The Definition of Points
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From: Tony Orlow on 23 Apr 2007 11:54 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. 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. 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. If I say 3x+3=3(x+1) is that true? Yes, it's true for all x. If I say a=not b, is that true? Not if a and b are both true. 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. If you want to talk about the truth values of individual facts used in deduction, by all means, go for it. 01oo
From: Tony Orlow on 23 Apr 2007 11:57 stephen(a)nomail.com wrote: > In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >> stephen(a)nomail.com wrote: >>> In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >>>> stephen(a)nomail.com wrote: >>>>> In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >>>>>> stephen(a)nomail.com wrote: >>>>>>> In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >>>>>>>> Mike Kelly wrote: >>>>>>>>> The point that it DOESN'T MATTER whther you take cardinality to mean >>>>>>>>> "size". It's ludicrous to respond to that point with "but I don't take >>>>>>>>> cardinality to mean 'size'"! >>>>>>>>> >>>>>>>>> -- >>>>>>>>> mike. >>>>>>>>> >>>>>>>> You may laugh as you like, but numbers represent measure, and measure is >>>>>>>> built on "size" or "count". >>>>>>> What "measure", "size" or "count" does the imaginary number i represent? Is i a number? >>>>>>> The word "number" is used to describe things that do not represent any sort of "size". >>>>>>> >>>>>>> Stephen >>>>>> Start with zero: E 0 >>>>>> Define the naturals: Ex -> Ex+1 >>>>>> Define the integers: Ex -> Ex-1 >>>>>> Define imaginary integers: Ex -> sqrt(x) >>>>>> i=sqrt(0-(0+1)), so it's built from 0 and 1, using three operators. It's >>>>>> compounded from the naturals. >>>>> That does not answer the question of what "measure", "size" or "count" i represents. >>>>> And it is wrong on other levels as well. You just pulled "sqrt" out of the >>>>> air. You did not define it. Claiming that it is a primitive operator seems >>>>> a bit like cheating. And if I understand your odd notation, the sqrt(2) >>>>> is an imaginary integer according to you? And sqrt(4) is also an imaginary integer? >>>> No, but sqrt on the negatives produces imaginary numbers. Besides, sqrt >>>> can be defined, like + or -, geometrically, through construction. >>> You cannot define the sqrt(-1) geometrically. You are never going to draw a line >>> with a length of i. >>> >>>>> You also have to be careful about about claiming that i=sqrt(-1). It is much safer >>>>> to say that i*i=-1. If you do not see the difference, maybe you should explore the >>>>> implications of i=sqrt(-1). >>>>> >>>>> So what is wrong with >>>>> Start with zero: E 0 >>>>> Define the naturals: Ex -> Ex+1 >>>>> Define omega: Ax >>>>> I did that using only one operator. >>>>> >>>>> >>>> Ax? You mean, Ax x<w? That's fine, but it doesn't mean that w-1<w is >>>> incorrect. >>> No, I meant Ax. All of the natural numbers. I know you are incapable of >>> actually imagining "all", but others do not have that limitation. >>> >>> Of course, who knows what your notation is really supposed to mean. >>> What is (Ax)-1 supposed to be? How do you subtract one from all the naturals? >>> >>>>>> A nice picture of i is the length of the leg of a triangle with a >>>>>> hypotenuse of 1 and a leg of sqrt(2), if that makes any sense. It's kind >>>>>> of like the difference between a duck. :) >>>>> That does not make any sense. There is no point in giving a nonsensical >>>>> answer, unless you are aiming to emulate Lester. >>>>> >>>>> Stephen >>>>> >>>>> >>>> It's not nonsensical, and may even apply to uses of imaginary numbers in >>>> practice, but you can ignore it as I knew you would. That's okay. >>>> Tony >>> It is nonsense. Such a triangle does not exist. >>> >>> A >>> # >>> # >>> # >>> # >>> # >>> C############B >>> >>> Are you claiming this is a triangle? Are you claiming the distance from A to C >>> is i? How exactly is this supposed to be a picture of i? What is i measuring >>> in this picture? >>> >>> If this is not what you meant, please draw your picture of i. >>> >>> And you still have not answered what "size", "count" or "measure" i represents. >>> Is i a number, or not? >>> >>> Stephen > >> i is a number produced by extending completeness to the multiplicative >> field, after introducing completeness to the additive field to produce >> -1, like I said above. > > >> 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. > > 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.
From: Virgil on 23 Apr 2007 12:48 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. 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)) > > > > > 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.
From: Tony Orlow on 23 Apr 2007 13:29 Virgil wrote: > In article <4628df20$1(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > >>> I do say so. Does this mean you're going to stop claiming that >>> relations in set theory aren't based solely on 'e'? >>> >>> -- >>> mike. >>> >> No, not when sequences are defined using a recursively defined successor >> function, which is a relation between two elements, as opposed 'e', a >> relation between an element and a set. The combination of the two is >> what produces an infinite set, no? > > So that x -> x u {x} does not depend on 'e' ? > > On can define successor entirely in terms of 'e'. > > Successor, in the x -> x u {x} sense, depends only on singleton sets, > subsets and unions, all of which are defined strictly in terms of 'e', > so what is left? Nothing. Only the implication from the existence of one element to the next, which defines the successor relation as one between two elements, and in this case, equivalent to 'e'. If we define successor in the x -> x+1 sense, then it's not really based on 'e', but it does define the naturals, in a quantitative way. Successor CAN be related to 'e', but really depends on recursive implication of existence.
From: Lester Zick on 23 Apr 2007 14:47
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. 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. 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~~ |