The Definition of Points
in [Math]
|
Prev: Guide to presenting Lemma, Theorems and Definitions
Next: Density of the set of all zeroes of a function with givenproperties
From: Lester Zick on 26 Apr 2007 15:39 On Thu, 26 Apr 2007 02:25:25 +0100, Ben newsam <ben.newsam.remove.this(a)gmail.com> wrote: >On Wed, 25 Apr 2007 12:24:17 -0700, Lester Zick ><dontbother(a)nowhere.net> wrote: > >>On Wed, 25 Apr 2007 01:39:21 +0100, Ben newsam >><ben.newsam.remove.this(a)gmail.com> wrote: >> >>>On Tue, 24 Apr 2007 15:15:45 -0700, Lester Zick >>><dontbother(a)nowhere.net> wrote: >>> >>>>On Tue, 24 Apr 2007 09:27:05 -0400, Tony Orlow <tony(a)lightlink.com> >>>>wrote: >>>> >>>>>>> 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. >>>>>> >>>>> >>>>>1 is true, 0 is false. If a is 0 or 1, then we have "0 or 1", or "1 or >>>>>0", respectively. Since or(a,b) is true whenever a is true or b is true, >>>>>or both, or(1,0) and or(0,1), the only possible values for the >>>>>statement, are both true. So, or(a,not(a)) is always true, in boolean >>>>>logic, or probability. >>>>> >>>>>Intuitively, if a is a subset of the universe, and not(a) is everything >>>>>else, then the sum of a and not(a) is very simply the universe, which is >>>>>true. >>>> >>>>Yeah but you still haven't proven that 1 is true and 0 false or what >>>>either of these terms has to mean in mechanically exhaustive terms. >>> >>>Try this: "1" is everything or anything. "0" is whatever "1" is not. >>>You may assign the terms "true" and "false" if you wish. Or vice >>>versa. >> >>Yeah and when you do, Ben, all you'll have are TvN binary 1 and 0 and >>not "true" and "false". A rose by any other name would still be binary >>1 and 0 because all you've done is develop 1 and 0 mathematically and >>not in terms of what true and false really mean and necessarily have >>to mean in mechanically reduced exhaustive universal terms. > >What you fail to realise is that binary 1 and 0 are synonymous with >the terms "true" and "false". And what you fail to realize, Ben, is that you haven't proven this is true. It's merely an assumption on your part. If the terms "true" and "false" were truly synonymous with binary 1 and 0 in this sense why wouldn't the same apply to "figs" and "ideas" or any other pair of synonyms? There is no reason here to suggest any true synonomy between TvN binary 1 and 0 and "true" and "false" except a desire to systematize descriptions of "true" and "false" in binary mathematical terms but without extrapolating the truth of "true" and "false" in mechanically exhaustive terms to begin with. > Also synonymous would seem to be the >terms "mathematical" and your odd phrase "mechanically reduced >exhaustive universal". If we were merely assigning arbitrary aliases I would agree. The fact however is that what we're doing is trying to ascertain the truth of "true" and "false" in mechanical terms and not just assigning aliases. Presuming we already understand TvN binary mathematical logic sufficiently, what's the purpose of assigning the aliases "true" and "false" to 1 and 0? Obviously it's to pretend real truth and falsehood share identical properties with mathematical binary 1 and 0 when in fact we know nothing of the kind until we can demonstrate they share identical properties. And the fact you call 1 and 0 by other names has no affect on the properties of 1 and 0 or on the properties associated with those other names. As for the phrase "mechanically reduced exhaustive universal terms" my purpose in using it was to illustrate my approach to the demonstration of the real or actual meanings of "true" and "false" whether or not we can draft any coincidence between those meanings and binary 1 and 0. ~v~~
From: Tony Orlow on 26 Apr 2007 16:03 Lester Zick wrote: > On Tue, 24 Apr 2007 19:08:46 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Tue, 24 Apr 2007 09:27:05 -0400, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>>> 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. >>>>> >>>> Again, please comment on my use of "not" to define the relationship >>>> between true() and false(). >>> Okay that's an improvement. But one of the things I don't see is how >>> you produce and "truth values" between 0 and 1 using not. >>> >>> ~v~~ >> If you already have truth values between 0 and 1, as in a probabilistic >> model, then not(x) is defined as 1-x, which is between 0 and 1 for x >> between 0 and 1. In that kind of system, not(x) is arithmetic. For >> uncorrelated x and y, and(x,y)=x*y, and or(x,y)=not(and(not(x),not(y))), >> or 1-(1-x)(1-y), or x+y-x*y. > > Well, Tony, my problem comes with your apparent inference that 1=not 0 > and 0=not 1. That's not an inference. It's the definition of what not not means, together with the fact that not(not(x))=x. That can only be true in the context of TvN assumptions > and mechanics where no intermediate values can exist and the physical > constraints on bit representations in the system assure it. No, I just showed how to express not(x) as 1-x so as to extend the operation to intermediate values. In any finite system, of course, there will be a finite