The Virgin Birth of Points
in [Physics]
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From: John Jones on 14 Nov 2007 15:35 On Nov 13, 1:42?pm, Venkat Reddy <vred...(a)gmail.com> wrote: > On Nov 13, 12:31 am, John Jones <jonescard...(a)aol.com> wrote: > > > > > > > On Nov 12, 6:05?pm, Dave Seaman <dsea...(a)no.such.host> wrote: > > > > On Mon, 12 Nov 2007 07:50:52 -0800, John Jones wrote: > > > > On Nov 12, 3:42?pm, Dave Seaman <dsea...(a)no.such.host> wrote: > > > >> On Mon, 12 Nov 2007 07:06:39 -0800, Hero wrote: > > > >> > Robert wrote: > > > >> >> Lester Zick wrote: > > > >> >> > The Virgin Birth of Points > > > >> >> > ~v~~ > > > > >> >> > The Jesuit heresy maintains points have zero length but are not of > > > >> >> > zero length and if you don't believe that you haven't examined the > > > >> >> > argument closely enough. > > > > >> >> In Euclidean space a set which has exactly one pont as a member has > > > >> >> measure zero. But you can take the union of an uncountable set of such > > > >> >> singleton sets and get a set with non-zero measure. > > > > >> > What measure will give a non-zero number/value? > > > > >> Lebesgue measure will do so, not for all possible uncountable sets, but > > > >> for some. For example, the Lebesgue measure of an interval [a,b] is its > > > >> length, b-a. > > > > >> -- > > > >> Dave Seaman > > > >> Oral Arguments in Mumia Abu-Jamal Case heard May 17 > > > >> U.S. Court of Appeals, Third Circuit > > > >> <http://www.abu-jamal-news.com/> > > > > An interval [a,b] is composed of positions, not points. But even > > > > positions are constructions, and it is not appropriate to analyse a > > > > construction in spatial terms. > > > > I think you need to learn some measure theory. This is a question about > > > mathematics, by the way, not philosophy. > > > > -- > > > Dave Seaman > > > Oral Arguments in Mumia Abu-Jamal Case heard May 17 > > > U.S. Court of Appeals, Third Circuit > > > <http://www.abu-jamal-news.com/>- Hide quoted text - > > > > - Show quoted text - > > > I think you need to distinguish between a position and a point before > > wildly conflating them in both a philosophical and mathematical > > confusion. > > The position of a point is relative to the reference coordinate > system. So, position is an attribute on a point to locate it with > reference to the given coordinate system. > > Does it make some sense? > > - venkat- Hide quoted text - > > - Show quoted text - Yes, but why do we need a position to indicate a point when a position is quite adequate by itself?
From: Dave Seaman on 14 Nov 2007 15:48 On Wed, 14 Nov 2007 12:33:53 -0800, John Jones wrote: > On Nov 12, 8:48?pm, Dave Seaman <dsea...(a)no.such.host> wrote: >> On Mon, 12 Nov 2007 11:31:22 -0800, John Jones wrote: >> > On Nov 12, 6:05?pm, Dave Seaman <dsea...(a)no.such.host> wrote: >> >> On Mon, 12 Nov 2007 07:50:52 -0800, John Jones wrote: >> >> > On Nov 12, 3:42?pm, Dave Seaman <dsea...(a)no.such.host> wrote: >> >> >> On Mon, 12 Nov 2007 07:06:39 -0800, Hero wrote: >> >> >> > What measure will give a non-zero number/value? >> >> >> Lebesgue measure will do so, not for all possible uncountable sets, but >> >> >> for some. For example, the Lebesgue measure of an interval [a,b] is its >> >> >> length, b-a. >> >> > An interval [a,b] is composed of positions, not points. But even >> >> > positions are constructions, and it is not appropriate to analyse a >> >> > construction in spatial terms. >> >> I think you need to learn some measure theory. This is a question about >> >> mathematics, by the way, not philosophy. >> >> - Show quoted text - >> > I think you need to distinguish between a position and a point before >> > wildly conflating them in both a philosophical and mathematical >> > confusion. >> In my statement that you quoted, I used neither of the terms "position" >> or "point". I mentioned only Lebesgue measure, uncountable sets, and >> intervals. Exactly what is your, er, point? Why do I need to distinguish >> between terms that I didn't use? >> Neither of those is a precise mathematical term, by the way. The meaning >> depends on context, but to me a "point" is a member of some abstract >> space (possibly a vector space, or a topological space, or a metric >> space, or a measure space, or a Banach space, or whatever). A >> "position", on the other hand, suggests a point that is given in some >> coordinate system. That doesn't always apply. Lots of times we talk >> about points in situations where there are no coordinates in sight. >> I consider "position" to be too limited a term for that reason. > It seems you cannot understand the things you write. Let me put you > straight. You said: > 'I think you need