The Definition of Points
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From: Tony Orlow on 21 Mar 2007 01:25 Virgil wrote: > In article <46001b67(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Bob Kolker wrote: >>> Tony Orlow wrote:> >>>> How do you know the conclusions are correct, >>> You mean how do you know conclusions correctly follow from the axioms. >>> You look. Proofs are precisely the evidence that the conclusion follows >>> from the premises. >> You know that's not what I 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. > > The only way to "check results" in an axiomatic system, whether > algebraic or geometrical, is to assure that all conclusions follow from > their explicitely stated premises, and that no hidden assumptions are > involved. > > Since axiomatic systems in mathematics say nothing about the physical > world, there is no physical way of checking any of ones "results". > > If a particular axiom system is supposed to reflect some physical > situation, one can also test how well the theory matches the facts, but > that has nothing to do with the derivations from the axioms, but only > with how well the axioms model the "reality". > > That was my point. > >> If some >> set of rules you define leads you to conclude that the volume of a >> sphere is equal to the volume of two spheres of the same radius as the >> first, well, you probably want to go back and reexamine your premises >> and make sure you didn't err somewhere in the derivation of your >> conclusions. > > In this case, one can easily show that the mathematical model and any > physical interpretation are inconsistent, but that does not establish > any inconsistency in the mathematics at all. No, it just points to a deficit in the starting axioms, if they are to pertain to anything at all. But I suppose that's not important... >>> Look any any standard treatise on first order logic for a definiton of >>> proof. Checking to see that a proof indeed shows the desired conclusion >>> follows from the axioms is a purely mechanical algorithmeic proceedure. >>> It does not involve intelligence. -Finding- a proof does. Checking a >>> proof can be done by a trained gorilla or Lester Zick on one of his >>> better days. >>> >>> Bob Kolker >>> >> Bob - wake up. How do we know relativity is correct? Because it follows >> from e=mc^2? > > Relativity is not a mathematical theory, it is a physical theory. > The mathematical model is consistent, but its consistency is not a > physics question, and whether it matches the physics is not at all a > mathemtical question. >> Oy! > > Indeed! Touche!
From: Tony Orlow on 21 Mar 2007 01:47 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.... > 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. :) > > Stephen > Tony
From: Tony Orlow on 21 Mar 2007 01:56 Virgil wrote: > 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. Yes, I'm sure. That's wonderful, as long as the conclusions don't contradict geometric ones. I'm not putting down axiomatization persay. I'm just suggesting that revisiting the root questions is not without merit. Do purely axiomatic considerations determine that the point is more basic than the line or segment? > > 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. So, the quest is for the basic axioms that give rise to all the theorems we want, and more. For you, that's ZFC, but for some, that's not satisfying. Oh well.
From: Virgil on 21 Mar 2007 03:25 In article <4600c900(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > 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. > > Yes, I'm sure. That's wonderful, as long as the conclusions don't > contradict geometric ones. I'm not putting down axiomatization persay. > I'm just suggesting that revisiting the root questions is not without merit. > > Do purely axiomatic considerations determine that the point is more > basic than the line or segment? > > > > > 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. > > So, the quest is for the basic axioms that give rise to all the theorems > we want, and more. For you, that's ZFC, but for some, that's not > satisfying. Oh well. While ZFC is a nice axiomatic system, I am not at all sure that it is what you claim for it. NBG is a nice system too, and there are a variety of others of considerable virtue.
From: Mike Kelly on 21 Mar 2007 05:15
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. |