The Definition of Points
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From: Lester Zick on 20 Mar 2007 14:58 On 20 Mar 2007 02:10:30 -0700, "hagman" <google(a)von-eitzen.de> wrote: >On 20 Mrz., 00:43, Lester Zick <dontbot...(a)nowhere.net> wrote: >> However I would like to suggest a couple of folk remedies for the >> unwary. First off if like Hilbert you don't intend to define points >> and lines don't use the terms. > >Well I define geometries and - in the context of such a geometry - the >terms >"point" and "line" *are* defined. >Therefore you won't keep me from using these terms in such a context. Well you can take the issue up with Hilbert and his disciples. All I'm interested in is whether implicit or explicit definitions are circular and whether you try to say things such as "lines are the set of all points on lines" or "tables are sets of all beer bottles on tables". >> And second if you intend to use terms >> don't pretend they're undefined because the predicates you use them in >> conjunction with define them quite effectively. > >Isn't that exactly what I and others stated? Not if the issue is circular definition. Then we're concerned not only with the predicates implicitly or explicitly defining terms but also the manner of their usage. I routinely see SOAP opera definitions cast in circular terms with geometric predicates, such as "a circle is the set of all points equidistant from any point on a plane" where the geometic predicates "equidistant" and "plane" remain undefined except in terms of other SOAP operas referencing other undefined geometric objects. >As a matter of speach, you start with previously undefined words, >cast a few axioms at them and - voila - they are defined. >And the definition as given is not circular. It is if the definition as given refers to geometric figures given in the definition such as "a line is the set of all points on a line". In other words if you intend to define geometric objects in terms of SOAP operas then all geometric objects have to be defined that way or you wind up begging the question. >> And third if you >> decide to use the terms don't like Bob try to justify their usage with >> nonsensical circular definitional regressions. > >I had not seen Bob's post. Well it's a recurrent theme and I couldn't point to any one post. >However, assuming you have a complete ordered field named R at hand, >one might >- define a point to be a pair (a,b) in RxR >- define a line to be a triple (a,b,c) in RxRxR such that a^2+b^2=1 >and a>0 or (a=0 and b>0) >- define the incidence relation as (x,y) is_on (a,b,c) iff a*x+b*y >+c=0 >One then proceeds to show that all of e.g. Hilbert's axioms hold and >thus we have a model of a Euklidean plain. >In this model, the definition of lines does not use points and vice >versa. >However, in the coursee of showing the validity of geometric axioms >for this model, >you have shown that "two lines determine a point", i.e. for two lines >(a,b,c) and (a',b',c') there is at most one point (x,y) such that >(x,y) is_on (a,b,c) and (x,y) is_on (a',b',c'). > >Here's a different model: >- define a point to be a pair (a,b) in RxR >- define a line to be a certain set of points(!), i.e. a subset of RxR >of the form {(x,y) in RxR | a*x+b*x+c=0 } for some a,b,c in R with >(a,b) != (0,0) >- define incidence as element containment >This is another model and is indeed isomorphic to the one above (how >surprising). > >Here, lines are composed of points. >And two (non-parallel) lines determine (not define!) a point, namely >the unique element of their intersection. >Thus: Lines are composed of points. And lines determine points. >Where is the problem? Well at a glance the major problem I see is that you are using terms like RxR, numbers, and numerical coordinates as if they were already defined and as if they defined geometrical objects. Thus the question itself of defining geometrical objects in such ways is already begged. Remember it isn't enough to show Hilbert's axioms and theorems are self consistent we also need to show that they result in what we claim which are the geometric objects we wish to define. Nor do we wish to define "models" of geometric objects because we don't really know what "models" amount to; we wish to define the objects themselves. And finally where did all these numbers and numerical coordinates come from? It is common today to pretend they're developed completely abstractly and independent of geometry. However there is nothing in the Peano and suc( ) axioms themselves I'm aware of which requires that successive integers to be colinear and lie together on any one straight line. That's merely a naive assumptions of truth. In point of fact successive integers could lie in any combination of directions on successive line segments and it's only tacit assumption which results in their lying together on a single straight line. Which brings up a final point that conventional SOAP operas cannot be used to define straight lines at all. The best we can assume is that they can be used to define straight line segments between successive points associated with integers but not straight lines as such which to the best of my knowledge are only definable through Newton's method of drawing tangents to curves. So we are forced to conclude that whatever geometrical objects may be described in terms of SOAP's they must be described through geometric subdivision instead of line segment addition and that differences and not addition represent the fundamental mathematical and arithmetic operation. ~v~~
From: Lester Zick on 20 Mar 2007 15:06 On 19 Mar 2007 16:13:38 -0700, "Hero" <Hero.van.Jindelt(a)gmx.de> wrote: > Lester Zick wrote: > >> ...However I'm of the >> opinion that if we can decipher what is actually happening to the >> point of a compass in dynamic terms of constant velocity and constant >> transverse acceleration we can nonetheless determine the mechanical >> nature and definition of a circle and other curvilinear forms exactly. >> However this still woudn't allow combination of dynamic and static >> measures. We couldn't just "roll out" a circular form on a straight >> line to "point out" pi this way. > >So You wouldn't accept the ropes and strings of the first geo-meters >in egypt? Of course I'd accept them. The question isn't whether I'd accept them but what I'd accept them for? And the answer is certainly not for the exact commensuration associated with rac construction. >Are You flexible enough to accept Origami-math for exact >transcendentals? The only exact commensuration for transcendentals lies on the curves themselves. >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. ~v~~
From: Lester Zick on 20 Mar 2007 15:17 On Tue, 20 Mar 2007 12:35:32 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Bob - wake up. How do we know relativity is correct? Because it follows >from e=mc^2? > >Oy! Not true, Tony. If relativity were correct e=mcc might follow from it and not vice versa but in the present instance we'll just have to find some other explanation. ~v~~
From: Lester Zick on 20 Mar 2007 15:18 On Mon, 19 Mar 2007 23:48:17 -0600, Virgil <virgil(a)comcast.net> wrote: >> How do you know the conclusions are correct, if not by comparing them >> with what one would expect from the original context? > >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. Especially when one just assumes what one is supposed to demonstrate. ~v~~
From: Lester Zick on 20 Mar 2007 15:21
On 20 Mar 2007 01:58:18 -0700, "Brian Chandler" <imaginatorium(a)despammed.com> wrote: > >Lester Zick wrote: >> On 19 Mar 2007 11:51:47 -0700, "Randy Poe" <poespam-trap(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: > ><snippety snoppety> Don't get snippy with me, Brian. I can always do transfinite zen whereas you can't always do science or math or much of anything else that I can tell. >> ... So why don't you just carry on a dialog with yourself... > >Ah, there speaks the Master... > >(I'm sure you'll respond to this, Lester, but excuse me if I don't >reply. After all, I advocate preaching what you practice.) Technically, Brian, you advocate preaching regardless of what you do or don't practice. Most true believers do. ~v~~ |