Re: Review of Mueckenheims book.

On 15 Mrz., 17:00, "*** T. Winter" <***.Win...@xxxxxx> wrote:
In article <1173961850.510535.178...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueck....@xxxxxxxxxxxxxxxxx writes:

> On 13 Mrz., 15:21, "*** T. Winter" <***.Win...@xxxxxx> wrote:
> > In article <1173723524.655982.61...@xxxxxxxxxxxxxxxxxxxxxxxxxxx> mueck...@xxxxxxxxxxxxxxxxx writes:
...
> > The Greek does *not* contain "all" at all in that position. "Oi protoi
> > arithmoi pleious eisi pantos tou protethentos plethous protoi arithmon."
> > "The prime numbers are more than all assigned multitude of prime numbers."
> > No assumption about "all prime numbers" at all.
>
> "The" covers the meaning of "all". The men have two feet means all men
> have two feet.

Oh, I wonder. Moreover, your statement about "men" is clearly false.

Then improve it by all men have a head.

But in a language like French: "les chiens ont quatre pieds" does *not*
mean that all dogs have foor feet, only that dogs, in general, have
four feet. (Similar in the Germanic languages I know, where however
the article is omitted.)

Die Menschen haben Köpfe means all men have heads in German and
certainly also in Dutch and Greek.

> > > That was not the knowledge at Euclid's times.
> >
> > In that case he did not proof his theorem at all.
>
> He did prove it by assuming the product of all ("the") prime numbers.

He did not assume such in his proof.

Otherwise the headline of his proof would be wrong. But Euclid was
extremely careful.

> > > Our account does not rob the mathematicians of their science, by
> > > disproving the actual existence of the infinite in the direction of
> > > increase, in the sense of the untraversable. In point of fact they do
> > > not need the infinite and do not use it. They postulate only that the
> > > finite straight line may be produced as far as they wish. (Aristotle)
> >
> > That is philosophy, not mathematics.
>
> That was not different at those times. Compare Pythagoras and Plato,
> for instance.

It is philosophy, not mathematics.

That was not different at those times.

> > > Euclid did know about potentially infinite sets and he did not believe
> > > inactually infinite sets.
> >
> > So he could not (and has not) spoken about the set of all primes.
>
> And he proved that one cannot have all of them.

Where did he prove that?

He claimed it in the headline of his originally written Greek proof.

> > > And he did not know about induction and
> > > recursion. Euclid did not know wheter the set of prime numbers is
> > > finite. In order to be sure that for any given set of prime numbers
> > > there is another prime number, Euclid had to contradict the assumption
> > > that a product of all prime numbers can be given. Therefore he assumed
> > > just this.
> >
> > He did not assume anything like that. He did not contradict the assumption
> > that a product of all prime numbers can be given.
>
> He did.

How did he?

He assumed that it was possible and then multiplied all of them and
then added the unit and then found out that the product contained
another prime. This consequence was obviously obvious to any reader of
his proof. Otherwise there would have been many minor know-alls, even
in ancient Greek, who would have pointed with great pleasure to an
error of this great.

Regards, WM


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