The Definition of Points
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From: Virgil on 20 Mar 2007 16:09 In article <46001c85(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > When analytic geometry was invented, in which all geometric ideas were > > replaced by algebraic ones, it turned out that one could prove purely > > algebraically what had previously only been provable geometrically. > > > > Right, but you already had the answers supplied geometrically, with > which to compare the algebraic results. > > > > >> You don't even have the symbolic language you so treasure, without > >> geometric differences between symbols. > > > > There have been blind mathematicians, whose symbols, whatever they may > > have been, were not geometric. > > > > There have even been blind geometers, such as Lev Pontryagin. > > They are geometric, even if not visual. If they are purely auditory, > then those symbols are not likely to lead to much progress in spatial > analysis. Read up on Pontryagin and his ilk before making such idiotic claims.
From: Wolf on 20 Mar 2007 17:31 Tony Orlow wrote: > Virgil wrote: [...] >> There have even been blind geometers, such as Lev Pontryagin. > > They are geometric, even if not visual. If they are purely auditory, > then those symbols are not likely to lead to much progress in spatial > analysis. Even sighted geometers have to leave their visual intuitions behind to do geometry. Why? Because our visual system is prone to error and illusion, is why. That's why the first thing we were taught in math class was not to trust our eyes. You want to show that two triangles are congruent? Your drawing will not prove it - only logic. And what's logic? The manipulation of symbols, is what. You don't have to be able to see those symbols, nor what they refer to (if anything.) Actually, being able to see the symbols may be a hindrance. I was taught geometry both in Austria and in England. In England, they were still using Euclid's notation: "the line AB", for example, instead of "the line a". I found the use of capital letters to refer to both vertices and lines less than transparent, shall we say. HTH -- Wolf "Don't believe everything you think." (Maxine)
From: Hero on 20 Mar 2007 17:09 Lester Zick wrote: > > >And another question: is the trace, left by a movement, not part of > >static geometry? It is an invariant of dynamic geometry. > > You know, Hero, there are some extraordinarily subtle considerations > involved here which need to be considered for any exact analysis of > static rac versus dynamic non rac construction methods. However I'd > rather not get into them just at present because they really aren't > germane to the basic topics we're considering here at the moment. > It's a pity. (just one more comment, when You can get to exact algebraic lengths on the real line and because of an open window the temperature did change a bit and with it Your compass, so You have a chance of an exact transcendental). Okay, back to points and lines. There's topology, just the simple beginning: A space (mathematical) is a set with structure. A point is a geometrical space without geometrical structure, but it can give structure to geometry. Think of a vertex or a center and so forth. A line is made up of points and sets of points ( the open intervalls between each two points),which obey three topological rules. What i learned recently: With adding a point to an open (open in standard topology) flexible surface one can enclose a solid, with adding a point to an open line one can enclose a figure, and two points are the boundary of an intervall on a line. But there is no point at or beyond infinity. With friendly greetings Hero
From: PD on 20 Mar 2007 18:20 On Mar 20, 2:33 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: > On 19 Mar 2007 11:51:47 -0700, "Randy Poe" <poespam-t...(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. 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 a physicist, Randy, not a psychologist. By whose standard and measure are you a physicist, Lester? There are certain categorizations, like "expert", which one does not self-attach. One earns certain labels through attribution by others. This is also a fact that one cannot circumvent by willful abstinence or simple refusal to abide. That you don't like that, is completely irrelevant. PD > > > 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. > > ~v~~- Hide quoted text - > > - Show quoted text -
From: hagman on 20 Mar 2007 19:15
On 20 Mrz., 20:24, Lester Zick <dontbot...(a)nowhere.net> wrote: > On 20 Mar 2007 03:11:41 -0700, "hagman" <goo...(a)von-eitzen.de> wrote: > > >On 20 Mrz., 00:30, Lester Zick <dontbot...(a)nowhere.net> wrote: > > >> > 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~~ > > >You are trying to walk the path in the wrong direction. > >E.g. 0:={}, 1:={{}}, 2:= {{},{{}}}, ... > >is a model of the naturals: The Peano axioms hold. > > However we don't have a model of straight lines except by naive > assumption. Wrong. The field of reals provides a very good model of a straight line, also in the form of an affine subspace of a higher dimensional real vector space. > > >However, in this model we have "0 is a set", which does not follow > >from Peano axioms. > >Thus the model has some additional properties. What's wrong with that? > > It isn't a model of what we wish to model. You made a general statement against models (or rather grossly misunderstood someone elses general statement about models). Thus I am allowed to pull out any model I like. > > >However, the model shows that the Peno axioms are consistent (provided > >the set theory we used to construct the model is). > > Consistency is only a prerequisite not a final objective for a model. > On the contrary. Producing a model for a theory is the common way to show that the theory is consistent (provided the theory used to construct the model is consistent) A: Let's consider a theory where 'lines' consist of 'points' and 'lines' determine 'points'... B: Hey, isn't that nonsense? Can such a theory exist? A: Of course. Take pairs of real numbers for points and certain sets of such pairs as lines... B: I see. |