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: >> > >> > >> >> > Can a piece of music be true? An >> >> > architect's plan for a building? A computer program? >> > >> >> 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.
>> >> > >> >> > 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.
> You seem to suggest that one can bootstrap from >nothing to something, ie, to demonstrate the truth of some set of >initial axiomatic assumptions with no a priori assumtions of any sort >about anything.
Well I've been corrected on the issue myself but apparently the term "a priori" doesn't mean historically prior but logically consistent with and derivative of. 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.
>And I know of no way to do that.
>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 gat 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.
>Where does your truth come from?
>> >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?
>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".
>> 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 nothing in set theory which shows that tautological alternatives are self contradictory, no "non set theory" which demonstrates alternatives to set theory are impossible. Hence there is no way to show that set theory itself is necessarily true to the exclusion of other possibilities.