The Definition of Points
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From: Tony Orlow on 17 Apr 2007 13:29 Lester Zick wrote: > On Fri, 13 Apr 2007 14:40:40 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Thu, 12 Apr 2007 14:29:22 -0400, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>> Lester Zick wrote: >>>>> On Sat, 31 Mar 2007 21:14:27 -0500, Tony Orlow <tony(a)lightlink.com> >>>>> wrote: >>>>> >>>>>> Yeah, "true" and "false" and "or" are kinda ambiguous, eh?" >>>>> They are where your demonstrations of their truth are concerned >>>>> because there don't seem to be any. You just trot them out as if they >>>>> were obvious axiomatic assumptions of truth not requiring any >>>>> mechanical basis whatsoever or demonstrations on your part. >>>>> >>>>> ~v~~ >>>> So, you're not interested in classifying certain propositions as "true" >>>> and others as "false", so each is either true "or" false? I coulda >>>> swored you done said that....oh nebbe mine! >>> It makes no difference how you classify proposition as true or false >>> when you can't demonstrate how it is they're true or false to begin >>> with. Just saying they're true or false is irrelevant unless you can >>> show why and how. That's what I'm trying to point out to you. You seem >>> stuck on merely assuming certain propositions are true or false. >>> >>> ~v~~ >> Look, Lester, if you're actually interested in the mechanics of logical >> truth, then you are looking for general rules. These rules cover the >> general case, all input combinations, all possibilities. When evaluating >> a statement based on assumptions, > > Well there are "assumptions" Tony, and then there are "assumptions". I > assume there are things which can be true or not whereas you assume > you already understand which things are true or not and how to work > with their truth in mechanical terms while I'm interesting in finding > out how to demonstrate their truth in mechanical terms to begin with > and how to work with that truth or lack of it in mechanical terms. > >> do please assess each of those >> assumptions for certitude when assessing the statement's truth, but when >> speaking of the "mechanics" of deduction or induction it doesn't help to >> worry about "assuming". > > Of course, Tony. You just take assumptions for granted. I don't. > That's the whole context of the discussion. Mathematikers and empirics > can't be bothered to demonstrate their assumptions. > Which class do you fall into? Lester e Mathematikers, or Lester e Empirics, or Lester e Bullshitters? What have you demonstrated, besides exhaustive somethingorother? >> That's inherent to the process, if your process >> has anything to do with science. You "assume" there is such a thing as >> truth. Is there? What's the alternative? > > Well it's about time you started to ask questions, Tony, instead of > making problematic proclamations of mathematical certitude. You don't answer questions, Lester. Did you this time? ... I don't > assume there is any such thing as truth. I just posit a mechanism of > universal tautological contradiction and show that one alternative is > self contradictory so I assume the other alternative is universally > characteristic of everything which is not self contradictory and that > is what I call true just as I call self contradiction false. In other > words it's the mechanics underlying the determination of true and > false that I'm trying to get at and not just the proclamation of true > and false and the binary mechanics of working with those results. Oh. You don't assume. You posit. Huh! Is that like depositing, as opposed to withdrawing? You're certainly not withdrawing, that I can see. What does "posit" mean? http://dictionary.reference.com/browse/posit Huh!!!! It means "assume"!!! Wow, that's strange.... > > You on the other hand seem hell bent on explaining the mechanics of > working with the results of true and false without explaining the > mechanics of determining those circumstances. Mathematikers can't > bring themselves to call the results "true" and "false" with straight > faces so they just call them "truth values" instead of true and false. That's because the premises are...posited and not proven. > Let me ask you something, Tony. When you send off for some truth value > according to "true(x)" and it returns a 1 or 0 or whatever, how is the > determination of that "truth value" made? From the truth values of the posited assumptions, of course, just like yours. And if it's just made in > accordance with the manipulation of other "truth values" how are those > "truth values" determined? That's an inductive matter, based on evidence, and in the case