Re: (OT) Re: Godel
On Thu, Jan 08, 2004 at 11:38:00AM -0500, hendrik@pooq.com wrote:
> On Sun, Nov 30, 2003 at 01:25:09AM -0800, Tom wrote:
> > On Sun, Nov 30, 2003 at 10:11:39AM +0100, Nicos Gollan wrote:
> > > On Sun, 30 Nov 2003 00:00:05 -0800
> > > Tom <tb.31123.nospam@comcast.net> wrote:
> > >
> > > > Once somebody disproves Godel I will rest easy...
> > >
> > > The one(s) doing this wouldn't survive the lynching parties the
> > > theoreticians would start, and AFAIK, the Goedel stuff is rather well
> > > proven. So don't hold your breath.
> > >
> >
> > Good. I was hoping someone smart would speak up. (I was a lousy
> > mathematician which is why I switched to computers. Much easier.)
> >
> > I accept Godel. Does it matter much in day to day life? Or is it just
> > something not to worry about.
>
Forgive me if I missed something as I am entering in, in the middle, but
Godel's theorem isn't of much use in everyday life. Its useful in that
it states how far you can, or more precisely can't go with formal logic.
It basically states that in any formal logic system you either can't
prove everything covered by that system, or you will be able to prove
paradoxes (and that includes the logic system used for modern day
mathematics. And there are paradoxes in mathematics, but as of now
nowhere important afaik (it gives things that you can usually assume
whichever way is comfortable to you and it won't matter)
It did come in the middle of an epic three volume work by Alfred
North Whitehead and Bertrand Russell called Principia Mathematica that
tried to build a paradox-less logical basis to mathematical proof, and
showed that what they were doing was actually impossible (that work has
very great significance all the same)
As for disproving it. If you disprove it then go ahead and check where
you were wrong. Since it was proven mathematically and not physically,
you can take it as truth.
> It's something not to worry about, unless your day-to-day life involves
> proving a lot of theorems, especially using computers to perform the
> proof steps absolutely rigorously, and furthermore you want to prove
> that the rules of proof are correct, and complete, and consistent,
> and so on and on and on...
>
> >
> > I mean, life goes on, but if Godel is true, I kind of just keep waiting
> > for the train to derail (and it usually does). Is it an important
> > result? How does one sleep at night :-) ?
>
> Trains aren't formal deductive systems, even thought they have schedules,
> they do or don;t run o time, and Godel's theorem has othing much to say
> about them.
>
> >
> > This is just one of those little things that I worry about...
> >
>
> In real life truth is much less absolute. It's determined by trial an error,
> and that's how science works. People who are sufficiently obsessive-compulsive
> about Truth end up retreating in mathematics where truth can be absolutely
> controlled -- after all, you decide on the axioms and rules of deduction, hey?
> BUT, what they lose when they crawl into the hole of formalist mathemetics (I've been there):
> Does their mathematics have anything to so with the real world?
> Are their rules true in any sensible sense, or do they define
> their own truth independent of any external purpose?
> The first wuestion can not be answered withing their system -- it
> requires reference to the real world.
>
> The second question is addressed by Godel's theorem, which says, in
> essence, that if their rules are such that all their deductions yield
> truth (whatever that is), then their rules are not complete.
>
> So there's always something more to discover, and no set of rules is
> enough to decide what is dicoverable.
>
> And another issue that mathematicians tend to sweep under the rug. If
> you look at the history of mathematics, you sill find that
> the rules of deduction, the axioms that they start with, the socially
> accepted norms of presenting mathematical results, have themselves been
> determined by trial and error through the ages.
> It's just astonishing that the mathematics that has eveolved is as powerful
> as it turns out to be, and apparently also consistent.
>
> Even now there are
> controversies. There is a whole school of thought (constructivism)
> that rejects proof by contradiction. There are other mathematicians
> that reject the axiom of choice altogether, and instead treat Solovay's
> conjecture as an axiom: that all real-calued functions on the reals are
> measurable. That's inconsistent with the axion of choice, but if you're
> doing analysus, it's a lot more useful tnan the axiom of choice.
>
> Yes, there's no set of rules to decide what is discoverable, what is true.
> There's only a sense of what convinces people and what is useful.
>
> It has nothing to do with trains derailing, though, as far as I know.
>
> -- hendrik
>
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