> Dear Claus-Peter,=0A=0AI answer to your mail concerning descent and r educti=
> on-descent. No problem if you were explicit, this is normal. Th e only think=
>  is that in your mail there were some strange c haracters which made the rea=
> ding non-easy, anyway I think to have und erstood the problems.

I have replaced Frankfurt Frakture font with Sans Serif Font now.

> Let's s=
> tart from Fermat, from what he say s, for the moment without interpretation =
> (even if of course this is n ot completely possible). First of all Fermat di=
> stinguishes between "a ffirmative" and "negative" propositions. This distinc=
> tion has nothing to do with the way in which a proposition is proved, but w=
> ith the li nguistic way in which a demonstration is posed. Example: "There i=
> s no Pythagorean triangle of which the area is equal to the square of an in=
> ; teger" is a negative proposition; "Every prime number of the form 4n+1 is t=
> he sum of two squares" is a n affirmative proposition. =0A=0AThe distinction=
>   between a pagogic and non-apagocic concerns the way of demonstrating: if a=
>   theorem is demonstrated by a reductio ad absurdum, the way of demonstrati o=
> n is apagogic, otherwise it is not. =0A=0AThe infinite or indefinite descen=
> t is always an apagogic way of demonstration both when it is a pplied to neg=
> ative propositions and to affirmative propositions. =0A= 0AFor the affirmati=
> ve propositions new principles are necessary in or der to apply the descent.=
> =0A=0AThis is what Fermat explicitely says.

I perfectly agree.

> Now let's come to the int=
> erpre tation: after Fermat the affirmative propositions of which he spoke ab=
> out (binary quadratic forms x^2+Ay^2, with A=3D1,2,3, and a set of preparat=> ory theorems for the study of these forms) were demonstrated by reduction-d=
> escent, while the negat ive propositions were demonstrated by ordinary desce=
> nt. Hence I thoug ht that one of the new principles of which Fermat spoke ab=
> out was: Fo r the affirmative propositions we have to suppose false the theo=
> rem a nd to construct a reductive algorithm such that for every step of the =
> reduction the theorem should be false. This algorithm reaches the initial v=> alues for which the theorem is true, this is absurd.=0AUseless to repeat wh=
> at happens with the ordinary descent. This way we have toe applica tion of t=
> he infinite or indefinite descent: 1) ordinary indefinite de scent; 2) reduc=
> tion-descent. Of course Fermat was a pioneer in this f ield and he did not=
>  take into account the fact that the red uction-descent could be applied to =
> negative propositions and the ordi nary descent to affirmative propositions,=
>  as well. In the work by Paolini, the 4n+1 pr imes theorem is demonstrate=
> d by a quite clear reduction-descent after having introduced a certain math=
> ematical structure that Fermat could know. I am not a logic, but the l=
> ogical part of my book has been bas ically written for a sense of dissatisfa=
> ction which derived from read ing that mathematical induction and indefinite=
>  descent are logically equivalent even if I have never found in the literat=
> ure a d ifficult demonstration by indefinite descent (for example the one co=
> n cerning the Pythagorean triangle) translated into a demonstration by mathe=
> matical induction. I mean: one uses the demonstration by descent to create =
> a demonstration by mathematical induction, but after that, the demon stratio=
> n by mathematical induction has to appear independent of the o ne by descent=
> , that is the demonstration by mathematical induction has to have the basis=
> &n bsp;and the inductive step, like every other demonstration non deduced by a de=
> monstration by indefinite descent. Furthermore this method must be "st andar=
> d", always the same for every transformation of a proof by desce nt in a pro=
> of by mathematical induction and vice versa. The law of
>  duality in projective geometry provides a clear example of wha t I mean. Ot=
> herwise the risk is that the mathematical logic creates a world that has no=
> thing to do with the work and the way of thinking o f the mathematicians, ev=
> en today, not only in the 17th century. =0AAl l of my considerations concern=
> ing the logical equivalence or non-equi valence between different mathematic=
> al methods (therefore even descen t  and reduction-descent) have no claim of=
>  comple teness or of complete correctness (on the other hand I have explicit=
> ely written in my book), they are doub ts rather than sure affirmations. Bec=
> ause of this, I would be very ha ppy to clarify all of these problems with a=
>  logic. Anyway, a thing is in my opinion sure: in the standard, academic lo=
> gical lite rature too many problems concerning this mathematical methods are=
>  ; given for granted, while a deeper analysis is necessary. And I think tha t =
> the main aim of our paper should be exactely this.

Sure! I d o perfectly agree with you.

> But, Claus-Peter, to b=
> e clear like you rightly write, if you do not see the sense in writing a jo=
> i nt paper, please inform me. We could continue to write our impressions and=
>  opinions by mail, and no problem for our personal relations.
No, of course not, we would stay friends anyway.
But I want to unders tand your point and to write the joint paper.

> If, on the co=
> ntrary, you want to continue our pap er, in few days I could send you the pa=
> rt concerning the demonstratio n by descent and my impressions on the first =
> six pages.

Send the demonstration!
I want to continue the paper, but first we have to synchr onize our views.
I have written you a surface Mail yesterday, but that will not help.
I contains this and that, such as the Unguru Fowler Induction War.
Did you read this?

Thus, my suggestion would be that I phone you Tue sday evening to
solve the problems. What would be a good time for this?

Meanwhile you may ask yourself, what the word "algorithm"
in your reduct ion-descent description means:
--- a   c o m p u t a b l e &n bsp; function which
     maps the assumed counterexampl e to the smaller counterexample,
--- a definable function instead
--- som e constructive way in which the smaller
     counterexample is to be obtained,
--- no axiom of choice required (or weak form thereof) to prove existence
  &n bsp;  of the smaller counterexample from that of the assumed counterexampl e,
--- arbitrary logical existence of smaller counterexample (i.e. nobody      knows how to find it, but we know that there must be one by some
     reasoning).

I will read in your book tonight and try to think more carefully than before.
My problem is tha t I do not see any difference between indefinite descent
and reduction-desce nt currently besides some feelings of the mathematician.
Especially I do not see any behavioral difference (i.e. difference in proof
steps) between the two. And without such a difference I do not consider it
worthwhile to mentio n it in history of mathematics, but maybe only in the
history of the psychol ogy of mathematicians, if such a discipline exists.
The thinking of mathematicians is part of the history of mathematics,
but the feelings they have on why a method should be sound do not, imho,
establish a different method. Same proof steps means same method, even if
the ideas on why the method is sound may differ.

Thus, let u s hope that I can understand it.
I thought we would agree in your library an d I hope we can get back
to that agreement.

I restate my central ques tion:

When I do reduction-descent and have to find a smaller counterexam ple
for an assumed counterexample of an affirmative theorem for n,
am I a dmitted to suppose that the n is not a small number?

When I do indefinit e descent and have to find a smaller example
for an assumed example of a neg ative theorem for n,
am I admitted to suppose that the n is not a small numb er?

I will phone you on Tuesday evening!!!! WHAT TIME???

Best,
CP