The Logic Off Topic list

Thoughts from the mind of GAmoore@..., 10-11-2001:

>  >A testimony to this is the fact that _any_ half-decent mathematician
>>knows what's meant when talking about a "beautiful proof".  The proof
>>of the 4-colour-map theorem (or whatever it's called in English) is
>>generally regarded to be an ugly proof, and people actually spend
>
>Thats about right. "Four Color Theorem" I think. It was proven about 1976
>by a team of mathematicians at the University of Illinois, using computer
>programs involving some 1,700 cases as I recall and could only be
>verified by other programs. That is not eloquent or beautiful.

I thought it was 4000+ cases.  But anyway: yes, that's the reason 
it's regarded as ugly.

>There are over 100 proofs of Pythorogus's theorem.

Hm, I'm not familiar with that one.  What theorem would that be? :-)

>  >No, this is not true.  What constituted a proof 500 years ago still
>>is a proof today.  What has changed is the scope of the field, and
>>hence the amount of "tools" you can use to construct a proof.  The
>>concept of what a valid proof is though has remained unchanged.
>>Otherwise: why would we still study Euclid's "Elements" today (and
>>not just for historical reasons, but to learn math from it)?
>
>Well the Greek proofs were valid as far as I know. But those topics were
>clearly laid out with the 10 axioms, and it was fairly straight forward.
>What passed for proof in Newton's day would not be considered rigorous
>today, at least in some cases. Even Gauss's original proof of the FTA was
>later seen to not be completely rigorous - which is ironic because that
>was one of Gauss's contributions - bringing greater rigor to mathemaical
>proof.

OK, that's true.  What I tried to argue however was that the change 
in proofs is for the larger part due to the fact that we have more 
"tools" now, and far less due to changes in perception of "what a 
good proof is".  Probably it would have been possible to explain to 
Gauss why his proof was not completely rigorous, and he would have 
agreed.  Again: what is and what is not a proof has been far less 
susceptible to changes than the tools that are used in proving stuff.

>The vice-president was complaining to the head of the physics department
>about their budget. Evidently they need a lot of expensive equipment. He
>said "look at the math department.. they do fine with pencils, paper, and
>trash cans"...." and look at the philosophy department, they do fine with
>pencil and paper."

LOL!! :-))

>  >>As you well know, many great scientists and mathematicians have been
>>>deeply religious, and have built impeccable proof structures built
>>>upon their religious assumptions whether they be Islamic, Jewish or
>>>whatever.
>
>I don't know about 'many' nor do I know about 'impeccable' proofs about
>the hereafter. I think I heard about one nut case physcist who wrote a
>book prooving God's existence. There is no way to prove anything about
>God.

In medieval philosophy, many "proofs" for the existence of God can be 
found -- and some of them are quite famous and clever.
For good order: I didn't write the 1st bit above.

>  >>What Poincare had done was change the game yet again by changing the nature
>>>of proof, and by changing what was an acceptable solution. Pretty much --
>>>creating a new branch of mathematics with new ways to tackle the solutions
>>>there was regarded as a 'correct' solution.
>
>What exactly are you talking about here?

Uhm, I think that's Kool Music speaking, not me.  Double quote, see...

>  >>I am more than happy to concede this point to the both of you and
>>>accept that 'most mathematicians who have lived' should take
>>>precedence over 'most of mathematical history'.
>
>What are we talking about?

That's what I was wondering about too.

Maybe, with all this seriousness, it's good to tell a joke that's 
somewhat critical of mathematicians:

A physicist, an engineer and a mathematician are each locked up in an 
empty room in which there's just a closed box containing 
life-supplies (food, drink).  It's agreed they'll stay there for a 
week before the rooms are opened.
After one week, they open the physicist's door.  He's sitting on the 
ground with the open box next to him, eating a sandwich and looking 
healthy.  The walls and ceiling are covered in calculations.  When 
asked about his solution, he says: "I happened to have a small stump 
of pencil.  I measured the box and was able to calculate its 
resonance frequency.  Then I tapped the pencil in the proper rythm on 
the box until it started to resonate and finally broke, after which 
all was fine."
Then they open the engineer's room.  He too is sitting on the ground, 
eating a sandwich and having a beer.  No calculations are seen 
anywhere, but clearly the box is open.  When asked about his trick, 
he says: "After a few days work, I managed to make this improvised 
can-opener from my trouser's zipper, and voila, I could open the box, 
no problem."
Finally they open the mathematicians door.  He too is sitting on the 
floor, looking completely exhausted and wasted.  The box is closed 
and the walls and ceilings are covered with formulas and 
calculations.  They hear him mutter: "OK, that doesn't work.  Let's 
try again.  Suppose the box is open. Then it folows that..."

:-)


cheers,
HJ
-- 
     Hendrik Jan Veenstra
     email: mailto:h@...
     www:   http://www.ision.nl/users/h/index.html