The Definition of Points
in [Math]
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From: Virgil on 19 Mar 2007 16:46 In article <45feb69a(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > > >>> That the cardinality of the symmetric group on n elements is n!. > >> Isn't that demonstrable geometrically? > > > > No. There is nothing geometrical in that statement. > > > > Arrange them in a convex polygon as vertices, and draw lines between > them.... Arrange what in a convex polygon? If you arrange the elements that way, how do you represent a permutation? > > >>> That a polynomial over the reals of degree n has no more than n real > >>> zeroes. > >> Also geometrically demonstrable. > > > > Please describe such a demonstration. > > > >>> That a square matrix always 'satisfies' its characteristic polynomial. > >>> Etc. > >> Not sure what that means, but excuse me if I take "square" to be a > >> geometrical concept.... > > > > It's not, in this context. > > > > - Randy > > > > > > Oh. Define the geometrical definition of square, that's not "analytical".... > > - Tony TO has it backward, as usual. One can "square" things analytically, without any geometric interpretation, whenever one has a multiplication operation that is analytic but not geometric. In this case the "square" is a Cartesian product of {1,2,3,...,n} with itself and the the matrix itself is a mapping from that "square" to some field of numbers.
From: Virgil on 19 Mar 2007 16:48 In article <45FE95CD.8010609(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 3/15/2007 7:07 PM, Bob Kolker wrote: > > > 9/10 + 9/100 + etc converges to 1.0 > > > > Bob Kolker > > > Not to 1.0, Oh! Since 1 = 1.0 = 1.00 = 1.000 = 1.000..., why not?
From: hagman on 19 Mar 2007 17:22 On 17 Mrz., 19:23, "Hero" <Hero.van.Jind...(a)gmx.de> wrote: > On 17 Mrz., 18:49, Bob Kolker <nowh...(a)nowhere.com> wrote:> SucMucPaProlij wrote: > > >>You can develop geometry based purely on real numbers and sets. You need not > > >>assume any geometrical notions to do the thing. One of the triumphs of > > >>mathematics in the modern era was to make geometry the child of analysis. > > > > And it means that lines, planes and points are defined in geometry. > > > Make up your mind, Bob! > > > Not true. One of the mathematical systems which satisfy Hilbert's Axioms > > for plane geometry is RxR , where R is the real number set. Points are > > ordered pairs of real numbers. Not a scintilla of geometry there. > > Left and right are geometrical concepts. > When You write down ( 3, 4 ) 3 is left in Your view and 4 is right. > > With friendly greetings > Hero No, (3,4) is {{3},{3,4}} and then 3 is the only element of the only singleton element and 4 is the other guy.
From: SucMucPaProlij on 19 Mar 2007 17:28 "hagman" <google(a)von-eitzen.de> wrote in message news:1174327593.061615.24560(a)y66g2000hsf.googlegroups.com... > On 16 Mrz., 15:35, "SucMucPaProlij" <mrjohnpauldike2...(a)hotmail.com> > wrote: >> "hagman" <goo...(a)von-eitzen.de> wrote in message >> >> news:1174053602.723585.89690(a)l77g2000hsb.googlegroups.com... >> >> >> >> > On 13 Mrz., 18:52, Lester Zick <dontbot...(a)nowhere.net> wrote: >> >> The Definition of Points >> >> ~v~~ >> >> >> In the swansong of modern math lines are composed of points. But then >> >> we must ask how points are defined? However I seem to recollect >> >> intersections of lines determine points. But if so then we are left to >> >> consider the rather peculiar proposition that lines are composed of >> >> the intersection of lines. Now I don't claim the foregoing definitions >> >> are circular. Only that the ratio of definitional logic to conclusions >> >> is a transcendental somewhere in the neighborhood of 3.14159 . . . >> >> >> ~v~~ >> >> > Please look up the difference between "define" and "determine". >> >> > In a theory that deals with "points" and "lines" (these are typically >> > theories about geometry), it is usual to leave these terms themselves >> > undefined >> > and to investigate an incidence relation "P on L" (for points P and >> > lines L) >> > with certain properties >> >> > Then the intersection of two lines /determines/ a point in the sense >> > that >> > IF we have two lines L1 and L2 >> > AND there exists a point P such that both P on L1 and P on L2 >> > THEN this point is unique. >> > This is usually stated as an axiom. >> > And it does not define points nor lines. >> >> This is interesting observation :)))) >> >> But how do you