The Definition of Points
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From: Brian Chandler on 21 Mar 2007 00:11 Randy Poe wrote: > On Mar 20, 3:30 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: > > On 20 Mar 2007 08:58:26 -0700, "Randy Poe" <poespam-t...(a)yahoo.com> > > wrote: > > >On Mar 19, 7:30 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: > > >> On 19 Mar 2007 11:51:47 -0700, "Randy Poe" <poespam-t...(a)yahoo.com> > > >> wrote: > > >> >On Mar 19, 2:44 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: > > >> >> On 19 Mar 2007 08:59:24 -0700, "Randy Poe" <poespam-t...(a)yahoo.com> > > >> >> wrote: > > > > >> >> >> > That the set of naturals is infinite. <snip> > 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. On the contrary, I think he has explained practically all of them, to his own satisfaction, mostly by simple repetition. He seems to be proceeding apace through his "course", and perhaps when he reaches the end of writing it, he will go back and take the course himself - no doubt passing with flying colours. Well, occasionally it provides bits of entertainment. (Yes, of course I know your meaning of "explain" is different, but you are talking to Lester, for goodness sake...) Brian Chandler http://imaginatorium.org
From: Tony Orlow on 21 Mar 2007 00:47 Bob Kolker 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. > > Math is about what follows from assumptions, not true statements about > the world. > > Bob Kolker If the algebraic portions of your mathematics that describe the geometric entities therein do not produce the same conclusions as would be derived geometrically, then the algebraic representation of the geometry fails. Hilbert didn't just pick statements out of a hat. Rather, he didn't do so entirely, though they could have been generalized better. In any case, they represent facts that are justifiable, not within the language of axiomatic description, but within the spatial context of that which is described. Tony Orlow
From: Virgil on 21 Mar 2007 00:59 In article <1174449993.369845.273040(a)o5g2000hsb.googlegroups.com>, "Brian Chandler" <imaginatorium(a)despammed.com> wrote: > Lester Zick wrote: > > On 19 Mar 2007 11:51:47 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> > > wrote: > > > > >On Mar 19, 2:44 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: > > >> On 19 Mar 2007 08:59:24 -0700, "Randy Poe" <poespam-t...(a)yahoo.com> > > >> wrote: > > <this section intentionally not present> > > > I'm a physicist, Randy, not a psychologist. > > Gosh, honto? Only the other day you were a mathematician, you said. > > Brian Chandler > http://imaginatorium.org But that was a thursday!
From: Virgil on 21 Mar 2007 01:06 In article <4600b8f8$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Bob Kolker 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. > > > > Math is about what follows from assumptions, not true statements about > > the world. > > > > Bob Kolker > > If the algebraic portions of your mathematics that describe the > geometric entities therein do not produce the same conclusions as would > be derived geometrically, then the algebraic representation of the > geometry fails. They do produce the same conclusions, and manage to produce geometric theorems that geometry alone did not produce until shown the way by algebra. Actually, if all the axioms of one system become theorems in another, then everything in the embedded system can be done in the other without any further reference to the embedded system at all.
From: Tony Orlow on 21 Mar 2007 01:06
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. "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. 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? :) > 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, 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? The fast chicken on the blue porch, don't you agree? I see no contradiction in that.... > > ~v~~ :D 01oo |