The Definition of Points
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From: Tony Orlow on 22 Mar 2007 17:42 Mike Kelly wrote: > 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. > So, CH doesn't have to do with whether c, the number of reals, is greater than aleph_1, the next transfinite cardinal after aleph_0? If there is a cardinal number less than the number of all reals, and greater than the number of all naturals, then c>aleph_1, right? If the number of reals overall is the number of reals per unit interval times the number of unit intervals (times aleph_0, that is), then the number of reals per unit interval qualifies as a value between the number of naturals and the number of reals. That's not cardinality, of course, so it's not an answer to CH within ZFC. The system I'm talking about doesn't require a hypothesis about this. The answer is clear. There are a full spectrum of infinities above and below c. > 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. > It goes a bit beyond Lebesgue measure, but I don't expect you to see that. >>> 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. CH is a question in ZFC. The answer lies outside ZFC. > > -- > mike. > tony.
From: Tony Orlow on 22 Mar 2007 18:11 Lester Zick wrote: > 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. > So this(blue) precludes that(blue)??? More than one thing can be blue, and any given thing can have more than one attribute. fluid(sky), blue(sky), fluid(liquid) and blue(liquid) can all be true. >> 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. > An algebraic expression does not give a quantitative result without quantitative parameters, and a logical expression does not give a truth value without logical parameters. Truth value calculations can be expressed as algebraic calculations, as well as using truth tables. After all, not every statement is 100% 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)". >>> >> 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. > So, "it" is "x", a variable parameter. Fill in that blank, and then you can evaluate the expression. >> 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. They simply express a certainty of truth, from 0 to 1. > > 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. I won't object to that. > > The whole situation would be a joke if it weren't so pathetic. > All humor is based on pain and foolishness. It's all pathetic. So, laugh a little. :) >>>>> 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. > Right. ASSUMING predicates a and b are true and false, respectively, a AND b is false, a OR b is true. The value of the expression depends on the values of the variables. Presumably, in real world situations, we can at least give a guesstimate of the certainty for any assertion involved in the logical statement, and therefore calculate at least a rough estimate of the certainty of the entire expression. That's science for you. ;) >>>> 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. > You can only assume truth, or calculate from other assumed truths. Some things are rather certain, but could still be wrong. The sun might not "rise" tomorrow, but I'll bet you it will. Assuming it does, chances are, it will become warmer during the first half of the day. >> 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". > Grammar lesson 101: Let's say the most basic elements of any expression are objects, that is, nouns. If we are speaking, we are speaking about something. Otherwise we should shut up. We can describe objects, which means we assign attributes to them, that is, adjectives describe nouns. When we assign attributes to nouns, or nouns to sets described by those attributes, we use an assumed predicate relation, either "describes" or "is a member of this class". The predicate is the verb of the statement. Real world predicates include actions and events, but unless you have rules pertaining to specific actions or events, those predicates do not come into play in the evaluation of a logical statement. In a predicate, the core is some verb, like "is". If there is no "is" or "does" or "has", then there is no predicate. A full statement also must have a noun, a subject, or it's not talking about anything. A statement does not require an attribute, such as "blue", but the most interesting, of course, do. Of course there are those like, "0 exists", which have their utility. So, there are objects, predicates involving an attribute or event, and statements that combine the two.:) >>>> 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. "true blue" is just a patriotic phrase. > > 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~~ Right. A full statement requires a subject and a predicate. "is blue" is just the predicate, and therefore is not a statement, and has no truth value in isolation. 01oo