define difference between "define" and "determine"? >> Can "definition" determine and can "determination" define? >> >> Lester Zick has problem with "circular definitions" and you used term "point" >> in >> your "determination" to determine it. Maybe you want to say that in >> definition >> you can't use term you define to define it and in termination you can use it >> to >> determine it. >> >> I think it's time to call Determinator :)))) >> He is the only one who can help us! hahahahahahaha > > All I wanted to do is go along with Lester on his path of arguments as > far as bearable - which is more than "as far as correct". > Let me start again with a bit more sleep: > In his OP, Lester talked about lines as being composed of points by > definition and that the intersection of two lines determined a point. > This is his private theory although he states that it were somehow > usual math folklore. > > First imtermezzo: What is a definition? > I won't define that term here rogorously, but a definition should be > useful (that's just a pragmatic aspect) and make the defined term > eliminable. > Example: In the context of natural number we can define > DEF. k is a divisior of n if there exists some m such that k*m=n. > DEF. p is prime if p has eactly two divisors. > These are good definitions as we can eliminate the terms "prime" and > "divisor" from statements like > "There are infinitely many primes" = "There are infinitely many > numbers p such that there are exactly two numbers k such that there > exists some m such that p=k*m" > > So: What is the definition of "point"? And what is the definition of > "line"? > Mathematical theories where both the terms "points" and "lines" are > used usually go like this: > A tuple G=(P,L,E) is called a "geometry" (the precise term may depend > on the precise set of axioms used below and might as well be > "euklidean plane" or the like) if P and L are disjoint sets and E is a > relation among them, i.e. E subset PxL, such that > 1. For two distinct p,q in P, there is exactly one g in L such that > (p,g) in E and (q,g) in E > 2. For two distinct g,h in L, there is at most one p in P such that > (p,g) in E and (p,h) in E > 3. For each p in P there is at least one g in L such that (p,g) not in > E > 4. ... > > Only the context of such a geometry and the complete set of axioms > listed define point (and line and incidence). > > Thus, "Let p be a point ..." should be writen via elimination as > "Assume G=(P,L,E) is a geometry and p in P ..." > Note that lines are not "composed of" points. > However, two (non-parallel) lines determine a point in the sense that > it is a theorem (in fact an axiom) in this theory that for these lines > there is a unique point incident with both lines. > > As lines are not composed of points, the very first sentence of the OP > is nonsense. > However, this nonsense can be escaped from: > Given a geometry G(P,L,E), we ca define a mapping from > pointson: L -> 2^P, g |-> {p in P: (p,g) in E} > This mapping is injective and allows us to replace L by {pointson(): g > on L} and E by element containment to obtain an isomorphic geometry. > Hence one may assume wlog that lines are sets of points. > > However, we could alternatively have used a mapping > goesthru: P -> 2^L, p |-> {g in L: (p,g) in E} > and would then interprete lines as "atomic" and points as sets of > lines. > > The first method is more useful as it is more readily generalized to > other sets of points (circles etc.) > > In view of the above, "lines are composed of points" and "lines > determine points" is no more miraculous than "sets contain elements" > and "sometimes two sets have exactly one element in common" > General problem with this kind of discussions is that people don't want to accept "alternative truth" or the fact that you don't have to use the same words to describe same thing. I do try to understand what other people say and I do try to recognize my thoughts in other people's words but sometimes I just fool around :))))
From: Hero on 19 Mar 2007 17:55
hagman wrote: > Hero wrote: > > Bob Kolker wrote: > > > ...Points are ordered pairs of real numbers. Not a scintilla of geometry there. > > > Left and right are geometrical concepts. > > When You write down ( 3, 4 ) 3 is left in Your view and 4 is right. > > No, (3,4) is {{3},{3,4}} and then 3 is the only element of the only > singleton element and 4 is the other guy. Congratulations, You win. With friendly greetings Hero |