The Definition of Points
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From: Eckard Blumschein on 19 Mar 2007 09:53 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, but why not to 1.00... as 1 + 0/10 + 0/100 + etc converges to 0.99...
From: Tony Orlow on 19 Mar 2007 11:30 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? > > Similarly algebraic systems (rings) can be derived from affine spaces > geometry by using similar triangles to develop products from proportions. Yes, one can geometrically represent multiplication of two reals as either a rectangle with those two side measures, in which case no other information is required fro the representation, or as "similar" triangles which represent the same ratio, applied to different extensions of that ratio. In that case, we can produce a linear representation of the product, but we need to know the length of the unit of measure. > > Bob Kolker > Tony Orlow
From: Tony Orlow on 19 Mar 2007 11:32 Virgil wrote: > In article <45fd9bfe$1(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <45fd6045(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Virgil wrote: >>>>> In article <45fc6fd6(a)news2.lightlink.com>, >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>> >>>>> >>>>>> Except that linear order (trichotomy) and continuity are inherent in R. >>>>>> Those may be considered geometric properties. >>>>> If one defines them algebraically, as one often does, are they still >>>>> purely geometric? >>>>>> Tony Orlow >>>> One may express them algebraically, but their truth is derived and >>>> justified geometrically. >>> How does one prove geometrically what is only defined algebraically? > >> example? > > 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. > > That the cardinality of the symmetric group on n elements is n!. Isn't that demonstrable geometrically? > > That a polynomial over the reals of degree n has no more than n real > zeroes. Also geometrically demonstrable. > > 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....
From: Tony Orlow on 19 Mar 2007 11:42 Lester Zick wrote: > On 18 Mar 2007 11:12:39 -0700, "VK" <schools_ring(a)yahoo.com> wrote: > >> On Mar 18, 8:33 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: >>> Oh I don't actually disagree; I just can't tell exactly what all these >>> qualifications amount to and mean. You've got "abstraction" and >>> "perception" and "equivalence" and all sorts of terms mixed up in here >>> that make me suspect none of us including you knows exactly what >>> you're talking about in mechanically exhaustive terms. >> If anyone of rivals (mathematics, philosophy, religion) would knew one >> day "in mechanically exhaustive terms" what is a "thing without sides" >> or say what is "infinity" - wow, the rest would come begging to clean >> their shoos :-) > > Personally I don't agree. I see the issue of points with or without > sides as almost frivolous as I've never known anyone who thought > points had sides. > You've met Ross, and I'm not adverse to the concept of points of various dimensions with zero measure, depending on their spatial context. :) >> <snip> >>> Well maybe that would be true if your initial predicates had any >>> specific and exhaustive value. But lots of things may be true of >>> points without being essential to their definition. I don't understand >>> what "ti en einai of infinity" is supposed to mean nor a "reversed >>> infinity". >> That was not a question which one of definition is correct, neither >> "in mechanically exhaustive terms" nor even by some intuitive feeling; >> well probably neither one. I was asking: do you believe that there is >> one and only one correct definition of the point (a point on a line) >> implied by the very nature of this entity? > > I think probably so. However my interest as I pointed out early on was > more directed at whether lines are made up of points or not if points > are in fact defined by the intersection of lines. > Can't lines be defined mutually as the set of intersections with other lines? Maybe, though that doesn't identify a root concept. I would say that the point is undoubtedly the atom of space, indivisible while every line or segment can be divided at any given internal point. In that sense, the point is more elementary than the line. The relevant question, as I see it, is whether useful mathematics can be developed by only looking at measureless points, or whether the concept of the line is more central to what really flowers in science and math. A line connects two points, and a point identifies the intersection between two lines. Is one more "important" than the other? >> The fact that maybe no one can bring it in some mechanically >> exhaustive terms right in this second does not change anything in the >> question. > > No of course not. However the issue I'm really interested in doesn't > require a mechanically exhaustive or any other kind of definition for > points apart from what is mentioned directly above. > Do you think ALL points are "intersections of lines"? I am not sure this is so, especially when you consider a 1-D space like the number line, where there are no other lines to intersect with. There are certainly still some infinite number of points. >> After all there is a number of unresolved problems not >> because they don't have any solution but simply because they are not >> solved yet due to different obstacles. >> But as long as we arrived to such entities as "point", "line", >> "infinite set", "natural number", "real number", "irrational number" >> etc. - as long that: do you believe that each of them there is one and >> only one proper mechanically exhaustive definition to find - coming >>from the very nature of these entities? > > Once again probably so. I just haven't spent a lot of time on those > issues as yet - if I ever do. > Oh, do! >> So once found we may expect >> them universally correct, so even for some civilization from another >> star they will be necessary either the same or wrong (so the said >> civilization did not find the proper definition yet)? > > Sure. The truth of what I'm after is universal in scope and not just a > particular or local truth in the sense it is demonstrably so. > > ~v~~ I have some ideas concerning universal truth, but you might not find them quite tautologically regressive enough. Or, you might. 01oo (Tucker's being snuggly on my back. Bad Puddy!)
From: Tony Orlow on 19 Mar 2007 11:48
PD wrote: > On Mar 18, 6:08 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: >> On 18 Mar 2007 10:36:04 -0700, "PD" <TheDraperFam...(a)gmail.com> wrote: >> >>>> All of a sudden you want to talk about original posts? I mean like the >>>> original post where in response to your specific questions I spell out >>>> the combined vector analysis pertinent to Michelson-Morley and you >>>> just ignore it but subsequently pretend there is no combined vector >>>> analysis relevant to Michelson-Morley? >>> Actually, no, I didn't ignore it. Others could see my posts, but you >>> (and to all evidence) you alone said you could not. Then you claimed >>> that I was "channeling" through someone else, who plainly could see my >>> posts and was responding to them. You, of course, assumed that the >>> problem was not yours, and that whatever was happening was by my >>> choice or design. >> Well if not by design a rather peculiar lacuna in any event since you >> subsequently asked me to repeat my analysis of Michelson-Morley. > > No, I asked you to do what you *claim* to do about your analysis of M- > M. > What you did in your "analysis" of M-M was propose (guess) a > polarization dependency of the speed of light, which you supposed > accounted for the null result. > But what you *claim* to do to establish truth of a proposal is to > catalog all alternatives and to demonstrate that they are false. This > you simply have not done in any explicit manner. If you have all those > in your notes somewhere in your bottom drawer, do please draw them out > and explicate them. > Of course, PD, you know it's impossible to enumerate all possible alternative explanations for a phenomenon, and that's why science works the way it does. It seems to me Lester wants to find a formula for truth, rather than a process to detect falsehoods. I don't see his vision in that respect, but I do agree with his disagreement regarding sets of points as full descriptions of geometric and physical objects, as far as he understands it himself. Right, Lester? >> Curiously I've never had the specific kind of problem you presented in >> this regard and I think you're being just a little too clever by half. >> >>>> Or the original post wherein I >>>> point out that points making up lines and the interesection of lines >>>> defining point is circular logic? Do tell which original posts exactly >>>> did you have in mind? >>> Yes, I believe I answered that post as well. In fact, mine was the >>> first response. Your memory is apparently dismal. >> I seem to recollect some kind of remarks but nothing I considered >> substantive. There's a huge difference between posting a reply and >> addressing the subject itself in terms responsive to those employed. >> >> However I have a further notion. I'd assumed when you said you were >> bored that I'd seen the last of you. If you wish to offer constructive >> criticism by all means do so. Just please get to the point. Brevity is >> the soul of wit and at this juncture you're neither. >> > > Brief enough for you? > > PD > > > My two cents worth, TO |