > >Okay. What's the derivative of 0? 46 ~v~~
From: Marcus on 17 Apr 2007 18:25 On Mar 19, 7:20 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: > On Mon, 19 Mar 2007 13:07:30 EDT, "G.E. Ivey" > > > > <george.i...(a)gallaudet.edu> wrote: > >> On Tue, 13 Mar 2007 20:24:01 +0100, "SucMucPaProlij" > >> <mrjohnpauldike2...(a)hotmail.com> wrote: > > >> >"PD" <TheDraperFam...(a)gmail.com> wrote in message > >> >news:1173810896.000941.35900(a)q40g2000cwq.googlegroups > >> .com... > >> >> On Mar 13, 12:52 pm, 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~~ > > >> >> Interestingly, the dictionary of the English > >> language is also > >> >> circular, where the definitions of each and every > >> single word in the > >> >> dictionary is composed of other words also defined > >> in the dictionary. > >> >> Thus, it is possible to find a circular route from > >> any word defined in > >> >> the dictionary, through words in the definition, > >> back to the original > >> >> word to be defined. > > >> >> That being said, perhaps it is in your best > >> interest to find a way to > >> >> write a dictionary that eradicates this > >> circularity. That way, when > >> >> you use the words "peculiar" and "definitional", > >> we will have a priori > >> >> definitions of those terms that are noncircular, > >> and from which the > >> >> unambiguous meaning of what you write can be > >> obtained. > > >> >> PD > > >> >hahahahahahaha good point (or "intersections of > >> lines") > > >> And it might be an even better point if it weren't > >> used to justify > >> mathematikers' claims that lines are made up of > >> points. > > >> ~v~~ > > > Could you give a reference in which a mathematician (not a > > high-school geometry book- I would accept a college geometry book) > > states that lines are made up of points? In every text I have seen > >"points" and "lines" are undefined terms. > > That's probably why you never ever see those terms used in relation to > any another because they're undefined except by predicates specified > in relation to predicates of other objects which don't define them. > > > I believe > > it was Hilbert who said that "If you replace points and lines by > > beer steins and tables, every statement should still be true." > > The difficulty is that the statements "lines are made up of points" > and "the intersection of lines" defines or determines points are a > definitive circular regression. I don't care whether Hilbert liked the > idea or not. If he proclaimed beer steins and tables are undefined but > tables define beer steins and tables are made up of beer steins the > problem is identical. It's the logic which defines tables and beer > steins in relation to one another and it's the logic that's definitive > and definitively circular. > > As for the contention that "lines are made up of points" I got that > from Bob Kolker and I kinda like think he made that up from some > notion that a line is the set of all points on a line. Pretty slippery > but there it is. If you disagree then I suggest you take it up with > him. I don't really care as long as the logic isn't circular and you > don't try to claim that objects which have specific relations with > other objects are not claimed to be undefined by Hilbert or whoever. > > (By the way I would appreciate it if you could keep your line length > around 60 or so.) > > ~v~~ Can you answer Ivey's challenge? If not, then it seems that your so- called "problem" may not be a problem after all. M
From: Lester Zick on 17 Apr 2007 18:38
On Tue, 17 Apr 2007 13:12:57 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Fri, 13 Apr 2007 16:11:22 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>>>>>>>>> Constant linear velocity in combination with transverse acceleration. >>>>>>>>>> >>>>>>>>>> ~v~~ >>>>>>>>> Constant transverse acceleration? >> >>>>>>>> What did I say, Tony? Constant linear velocity in combination with >>>>>>>> transverse acceleration? Or constant transverse acceleration? I mean >>>>>>>> my reply is right there above yours. >>>>>>>> >>>>>>>> ~v~~ >>>>>>> If the transverse acceleration varies, then you do not have a circle. >>>>>> Of course not. You do however have a curve. >>>>>> >>>>>> ~v~~ >>>>> I thought you considered the transverse acceleration to vary >>>>> infinitesimally, but that was a while back... >>>> Still do, Tony. How does that affect whether you have a curve or not? >>>> Transverse a produces finite transverse v which produces infinitesimal >>>> dr which "curves" the constant linear v infinitesimally. >>>> >>>> ~v~~ >>> Varying is the opposite of being constant. Checkiddout! >> >> I don't doubt "varying" is not "constant". So what? The result of >> "constant" velocity and "varying" transverse acceleration is still a >> curve. >> >> ~v~~ > >I asked about CONSTANT transverse acceleration. Oy! So what? Constant transverse acceleration produces a curve. So does non constant transverse acceleration. >More exactly, linearly proportional velocity and transverse acceleration >produce the circle. It can speed up and slow down, as long as it changes >direction at a rate in proportion with its change in velocity. Close >your eyes, and watch.... I don't understand what the problem is here, Tony. I don't understand the qualification "linearly proportional" is intended for. Anyway what is the purpose of your question? ~v~~ |