The Virgin Birth of Points
in [Physics]
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From: John Jones on 14 Nov 2007 16:15 On Nov 14, 9:07?pm, lwal...(a)lausd.net wrote: > 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. >One can prove, for > example, that the intersection of two lines is a point, > even without referring to its coordinate/position. Ok, but I don't believe that there is, or can be, such a proof.
From: John Jones on 14 Nov 2007 16:20 On Nov 14, 8:48?pm, Dave Seaman <dsea...(a)no.such.host> wrote: > 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/>- Hide quoted text - > > - Show quoted text - It's not good enough to say that your reference to measure theory had career guidance value rather than argumentative value. Steenrod must be big in some arena, but to use the term 'abstract' in any context except art and everyday conversation is fudge.
From: Dave Seaman on 14 Nov 2007 16:27 On Wed, 14 Nov 2007 13:20:38 -0800, John Jones wrote: > On Nov 14, 8:48?pm, Dave Seaman <dsea...(a)no.such.host> wrote: >> 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/>- Hide quoted text - >> >> - Show quoted text - > It's not good enough to say that your reference to measure theory had > career guidance value rather than argumentative value. Not good enough for what? I answered Hero's question, and I did so without being argumentative. You are the one who is determined to start an argument. > Steenrod must be big in some arena, but to use the term 'abstract' in > any context except art and everyday conversation is fudge. <http://en.wikipedia.org/wiki/General_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: Robert J. Kolker on 14 Nov 2007 16:37 John Jones wrote: > > 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. A distinction without a difference. O.K. call the damned things positions. Substitute the word position for "point" in all the axioms. You still get the same system. You have raised a trivial issue of designation, names and semantics. Nothing mathematical is altered. Bob Kolker >
From: Randy Poe on 14 Nov 2007 16:51
On Nov 14, 4:20 pm, John Jones <jonescard...(a)aol.com> wrote: > Steenrod must be big in some arena, but to use the term 'abstract' in > any context except art and everyday conversation is fudge. You are decreeing that we aren't allowed to use the abstract concept of a number? We aren't allowed to say "2"? We have to only refer to concrete things, like 2 apples or 2 worms? - Randy |