Re: JSH: The "Published" paper he dosen't what you to know about.

[Tim Peters]
...

I'm not sure what he wants from his "object ring", and he doesn't have
the
technical vocabulary to explain it,

[Proginoskes]
Actually, he describes it at
http://mymath.blogspot.com/2005/03/object-ring.html .

Right, but that's only his attempt to define it -- it doesn't reveal what he
/wants/ from it. In that sense it's like his "pile of equations" surrogate
factoring methods: you can see what they are, but there's rarely a visible
reason for why those particular equations were picked. What does he think
/follows from/ this "definition" that he thinks he needs? My guess is
below.

"The object ring is defined by two conditions, and includes all numbers
such that these conditions are true:

1. 1 and -1 are the only rationals that are units in the ring.

2. Given a member m of the ring there must exist a non-zero member n
such that mn is an integer, and if mn is not a factor of m, then n
cannot be a unit in the ring."

The first red flag is the fact that the last part of (2.) is vacuuously
true, so JSH might not literally mean these properties.

I've lost count of how many times that's been pointed out to JSH, but he
keeps the vacuous clause in. OTOH, he went thru many other definitions and
kept silly things in those for long stretches of time too. Perhaps he
thinks everyone is lying, and is too lazy to follow one of the easy proofs
given to him?

You can also search for "object ring" at Google Groups. Other people
have tried to make sense of his "object ring".

Indeed ;-). As given, it's a senseless definition because the "the" in "the
object ring" can't be satisfied (i.e., uniqueness can't hold) unless we read
"number" in a trivial way. For example, if by "number" we think he means
integers, then Z is the (unique) object ring.

He almost certainly intends a subring of the complex numbers. The best
analysis of what follows then I've seen appears in a long thread March/April
2005; this is just a starting point:

Subject: Re: JSH: Critique means slow, and thorough
From: Arturo Magidin <magidin@xxxxxxxxxxxxxxxxx>
Date: Tue, 29 Mar 2005 18:33:26 +0000 (UTC)
Newsgroups: sci.math
Message-ID: d2c71m$od0$1@xxxxxxxxxxxxxxxxxx

That thread was remarkable for several reasons. First is that William
Hughes finally got it into James's head that a theorem of Dedekind's implies
that if two algebraic integers (AI) a and b are coprime, they remain coprime
in any larger ring.

Second is that the application of Dedekind's theorem is a bit indirect here:
a & b coprime -> [Dedekind] there exist AI x & y s.t. a*x + b*y = 1 -> the
same equality holds in any larger ring -> a & b are also coprime in any
larger ring. The notable thing is that James /followed that argument/
despite that it took a bit of clear thinking, which I believe is the last
time I've seen him understand what was being spelled out for him in this
area (and despite that in all subsequent reincarnations of this topic,
people have generally been trying to spell out simpler things).

Third is that this is in fact the source of James's belief that he's
"proved" ideal theory is wrong:

[William Hughes, quoting Arturo Magidin]
In "The Theory of Algebraic Integers, by Richard Dedekind
(translated by John Stillwell, Cambridge University Library,
Cambridge University Press 1996), Dedekind states in Chapter
3, "General properties of algebraic integers", section 14
"divisibility of integers", final paragraph (pp. 106 in my
edition):

"A deeper investigation will enable us to see that two
nonzero [algebraic] integers a and b have a ->greatest
common divisor<- [emphasis in the original], which can be
put in the form aa'+bb', where a' and b' are [algebraic]
integers. This important theorem is NOT AT ALL EASY TO
PROVE [emphasis added] with the help of the principles
developed thus far, but we shall later (section 30) be
able to derive it very simply from the theory of ideals."

Note the "theory of ideals" in the last sentence. Now try to react like JSH
;-) His response:

[JSH]
Thanks for the citation.

Well for the question about when the problem entered into the
field of mathematics, that's when, and I guess I didn't figure
on Dedekind having made the mistake, but I guess I should have.

That does make it a little more problematic in continuing to
critique in this direction as now it will be necessary to
include Dedekind's work and the theory of ideals.

So the critique will continue there from this side.

He's never been clear about this again, but /that's/ what it's all about:
James refuses to acccept that he can't magically make coprime algebraic
integers non-coprime by moving to a larger ring, and the only thing he knows
about ideal theory is that he's been told it's one way of proving he can't
(and best guess is that he doesn't know it's not the only way). This is
what I believe "he wants" (but can't have) from his object ring incantations
(although even then I remain unclear on what would follow from /that/ that
he thinks he needs in his FLT thrashing).

Finally, perhaps most remarkable in retrospect is that only Jesse Hughes
called him on his remarkable response at the time. /Most/ of the tail end
of that thread is James insisting that if you adjoin 1/2 to Z, you get the
reals, because "mathematically, there's no way to block convergent infinite
sums" -- and reams of futile replies trying to explain the obvious.


.

Relevant Pages

  • Re: JSH: The "Published" paper he dosent what you to know about.
    ... I'm not sure what he wants from his "object ring", ... so the other factors needed to be coprime to f --- in the ring ... happen in the ring of algebraic integers. ... Dedekind's work and the theory of ideals. ...
    (sci.math)
  • Re: JSH: Understanding the rebuttal
    ... the ring of algebraic integers. ... necessity then that solution will ... group via multiplication of ideals, ...
    (sci.math)
  • Re: JSH: Understanding the rebuttal
    ... the ring of algebraic integers. ... For instance, if g is a multiple of 5, then one can solve ... group via multiplication of ideals, ...
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  • Re: JSH: Simpler way to the exclusionary rings with asymptotic series
    ... non- convergence and what it means with rings like the ring of algebraic ... BUT in the ring of algebraic integers, ... 'object ring' cannot be closed under infinite sums. ...
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  • Re: JSH: Now the fun part
    ... and remain in that ring. ... To be a unit in the ring of algebraic integers a number must be the ... root of some monic polynomial with integer coefficients and a last ... But that requirement does not exist in the object ring for units. ...
    (sci.math)
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