number of intermediate values, but so what? Otherwise > you have no strict binary mechanics and representations both within > and outside the limits 0-1 are possible. So either your mechanics is > binary and non probabalistic or probabalistic and non binary. I don't > see there is any other possibility. > Right, it's either binary or continuous and probabilistic, but what makes not(x) a logical function is that, given the value of 0 or 1, it produces a value of 0 or 1. In a probabilistic model, 1-x produces the same results for 0 and 1, and also produces values between 0 and 1 for any input between 0 and 1. It's a proper extension, on every level. >> Where there is a correlation between x and >> y, it becomes hairier. I think that's what you're trying to address with >> your large and/or green apples? > > Not really, Tony. I'm just trying to point out that the truth of > combinatorial predicates is inherently problematic and not > probablistic one way or the other. You haven't demonstrated anything specific about that. Have you some method in mind? > > Let's suppose we ask whether the proposition "large green apples are > large red apples" is true? The proposition itself is obviously false > and on that basis should warrant a truth value of 0. But when we > assign a truth value to a proposition the question is how do we do it? If there is an inherent contradiction, "red things are green", because red implies not green and vice versa, then we know the statement is false. We have partitioned the world into red things and green things and things of other colors, mutually exclusive sets of things. > > The fact is two of the predicates are true and one false. Ummm, you mean two of the three attributes defining the two sets are common between them. They are distinguished by the nature of their third attribute, color. Red and green are considered mutually exclusive. Once we see "large red..." and "large green..." we know we're talking about different things. It doesn't matter at that point that they are apples. Does that > mean t=0.000 or t=0.667? The same would apply to combinations of > propositions. Are we supposed to be taking an arithmetic average or > exercising some kind of intuitional insight? Not even to mention the > weighting of predicates. I just can't imagine that all predicates have > the same significance in terms of probablistic truth. Are we supposed > to just adopt someones weighting opinions on the subject of truth? > > ~v~~ Our brains weigh things all the time. That's how it works. If I say a word, it adds weight to all associated concepts. If I say another word it does the same, and if some concept is related to both, then it has become especially "weighted". When it gets too "heavy", depending on the activity level of consciousness, it "tips" and spills into consciousness, adding weight to all related concepts. Fruit Red George Washington What am I thinking of? Car Yellow Can you guess? "Logic" is an extension of association, as implemented in the human head. 01oo
From: Tony Orlow on 26 Apr 2007 16:12 Lester Zick wrote: > On Tue, 24 Apr 2007 19:10:15 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Tue, 24 Apr 2007 09:27:05 -0400, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>>> 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. >>>>> >>>> If you're discussing logic, you have the additional recourse to the >>>> mechanics of logic itself, the basics of which are well understood, if >>>> not widely. >>> What kind of logic do you have in mind? Boolean conjunctive logic, >>> truth value logic or what? I don't see these as mechanical. >>> >>> ~v~~ >> Well, machines can perform those operations just fine, so they seem >> pretty mechanical to me. Are you trying to determine the mechanics of >> induction rather than deduction? > > Except, Tony, your references to logic are all over the place. > > Where are these boolean conjunctions supposed to be? I've already > shown there are no boolean conjunctions in strict mechanical terms and > the only possible conjunction is "not" and compounding of "not". No, you didn't. You started with "not a not b", but interpreted as what most people would call "not a or not b". Then you compounded that with not to get a and b. But, you started, really, with "or" implicit. You notice I like to write these operators as functions, and that's for a reason. When you say "not(a) not(b)" those are two different truth values WITHOUT a conjunction. A single truth value has one operator outside parentheses. What you are actually talking about is or(not(a),not(b)). And you're right, not(or(not(a),not(b))) is the same as and(a,b). But it is not solely built upon not. not(x) can only take one parameter, so you cannot form an expression of any more than one parameter with not. You must have at least one of the non-trivial two-place operators, most commonly or(x,y) or and(x,y) in discussion, though NAND and NOR gates are used too. You started with an "or". If you disagree, then answer the question I asked forever ago about my simple truth table. > > Then when you willy-nilly appeal to TvN binary logic you can't even > show how you can accommodate both unambiguous truth values and > probabalistic values in one scheme. Just did. > > I mean you can't have it both ways, Tony. Either your mechanics is TvN > binary and non probablistic or probabalistic and non TvN binary. And > just saying machines do it just fine doesn't mean you can have it both > ways. > > ~v~~ One is a subset of the other. Duh. 01oo
