The Definition of Points
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From: Lester Zick on 25 Apr 2007 15:03 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. 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. 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~~
From: Lester Zick on 25 Apr 2007 15:11 On Tue, 24 Apr 2007 19:17:01 -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 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. >> >>> Well, if the axiom systems we develop produce the results we expect >>> mathematically, then we can be satisfied with them as starting >>> assumptions upon which to build. My issue with transfinite set theory is >>> that it produces a notion of infinite "size" which I find >>> unsatisfactory. I accept that bijection alone can define equivalence >>> classes of sets, but I do not accept that this is anything like an >>> infinite "number". So, that's why I question the axioms of set theory. >>> Of course, one cannot do "experiments" on infinite set sizes. In math, >>> one can only judge the results based on intuition. >> >> I was discussing Aristotelian syllogistic inference and truisms here, >> Tony. So I don't know why you're talking transfinite sets and so on. >> >> ~v~~ > >You're talking about logical deduction and starting assumptions, right? Well actually I was talking about the logic of deduction and the deductive demonstration of the truth of starting assumptions. >I am responding with measuring starting assumptions by the conclusions >that can be deduced from them. Which is known as empiricism. Postulate a set of starting assumptions and validate those assumptions based on the apparent utility of their predictions. > Perhaps you are questioning the nature of >logical implication itself. If so, how do you propose to derive logical >implication? Just the way I have: by regressing the nature of truth in general to alternatives to self contradiction in universal terms and application of the principle compounded in terms of itself to boolean conjunctions and mathematics as well as science of all kinds. ~v~~
From: Lester Zick on 25 Apr 2007 15:11 On Tue, 24 Apr 2007 19:21:19 -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: >> >>>>> If I say 3x+3=3(x+1) is that true? >>>>> Yes, it's true for all x. >>>> How about for x=3/0? >>>> >>> Division by pure 0 is proscribed because it produces an unmeasurable oo. >>> If x is any specific real, or hyperreal, or infinitesimal, then that >>> statement is true for all x. 3/0 is not a specific number. >> >> In other words the statement isn't true for all x. >> >> ~v~~ > >It's true for all x in R, all real numbers. I kind of thought that was >understood. I suppose if you want to say x is a chicken, or a tissue, or >a 3/0, then in a sense it's still true, but I meant it as an arithmetic >expression, with x as a real number. Trick question, Tony. Not really an issue. ~v~~
From: Lester Zick on 25 Apr 2007 15:12 On Wed, 25 Apr 2007 01:36:01 +0100, Ben newsam <ben.newsam.remove.this(a)gmail.com> wrote: >On Tue, 24 Apr 2007 15:12:19 -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 3x+3=3(x+1) is that true? >>>>> Yes, it's true for all x. >>>> >>>> How about for x=3/0? >>>> >>> >>>Division by pure 0 is proscribed because it produces an unmeasurable oo. >>>If x is any specific real, or hyperreal, or infinitesimal, then that >>>statement is true for all x. 3/0 is not a specific number. >> >>In other words the statement isn't true for all x. > ><Sigh> Tap * Dance = Porridge What is this preoccupation you seem to have with oatmeal, Ben? ~v~~
From: Lester Zick on 25 Apr 2007 15:18
On Tue, 24 Apr 2007 19:25:44 -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: >> >>>>> 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. >> >> ~v~~ > >I declare that logical statements have values of 0 or 1, or perhaps in >between those two. "False" and "True" are just names for 0 and 1. I >showed how they define each other using not(x), and how not(x) is the >only 1-place operator. What's left? What's left is that "true" and "false" are only your names for 1 and 0, Tony. I don't care what names you use you're still talking 1 and 0 and that's not the same as true and false. If you developed the ideas of true and false independently and demonstrated their truth in mechanically reduced exhaustive terms then you could name them anything you want and they would still mean true and false. But developing the idea of TvN binary 1 and 0 mathematically doesn't mean they're true and false regardless of the names you give them. ~v~~ |