of math, the acceptability of the conclusions derived from the posited assumptions. Or is it all just a bunch of running "truth > value" manipulations with no beginning or end? If that's all they are > then you have no reason to call "truth values" "truth" values and you > might just call them what they are 1's and 0's because that's all they > really are. You cannot deduce conclusions without inducing assumptions, for the sake of logical consideration. > > So what all this nonsense comes down to is that your "truth values" > have no beginning in actual mechanical terms because they're given to > you by assumption and not demonstrated in mechanical terms and all > those conjunctions and conjunctive manipulations you describe are just > so many arbitrary translation rules to work with otherwise meaningless > 1's and 0's. > > ~v~~ 0 and 1 are meaningful. They are nothing and all. 01oo
From: Tony Orlow on 17 Apr 2007 13:33 Lester Zick wrote: > On Fri, 13 Apr 2007 16:19:04 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On 13 Apr 2007 11:24:48 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >>> >>>> On Apr 13, 10:56 am, Tony Orlow <t...(a)lightlink.com> wrote: >>>>> Well, of course, Moe's technically right, though I originally asked >>>>> Lester to define his meaning in relation to his grammar. Technically, >>>>> grammar just defines which statements are valid, to which specific >>>>> meanings are like parameters plugged in for the interpretation. >>>> That is completely wrong. You have it completely backwards. What you >>>> just mentioned is part of semantics not grammar. Grammar is syntax - >>>> the rules for formation of certain kinds of strings of symbols, >>>> formulas, sentences, and other matters related purely to the >>>> "manipulation" of sequences of symbols and sequences of formulas, and >>>> of such objects. On the other hand, semantics is about the >>>> interpretations, the denotations, the meanings of the symbols, strings >>>> of symbols, formulas, sentences, and sets of sentences. Mathematical >>>> logic includes the study of these two things - syntax and semantics - >>>> both separately and in relation to each other. >>>> >>>>> I asked >>>>> the question originally using truth tables to avoid all that, so that we >>>>> can directly equate Lester's grammar with the common grammar, on that >>>>> level, and derive whether "not a not b" and "not a or not b" were the >>>>> same thing. They seem to be. >>>> Truth tables are basically a semantical matter. Inspection of a truth >>>> table reveals the truth or falsehood of a sentential formula per each >>>> of the assigments of denotations of 'true' or 'false' to the sentence >>>> letters in the formula. >>> If any and all these things are not demonstrably true and merely >>> represent so many assumptions of truth why would anyone care what you >>> think about what they are or aren't? I mean it really isn't as if >>> truth is on your side to the exclusion of what others claim, Moe(x). >>> >>> ~v~~ >> Define "assumption". > > Any declarative judgment not demonstrated in mechanically exhaustive > terms. > >> Do you "believe" that truth exists? > > Of course. > Prove it in "mechanically exhaustive terms". >> Is there a set >> of statements S such that forall seS s=true? > > No idea, Tony. There looks to be a typo above so I'm not sure exactly > what you're asking. > I am asking, in English, whether there is a set of all true statements. >> Is there such a thing as >> truth, or falsity? > > Of course. > "Prove" logic exists, in terms that precede logic. >> Does logic "exist". > > Yes. > Prove it from first principles. Unless, of course, you're just "positing". >> There exists a set of assumptions, A, which are true. >> >> True? > > Yes. The difference is that to the extent they're undemonstrated > assumptions we can have no idea which assumptions are true. > > ~v~~ The difference between which duck? Quack, Lester. Quack. 01oo
From: Tony Orlow on 17 Apr 2007 13:39 Lester Zick wrote: > On Fri, 13 Apr 2007 14:36:12 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Thu, 12 Apr 2007 15:30:57 -0400, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>> Lester Zick wrote: >>>>> On Mon, 2 Apr 2007 16:12:46 +0000 (UTC), stephen(a)nomail.com wrote: >>>>> >>>>>>> It is not true that the set of consecutive naturals starting at 1 with >>>>>>> cardinality x has largest element x. A set of consecutive naturals >>>>>>> starting at 1 need not have a largest element at all. >>>>>> To be fair to Tony, he said "size", not "cardinality". If Tony wishes to define >>>>>> "size" such that set of consecutive naturals starting at 1 with size x has a >>>>>> largest