Dear Claus-Peter,

 

I send you the first part of 3. Historical Problems, for our article. I have made some modifications to the original plane. Please, tell me if, in your opinion what I wrote works. Of course change the English like you think because you know English quite better than me. If for you this part suits, I will continue.

 

I read with attention your introduction. I like it, but some aspects should be clarified. In particular: when you propose the scheme of the "Descente Infinie" and when you associete "without expections" with "indefinite descent", I would specify "without exceptions apart from trivial initial cases". I explain: Euler proved Fermat's last theorem for the cubes by indefinite descent. But this theorem is false for some trivial initial cases because obviously 0^3+0^3=0^3 and 1^3+0^3=1^3. I think this little addiction - I perfectly realize that it is a minor problem - is necessary to avoid criticisms.

In the last proposition immediately before 3. hitorical problems, you write that the reduction-descent is rarely applied apagogically. But this is not the case, the reductio-descent is always applied apagogically (I do not know if here there is a linguistic misunderstang, but in Italian "apagogico" means "by reductio ad absurdum" and I think that the English word "apagogic" has the same meaning). The difference between descent and reduction descent is that in this last case, the theorem we have to prove is true for a set of nontrivial initial values, if we suppose the theorem false for a value p, we have a reduction which should imply that the theorem is false for the initial values, as well. But this is contradictory. Here we have a reduction which reaches the initial values. No infinity, properly speaking. While in the descent there is a less limit m and a reduction that starts from n>m. The reduction can be formally continued infinitely, but for specific mathematical reason, it cannot reach m, but this implies an infinite quantity of numbers between m and n.

Returning to our paper, maybe you can simply eliminate the words "for instance...applied apagogically". All the rest is OK.

 

Claus Peter, I wish once again marry Christmas and happy new hear. I am waiting for your answer because I am very interested in writing our paper.

 

Hear you soon,

 

Paolo

----- Original Message ----
From: Claus-Peter Wirth <cp@ags.uni-sb.de>
To: bussottipaolo@yahoo.com
Cc: wirth@logic.at
Sent: Sunday, December 17, 2006 6:40:03 PM
Subject: Merry Xmas