The Definition of Points
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From: Ben newsam on 23 Apr 2007 19:54 On Mon, 23 Apr 2007 11:47:26 -0700, Lester Zick <dontbother(a)nowhere.net> wrote: >On Mon, 23 Apr 2007 11:54:05 -0400, Tony Orlow <tony(a)lightlink.com> >wrote: >> Truth tables and logical statements involving variables are >>just that. If I say, 3x+3=15, is that true? No, we say that IF that's >>true, THEN we can deduce that x=4. > >But here you're just appealing to syllogistic inference and truisms >because your statement is incomplete. You can't say what the "truth" >of the statements is or isn't until x is specified. So you abate the >issue until x is specified and denote the statement as problematic. > >The difficulty with syllogistic inference and the truism is Aristotle >never got beyond it by being able to demonstrate what if anything >conceptual was actually true. The best he could do was regress >demonstrations of truth to perceptual foundations which most people >considered true, even if they aren't absolutely true. But even based >on that problematic assumption he could still never demonstrate the >conceptual truth of anything beyond the perceptual level. Here we go, tap dancing in porridge again.
From: Bob Kolker on 23 Apr 2007 21:27 Ben newsam wrote: > > Here we go, tap dancing in porridge again. I thought it was word stew. Bob Kolker
From: Tony Orlow on 23 Apr 2007 21:31 Virgil wrote: > In article <462cd75a(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> stephen(a)nomail.com wrote: >>> In sci.math Tony Orlow <tony(a)lightlink.com> wrote: > >>>> Consider that you have a right triangle so A^2+B^2=C^2. Now, if you swap >>>> the values between the hypotenuse and one of the legs, that formula will >>>> produce a second leg of the original length times i. So, in a way it is >>>> like picturing an impossible triangle. You have to "imagine" the other >>>> leg of length i. :) >>> Swap the values between the hypotenuse and one of the legs? What >>> exactly does that mean? >>> >>>> A >>>> |\ >>>> | \ >>>> | \ 1 >>>> sqrt(2) | \ >>>> | \ >>>> | \ >>>> B i C >>> >>> So you are claiming that the obviously longer line from A to C >>> is shorter than the line from A to B? So you can just >>> assign any old "length" to a line regardless of how long >>> it actually is? >> If the distance from B to C is imaginary, then the hypotenuse is shorter >> than the other leg. A strange concept, but perhaps the best >> visualization one can get of a square root of a negative. > > Distances are not imaginary. Points in the complex planes may have > imaginary as well as real parts, but the distances between them are > still real. Distances didn't used to be negative, but the distance to the left of any point on a real line is a negative vector. Is distance an "absolute value". Is distance a "real value"? Is that a valid question? Yepperdoodles. > > And one can "visualize" the square roots (plural) of complex numbers > best in polar representation > If x + i*y = r*(cos(theta + 2*n*pi) + i*sin(theta + 2*n*pi)), > n in N and r >= 0, then the square roots (plural) of x + i*y are > sqrt(r)*(cos(theta/2 + n*pi) + i*sin(theta/2 + n*pi)) "Best" in some ways. Kind of like "quantitative order". :) <3 > >>> There is a way to make sense of this, but it is quite at odds >>> with traditional geometry, which seems to be the basis for all >>> your arguments. >>> >>> Stephen >>> >> It's somewhat at odds with traditional geometry. It's not a very well >> fleshed out idea. Just thought I'd mention the image. > > TO would do better to flush it out that to try to flesh it out. Or, flash it up. TADA!!! (whoosh) teedles
From: Tony Orlow on 23 Apr 2007 21:47 Lester Zick wrote: > On Mon, 23 Apr 2007 11:54:05 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Fri, 20 Apr 2007 12:00:37 -0400, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>>>> What truth have you demonstrated without positing first? >>>>> And what truth have you demonstrated at all? >>>>> >>>>> ~v~~ >>>> Assuming two truth values as 0-place operators, I demonstrated that >>>> not(x) is the only functional 1-place operator, and then developed the >>>> 2-place operators mechanically from there. That was truth about truth. >>> And how have you demonstrated the truth of the two "truth values" you >>> assumed? >>> >>> ~v~~ >> The "truth" of the two truth values is that I've declared them a priori >> as the two alternative evaluations for the truth of a statement. > > Well there are two difficulties here, Tony. You declare two a priori > alternatives but how do you know they are in fact alternatives? In > other words what mechanism causes them to alternate from one to the > other? It looks to me like what you actually mean is that you declare > two values 1 and 0 having nothing in particular to do with "truth" or > anything except 1 and 0. > > The second problem is what makes you think two "truth" alternatives > you declare are exhaustive? This is related to the first difficulty. > You can certainly assume one thing or alternative a priori but not > two. And without some mechanism to produce the second from the first > and in turn the first from the second exclusively you just wind up > with an non mechanical dualism where there is no demonstration the two > are in fact alternatives at all or exhaustive alternatives either. > > We already know you think there are any number of points in the > interval 0-1 so apriori declarations do not erase that inconsistency > between different sets of assumptions. > > A few days ago you asked about what I mean by "mechanics" or an > "exhaustive mechanics" and this is what I mean. You're welcome to > assume one mechanism but then you have to show how that one mechanism > produces every other aspect of the "mechanics" involved. > > And I believe it obvious that the one "mechanism" has to be the > process of "alternation" itself or there is no way to produce anything > other than our initial assumption. A chain is no stronger than its > weakest link and if you make dualistic apriori assumptions neither of > which is demonstrably true of the other in mechanically exhaustive > terms you already have the weakest link right there at the foundation. > >> That's >> not a matter