The Definition of Points
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From: Virgil on 21 Mar 2007 14:33 In article <46015390(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > On 21 Mar, 05:47, Tony Orlow <t...(a)lightlink.com> wrote: > >> step...(a)nomail.com wrote: > >>> In sci.math Tony Orlow <t...(a)lightlink.com> wrote: > >>>> PD wrote: > >>>>> No one says a set of points IS in fact the constitution of physical > >>>>> object. > >>>>> Whether it is rightly the constitution of a mentally formed object > >>>>> (such as a geometric object), that seems to be an issue of arbitration > >>>>> and convention, not of truth. Is the concept of "blue" a correct one? > >>>>> PD > >>>> The truth of the "convention" of considering higher geometric objects to > >>>> be "sets" of points is ascertained by the conclusions one can draw from > >>>> that consideration, which are rather limited. > >>> How is it limited Tony? Consider points in a plane, where each > >>> point is identified by a pair of real numbers. The set of > >>> points { (x,y) | (x-3)^2+(y+4)^2=10 } describes a geometric object. > >> That's a very nice circle, Stephen, very nice.... > >> > >>> In what way is this description "limited"? Can you provide a > >>> better description, and explain how it overcomes those limitations? > >> There is no correlation between length and number of points > > > > Oh. And I suppose there *is* such a correlation in "real" geometry? > > > > -- > > mike. > > > > In my geometry |{x in R: 0<x<=1}| = Big'Un (or oo). Which is one of many reasons why your geometry sucks.
From: Virgil on 21 Mar 2007 14:35 In article <46015463(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Bob Kolker wrote: > > Tony Orlow wrote: > >> > >> There is no correlation between length and number of points, because > >> there is no workable infinite or infinitesimal units. Allow oo points > >> per unit length, oo^2 per square unit area, etc, in line with the > >> calculus. Nuthin' big. Jes' give points a size. :) > > > > Points (taken individually or in countable bunches) have measure zero. > > > > Bob Kolekr > > > > They certainly have measure less than any finite measure, which in > standard mathematics means 0, however, extensions which allow an > infinite unit may define the size of the point as the infinitesimal > reciprocal of that unit. Given that, the sum of this infinite number of > infinitesimal atoms gives the actual measure of the object. :) The only infinitesimals evident here have to do with TO's understanding.
From: Virgil on 21 Mar 2007 14:39 In article <4601555c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: > > In sci.math Tony Orlow <tony(a)lightlink.com> wrote: > >> stephen(a)nomail.com wrote: > >>> In sci.math Tony Orlow <tony(a)lightlink.com> wrote: > >>>> PD wrote: > >>>>> No one says a set of points IS in fact the constitution of physical > >>>>> object. > >>>>> Whether it is rightly the constitution of a mentally formed object > >>>>> (such as a geometric object), that seems to be an issue of arbitration > >>>>> and convention, not of truth. Is the concept of "blue" a correct one? > >>>>> > >>>>> PD > >>>>> > >>>> The truth of the "convention" of considering higher geometric objects to > >>>> be "sets" of points is ascertained by the conclusions one can draw from > >>>> that consideration, which are rather limited. > >>> How is it limited Tony? Consider points in a plane, where each > >>> point is identified by a pair of real numbers. The set of > >>> points { (x,y) | (x-3)^2+(y+4)^2=10 } describes a geometric object. > > > >> That's a very nice circle, Stephen, very nice.... > > > > Yes, it is a circle. You knew exactly what this supposedly > > "limited" description was supposed to be. > > > >>> In what way is this description "limited"? Can you provide a > >>> better description, and explain how it overcomes those limitations? > > > >> There is no correlation between length and number of points, because > >> there is no workable infinite or infinitesimal units. Allow oo points > >> per unit length, oo^2 per square unit area, etc, in line with the > >> calculus. Nuthin' big. Jes' give points a size. :) > > > > How is that a limitation? You knew exactly what shape the > > set of points described. There is no feature of the circle > > that cannot be determined by the above description. There is > > no need to correlate length and number of points. Neither > > Euclid or Hilbert ever did that. > > Gee, I guess it's a novel idea, then. That might make it good, and not > necessarily bad. Hilbert also didn't bother to generalize his axioms to > include anything but Euclidean space, which is lazy. He's got too many > axioms, and they do too little. :) > > There's no reason the circumference of the unit circle can't be > considered to have 2*pi*oo points. There is equally no reason why it can't be considered to have log(pi)*oo points either. > > > > > So where is your better description, and where is the explanation > > as to why it is better? What more can you say about a circle > > centered at (3,-4) with a radius of sqrt(10)? > > It's got 2*pi*sqrt(1)*oo points, as a set, which is greater than the > number of points in the unit interval. :) By what measure? Certainly not by cardinality.
