Re: James' object ring

From: David Kastrup (dak_at_gnu.org)
Date: 10/16/04 

Date: Sat, 16 Oct 2004 22:50:03 +0200

jstevh@msn.com (James Harris) writes:

> It turns out that the story here is *really* old as years back I
> started talking about "flat rings" which drew a lot of derision on
> sci.math and later I learned of algebraic integers and thought for a
> while they were my "flat ring".
>
> The simplest way to understand the ring in complex numbers is that if
> you have any number z=x/y in the ring, where x, y and z are in that
> ring, and, of course, y is not 0, then x and y cannot be factors of
> any integers that are coprime in the ring of integers.
>
> So it is a flat ring, in that given z = x/y, the y must be a factor of
> x without contradicting factorizations in the ring of integers.
>
> Some have attempted to add numbers like pi to the ring, but it's an ad
> hoc thing where they basically just *say* add pi to the ring.
>
> However, it is easy enough to show that you can construct all numbers
> in the ring starting with 1 and -1 and a few operations, but cannot
> construct pi from within the ring.
>
> If you add in what I call operators, like 1/2, which means 1 of 2,
> then you can construct pi as an operator.
>
> So you can build the entire ring of complex numbers using these ideas.
>
> Note that coprimeness in the ring then is a simple matter of not
> sharing non-unit factors.

You are drunk, talking incoherently. More importantly, you are
reverting to talking nonsense that is already dated several years.

> At this point in time I've shown how the older ideas that have
> dominated the math world can lead to rather simplistic errors and a
> muddled view of coprimeness, like saying that if 1/2 is in the ring
> that 2 is coprime to 3, though it is a factor of 3.

Coprimeness is defined quite simple. Units are factors of
_everything_. Talking about coprimeness with regard to them is
useless.

So what is your unmuddled view of coprimeness? Give a clear
definition. The usual definition is
a and b are coprime in a ring if there exist values c and d in the
ring such that
a c + b d = 1.
Very clear and unmuddled. What's your definition? I didn't hear you,
speak up.

> I think it's time humanity began to catch-up.

Well, start catching up to the state of mathematics 200 years ago
first, then complain about humanity. 

-- 
David Kastrup, Kriemhildstr. 15, 44793 Bochum

Relevant Pages

  • Re: My paper, and the cheaters
    ... > coprimeness result to handle the problem with the ring of algebraic ... > the problem with the ring of algebraic integers. ... we need to discuss the Galois theory ...
    (sci.math)
  • Re: JSH: Critique means slow, and thorough
    ... coprimeness in the more inclusive ring. ... can have an odd thing, where you can prove "coprimeness" by relying on coprimeness in the ring of algebraic integers, and algebraically ... particular field extension called the Hilbert class field of F. ...
    (sci.math)
  • Re: New paper, algebraic integers, Galois Theory
    ... > ring of algebraic integers. ... The coprimeness claim is one and the same ... special claim based on their special, arbitrary rule. ...
    (sci.math)
  • Re: Attacking my algebraic integer work
    ... 1 and -1 are the only rationals that are units in the ring. ... and the ring of algebraic integers. ... where x and y are members of that ring, ... Then coprimeness in that ring would not mean coprimeness in the more ...
    (sci.math)
  • Re: JSH: Critique means slow, and thorough
    ... Given a member m of the ring there must exist a non-zero member n ... > where x and y are members of that ring, but not algebraic integers. ... > coprimeness in the more inclusive ring. ...
    (sci.math)