The Definition of Points
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From: Bob Kolker on 20 Mar 2007 14:02 Tony Orlow wrote: > > You know that's not what I mean. I do? Then what do you mean. How do you measure the accuracy of the > premises you use for your arguments? You check the results. That's the > way it works in science, and that's the way t works in geometry. If some But not in math. The only thing that matters is that the conclusions follow from the premises and the premises do not imply contradictions. Matters of empirical true, as such, have no place in mathematics. Math is about what follows from assumptions, not true statements about the world. Bob Kolker
From: Lester Zick on 20 Mar 2007 14:17 On Mon, 19 Mar 2007 18:04:27 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >> No one says a set of points IS in fact the constitution of physical >> object. >> Whether it is rightly the constitution of a mentally formed object >> (such as a geometric object), that seems to be an issue of arbitration >> and convention, not of truth. Is the concept of "blue" a correct one? >> >> PD >> > >The truth of the "convention" of considering higher geometric objects to >be "sets" of points is ascertained by the conclusions one can draw from >that consideration, which are rather limited. > >"blue" is not a statement with a truth value of any sort, without a >context or parameter. blue(sky) may or may not be true. I disagree here, Tony. "Blue" is a predicate and like any other predicate or predicate combination it is either true or not true. However the difference is that a single predicate such as "blue" cannot be abstractly analyzed for truth in the context of other predicates. For example we could not analyze "illogical" abstractly in the context of "sky" unless we had both predicates together as in "illogical sky". But that doesn't mean single isolated predicates are not either true or false. ~v~~
From: Tony Orlow on 20 Mar 2007 13:40 Virgil wrote: > In article <45ff1d67$1(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <45feac8a(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Bob Kolker wrote: >>>>> Tony Orlow wrote: >>>>> >>>>>> One may express them algebraically, but their truth is derived and >>>>>> justified geometrically. >>>>> At an intuitive level, but not at a logical level. The essentials of >>>>> geometry can be developed without any geometric interpretations or >>>>> references. >>>> But how do you know they are essentials of anything without a reference >>>> to that to which they refer? >>> If a system isolated from those references allows one to produce exactly >>> the same set of theorems as one can get using those references, then the >>> the references themselves are irrelevant to the theory. >> 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. > > 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.
From: Bob Kolker on 20 Mar 2007 14:26 Lester Zick wrote: > > I disagree here, Tony. "Blue" is a predicate and like any other Blue as in sad? Blue as in color? Blue as in puritanical? Bob Kolker
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~~ |