From: Lester Zick on 21 Mar 2007 14:48 On Wed, 21 Mar 2007 09:44:02 -0500, Wolf <ElLoboViejo(a)ruddy.moss> wrote: >Tony Orlow wrote: >[...]I'm not putting down axiomatization persay. >[...] > >I'm about to pick a nit: > >It's not "persay", it's "per se." Latin. Means "by means of itself." The >usual English rendering is "in and of itself." > >OK? ROTFL - Wolf - More than OK. As a matter of fact way more than OK. But I consider Tony's malapropism may have a kernel of folksy wisdom and Lincolnesque truth to it that you may have escaped your notice. For if we allow that the experiments performed by empiricists are a benefit to all mankind, we must also allow that commentaries on the significance of those experiments are what distinguish empirics from empiricists. Thus whereas empirical experiments are true "per se" the commentaries on the significance of those experiments by empirics are only true "per say". And of course I fully intend to canonize Tony's malapropism in the course of cannonizing empirics, mathematikers, and others who insist on abusing naive assumptions of truth only per say instead of per se. For my own part I'm trying to enliven conversational discourse with a variety of mixed metaphors and yogiisms (grist for another windmill to tilt at, pull the wool over sheeps clothing, et al.). However I'm nonetheless forced to ask how you managed to arrive at your insight here? I mean was it logical and did you just rattle some bones around or other symbols detached from significance? Or did you arrive at it through finite tautological regression of word meanings to self contradictory alternatives as the rest of us plebeians do? In any event much grass to both you and Tony and keep up the good work. The two of you have made my day. ~v~~
From: Lester Zick on 21 Mar 2007 15:09
On Wed, 21 Mar 2007 00:06:40 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Mon, 19 Mar 2007 18:04:27 -0500, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>>> No one says a set of points IS in fact the constitution of physical >>>> object. >>>> Whether it is rightly the constitution of a mentally formed object >>>> (such as a geometric object), that seems to be an issue of arbitration >>>> and convention, not of truth. Is the concept of "blue" a correct one? >>>> >>>> PD >>>> >>> The truth of the "convention" of considering higher geometric objects to >>> be "sets" of points is ascertained by the conclusions one can draw from >>> that consideration, which are rather limited. >>> >>> "blue" is not a statement with a truth value of any sort, without a >>> context or parameter. blue(sky) may or may not be true. >> >> I disagree here, Tony. "Blue" is a predicate and like any other >> predicate or predicate combination it is either true or not true. > >No, Lester. I hate to put it this way, but here, you're wrong. No need for any regret, Tony. I certainly don't mind being called on errors or even called on non errors when there's something you think is wrong. But we shall just have to see what we shall see here, Tony. > "Blue" is >a descriptor for an object, a physical object as perceived by a human, >if "blue" is taken to mean the color. It's an attribute that some >humanly visible object may or may not have. The "truth" of "blue" >depends entirely on what it is attributed to. Blue(moon) is rarely true. >Blue(sky) is often true in Arizona, and not so often around here. So "blue" is not a predicate, Tony, as in "it is blue"? And this predicate cannot be true or false? >One can assign an attribute to an object as a function, like I just did. >One can also use a function to include an object in a set which is >described by an attribute, like sky(blue) or moon(blue) - "this object >is a member of that set". The object alone also doesn't constitute an >entire statement. "Sky" and "moon" do not have truth values. Blue(sky) >might be true less than 50% of the time, and blue(moon) less than 1%, >but "blue" and "sky" and "moon" are never true or false, because that >sentence no verb. There is no "is" there, eh, what? :) Tony, I think you're confusing the supposed truth or falsity of a single predicate with the supposed truth or falsity of an abstract proposition. For that matter we can always make a proposition out of a single predicate by saying "it is blue (or whatever)". >> However the difference is that a single predicate such as "blue" >> cannot be abstractly analyzed for truth in the context of other >> predicates. For example we could not analyze "illogical" abstractly in >> the context of "sky" unless we had both predicates together as in >> "illogical sky". But that doesn't mean single isolated predicates are >> not either true or false. > >But, it does. In order for there to be a statement with a logical truth >value, I'm not sure what the significance of "truth value" is, Tony. I deal with truth and finite tautological regressions to self contradictory alternatives as the basis for truth and not merely the assignment and evaluation of truth values whatever that may mean. > there must be buried within it a logical implication, "this >implies that". The only implication for "blue" alone is that such a >thing as "blue" exists. Does "florange" exist, by virtue of the fact >that I just used the word? > >If "blue" and "fast" are predicates, is "blue fast" a predicate? Does >that sound wrong? How about "chicken porch"? Is that true or false? Well all predicates and predicate combinations are what they are, Tony. And they can range from true to false to self contradictory. I don't see the problem with that whether we're considering things literally or even metaphorically for that matter. It's all one system of predicates and predicate combinations which are either true or false in combinations. Some predicate combinations appear silly because we already understand the combinations are false and self contradictory. But that isn't to say the predicates themselves cannot be true or false alone or in other combinations in given instances. >The fast chicken on the blue porch, don't you agree? I see no >contradiction in that.... Nor do I. Nor do I see any problem in evaluating the truth or falsity of "it is blue". No special mystery there that I can see. ~v~~ |