to learn some measure theory. This is a question > about mathematics, by the way, not philosophy.' > Now then. Either you are talking about my concerns about positions or > points, or you are implying by the term 'this' that Lebesque measures > are really all we need to know about when doing mathematics! Play > games with me and you will come off worse. If you look at the context quoted above, you will find that: 1) I answered a question that was posed by Hero, namely, "what measure will give a non-zero number/value?" (i.e., what measure fails to be uncountably additive?). As far as I was aware, you were not even a part of the discussion up to that time. I certainly was not attempting to address any point raised by you. 2) I answered Hero's question by offering Lebesgue measure as an example. I did not use the terms "position" or "point" at all. 3) You responded to this with an irrelevant discourse about "positions" and "points". I thought your response had nothing to do with what I said, and furthermore betrayed a lack of knowledge of measure theory. Hence, my response that you need to learn some measure theory. 4) I certainly do not mean to imply that measure theory is all we need to know when doing mathematics. However, the question that Hero asked and I answered was a question specifically about measure theory, and I answered appropriately. > Let me put you right on another point. Please, don't use the term > 'abstract' in description. It indicates only the limit of > understanding. I suppose you think Steenrod was betraying his limit of understanding when he referred to category theory as "abstract nonsense"? -- Dave Seaman Oral Arguments in Mumia Abu-Jamal Case heard May 17 U.S. Court of Appeals, Third Circuit <http://www.abu-jamal-news.com/>
From: John Jones on 14 Nov 2007 15:51 On Nov 13, 4:06?pm, "Robert J. Kolker" <bobkol...(a)comcast.net> wrote: > Venkat Reddy wrote:> > > The position of a point is relative to the reference coordinate > > system. So, position is an attribute on a point to locate it with > > reference to the given coordinate system. > > > Does it make some sense? > > In a way. Position is a name we give to points to identify them uniquely. > > Bob Kolker You would then, propose the notion of unpositioned points, and lines composed of unpositioned points. 'Points' are superfluous.
From: John Jones on 14 Nov 2007 16:00 On Nov 12, 10:39?pm, "Robert J. Kolker" <bobkol...(a)comcast.net> wrote: > John Jones wrote: > > > The intersections of lines are positions, not points. There is no > > precedent for creating a new metaphysical entity from the arbitrary > > arrangement of lines. I would have thought it obvious. But plainly I > > was mistaken. > > Take a pair of linear equations in two variables each of which define a > line. If the equations are not linearly dependent they determine a > unique solution (x,y) which is --- aha!---- the point of intersection. > > In a Euclidean Plane lines when they intersect at all, have a unique > point of intersection. And mathematical objects are not metaphysical > entities. They are brain farts the blow about in our skulls. > > Bob Kolker > > > > - Hide quoted text - > > - Show quoted text - No, in a euclidean plane lines intersect at a unique position of intersection. 'Intersection' and 'point' are not mathematical entities but loose images and symbols.
From: lwalke3 on 14 Nov 2007 16:07
On Nov 14, 12:51 pm, John Jones <jonescard...(a)aol.com> wrote: > On Nov 13, 4:06?pm, "Robert J. Kolker" <bobkol...(a)comcast.net> wrote: > > > Venkat Reddy wrote:> > > > The position of a point is relative to the reference coordinate > > > system. So, position is an attribute on a point to locate it with > > > reference to the given coordinate system. > > > > Does it make some sense? > > > In a way. Position is a name we give to points to identify them uniquely. > You would then, propose the notion of unpositioned points, and lines > composed of unpositioned points. 'Points' are superfluous. In the model of Hilbert that I've been using in these posts, the primitive "points" are mapped to singleton _subsets_ of R^3, while the positions are the _elements_ of R^3. Then again, your comment about "unpositioned" points does hearken back to the distinction between "synthetic" geometry and "coordinate" geometry. One can prove, for example, that the intersection of two lines is a point, even without referring to its coordinate/position. It is therefore a theorem of Hilbert and thus provable using only the axioms of Hilbert, without referring to the model (where the positions are) at all. Sometimes, coordinate proofs may make a proof easier, but most theorems of geometry can be proved without them. |