of deduction until you specify what statements you are >> assuming true or false, and on what basis you surmise them to be one or >> the other. > > Well this comment is pure philosophy, Tony, because we only have your > word for it. You can certainly demonstrate the "truth" of "truth" by > regression to alternatives to "truth" by the mechanism of alternation > itself and I have no difficulty demonstrating the "truth" of "truth" > by regression to a self contradictory "alternatives to alternatives". > Of course this is only an argument not a postulate or principle but > then anytime you analyze "truth" you only have recourse to arguments. > >> Truth tables and logical statements involving variables are >> just that. If I say, 3x+3=15, is that true? No, we say that IF that's >> true, THEN we can deduce that x=4. > > But here you're just appealing to syllogistic inference and truisms > because your statement is incomplete. You can't say what the "truth" > of the statements is or isn't until x is specified. So you abate the > issue until x is specified and denote the statement as problematic. I don't consider it a problem, but a state of affairs. If one doesn't know what the state of affairs IS, they can say, IF it's THIS then THIS, and IF THAT then THAT. I don't know if THIS or THAT, but once I know, then I'll know. That how logic is. This comes back to axioms. Surely (or probably, or maybe) you see that. > > The difficulty with syllogistic inference and the truism is Aristotle > never got beyond it by being able to demonstrate what if anything > conceptual was actually true. May I address that? When I say that true() and false() are "zero-place" logical operators, that really means that they take no "logical parameters". That is, what is passed to the function for evaluation includes neither "true" nor "false" but is rather a statement which is to be evaluated, in the context of whatever "theory" we are considering, as either "true(x)" or "false(x)". The truth of x or y depends on what rules we consider valid in out language. The best he could do was regress > demonstrations of truth to perceptual foundations which most people > considered true, even if they aren't absolutely true. But even based > on that problematic assumption he could still never demonstrate the > conceptual truth of anything beyond the perceptual level. I'll get back "at" cha Lester. Gotta go. > > And here the matter has rested for mathematics and science in general > ever since. Empiricism benefitted from perceptual appearances of truth > in their experimental results but the moment empirics went beyond them > to explain results in terms of one another they were hoist with the > Aristotelian petard of being unable to demonstrate what was actually > true and what not. The most mathematicians and scientists were able to > say at the post perceptual conceptual level was that "If A then B then > C . . ." etc. or "If our axiomatic assumptions of truth actually prove > to be true then our theorems, inferences, and so forth are true". But > there could never be any guarantee that in itself was true. > > > >> If I say 3x+3=3(x+1) is that true? >> Yes, it's true for all x. > > How about for x=3/0? > >> If I say a=not b, is that true? Not if a and b >> are both true. > > How do you arrive at that assumption, Tony? If a and b are both true > of the same thing they can still be different from each other. > >> If I say a or not a, that's true for all a. a and b are >> variables, which may each assume the value true or false. > > Except you don't assign them the value true or false; you assign them > the value 1 or 0 and don't bother to demonstrate the "truth" of either > 1 or 0. > >> If you want to talk about the truth values of individual facts used in >> deduction, by all means, go for it. > > I don't; I never did. All I ever asked was how people who assume the > truth of their assumptions compute the truth value of the assumptions. > > ~v~~
From: Virgil on 23 Apr 2007 21:48
In article <462d5e09(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <462cd75a(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> stephen(a)nomail.com wrote: > >>> In sci.math Tony Orlow <tony(a)lightlink.com> wrote: > > > >>>> Consider that you have a right triangle so A^2+B^2=C^2. Now, if you swap > >>>> the values between the hypotenuse and one of the legs, that formula will > >>>> produce a second leg of the original length times i. So, in a way it is > >>>> like picturing an impossible triangle. You have to "imagine" the other > >>>> leg of length i. :) > >>> Swap the values between the hypotenuse and one of the legs? What > >>> exactly does that mean? > >>> > >>>> A > >>>> |\ > >>>> | \ > >>>> | \ 1 > >>>> sqrt(2) | \ > >>>> | \ > >>>> | \ > >>>> B i C > >>> > >>> So you are claiming that the obviously longer line from A to C > >>> is shorter than the line from A to B? So you can just > >>> assign any old "length" to a line regardless of how long > >>> it actually is? > >> If the distance from B to C is imaginary, then the hypotenuse is shorter > >> than the other leg. A strange concept, but perhaps the best > >> visualization one can get of a square root of a negative. > > > > Distances are not imaginary. Points in the complex planes may have > > imaginary as well as real parts, but the distances between them are > > still real. > > Distances didn't used to be negative, but the distance to the left of > any point on a real line is a negative vector. Distance is a scalar, not a vector. otherwise the distance from Chicago to New York wold be the negative of the distance from New Youk to Chicago. There is such a thing as a /directed/ distance, which, on a line, is like a one-dimensional vector. > Is distance an "absolute > value". Yup! > > And one can "visualize" the square roots (plural) of complex numbers > > best in polar representation > > If x + i*y = r*(cos(theta + 2*n*pi) + i*sin(theta + 2*n*pi)), > > n in N and r >= 0, then the square roots (plural) of x + i*y are > > sqrt(r)*(cos(theta/2 + n*pi) + i*sin(theta/2 + n*pi)) > > "Best" in some ways. Kind of like "quantitative order". :) <3 TO suggests that there are ways in which it is not best, but one of them not appear to exist within mathematics. |