The Definition of Points
in [Math]
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From: Hero on 19 Mar 2007 19:13 Lester Zick wrote: > ...However I'm of the > opinion that if we can decipher what is actually happening to the > point of a compass in dynamic terms of constant velocity and constant > transverse acceleration we can nonetheless determine the mechanical > nature and definition of a circle and other curvilinear forms exactly. > However this still woudn't allow combination of dynamic and static > measures. We couldn't just "roll out" a circular form on a straight > line to "point out" pi this way. So You wouldn't accept the ropes and strings of the first geo-meters in egypt? Are You flexible enough to accept Origami-math for exact transcendentals? And another question: is the trace, left by a movement, not part of static geometry? It is an invariant of dynamic geometry. With friendly greetings Hero
From: Lester Zick on 19 Mar 2007 19:20 On Mon, 19 Mar 2007 13:07:30 EDT, "G.E. Ivey" <george.ivey(a)gallaudet.edu> wrote: >> On Tue, 13 Mar 2007 20:24:01 +0100, "SucMucPaProlij" >> <mrjohnpauldike2006(a)hotmail.com> wrote: >> >> > >> >"PD" <TheDraperFamily(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~~
From: Tony Orlow on 19 Mar 2007 19:25 Randy Poe 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. >>>> Geometrically incorrect. Unless there is a natural infinitely greater >>>> than the origin, there is no infinite extent involved. >>> The naturals don't have physical positions, since they are not >>> defined geometrically. >> They are if they're associated with points and points define line >> segments. > > By "associated with points" I assume you mean something > like using points to model the naturals. In that case the points > in your model have positions, but nevertheless the naturals > themselves don't have physical positions or exist as geometric > entities. > > Do you have any idea what I'm saying? I'm saying that a > model is just a model. The properties of the model do not cause > the thing it's modeling to have those properties. > > - Randy > Oh. Then your syntactical definitions also may exhibit properties that do not pertain to the naturals they model, eh? Look, either the definitions and the model are one, or they are different things. Devise THE model, and the properties are apparent. There is no good reason to say something applies to the real number line, but not the real numbers, or any subset thereof. - Tony
From: Lester Zick on 19 Mar 2007 19:30 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: >> >> >> > That the set of naturals is infinite. >> >> >> Geometrically incorrect. Unless there is a natural infinitely greater >> >> than the origin, there is no infinite extent involved. >> >> >The naturals don't have physical positions, since they are not >> >defined geometrically. >> >> They are if they're associated with points and points define line >> segments. > >By "associated with points" I assume you mean something >like using points to model the naturals. Why do you assume that, Randy? I mean if I raise an issue and you willy-nilly recast it in terms amenable to you then it becomes your issue instead of mine and you can thus congratulate yourself on addressing and answering your question instead of mine. So why don't you just carry on a dialog with yourself and forget I mentioned it? > In that case the points >in your model have positions, but nevertheless the naturals >themselves don't have physical positions or exist as geometric >entities. Good. Then maybe modern math can't model geometry after all. >Do you have any idea what I'm saying? What do you think I am, a brain surgeon? > I'm saying that a >model is just a model. The properties of the model do not cause >the thing it's modeling to have those properties. Oh great. So now the model of a thing has properties which don't model the properties of the thing it's modeling. So why model it? ~v~~
From: Tony Orlow on 19 Mar 2007 19:31
Virgil wrote: > In article <45feac8a(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Bob Kolker wrote: >>> Tony Orlow wrote: >>> >>>> One may express them algebraically, but their truth is derived and >>>> justified geometrically. >>> At an intuitive level, but not at a logical level. The essentials of >>> geometry can be developed without any geometric interpretations or >>> references. >> But how do you know they are essentials of anything without a reference >> to that to which they refer? > > If a system isolated from those references allows one to produce exactly > the same set of theorems as one can get using those references, then the > the references themselves are irrelevant to the theory. How do you know the conclusions are correct, if not by comparing them with what one would expect from the original context? Did Hilbert just postulate that every two points are contained in some line? No, that's a geometrical fact, EXPRESSED in language. He did not pull his axioms out of a hat, but from pictures. You don't even have the symbolic language you so treasure, without geometric differences between symbols. Don't spurn the mother of your verbiage. You might as well kill all plants because animals are better. The last surviving will be the ugliest carnivores. |