On Tue, 29 Jul 2008 18:17:34 +0100, Orchid XP v8 wrote:

>>> Scientific facts have been found to be incorrect. There are far fewer
>>> examples of mathematical truths which have needed to be adjusted. And
>>> there are vanishingly few examples of widely accepted *proofs* that
>>> turn out to be wrong - it tends to be things lots of mathematicians
>>> "think" are true that eventually turn out to be disproven.
>> 
>> Exactly my point, but with a narrower focus.  Things lots of *people*
>> "think" are true sometimes/frequently/often turn out to be disproven.
> 
> Show me one single mathematical result which was *proven* to be true,
> and verified independently by a large number of mathematicians, and
> subsequently turned out to actually be false.

The most obvious answer I can provide (but I don't know how many 
confirmed the proof) is Fermat's Last Theorem - at least from my read, it 
was proven for n=3 in the 10th century, but the proof was later 
invalidated.

A correct proof was later constructed in the mid-20th century, again, 
from what I understand.

>> It's simple logical deduction that unless I have a screwdriver, I can't
>> drive a screw.
>> 
>> Until you realise that the screw has a hex head and an allen wrench
>> will do the job just as nicely.
>> 
>> *Sometimes* all you need is a new tool.  Sometimes the new tool hasn't
>> been invented yet.
> 
> And I suppose next you'll be telling me that some day, some future
> technology might enable us to find a sequence of chess moves whereby a
> bishop can get from a black square to a white square, despite it being
> trivially easy to mathematically prove the impossibility of this...

You're still missing my point....

>> I think it's a mistake to say "we know all there is to ever know about
>> 'x'".  There have been many points in history where humankind has made
>> such declarations about many things - including mathematics - and it
>> has turned out that we'd only scratched the surface.  It's the height
>> of hubris to assume we can't learn anything new.
> 
> I'm not claiming that nothing new can be learned - I am saying that, at
> least in mathematics, learning new things doesn't invalidate what we
> already know.

Except that it can.  That was demonstrated in the example I provided 
above - proof done, proof invalidated, new proof done again.  Will the 
new proof be invalidated?  Probably not, but who's to say it never will 
be?

>> And yet you agreed with another post in this thread that said that
>> something was possible.  Look at the refocusing capabilities of some of
>> the tools for that to reconstruct detail in blurred images.  Blurring
>> is lossy compression, yet being able to recover that data isn't
>> impossible; that's been proven.
> 
> Hey, guess what? Blurring isn't compression. It might *look* like it is,
> but it isn't.

My point is that there's plenty of examples where raw data is lost but it 
can be reconstructed.

> Sure. And no doubt some day we'll discover that 2+2 isn't actually 4. I
> won't hold by breath for that though. :-P

Well, who knows?  There are ancient civilizations that had no concept of 
zero.  The introduction of imaginary numbers didn't come along until the 
late 1500s.  Up until that point, sqrt(-1) was undefined.

Who knows what we don't know about mathematics even today?

Jim

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