The Definition of Points
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From: PD on 22 Mar 2007 09:24 On Mar 21, 6:00 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: > > Well I can tell as soon as boobirds like Stephen come out the cause is > hopeless. At least I'm witty. Just keep telling yourself that, Lester. Oh, and perhaps you might ask yourself why you are exercising your self-acclaimed wit on a *science* group, where wit is not particularly of value. Wouldn't you be having more fun at rec.humor.arent.I.clever or alt.witty.perceived.self? PD
From: Mike Kelly on 22 Mar 2007 10:59 On 22 Mar, 13:03, Tony Orlow <t...(a)lightlink.com> wrote: > Mike Kelly wrote: > > On 22 Mar, 03:28, Tony Orlow <t...(a)lightlink.com> wrote: > >> Mike Kelly wrote: > >>> On 21 Mar, 19:20, Tony Orlow <t...(a)lightlink.com> wrote: > >>>> step...(a)nomail.com wrote: > >>>>> In sci.math Tony Orlow <t...(a)lightlink.com> wrote: > >>>>>> step...(a)nomail.com wrote: > >>>>>>> In sci.math 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.... > >>>>>>> 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. :) > >>>>> Then you should not be complaining about the "set of points" approach > >>>>> to geometry and instead should be complaining about all prior approaches > >>>>> to geometry. Apparently they are all "limited" to you. Of course > >>>>> you cannot identify any actual limitation, but that is par for the course. > >>>> I wouldn't say geometry is perfected yet. > >>> And yet you remain incapable of stating what these "limitations" are. > >>> Why is that? Could it be because you can't actually think of any? > >>> -- > >>> mike. > >> I already stated that the divorce between infinite set size and measure > >> of infinite sets of points is a limitation, and indicated a remedy, but > >> I don't expect you to grok that this time any better than in the past. > >> Keep on strugglin'.... > > >> tony. > > > What theorems can't be proved with current axiomatisations of geometry > > but can be with the addition of axiom "there are oo points in a unit > > interval"? Anything more interesting than "there are 2*oo points in an > > interval of length 2"? > > > -- > > mike. > > Think "Continuum Hypothesis". If aleph_1 is the size of the set of > finite reals, is aleph_1/aleph_0 the size of the set of reals in the > unit interval? Is that between aleph_0 and aleph_1? Uh huh. Firstly, the continuum hypothesis is nothing to do with geometry. Secondly, division is not defined for infinite cardinal numbers. -- mike.
From: Tony Orlow on 22 Mar 2007 12:38 Mike Kelly wrote: > On 22 Mar, 13:03, Tony Orlow <t...(a)lightlink.com> wrote: >> Mike Kelly wrote: >>> On 22 Mar, 03:28, Tony Orlow <t...(a)lightlink.com> wrote: >>>> Mike Kelly wrote: >>>>> On 21 Mar, 19:20, Tony Orlow <t...(a)lightlink.com> wrote: >>>>>> step...(a)nomail.com wrote: >>>>>>> In sci.math Tony Orlow <t...(a)lightlink.com> wrote: >>>>>>>> step...(a)nomail.com wrote: >>>>>>>>> In sci.math 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.... >>>>>>>>> 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. :) >>>>>>> Then you should not be complaining about the "set of points" approach >>>>>>> to geometry and instead should be complaining about all prior approaches >>>>>>> to geometry. Apparently they are all "limited" to you. Of course >>>>>>> you cannot identify any actual limitation, but that is par for the course. >>>>>> I wouldn't say geometry is perfected yet. >>>>> And yet you remain incapable of stating what these "limitations" are. >>>>> Why is that? Could it be because you can't actually think of any? >>>>> -- >>>>> mike. >>>> I already stated that the divorce between infinite set size and measure >>>> of infinite sets of points is a limitation, and indicated a remedy, but >>>> I don't expect you to grok that this time any better than in the past. >>>> Keep on strugglin'.... >>>> tony. >>> What theorems can't be proved with current axiomatisations of geometry >>> but can be with the addition of axiom "there are oo points in a unit >>> interval"? Anything more interesting than "there are 2*oo points in an >>> interval of length 2"? >>> -- >>> mike. >> Think "Continuum Hypothesis". If aleph_1 is the size of the set of >> finite reals, is aleph_1/aleph_0 the size of the set of reals in the >> unit interval? Is that between aleph_0 and aleph_1? Uh huh. > > Firstly, the continuum hypothesis is nothing to do with geometry. It does, if sets are combined with measure and a geometrical representation of the question considered. Is half an infinity less than itself? Geometrically, yes, the reals in (0,1] are half the reals in (0,2]. > > Secondly, division is not defined for infinite cardinal numbers. I'm not interested in cardinality, but a richer system of infinities, thanks. > > -- > mike. > tony.