element x, he can, but an immediate consequence of that definition >>>>>> is that N does not have a size. > >>>>> Is that true? >>>>> >>>>> ~v~~ >>>> Yes, Lester, Stephen is exactly right. I am very happy to see this >>>> response. It follows from the assumptions. Axioms have merit, but >>>> deserve periodic review. >>> What follows from the assumptions, Tony? Truth? > >> "that N does not have a size." > > I wasn't commenting on whether your assumptions are consistent with > your axioms, Tony. I was asking whether your assumptions were true. > So, then. it's not true that every statement is either true or false. What about the statement that every statement is true or false? That's false? Perhaps it's not possible to determine the root of truth in any deductive manner, but that determining truth of statements is an infinite regress called "science". Have you considered that notion? >> If the assumptions >>> were true and could be demonstrated they wouldn't have to be assumed >>> to begin with. >> Can we assume that a statement is either true, or it's false? > > Sure. Happens all the time. However if you're asking whether a > statement must be one or the other the answer is no. There are > problematic exceptions to the so called excluded middle. > Please eloborate. >> Is that >> too much of an assumption to make, when exploring the meaning of truth? >> In ways yes, but for a start, no. > > Well your phrase "exploring the meaning of truth" is ambiguous, Tony, > because what you're really doing is exploring consequences of truth or > falsity given assumptions of truth or falsity to begin with, which is > an almost completely trivial exercise in comparison with the actual > determination of truth in mechanically exhaustive terms initially. > I am exploring the mechanics of truth, and its pursuit, which you are not, really, as far as I can tell. >> Mathematikers and empirics expect their students to use >>> the most rigorous, exhaustive mechanics in extrapolating theorems and >>> experimental methods from foundational assumptions. But the minute the >>> same requirements of rigorous mechanics are laid on them and their own >>> axioms and foundational assumptions they cry foul and claim no one can >>> prove their assumptions and that even their definitions are completely >>> arbitrary and can be considered neither true nor false. >>> >>> ~v~~ >> The question about axioms is whether each one is justifiable and >> sufficiently general enough to be accepted as "true" in some universal >> sense. > > No the actual question is whether each and every axiom is actually > true and demonstrably so in mechanically exhaustive terms. Otherwise > there's not much point to the exhaustively rigorous demonstration of > theorems in terms of axioms demanded of students if axioms themselves > are only assumed true. > > ~v~~ I am saying that one can assume axioms for the sake of deduction, but that the conclusions derived are only as reliable as the starting axioms, and so there is an inductive process in deciding which axioms to accept for the sake of one's "theory", expecially when looking for universal truths that serve as axions in a TOE, depending on whether the conclusions drawn fit the empirical evidence. 01oo
From: Tony Orlow on 17 Apr 2007 13:45 Virgil wrote: > In article <462117d1(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Mike Kelly wrote: > >>> Blind axiomatics? So you think ZFC was developed by blindly? People >>> picked the axioms randomly without any real consideration for what the >>> consequences would be? Please. ZF(C) provides a foundation for >>> virtually all modern mathematics. This didn't happen by accident. >>> >>> What's "blind" about ZF(C)? What great insight do you think is missed >>> that you are going to provide, oh mighty revolutionary? What >>> mathematics can be done with your non-existant foundation that can't >>> be done in ZF(C)? >>> >> Axiomatically, I think the bulk of the burden lies on Choice in its full >> form. Dependent or Countable Choice seem reasonable, but a blanket >> statement for all sets seems unjustified. > > Since it has been shown that if ZF is consistent then ZFC must be > consistent as well, what part of ZF does TO object to? > Well, I also put the onus on the Extensionality, as far as equating sets with the same general membership, but different rates of growth per iteration, but I haven't quite figured out how to formalize that statement, or at least, am not in a position to do so now. > >>>> Then how do you presume to declare that my statement is "not true"? >>>> >> No answer? Do you retract the claim? >> Yes? >>>>> It's very easily provable that if "size" means "cardinality" that N >>>>> has "size" aleph_0 but no largest element. You aren't actually >>>>> questioning this, are you? >>>> No, have your system of cardinality, but don't pretend it can tell >>>> things it can't. Cardinality is size for finite sets. For infinite sets >>>> it's only some broad classification. > > It is one form of size for all sets. One might use the physical analogy > that volume, surface area, and maximum linear dimension are all measures > of the size of a solid. So implying that one "size" fits all is false. > > Each of those is derivative of the last, given the proper unit of measure. Ask me more about that, if you're interested. :) >>> OK so all of the above comes down to you demanding that we don't call >>> cardinality "size". If we don't call cardinality "size" then all your >>> objections to cardinality disappear. >> It would also be nice to have an alternative to cardinality > > Why? > Because intuitions and details are not satisfied by cardinality. >> So, what's your opinion of infinite-case induction, IFR and N=S^L, and >> multilevel logics, again? I forget. > > AS presented by TO, garbage, garage and garbage. There is a transfinite > induction but it doesn't work the way TO would have it work, and there > are a wide variety of logics none of which seems to work at all like TO > would have them work. Hmmm.... Garbage? I don't think you've actually considered them, but you may be too old to do so at this point. That's okay. I appreciate you, Virgil. Have a nice day. (no signature, but at least my name is real)
From: Tony Orlow on 17 Apr 2007 13:58
Virgil wrote: > In article <46211c0a(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <461fd938(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Virgil wrote: >>>>> Nothing in TO's definition of "<" prohibits '(x>y) and (y>x)' from being >>>>> true, so if he wishes to require such a prohibition, he must >>>>> specifically add it to his transistivity requirement. >>>>> >>>> Yeah, actually, I misspoke, in a way. Your statement is still blatantly >>>> false, in any case. It's possible for x<y and y<x in a cyclical-type >>>> system, but those two facts together do not imply x=y. >>> >>> But a "cyclical-type system" is not an "ordered system" in any standard >>> mathematical sense. >> Times of day have no order? Pulllease!!! > > Does 12 midnight on a clock come before or after 12 noon on that clock? > > There is an assumed local order in the sense that if two clock times are > close enough together one usually assumes that one of them is "before" > and the other "after", but is one minute before midnight clocktime > before or after one minute after midnight clocktime? it could be either. Yes, it can be that x<y and y<x and y<>x. >>> For any in which "<" is to represent the mathematical notion of an order >>> relation one will always have >>> ((x<y) and (y<x)) implies (x = y) >>> >> Okay, I'm worried about you. You repeated the same erroneous statement. >> You didn't cut and paste without reading, did you? Don't you mean "<=" >> rather than "<". The statement "x<y and y<y" can only be true in two >> unrelated meanings of "<", or else "=" doesn't have usable meaning. > > TO betrays his lack of understanding of material implication in logic. > For "<" being any strict order relation, "(x<y) and (y<x)" must always > be false so that any implication with "(x<y) and (y<x)" as antecedent > for such a relation, regardless of conseqeunt, is always true. Oh, yes, well. Any false statement implies any statement, true or false, as long as you're not an intuitionist. If (x<y) -> ~ (y<x), then x<y ^ y<x is of the form P ^ ~P, or ~(P v ~P) which is false in classical logic, but not intuitionistically. There is debate on this topic. > > So that I have better cause to be worried about TO than he has to worry > about me. > I won't take it personally. >>>> The rationals are defined by NxN, minus the redundancies in >>>> quantity within the matrix. >>> That "matrix" is a geometric interpretation, which is quite irrelevant. >>> >>> A better definition for the rationals, based on I as the set of integers >>> and P as the set of strictly positive integers is the set IxP modulo the >>> "==" relation defined by (a,b) == (c,d) iff a*d = b*c. >>> >>> >>> >>>> Equinumerous to those redundancies, which >>>> are the vast majority of cells, are the irrationals. That's how it >>>> actually works. >>> Is TO actually claiming that the irrationals form a subset of the >>> countable set NxN. >>> >>> That is NOT how it works in any standard mathematics. > > Then why does TO claim it? I am not. I am saying it is a set equal in magnitude to the redundancies in NxN. |