The Definition of Points
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From: Tony Orlow on 24 Apr 2007 19:08 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. 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? 01oo
From: Tony Orlow on 24 Apr 2007 19:10 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? 01oo
From: Tony Orlow on 24 Apr 2007 19:12 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? 01oo
From: Tony Orlow on 24 Apr 2007 19:17 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? I am responding with measuring starting assumptions by the conclusions that can be deduced from them. Perhaps you are questioning the nature of logical implication itself. If so, how do you propose to derive logical implication? 01oo
From: Tony Orlow on 24 Apr 2007 19:21
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. 01oo |