The Definition of Points
in [Math]
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From: Tony Orlow on 31 Mar 2007 19:48 Virgil wrote: > In article <460e82b1(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > >> As I said to Brian, it's provably the size of the set of finite natural >> numbers greater than or equal to 1. No, there is no last finite natural, >> and no, there is no "size" for N. Aleph_0 is a phantom. > > All numbers are equally phantasmal in the physical world and equally > real in the mental world. Virgule, you don't really believe that, do you? You're way too smart for that... :) Tony
From: Lester Zick on 31 Mar 2007 19:50 On Fri, 30 Mar 2007 11:39:53 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> Hi Lester - >>> >>> Glad you responded. I was afraid I put you off. This thread seems to >>> have petered unlike previous ones I've participated in with you. I hope >>> that's not entirely discouraging, as I think you have a "point" in >>> saying points don't have meaning without lines, and that the subsequent >>> definition of lines as such-and-such a set of points is somewhat >>> circular. Personally, I think you need to come to grips with the >>> universal circularity, including on the level of logic. Points and lines >>> can be defined with respect to each other, and not be mutually >>> contradictory. But, maybe I speak too soon, lemme see... >> >> Hey, Tony - >> >> Yeah I guess I'm a glutton for punishment with these turkeys. The >> trick is to get finite regressions instead of circular definitions. We >> just can't say something like lines are the set of all points on lines >> because that's logically ambiguous and doesn't define anything. I >> don't mind if we don't know exactly what points are in exhaustive >> terms just that we can't use them to define what defines them in the >> intersection of lines and in the first place. >> >> The problem isn't mathematical it's logical. In mathematics we try to >> ascertain truth in exhaustively demonstrable terms. That's what >> distinguishes mathematics from physics and mathematicians from >> mathematikers and empirics. >> >> (By the way, Tony, I'm chopping up these replies for easier access and >> better responsiveness.) >> >> ~v~~ > >Sounds like a good idea. Well sure. It fertilizes the ground. At 800+ lines I don't know too many mathematikers capable of that level of reading comprehension. >I've certainly taken my licks around here too, but that's a big part of >this process - debate. You win some and lose some. Or rather, some lose >and some win, when it comes to ideas. And some just spar forever before >they finally realize they're actually dancing together, like waves and >particles, in a fluid universe. > >When you say it's not necessary to know exactly what points "are", >that's somewhat true. We don't even know what mass "is", but as in >science, we define objects by their properties and the predictions we >can make. So, if we can characterize the relationship between points and >lines, then we can define them relative to each other, which may be the >best we can do. But, that is not what you desire. You want a "ground >zero" starting point upon which all else is built. This is akin to set >theorists' e operator: "is an element of". They start with that one >operator, then supposedly measure is built upon that. Well, they do the >same thing you are doing when you assume an implied OR in "not a not b", >and then derive OR from AND, which you define as not(not a not b). They >introduce the von Neumann ordinals defined solely by set inclusion, and >yet, surreptitiously introduce the notion of order by means of this set. >Order, '<', is another operator and should be recognized as such. One >should allow that there are always two first elements or operators, >whose interplay produces the whole we're considering. That's the Tao. >It's not wrong. There is no single perspective, and there is no straight >line. It's all circles. You want tao, Tony, talk to Brian. You want mechanics talk to me. ~v~~
From: Mike Kelly on 31 Mar 2007 19:51 On 31 Mar, 16:47, Tony Orlow <t...(a)lightlink.com> wrote: > Mike Kelly wrote: > > On 31 Mar, 13:48, Tony Orlow <t...(a)lightlink.com> wrote: > >> step...(a)nomail.com wrote: > >>> In sci.math Virgil <vir...(a)comcast.net> wrote: > >>>> In article <460d4...(a)news2.lightlink.com>, > >>>> Tony Orlow <t...(a)lightlink.com> wrote: > >>>>> An actually infinite sequence is one where there exist two elements, one > >>>>> of which is an infinite number of elements beyond the other. > >>>> Not in any standard mathematics. > >>> It is not even true in Tony's mathematics, at least it was not true > >>> the last time he brought it up. According to this > >>> definition {1, 2, 3, ... } is not actually infinite, but > >>> {1, 2, 3, ..., w} is actually infinite. However, the last time this > >>> was pointed out, Tony claimed that {1, 2, 3, ..., w} was not > >>> actually infinite. > >>> Stephen > >> No, adding one extra element to a countable set doesn't make it > >> uncountable. If all other elements in the sequence are a finite number > >> of steps from the start, and w occurs directly after those, then it is > >> one step beyond some step which is finite, and so is at a finite step. > > > So (countable) sequences have a last element? What's the last finite > > natural number? > > > -- > > mike. > > As I said to Brian, it's provably the size of the set of finite natural > numbers greater than or equal to 1. Provable how? > No, there is no last finite natural, You keep changing your position on this. > and no, there is no "size" for N. Aleph_0 is a phantom. When we say that a set has cardinality Aleph_0 we are saying it is bijectible with N. Are you saying it's impossible for a set to be bijectible with N? Or are you saying N does not exist as a set? Something else? I find it very hard to understand what you are even trying to say when you say "Aleph_0 is a phantom". It seems a bit like Ross' meaningless mantras he likes to sprinkle his posts with. -- mike.
