On Sun, 16 Jul 2006 00:56:05 -0600, Virgil <email@example.com> wrote:
>In article <firstname.lastname@example.org>, > Lester Zick <DontBother@nowhere.net> wrote: > >> On Sat, 15 Jul 2006 15:16:06 -0600, Virgil <email@example.com> wrote: >> >> >In article <firstname.lastname@example.org>, >> > Lester Zick <DontBother@nowhere.net> wrote: >> > >> >> On Fri, 14 Jul 2006 21:36:09 -0600, Virgil <email@example.com> wrote: >> >> >> >> >In article <firstname.lastname@example.org>, >> >> > Lester Zick <DontBother@nowhere.net> wrote: > >> >> >> Are any of these things constructed theoremetrically from >> >> >> demonstrably true assumptions? >> >> > >> >> >What does a "constructed theoremetrically" mean in English? >> >> >> >> Probably exactly what it says. >> > >> >As I do not find any definition of "theoremetrically" anywhere, I assume >> >that it means nothing at all. >> >> Well, hell, Virgil, I just coined the term to describe the logical >> metric associated with the demonstration of theorems more artfully >> with considerably fewer words. > >In other words, I assumed correctly.
No actually you assumed incorrectly. But the primary problem is that you assumed to begin with without being able to demonstrate the truth of your assumption. You just assumed because you couldn't find the term anywhere it was meaningless.
>> >> >> > I don't see any >> >> >> >sense in asserting that the axioms of Euclidean geometry or of group >> >> >> >theory are true or wrong. >> >> >> >> >> >> But you see plenty of sense in assuming the truth of what you can't >> >> >> demonstrate true or false? >> >> > >> >> >If one cannot assume anything without proofs or demonstrations, one >> >> >can never get started at all. >> >> >> >> Getting started is way different from getting finished. What you're >> >> talking about is getting started; what I'm talking about is getting >> >> finished by demonstrating the truth of initial axiomatic assumptions. >> > >> >How does one come up with the ur-truths on which to found one's initial >> >axiomatic assumptions? >> >> Well here if the ur-truths are actually demonstrably true they >> represent more than mere assumptions. But getting from mere >> assumptions to ur-truths is an interative regressive process of >> determining what and how things can be proven true other than by >> consistency with aximatic assumptions which of course don't prove them >> true at all absolutely but only relatively. > >In other words, and more briefly, there are no absolute ur-truths in >mathematics.
None at all in axiomatic mathematics.
>> However apart from this I have to suggest it >> is possible to stumble on an initial assumption which must be true >> because its tautological alternatives must be false and its >> tautological alternatives must be false on thoroughly mechanical >> grounds. > >In what sense are you using the word "tautological? none of the meanings >I can find in any dictionaries make sense in your context.
Tautology "p, not p". If "not p" is false then "p" is perforce true.
>> >And I know of no way to do that. >> >> That's true. >> >> >I have always regarded something for nothing as pie in the sky. >> > >> >> Conventional mathematical methodologies just get started and then >> >> assume everything is honky dorry for no better reason than they got >> >> started. >> > >> >One has to start somewhere to get anywhere. You seem to think it >> >possible to get something from nothing. I do not. >> >> >> >> >> > And this does not mean that axioms of this kind are arbitrary >> >> >> > assumptions. There seem to be very few axiom systems that lead to >> >> >> > interesting theories. >> >> >> >> >> >> Arbitrary or not they're still assumptions. >> >> > >> >> > >> >> >Without any of which one has nothing to start with. >> >> >> >> Once again getting started is not the same as getting finished. >> > >> >When you claim to be able to justify an initial set of axioms as somehow >> >being "true", I have to ask "How do you know they are true?" >> >> Because mechanically possible alternatives are false. > >For example? Outside of the trivially tautologous statements of formal >logic like "if P then P" or "P or not P", I was not aware that there >were any such axioms possible.
I agree you were not aware that if "not p" is false "p" is true. I don't quite understand why you were not aware of this. Possibly because you've spent your time studying set theory instead of the demonstration of truth in universal terms.
>> >Where does your truth come from? >> >> The tautology. > >Which one?
"p, not p".
>> >> >Unless, like Jefferson, et al, you take certain truths to be self >> >> >evident. >> >> >> >> Exactly my point. Jefferson had his self evident truth and every >> >> dictator in history theirs. Hard to tell the chaff from the wheat >> >> without a scorecard. >> >> >> >> >But then, again like Jefferson, et al, you must state what those self >> >> >evident truths are, so that we can know what we start with. >> >> > >> >> >I find that in matters mathematical there are no self evident truths. >> >> >One must assume something in order to get anywhere. >> >> >> >> And if what one assumes is not true? >> > >> >That is precisely my point! By what criteria can one determine that? >> >> Self contradiction. > >Do you mean that every statement that is not self-contradictory is true?
Yes. The difficulty comes in figuring out which statements are not self contradictory. Just because we might say "A is B" doesn't mean there is no self contradiction involved.
>That would mean that two statements which are not self contradictory but >are mutually so, could both be "true".
Lots of statements can be true or self contradictory. There is the obvious appearance of self contradiction which is trivial. But there are also embedded self contradictions which are non trivial since they are not apparent.
>> >What I say that there are no absolute a priori criteria. >> >> Well if you're talking prior to any experience at all I'd agree. But >> the sense in which I use "a priori" is strictly "logically consistent >> with and demonstrable of". > >But there is nothing at the very beginning to be logically consistent >with or demonstrable of.
Of course there is: "logically consistent with or demonstrable of itself". If alternatives to what is predicated by initial assumption are demonstrably false then what is predicated by initial assumption must perforce be true.
>> >> This whole discussion is about >> >> whether contemporary mathematical set theory is true or false not >> >> whether it's assumed to be true. >> > >> >And I am asking once more, how one tells the difference. >> >> And the answer is that tautological alternatives to what's true are >> self contradictory. > > There is that ambiguous "tautological alternatives" again which as far >as I can see means nothing. So I find your response non-responsive.
Claim "p". Tautological alternatives "not p". What's non responsive about that? This isn't rocket science for chrissakes. It's only basic logic. Pull your head outta your ass for a change.
>> There is nothing in set theory which shows that >> tautological alternatives are self contradictory > >There is nothing in set theory or logic which explains what >"tautological alternatives" even are, so it is not surprising one cannot >show much of anything about them.
Yeah. So you say. Then you run around proclaiming definitions are arbitrary.
>> no "non set theory" >> which demonstrates alternatives to set theory are impossible. > >Some versions of category theory are a sort of non-set theory, and there >are those who expect eventually to develop viable alternatives to set >theory along lines suggested by abstract category theory.
Yadda yadda whatever. Then set theorists can all sit around jacking off and extrapolating string theories of math because they're too lazy or stupid to figure out what's actually true and false. Set theorists proclaim they're mathematical platonists but can't even come up with a correct definition for that most platonic of all figures: the circle.