The Definition of Points
in [Math]
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From: Tony Orlow on 20 Mar 2007 13:35 Bob Kolker wrote: > Tony Orlow wrote:> >> >> How do you know the conclusions are correct, > > You mean how do you know conclusions correctly follow from the axioms. > You look. Proofs are precisely the evidence that the conclusion follows > from the premises. You know that's not what I 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 set of rules you define leads you to conclude that the volume of a sphere is equal to the volume of two spheres of the same radius as the first, well, you probably want to go back and reexamine your premises and make sure you didn't err somewhere in the derivation of your conclusions. > > Look any any standard treatise on first order logic for a definiton of > proof. Checking to see that a proof indeed shows the desired conclusion > follows from the axioms is a purely mechanical algorithmeic proceedure. > It does not involve intelligence. -Finding- a proof does. Checking a > proof can be done by a trained gorilla or Lester Zick on one of his > better days. > > Bob Kolker > Bob - wake up. How do we know relativity is correct? Because it follows from e=mc^2? Oy! Tony
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 |