Dear Paolo!

Thank you very much indeed for your interesting email.

I am sorry to say that I see a major problem in understanding 
of the method of reduction-descent between the two of us. 
We must clarify this first.
Moreover, if my description of reduction-descent 
in SWP--2006--02 is wrong, I would like to stop the printing,
which is probably to take place at Jan 2, 2007.
Thus, we should clarify it urgently. Please!
To this end, this email organizes as follows:

1. There is an attachment in pdf of the relevant
   part of our paper in its current state. 
   Note that I have made some
   little changes in your text, not to correct your style
   but to help us find out efficently on whether and where we agree.
   The part set in Frankfurt Fracture is my editing notes.
   You can still find the whole pdf at the old place:
   http://www.ags.uni-sb.de/~cp/p/bussottiwirth/pdf.pdf

2. In the postscript, 
   there are some minorly important remarks on 
   "apagogic" (and why I wrote something wrong),
   and on ASCII (and how happy it makes me).

3. The source ASCII file of the attached pdf file.

Please read the pdf (1) or its ASCII first and answer it
as fast as possible. Thank you very much indeed in advance!!

Anyway, I am glad to have you.
I wish you a happy new year.
In German we wish ``einen guten Rutsch'',
which means literally a good slide (into the new year),
but actually this is Jiddish and means a good start,
from Jiddish Rotsch=Head, but the German Jews are too few now
to remind them stupid Germans.

Best,
CP

PS

APAGOGIC:
When I wrote that reduction descent is rarely applied
apogocially there was not a problem with words.
According to the second of the sheets of paper that we scribbled in
your splendid private library, 
``neg. reduction-descent (hardly occurs)''.
Since this means that a negative demostration, 
\ie\ an apagogic demonstration, hardly occurs,
we must have done something wrong in your library,
and I think we should have written "(hardly occurs)"
into the upper right box instead of the lower right box.
Then, I think we can complete solve this problem if you answer the
following questions:
The indefinite descent (top-down, is it apagogic?) 
    is to Noetherian Induction (bottom-up)
as
the reduction-descent (top-down, apagogic) 
    is to WHAT (bottom-up)?
The indefinite descent prefers the \neg\exists-form,
the Noetherian Induction and the reduction-descent prefer the \forall-form,
so the WHAT prefers the \neg\exists-form?

ASCII I LOVE YOU
Your email was in some horrible non-standard MSWord (?) Format 
and not in ASCII as promised. 
I would prefer ASCII a lot next time!!!
This time it was quite some work to remove all the non-standard
characters (e.g. to write \`a in ASCII for a non-standard
``a'' with accent agiu, \'a for grave, \^a for circonflexe.
I would be nice, if you could write `` for opening and 
'' for closing parentheses, enclose mathematical formulas
in \math{  ....  } and write p with a prime, index n and
power m simply as p'_n^m. Thus p^m^n is illegal, but
p^{m^n} is p to the power of the power of m to n.
But this is all unimportant. 

ASCII SOURCE
More general: 
Let us suppose that we must prove the theorem 
\nlbmath{\forall x\stopq\app T x}. 
Suppose that \nlbmath{\app T x} is true for \math x among some initial values.
It is possible to use the expression 
``\math{\app T x} \nolinebreak 
is true for \math x being a small number\closequotefullstop
For \math S being the predicate that holds exactly for these small numbers,
we then have \bigmaths{\forall x\stopq\inpit{\app S x\implies\app T x}}.
Now let us suppose that \math{\app T n} is false for an arbitrary natural
number \nlbmath n,
and that we are able to construct an algorithm such that, 
if \math{\app T n} \nolinebreak is false, 
then \math{\app T m} \nolinebreak is false for a number \nlbmath m smaller 
than \nlbmath n. 
\ednote{Thi\es\ description lack\es\ generality: ``algorithm'' mean\es\ 
computability. But we do not even need constructivene\ses, not even 
describability, we just need exi\esi tence.}

\ednote{The following question i\es\ most seriou\es\ imho:
Regarding the actual theorem proving activity of the mathematician,
thi\es\ method of reduction descent 
differ\es\ from the method of indefinite descent only if
he may additionally assume that \math n i\es\ not a small number.
So, please, Poalo, answer the following question to me:
Doe\es\ \bigmaths{\inparenthesesoplist{       
            \neg\app S n\und\neg\app T n
          \oplistimplies
            \exists m\tightprec n\stopq\neg\app T m}
}{}
suffice or i\es\ the mathematician required to show
\bigmaths{\inparenthesesoplist{     
            \neg\app T n
          \oplistimplies
            \exists m\tightprec n\stopq\neg\app T m}
}{}
for the method of reduction descent actually?
In the later case I would be very unhappy for two reason\es:
\begin{enumerate}\item
I did not understand you and your book properly up to now
and what I wrote about you in SWP--2006--02
i\es\ wrong although you counter-checked it. It i\es\ just now 
in the printing factory, and I would like to stop it if possible and
if to prove the former would not suffice for the method of reduction descent.
\item I do not think the distinguishing of the two concept\es\
``indefinite descent'' and ``reduction descent'' to be appropriate\emph
{in any mathematical sense, even not in the hi\esi torical one}, 
but judge thi\es\ a\es\ a sophistication which i\es\ ``over the top''
similar to
what I wrote in \litsectref{2.4.1} on \unguru\ and \acerbi\ in 
SWP--2006--02. Sorry for being so horribly explicit, but I am a German, 
after all, even if I do not like the German\es.\end{enumerate}
Even if you think that the latter i\es\ actually required,
maybe \fermat\ thought differently? What do the many reconstruction\es\
of Sergio Paolini say about thi\es? I\es\ there any example of a 
reduction descent where \math{\neg\app S n} i\es\ used in the proof,
\ie\ the reduction assume\es\ that then \nlbmath n i\es\ not a small number?
A single example would show that the former alternative i\es\ right and
that we all could be happy!}