Re: (OT) Re: Godel

[hendrik@pooq.com]

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.

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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