The Virgin Birth of Points
in [Physics]
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From: John Jones on 13 Nov 2007 16:34 On Nov 13, 10:26?am, Amicus Briefs <ami...(a)curiae.net> wrote: > On Mon, 12 Nov 2007 14:45:25 -0800, John Jones <jonescard...(a)aol.com> > wrote: > > >A position may well not be a primitive, but the intersections of lines > >construct positions, not points > > So if we change the name of "points" to "positions" we'll solve the > problem? I think many problems could be solved by not conflating point and position. A position is not a point, nor a point a position. A position is an object in a framework or construction; while a point, like a line, is a framework for the construction of objects such as positions. Frameworks are incommensurables while objects (positions) are not. So the rules for frameworks are quite different to that of the rules for objects. Perhaps Hibert rudimentally envisaged this when he described points and lines as 'primitives'.
From: Lester Zick on 13 Nov 2007 17:45 On Tue, 13 Nov 2007 06:08:03 -0800, Venkat Reddy <vreddyp(a)gmail.com> wrote: >On Nov 13, 4:41 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: >> On Mon, 12 Nov 2007 20:56:04 -0500, "Robert J. Kolker" >> >> <bobkol...(a)comcast.net> wrote: >> >Lester Zick wrote: >> >> >> Well I know enough of mathematics to have convinced you there is no >> >> real number line. >> >> >So what. The theory of real numbers can and is developed without any >> >geometric content of all. Any geometrical associations with real numbers >> >are merely aids to intuition, not logical necessity. >> >> Sure, sure, Bobby. That's why the expression "real number line" pops >> up all over the place. That's why you talk incessantly about lines, >> points, and lattices etc. >> >> >In the nineteenth century a purely analytic foundations for the theory >> >of real and complex variables was developed. Geometry was purged as a >> >logical necessity. Of course, geometry can be very helpful for the >> >right-brain operations associated with discovering new theorems to prove >> >or new mathematical systems. >> >> Of course geometry can be very helpful because there is a geometry of >> arithmetic but no arithmetic of geometry. The problem is there is no >> set of real numbers. If there were you could dance on geometry. But >> you can't define any set of real numbers because there is no single >> modality for real numbers for the definition of any single set.Pi lies >> on circular arcs as Archimedes showed and doesn't lie anywhere else. > >Values, quantities do not need any geometrical entities to lie in or >lie at. I can count 5 apples without using any geometrical space. So >where does the value "5" lie? > >We should not try to model the continuum with the patch work of >different kinds of numbers and keep on worrying about the holes they >leave. Basically we faltered somewhere while moving from counting >numbers to real numbers and then applying them to continuum. Well the problem here, venkat, is that we can model arithmetic exactly in terms of the difference between rational numbers.But we can't model transcendental numbers using the same technique because the only things we can model "between" rationals are straight line segments and square roots of rationals. Where geometry becomes essential to the grasp of mathematics lies in the comprehension of curves. For example pi lies on circular curves as shown by Archimedes and not on any straight line segment. And such curves are what correspond to transcendental numbers. And without them all we can do is arithmetic. Consequently what we find is geometry is the basic progenitor to arithmetic. ~v~~
From: Amicus Briefs on 13 Nov 2007 17:48 On Tue, 13 Nov 2007 16:05:12 +0000 (UTC), Dave Seaman <dseaman(a)no.such.host> 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? > >What if no coordinate system is specified? The definition of a measure >space says nothing about a coordinate system. Well then there is no measure space to measure against. > For that matter, the >important elements of a measure space are not the points (elements of the >space itself), but rather the measurable sets (members of the specified >signma-algebra). "Signma-algebra"? I must have missed that one in ninth grade algebra.
From: Lester Zick on 13 Nov 2007 17:49 On Tue, 13 Nov 2007 11:06:48 -0500, "Robert J. Kolker" <bobkolker(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. Identify them uniquely? Why would we want to identify them uniquely? Why not just wing it and unionize them instead? ~v~~
From: Schlock on 13 Nov 2007 17:51
On Tue, 13 Nov 2007 08:00:42 -0500, "Robert J. Kolker" <bobkolker(a)comcast.net> wrote: >Amicus Briefs wrote: > >> On Mon, 12 Nov 2007 14:45:25 -0800, John Jones <jonescardiff(a)aol.com> >> wrote: >> >> >>>A position may well not be a primitive, but the intersections of lines >>>construct positions, not points >> >> >> So if we change the name of "points" to "positions" we'll solve the >> problem? > >Better still call them potatoes. Two potatoes determine a kugel. Two >kugels intesect on a potatoe. Well that's okay as long as we're dealing with straight kugels. Curved kugels are a whole nuther kettle of fish however. |