From: Tony Orlow on 26 Apr 2007 16:14 Lester Zick wrote: > On Tue, 24 Apr 2007 19:12:48 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Tue, 24 Apr 2007 09:27:05 -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. >>>> Right. The truth of the statement 3x+3=15 cannot be determined without >>>> specifying x. That's my point. >>> But my point is that even with x you still haven't established the >>> truth of the axioms on which such statements are based. >>> >>> ~v~~ >> My empirical evidence gives me no reason to doubt that the system we're >> referring to models all finite numbers quite well. I think the truth of >> the axioms is measured by the truth of the facts it produces. You don't >> really doubt that x must be 4, do you? > > What I doubt is that your "no reason to doubt" is not the same as the > truth you claimed to have proven. I don't doubt that x can be 4 but I > doubt that you've shown x is 4 or x must necessarily be 4 when all > you've shown is that x can be 4 under certain assumptions of truth > when you haven't demonstrated the truth of those assumptions of truth. x is a variable! It could be "banana", but that won't solve the equation. Sheesh! > > I wonder if you really understood what I was getting at with my essay > on truisms and the nature of Aristotelian syllogistic inference? When > we have problematic circumstances we can certainly say "If A then B". > But that doesn't allow us to conclude "A" definitely is. And Aristotle > had a great deal useful to say about the evaluation of truth given the > facts of truth to begin with but he could never establish the fact of > truth itself to begin with nor why and how facts of truth were true. You do that by testing the predictions of your deductions. If they don't work, you got something wrong. > > And when I say "truth" and "demonstrations of "truth" I'm talking > about "truth" and not merely "truisms" such as "If A then B" whereas > what you and the rest of mathematics insist on talking about are > truisms such as "If axioms are true and our assumptions regarding > logic are true then theorems are true" and "If boolean assumptions > regarding truth and conjunctions and so forth are true then truth > values etc. are true" and so on. > > ~v~~ And so on...... 01oo
From: Ben newsam on 26 Apr 2007 18:53
On Thu, 26 Apr 2007 12:39:35 -0700, Lester Zick <dontbother(a)nowhere.net> wrote: >On Thu, 26 Apr 2007 02:25:25 +0100, Ben newsam ><ben.newsam.remove.this(a)gmail.com> wrote: >>What you fail to realise is that binary 1 and 0 are synonymous with >>the terms "true" and "false". > >And what you fail to realize, Ben, is that you haven't proven this is >true. It's merely an assumption on your part. If the terms "true" and >"false" were truly synonymous with binary 1 and 0 in this sense why >wouldn't the same apply to "figs" and "ideas" or any other pair of >synonyms? If figs and ideas are mutually exlusive, and everything is either a fig or an idea, then yes that would be fine. >There is no reason here to suggest any true synonomy between TvN >binary 1 and 0 and "true" and "false" except a desire to systematize >descriptions of "true" and "false" in binary mathematical terms but >without extrapolating the truth of "true" and "false" in mechanically >exhaustive terms to begin with. > >> Also synonymous would seem to be the >>terms "mathematical" and your odd phrase "mechanically reduced >>exhaustive universal". > >If we were merely assigning arbitrary aliases I would agree. The fact >however is that what we're doing is trying to ascertain the truth of >"true" and "false" in mechanical terms and not just assigning aliases. You are. I am not. To me, "1" and "0" are sufficient, and "true" and "false" are adequate aliases for them. >Presuming we already understand TvN binary mathematical logic >sufficiently, what's the purpose of assigning the aliases "true" and >"false" to 1 and 0? Obviously it's to pretend real truth and falsehood >share identical properties with mathematical binary 1 and 0 when in >fact we know nothing of the kind until we can demonstrate they share >identical properties. And the fact you call 1 and 0 by other names has >no affect on the properties of 1 and 0 or on the properties associated >with those other names. They are both mutually exclusive, and everything must be either one or the other. If you think they are not synonymous, perhaps you could point out how they are not? >As for the phrase "mechanically reduced exhaustive universal terms" my >purpose in using it was to illustrate my approach to the demonstration >of the real or actual meanings of "true" and "false" whether or not we >can draft any coincidence between those meanings and binary 1 and 0. |