The Definition of Points
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From: Lester Zick on 29 Mar 2007 14:34 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~~
From: Lester Zick on 29 Mar 2007 14:58 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. There is a reason division of finites like n by zero are not defined. It's because any n*0=0 so that finite division by zero is ambiguous. In other words any n*0=0 so we can't just reverse the operation concluding n/0= any specific value. Infinites mean in-finite or not defined with respect to magnitude. 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~~
From: Lester Zick on 29 Mar 2007 15:06 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~~
From: Lester Zick on 29 Mar 2007 15:20 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 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~~
From: Lester Zick on 29 Mar 2007 15:23
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~~ |