From: Lester Zick on 31 Mar 2007 19:53 On Fri, 30 Mar 2007 12:04:33 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>>>>> Okay, Tony. You've made it clear you don't care what anyone thinks as >>>>>> long as it suits your druthers and philosophical perspective on math. >>>>>> >>>>> Which is so completely different from you, of course... >>>> Difference is that I demonstrate the truth of what I'm talking about >>>> in mechanically reduced exhaustive terms whereas what you talk about >>>> is just speculative. >>> You speculate that it's agreed that not is the universal truth. It's not. >> >> No I don't, Tony. It really is irritating that despite having read >> E201 and E401 you call what I've done in those root threads >> "speculation". What makes you think it's speculation? I mean if you >> didn't understand what I wrote or how it demonstrates what I say then >> I'd be happy to revisit the issue. However not questioning the >> demonstration and still insisting it's speculation and no different >> from what you say is just not okay. > >I've questioned that assumption all along. We've spoken about it plenty. What assumption, Tony?You talk as if there is some kind of assumption. >> I don't speculate "it's agreed" or not. I don't really care whether >> it's agreed or not and as a practical matter at this juncture I'd have >> to say it's much more likely not agreed than agreed. What matters is >> whether it's demonstrated and if not why not and not whether it's >> agreed or not since agreements and demonstrations of truth are not the >> same at all. Agreements require comprehension and comprehension >> requires study and time whereas demonstrations of truth only require >> logic whether or not there is comprehension. >> >> ~v~~ > >Demonstrate what the rules are for producing a valid one of your logical >statements from one or more other valid ones of your logical statements, >because "not not" and "not a not b" are not standard valid logic >statements with known rules of manipulation. What are the mechanics? As >far as I can tell, the first is not(not(true))=true and the second is >or(not(a),not(b)), or, not(and(a,b)). Or you could demonstrate why the standard valid logic you cite is standard and valid. ~v~~
From: Tony Orlow on 31 Mar 2007 19:53
Lester Zick wrote: > On Fri, 30 Mar 2007 12:24:12 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>>>> Add 1 n >>>>>> times to 0 and you get n. If n is infinite, then n is infinite. >>>>> This is reasoning per say instead of per se. >>>>> >>>> Pro se, even. If the first natural is 1, then the nth is n, and if there >>>> are n of them, there's an nth, and it's a member of the set. Just ask >>>> Mueckenheim. >>> Pro se means for yourself and not for itself. >> In my own behalf, yes. >> >>> I don't have much to do >>> with Mueckenheim because he seems more interested in special pleading >>> than universal truth. At least his assumptions of truth don't seem >>> especially better or worse than any other assumptions of truth. >>> >>> ~v~~ >> He has some valid points about the condition of the patient, but of >> course he and I have different remedies. > > Some of which may prove deadly. > > ~v~~ Well, his mostly consist of amputation and leeches, but as long as he sticks to the extremities, I don't think death is inevitable... Mine don't actually break anything, except for the leeches, and some bones... 01oo |