The Definition of Points
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From: Lester Zick on 29 Mar 2007 22:26 On Thu, 29 Mar 2007 16:56:18 -0600, "nonsense(a)unsettled.com" <nonsense(a)unsettled.com> wrote: >Lester Zick wrote: >> On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >> >>>>Finite addition never produces infinites in magnitude any more than >>>>bisection produces infinitesimals in magnitude. It's the process which >>>>is infinite or infinitesimal and not the magnitude of results. Results >>>>of infinite addition or infinite bisection are always finite. >>>> >>>> >>>>> Wrong. >>>> >>>>Sure I'm wrong, Tony. Because you say so? >>>> >>> >>>Because the results you toe up to only hold in the finite case. >> >> >> So what's the non finite case? And don't tell me that the non finite >> case is infinite because that's redundant and just tells us you claim >> there is a non finite case, Tony, and not what it is. >> >> >>> You can >>>start with 0, or anything in the "finite" arena, the countable >>>neighborhood around 0, and if you add some infinite value a finite >>>number of times, or a finite value some infinite number of times, you're >>>going to get an infinite product. If your set is one of cumulative sets >>>of increments, like the naturals, then any infinite set is going to >>>count its way up to infinite values. >> >> >> Sure. If you have infinites to begin with you'll have infinites to >> talk about without having to talk about how the infinites you >> have to talk about got to be that way in the first place. > > >Confused about absolute infinity? :-) Someone is. ~v~~
From: Lester Zick on 29 Mar 2007 22:52 On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> wrote: >Your "not a not b" has an assumed OR in it. Tony, let me ask you something: without an AND or any other conjunction how would you mechanize the OR conjunction you claim I assume? And if it's just there as a basic circumstance of nature how do you get from there to other conjunctions and logic especially when you consider there's no necessity for conjoined components to be present together at the same time? ~v~~
From: Tony Orlow on 30 Mar 2007 12:39 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. 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. :) Tony 01oo
From: Tony Orlow on 30 Mar 2007 12:50 Lester Zick wrote: > On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >>> Problem is that 0 and 1 are finites but so are lots of other numbers >>> and your original contention was that 00*0=finites not 1 so that you >>> still haven't clarified the process involved in 00*0 that makes it 1 >>> and not some other finite or which makes the process reversible. >>> >> 0 is not really finite, but The Origin. It is not a finite distance from >> The Origin, because there is no distance between it and itself. 0 is >> less than any finite, or infinitesimal, distance. What I meant is I*i=1. >> Oh, that's much better. Infinity times iota equals 1. :) >> >> So, here's how it hangs. In any interval of the real line, [x, x+1) or >> (x,x+1], we have oo reals. Each real will then be assumed to occupy 1/oo >> of this line, and if the length of this line is oo, then there will be >> oo^2 reals on the line, instead of 2^aleph_0, as if aleph_0 means >> anything anyway. There's no smallest infinite any more than a smallest >> nonzero finite, or infinitesimal. >> >> It is a simple assumption that subtracting a positive number from any >> other decreases it. One thing you may notice is that somehow >> aleph_0-1=aleph_0. There is no smallest infinity, though that's what >> aleph_0's supposed to be. Aleph_0 is a phantom. >> >> Of course, as WM correctly insists, asserting that there are oo naturals >> starting with number 1 directly implies that there is a natural oo, >> since the nth is always equal to n, and saying there are n many is >> equivalent to saying there is an nth one in any sequential ordering, >> which is the last. >> >> In any case, it's quite reversible, and well-defined. > > Sure, Tony. But only because you say it is and not because you show > how any of the mechanics associated with subtraction, addition, and > multiplication, and division are the same as those in ordinary finite > mathematics. In other words it's a lot more than just saying 00*0=1 > and presumably that 00=1/0 and 0=1/00. You're mixing up finites > and things you call infinites without defining them in terms which are > mechanically reciprocally exhaustive and true of each other. In this > regard you can't just say 00*0=1 without showing the extent to which > infinites like 00 and 0 can participate in ordinary finite arithmetic > operations with 1 and other finites and do so unambiguously. > We are free to discuss that. The T-riffic number system handles rational portions of uncountable infinities quite well, digitally, thanks. If you have specific questions, they can't be more difficult than many many that I've received while developing them. Then we can get into the H-riffics, if you have the stomach for it... > There is a reason division of finites like n by zero are not defined. Because infinity is all the same. But, it's not. > It's because any n*0=0 so that finite division by zero is ambiguous. n*0=0 for finite n. How many points are in a 1-unit line segment, and what else is in there? > In other words any n*0=0 so we can't just reverse the operation > concluding n/0= any specific value. n*0=0 is an axiomatic statement. Can you demonstrate, through exhaustive mechanical elimination of self-contradictory alternatives, or by some other method, the truth of such a statement? Infinites mean in-finite or not > defined with respect to magnitude. Or, larger than any finite number, as defined by containment within an interval defined by finite naturals, which are sizes of finite sets, non-bijectible with any subset thereof. And the only way we can address > relations between zeroes and in-finites is through L'Hospital's rule > where derivatives are not zero or in-finite. And all I see you doing > is sketching a series of rules you imagine are obeyed by some of the > things you talk about without however integrating them mechanically > with others of the things you and others talk about. It really doesn't > matter whether you put them within the interval 0-1 instead of at the > end of the number line if there are conflicting mechanical properties > preventing them from lying together on any straight line segment. > > ~v~~ Well, if you actually paid attention to any of my ideas, you'd see they are indeed mostly mechanically related to each other, but you don't seem interested in discussing the possibly useful mechanics employed therein. L'Hospital's Rule (actually one of the Bernoullis', but whatever) may be deeply significant. I've heard differing opinions on that. 01oo
From: JAK on 30 Mar 2007 12:58
On Mar 29, 7:52 pm, Lester Zick <dontbot...(a)nowhere.net> wrote: > >Your "not a not b" has an assumed OR in it. And so, too, does your quest for a definition of "point." Just as boolean ANDs and ORs entail comparisons, definitions also entail comparisons. We live in a universe of limits. The act of "definition" is a search for a limit, whatever it may be. Thus, by attempting to define "a point" (by topology or otherwise), an attempt to find "limits" is inherent in the behavior. And any limit is identified through our use of the term "not" (or "false"). "Not" is the foundation of sentential logic and the foundation of any definition. A desk is not a chair. A parabola is not a trapezoid. A point is not a 72 piece orchestra. "Not" defines the limit of any concept. And such definition is made through a comparison, a relationship comparing "that which is" with "that which is not". If no limit is found, then it becomes universal. A "point" without limits has no size, no relationships. It is ubiquitous, eternally omnipresent, infinite, without limits. However, without limits, a "point" can never have multiples. There could never be 2 points. Once you have 2 points, each point limits and defines the other. How they limit each other creates their definition. This is done through comparison (as noted): "not a not b". In other words, "point a" is "not point b" and vice versa. I believe a fine answer was posted earlier (by Eric, as I recall) noting that points are relative. And the posting of "not a not b" (Tony?) is also excellent. Either response was great. Combined, they are superb. To define the term "point" (or any term for that matter) without the use of "not" (or "false") is futile. Once a "not" is introduced, a comparison is inherent (boolean or otherwise). The comparison is a relationship between "that which is" and "that which is not". A "point" is a concept in abstract space. Its definition relies upon the placement of other concepts sharing its abstract space - lines, planes, spheres, other points, etc. |