The Definition of Points
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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.
From: Tony Orlow on 30 Mar 2007 13:00 Lester Zick wrote: > On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >>>>> Whatever. Is a point really nothing? Is a zero really nothing? Who >>>>> cares. If you want to fudge things why not just say 1 is zero? Then we >>>>> can all stop worrying about it one way or the other and go home. >>>>> >>>> That would cause inconsistencies. :) >>> And 00*0=1 wouldn't cause inconsistencies? >>> >> I*i=1 doesn't. Well, as long as you know it's not the imaginary i.... > > Well that's only one potential inconsistency. You still haven't shown > why 00*0=1 and not some other finite. It looks to me like you're just > trying to axiatomize zero and points without being able to show why > infinitesimal bisective subdivision could never reach and surpass such > atomic points without reaching zero. > > ~v~~ I would assert that a number, I, exists which is greater than any finite number, and justify that axiom by considering the interval [0,1), where 0<1, and for x, y and z in [0,1), x<z -> Ey x<y ^ y<z. If there is a measurable interval, and any measurable interval can be divided into two measurable intervals by defining an additional point between the endpoints, then this process of defining additional points in the interval never ends. The number of individual points in the interval can be said to be greater than any finite number, because any finite number n, producing 2^n-1 intermediate points, still leaves points between those unincluded. So, we say that some infinite number of points exists in [0,1), called I, and that each point therefore occupies 1/I=i space on that line, an infinitesimal. 01oo
From: Tony Orlow on 30 Mar 2007 13:04 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. > > 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)). 01oo
From: Tony Orlow on 30 Mar 2007 13:06
Lester Zick wrote: > On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >>>> You might be surprised at how it relates to science. Where does mass >>>> come from, anyway? >>> Not from number rings and real number lines that's for sure. >>> >> Are you sure? > > Yes. > >> What thoughts have you given to cyclical processes? > > Plenty. Everything in physical nature represents cyclical processes. > So what? What difference does that make? We can describe cyclical > processes quite adequately without assuming there is a real number > line or number rings. In fact we can describe cyclical processes even > if there is no real number line and number ring. They're irrelevant. > > ~v~~ Oh. What causes them? 01oo |