From: Tony Orlow on 22 Mar 2007 18:14 Lester Zick wrote: > On Wed, 21 Mar 2007 22:45:54 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Wed, 21 Mar 2007 14:17:16 -0500, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>> It states the specific infinite number of points in the unit interval, >>>> say, on the real line. >>> And what real line would that be, Tony? >>> >>> ~v~~ >> The one that fully describes the real numbers. > > You mean a straight line that describes curves exactly? Or some curve > that describes straight lines exactly? > There is no straight line, but only the infinitesimally curved. :D >> Like, duh! The one that >> exists. > > Except there is no such line, Tony. At least none that describes both > curves and straight lines together exactly.And if you don't believe me > Bob Kolker has acknowledged the point previously. > Well, if Bob says so, then.... >> E R >> 0eR >> 1eR >> 0<1 >> xeR ^ yeR ^ x<y -> EzeR x<z ^ z<y > > Very fanciful, Tony. You mean if you know the approximation for pi > lies between 3 and 4 on a straight line pi itself does too? For instance. If you can get arbitrarily close to pi without leaving the line, then it resides on the line. Otherwise it would be some distance from the line, and there would be a lower limit to your approximation. > > You see, Tony, this is the basic reason I refuse to be drawn into > discussion on collateral mathematical issues as interesting as they > might be. I can't even get the most elementary point across even to > those supposedly paying attention to what I say. > > ~v~~ Try me. 01oo
From: Tony Orlow on 22 Mar 2007 18:15 Lester Zick wrote: > On Wed, 21 Mar 2007 22:24:22 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Wed, 21 Mar 2007 07:40:46 -0400, Bob Kolker <nowhere(a)nowhere.com> >>> 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. >>> They probably also have zero measure in uncountable bunches, Bob. At >>> least I never heard that division by zero was defined mathematically >>> even in modern math per say. >>> >>> ~v~~ >> Purrrrr....say! Division by zero is not undefinable. One just has to >> define zero as a unit, eh? > > A unit of what, Tony? > >> Uncountable bunches certainly can attain nonzero measure. :) > > Uncountable bunches of zeroes are still zero, Tony. > > ~v~~ Infinitesimal units can be added such that an infinite number of them attain finite sums.
From: Lester Zick on 22 Mar 2007 18:59
On Thu, 22 Mar 2007 17:14:38 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Wed, 21 Mar 2007 22:45:54 -0500, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> Lester Zick wrote: >>>> On Wed, 21 Mar 2007 14:17:16 -0500, Tony Orlow <tony(a)lightlink.com> >>>> wrote: >>>> >>>>> It states the specific infinite number of points in the unit interval, >>>>> say, on the real line. >>>> And what real line would that be, Tony? >>>> >>>> ~v~~ >>> The one that fully describes the real numbers. >> >> You mean a straight line that describes curves exactly? Or some curve >> that describes straight lines exactly? >> > >There is no straight line, but only the infinitesimally curved. :D What evidence do you have to support your opinion, Tony? Straight lines have zero curvature. >>> Like, duh! The one that >>> exists. >> >> Except there is no such line, Tony. At least none that describes both >> curves and straight lines together exactly.And if you don't believe me >> Bob Kolker has acknowledged the point previously. >> > >Well, if Bob says so, then.... I might agree if Bob hadn't gone out of his way to say so. It was uncharacteristic of him to say so. But the very fact that he could see so and said so inclines me to his opinion rather than yours. >>> E R >>> 0eR >>> 1eR >>> 0<1 >>> xeR ^ yeR ^ x<y -> EzeR x<z ^ z<y >> >> Very fanciful, Tony. You mean if you know the approximation for pi >> lies between 3 and 4 on a straight line pi itself does too? > >For instance. If you can get arbitrarily close to pi without leaving the >line, then it resides on the line. Otherwise it would be some distance >from the line, and there would be a lower limit to your approximation. What makes you think pi resides on straight lines instead of circles? Or do you think straight lines reside on curves? Or do you think pi resides on both and a circle is equal to straight line approximations? Otherwise it would indeed be at some distance from the straight line and there would be no point on the straight line corresponding to pi. >> You see, Tony, this is the basic reason I refuse to be drawn into >> discussion on collateral mathematical issues as interesting as they >> might be. I can't even get the most elementary point across even to >> those supposedly paying attention to what I say. >> >> ~v~~ > >Try me. Already have. Don't know what else to say. ~v~~ |