The Logic Off Topic list

Hendrik Jan Veenstra wrote:
> >>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

GA Moore responded:
> >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.

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

Heesch, when he sketched out what was eventually to be the successful 
approach, thought that it would be necessary to consider 8900 
configurations. He got his idea from Kempe who suggested it in the late 
1870's, but whose own proof was in fact erroneous. It was proved Appel and 
Haken, '76, . They took Heesch's basic idea, but showed that there was an 
unavoidable set containing a mere!! 1500 configurations. 1200 hours of 
computer time.

May we be protected from more such things. Personally, I don't see how 
something that's basically a vast collection of computations that no human 
can check constitutes a 'proof', but that's just my prejudiced opinion. I 
really and honestly hope that that kind of thing doesn't work its way into 
the concept of mathematical beauty, but doubtless if I lived in 200 years 
time I would have a very different opinion.

GAM wrote:
> >
> >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.

HJV wrote:
>OK, that's true.
Agreed.

>   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".
Sorry, don't think so.
This is one of the points I was trying to make.
For example, the Greeks did not really do algebra, and did not think it was 
'real mathematics'. In that respect, they really set the course of 
mathematics back by centuries. This was largely because of Hippassus' 
discovery (he is the one usually credited) that the square root 2 was 
irrational.

To the Greeks, mathematics was a matter of number AND SHAPE. It was only 
possible to be logical, and thus provide proofs for things that met those 
criteria. That influenced their whole aesthetic regarding 'proof'. 
Diophantus' excellent work in algebra, although very very late period 
Greek, was largely ignored because it was still not considered 'real 
mathematics', and that held until the Renaissance, really. Just pushing 
numbers around and trying to 'prove' things that way was not really 
mathematics because it did not have 'shape'. It someone of the calibre of 
Fermat, who discovered a copy of Diophantus' Algebra and then translated 
and published it, to thrust algebra firmly into the mathematical discourse 
of contemporary Europe.

There was no overall change in the nature of mathematics until Leibniz and 
Newton (also Boole) forced it upon the subject through the discovery of 
calculus and the immense rise in the understanding of the possibilities of 
algebra. After Newton and Leibniz, mathematics, and proof, extended out to 
become the study of number, shape, motion, change and space. It was not 
until the end of the nineteenth century that mathematics broadened still 
further to become the study of number, shape, motion, change, space -- and 
also of the mathematical tools that are used to study what mathematicians 
have chosen to study.

And THAT is truly abstract. Or at least ... that's my personal opinion.

>  Probably it would have been possible to explain to
>Gauss why his proof was not completely rigorous, and he would have
>agreed.
I agree. Same goes for e.g. Archimedes IMO. Aristotle and Plato I'm less 
sure about. (JUST A JOKE GUYS!!!!!!!!)

>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.
I.just don't agree with this. I have tried hard to explain why ... and it 
would be fruitless to try further.

> > 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.

You make a very nice point there, GAM. It quite makes me smile to think of 
how happy Gauss would have been to see how things have come along since he 
opened that particular doorway.

Kool Musick wrote:
> >  >>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.
I wrote the above. And ... what I meant is what you, Hendrik Jan, 
indicated, although I do in principle agree with what GAM said about the 
provability of God. I think the arguments bogus because I just don't accept 
the premises ... but given that I am very interested in 'the history of 
thought', I do have a sneaking admiration for the flow, and sometimes the 
simplicity and sometimes the complexity, of those 'proofs'. Rightly, some 
of them ARE famous; and some of them are also very clever. They don't 
convince me, but I do think they are clever. Beautiful in their own way, 
but wryly beautiful. Kind of like the way a spider must smile when it sees 
a fly making a noble but doomed effort to escape the web. As long as the 
fly doesn't escape, one can admire the beauty and nobility of that 
struggle. If that fly escapes, though, the spider's in big trouble!!

Kool Musick said:
> >  >>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.

GA Moore (quite rightly!!) said:
> >What exactly are you talking about here?

Hendrik Jan said:
>Uhm, I think that's Kool Music speaking, not me.  Double quote, see...

I was trying to say that Poincare had initiated a completely new way of 
studying dynamical systems by treating, in a more combinatorial way, the 
equations used to describe them. I just got my words mixed up (too many 
'solutions') when writing.

> >>>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.

I have tried to make my point several times. I just don't think it worth 
expending any further effort in trying to make it.

Kool Musick
Keep Musick Kool


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