On Sat, 15 Jul 2006 19:00:09 -0600, Virgil <email@example.com> wrote:
>In article <firstname.lastname@example.org>, > Lester Zick <DontBother@nowhere.net> wrote: > > >> >If what one starts by assuming were demonstrably true, one would not >> >need to assume it, one could demonstrate it instead. >> >> One always needs to assume whether assumptions ultimately turn out to >> be true or false. This is true both of axioms and theorems. > >How does one ever find out that something "ultimately turn out to be >true or false" unless there is some ultimate source of truth?
It's called reasoning.
>I know of no such source.
Apparently. That's why you're a modern mathematiker.
>> >I am asking whether one can start by assuming nothing and ever get >> >something from that nothing. >> >> But you're misrepresenting the problem which is not whether one starts >> out with assumptions but whether the assumptions one starts out with >> are subject to proof or not. > >If what one takes as unproved assumptions could be proved, why wouldn't >one start by including the proofs themselves, and so avoid having to >make any assumptions at all?
Then what would you be proving?
>> Not only axioms but theorems as well start >> as speculative assumptions in the human mind. The difference is that >> theorems are held to standards of proof of consistency with axioms but >> axioms are not held to standards of proof of consistency with anything >> at all except dialectical plausibility. > > >But you keep appealing to some "ultimate truth" by which axioms and >axiom systems are ultimately to be tested, but give no source or >justification for these alleged "ultimate truths".
An ultimate truth is that whose alternatives are false. Is there something mysterious here?
>Do you just mean that an axiom system which is both useful and not known >to be self-inconsistent is "true". If so why not simply say so.
No I mean that a claim whose alternatives are demonstrably false is demonstrably true. That doesn't include set theory. Nor does it include real number lines. Nor does it include transcendental numbers as irrationals. Nor does it include people who make up definitions as suits their whims.