The Definition of Points
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From: Alan Smaill on 14 Apr 2007 14:56 Lester Zick <dontbother(a)nowhere.net> writes: > On Sat, 14 Apr 2007 13:56:37 +0100, Alan Smaill > <smaill(a)SPAMinf.ed.ac.uk> wrote: > >>Lester Zick <dontbother(a)nowhere.net> writes: >> >>> On Fri, 13 Apr 2007 16:10:39 +0100, Alan Smaill >>> <smaill(a)SPAMinf.ed.ac.uk> wrote: >>> >>>>Lester Zick <dontbother(a)nowhere.net> writes: >>>> >>>>> On Thu, 12 Apr 2007 14:23:04 -0400, Tony Orlow <tony(a)lightlink.com> >>>>> wrote: >>>>>> >>>>>>That's okay. 0 for 0 is 100%!!! :) >>>>> >>>>> Not exactly, Tony. 0/0 would have to be evaluated under L'Hospital's >>>>> rule. >>>> >>>>Dear me ... L'Hospital's rule is invalid. >>> >>> What ho? Surely you jest! >> >>Who, me? >> >>> Was it invalid when I used it in college? >> >>If you used it to work out a value for 0/0, then yes. > > Well the problem is that you didn't claim my application of > L'Hospital's rule was invalid. You claimed the rule itself was > invalid. So perhaps you'd like to show how the rule itself is invalid > or why my application of the rule is? Or both: The rule is invalid because that's what you find in Hospitals. Your use is invalid because the rule says nothing about the value of 0/0. > ~v~~ -- Alan Smaill
From: Lester Zick on 14 Apr 2007 14:57 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. > 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. > Is there such a thing as >truth, or falsity? Of course. > Does logic "exist". Yes. >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~~
From: Lester Zick on 14 Apr 2007 15:16 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. >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. > 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. >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~~
From: Lester Zick on 14 Apr 2007 15:20 On 13 Apr 2007 11:36:03 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >> I've discussed that with you and others. It doesn't cover the cases I am >> talking about. The naturals have a "measure" of 0, no? So, measure >> theory doesn't address the relationship between, say, the naturals and >> the evens or primes. It's not as general as it should be. So, what do >> you want me to say? > >Nothing, really, until you learn the mathematics you're pretending to >know about. Whereas you yourself should say whatever you want, Moe(x), about the mathematics you're pretending to know about? What mathematics would that be anyway, SOAP operas? ~v~~
From: Lester Zick on 14 Apr 2007 15:26
You know, Moe(x), I've never read quite the collection of buzzwords you provide here in any one location before. Is mathematics to you just a set of all slogans in any given domain of discourse? On 13 Apr 2007 14:32:45 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >On Apr 13, 12:51 pm, Tony Orlow <t...(a)lightlink.com> wrote: >> MoeBlee wrote: >> > On Apr 13, 11:25 am, Tony Orlow <t...(a)lightlink.com> wrote: >> >> >> I've discussed that with you and others. It doesn't cover the cases I am >> >> talking about. The naturals have a "measure" of 0, no? So, measure >> >> theory doesn't address the relationship between, say, the naturals and >> >> the evens or primes. It's not as general as it should be. So, what do >> >> you want me to say? >> >> > Nothing, really, until you learn the mathematics you're pretending to >> > know about. >> >> I didn't bring up "measure theory". > >Where do I begin: transitivity, ordering, recursion, axiom of >infinity, non-standard analysis...on and on and on... > >> > Nothing to which you responded "pretends" that cardinality "can tell >> > things it can't". What SPECIFIC theorem of set theory do you feel is a >> > pretense of "telling things that it can't"? >> >> AC > >If you mean non-constructivity, then no one disputes that the axiom of >choice is non-constructive. No one says that the axiom of choice >proves the existence of a definable well ordering. > >But if you require constructivity then you can't without contradiction >endorse Robinson's non-standard analysis. > >> >>> It has CARDINALITY aleph_0. If you take "size" to mean cardinality >> >>> then aleph_0 is the "size" of the set of naturals. But it simply isn't >> >>> true that "a set of naturals with 'size' y has maximum element y" if >> >>> "size" means cardinality. >> >> I don't believe cardinality equates to "size" in the infinite case. >> >> > Wow, that is about as BLATANTLY missing the point of what you are in >> > immediate response to as I can imagine even you pulling off. >> >> > MoeBlee >> >> What point did I miss? > >The MAJOR point - the hypothetical nature of mathematical reasoning >(think about the word 'if' twice in the poster's paragraph) and the >inessentiality of what words we use to name mathematical objects and >their properties. > >I've been trying to get you to understand that for about two years >now. > >> I don't take transfinite cardinality to mean >> "size". You say I missed the point. You didn't intersect the line. > >You just did it AGAIN. We and the poster to whom you responded KNOW >that you don't take cardinality as capturing your notion of size. The >point is then just for your to recognize that IF by 'size' we mean >cardinality, then certain sentences follow and certain sentences don't >follow and that what is important is not whether we use 'size' or >'cardinality' or whatever word but rather the mathematical relations >that are studied even if we were to use the words 'schmize' or >'shmardinal' or whatever. > >MoeBlee > ~v~~ |