From: Mike Kelly on 22 Mar 2007 12:57 On 22 Mar, 16:38, Tony Orlow <t...(a)lightlink.com> wrote: > Mike Kelly wrote: > > On 22 Mar, 13:03, Tony Orlow <t...(a)lightlink.com> wrote: > >> Mike Kelly wrote: > >>> On 22 Mar, 03:28, Tony Orlow <t...(a)lightlink.com> wrote: > >>>> Mike Kelly wrote: > >>>>> On 21 Mar, 19:20, Tony Orlow <t...(a)lightlink.com> wrote: > >>>>>> step...(a)nomail.com wrote: > >>>>>>> In sci.math Tony Orlow <t...(a)lightlink.com> wrote: > >>>>>>>> step...(a)nomail.com wrote: > >>>>>>>>> In sci.math 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.... > >>>>>>>>> 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. :) > >>>>>>> Then you should not be complaining about the "set of points" approach > >>>>>>> to geometry and instead should be complaining about all prior approaches > >>>>>>> to geometry. Apparently they are all "limited" to you. Of course > >>>>>>> you cannot identify any actual limitation, but that is par for the course. > >>>>>> I wouldn't say geometry is perfected yet. > >>>>> And yet you remain incapable of stating what these "limitations" are. > >>>>> Why is that? Could it be because you can't actually think of any? > >>>>> -- > >>>>> mike. > >>>> I already stated that the divorce between infinite set size and measure > >>>> of infinite sets of points is a limitation, and indicated a remedy, but > >>>> I don't expect you to grok that this time any better than in the past. > >>>> Keep on strugglin'.... > >>>> tony. > >>> What theorems can't be proved with current axiomatisations of geometry > >>> but can be with the addition of axiom "there are oo points in a unit > >>> interval"? Anything more interesting than "there are 2*oo points in an > >>> interval of length 2"? > >>> -- > >>> mike. > >> Think "Continuum Hypothesis". If aleph_1 is the size of the set of > >> finite reals, is aleph_1/aleph_0 the size of the set of reals in the > >> unit interval? Is that between aleph_0 and aleph_1? Uh huh. > > > Firstly, the continuum hypothesis is nothing to do with geometry. > > It does, if sets are combined with measure and a geometrical > representation of the question considered. Is half an infinity less than > itself? Geometrically, yes, the reals in (0,1] are half the reals in (0,2]. Everything you just said has nothing to do with the continuum hypothesis. You're terminally confused. As near as I can tell, your mish-mash of ideas all basically boil down to asserting "the 'number' of reals/points in an interval/line segment = the Lebesgue measure". What does asserting that do for us? Not much. > > Secondly, division is not defined for infinite cardinal numbers. > > I'm not interested in cardinality, but a richer system of infinities, > thanks. You brought up the continuum hypothesis. Next post, you say you don't want to talk about cardinality. Risible. -- mike.
From: Lester Zick on 22 Mar 2007 13:07
On Wed, 21 Mar 2007 22:21:25 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> 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. >> > >Very good. :) > >>> "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? >> > >Right. It(blue) and blue(it) have truth values between 0 and 1 >inclusive, but "blue" has no more truth value than "it". But "it" can be true of things other than "blue" whereas "blue" cannot be true of things other than "it" whatever the "it" is taken to mean. > Truth is a >matter of how much one idea applies to another, or perhaps something >less vague than that. I still don't understand how you arrive at "truth values" Tony. As far as I can tell they're just assigned arbitrarily or by assumption of truth. They're meaningless in this sense. They're nothing more than a mechanization of truisms defined by Aristotle's syllogistic inference like "if A then B then C" etc. They don't tell us anything about what's actually true or why. Finite tautological reduction to self contradictory alternatives on the other hand tells us what's actually true and necessarily and universally true of everything. >>> 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)". >> > >Without defining "it", there is no measure of the truth value of the >statement, "it is blue". Well "it" is a universal subject, Tony. It has no definition of its own apart from that. "It" can and does stand for any subject. And any predicate can be true or not of "it". "It" just gives us a universal subjective reference point for predicates just as "not" gives a universal predicate reference point for all predicates. > Likewise, if I say, "the moon is thus", one >cannot ascertain the truth value of that statement without knowing which >"thus" the moon is asserted to satisfy. Of course not. One can never ascertain actual "truth values" for anything as far as I can tell. They just represent an arithmetic mechanization of syllogistic inference. In other words "truth values" just mechanize arithmetically something that wasn't true to begin with. When you begin to talk about problematic intermediate "truth values" the picture becomes even clearer. Syllogistic inference offers no way to determine actual truth. All it produces are truisms not truth. Thus the truth of given predicates or predicate combination is problematic and we are left with no recourse but to express truth mathematically in terms of some kind of degree of confidence akin to a probability. The whole situation would be a joke if it weren't so pathetic. >>>> 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. >> > >Truth assumes values between 0, commonly known as "false", and 1, or >"true". In Boolean logic there are only these two values, but statistics >allows for the full spectrum. So, a statement has a truth value, which >can perhaps be evaluated, and which will always fall within these >bounds, quantitatively. Except you can't actually assign a "truth value" to any predicate as far as I can tell except by a naive assumption of truth to begin with. >>> 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. > >They are expressions of logical implication. > >And they can range from true to false to self contradictory. Which may or may not be true but you can't assign any "truth value" to any predicate as far as I can see except by naive assumption of truth. >Self-contradictory = automatically false. > >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. > >I think you need to distinguish between predicates and attributes and >objects. Predicates are assignments of attributes to objects, and only >predicates have truth values, true, fals, or somewhere in between. If you could demonstrate the difference between "attributes" and "predicates" perhaps I would. Both "attributes" and "predicates" are "predicated" of something or "it" through "predication". >>> 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~~ > >Only, just, what is it? What is what? I can't see much that it matters exactly what "blue" may be if it is true or false. The only thing that matters is whether it's true or not and whether it's true combined with other predicates. In the same regard what is "sky"? Who cares what "sky" is? The only thing that matters in this particular context is whether a combination of true "blue" and true "sky" remains true. That's how combinations of predicates acquire meaning and definition. Definitions and meanings of words are not just abbreviations as David Marcus would have us believe. They are true or not considered in isolation or combination with other words. ~v~~ |