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

[Tim Peters]
...
Arturo gave an excellent account of historical details, but at the
risk of simplifying ;-), the bottom line here is that James seems
to have learned "factorization by inspection" in high school, never
went beyond that, and is endlessly frustrated by that not all rings
"act like" Z wrt factorization.

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

[marcus_b]
It goes back to one of his FLT 'proofs'. He started with a
'tautology' and derived a polynomial which was essentially the
same as the now-hoary old P(m). The factors were of the
form (a_i(m) x + uf). He needed desperately to show that
two of the factors were divisible by f. Of course P(m) itself
was divisible by f^2 but not by any further bits of f,
so the other factors needed to be coprime to f --- in the ring
of algebraic integers. So eventually - it took literally years
of arguing - it got through to him that this was not going to
happen in the ring of algebraic integers. So he needed a
slightly bigger ring - not so big that it contained any rationals
other than 1 or -1, but bigger than the ring of algebraic
integers - if it contained 1/f, everything fell apart - so
he invented "the" object ring.

Incidentally, I think the key thing here is *not* the fact
that if two numbers are coprime in a ring R, then they remain
coprime in any larger ring S. It is the converse - if two
numbers are not coprime in R, they may be coprime in
S.

That's fine, except you're trying to make sense, while I'm trying to
understand what James thinks <0.3 wink>. This is the post in which he first
concluded ideal theory is wrong:

From: jstevh@xxxxxxx
Subject: Re: JSH: Critique means slow, and thorough
Date: 30 Mar 2005 16:48:06 -0800
Newsgroups: sci.math
Message-ID: 1112230086.528150.184860@xxxxxxxxxxxxxxxxxxxxxxxxxxxx

I'm not going to repeat it here, because you need to read the thread of
which it's a part. A few messages back, he challenged William Hughes:

[JSH]
...
but can you now prove that for every case

a/b

where a is an algebraic integer and b is an algebraic integer,
and a is coprime to b that you can find a construction

ax + by = 1

where x and y are algebraic integers?

If so I'd think that'd be a powerful argument against my claims.

William replied with the obvious short proof based on that the algebraic
integers are a Bezout domain. James doubted it, so William reproduced a
detailed citation of Dedekind's "The Theory of Algebraic Integers" earlier
posted by Arturo Magidin, and repeated to James several times already. He
either never read it before, or never understood it before. It included a
very clear quote from Dedekind, ending with:

[Richard Dedekind -- who used to post here a lot more ;-)]
...
"two nonzero [algebraic] integers a and b have a greatest
common divisor, which can be put in the form aa'+bb', where
a' and b' are [algebraic] integers.
...
we shall later (section 30) be able to derive it very simply
from the theory of ideals."

In any case, that's the exact point at which James decided ideal theory is
wrong:

[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.

In context, William was actually proving that AI coprimeness extends to
containing rings, in reponse to an even farther-back bogus claim of James's:

[JSH]
> Then coprimeness in the ring of algebraic integers does not
> mean coprimeness in the more inclusive ring [which, in context,
> meant the algebraic numbers].

The thread can be hard to follow, but its essence is clear enough:
regardless of what makes /sense/ here, as clearly as James can say anything
he said it was possible to get away from AI coprimeness by moving to a
larger ring, and he even correctly recognized that Dedekind's quote said he
was wrong about that.

That is why he had to contemplate larger rings - but not
so large (like, e.g., a field) that the coprimeness he
wanted became a triviality. You say in a reply to Proginoskes
that "James refuses to acccept that he can't magically make
coprime algebraic integers non-coprime by moving to a larger
ring" - but that's not what he wants to do -

As above, he certainly said that's what he wants to do, and he certainly
refused to believe that he can't ;-) More, that's the only explanation I've
ever seen from him for /why/ ideal theory "is wrong". For more than a year
now he just mechanically repeats that it's wrong, without giving any reason
whatsoever.

he wants to go the other way and make non-coprime integers
coprime by moving to a larger ring - and he can. Hence "the" object
ring.

Right, and that makes /sense/ to me too. He complains endlessly that, in
the AIs, none of the terms in his 3-term factorizations are coprime to 'f',
so it makes sense that he'd want a larger ring in which one of them is
coprime to 'f' -- but no so large that all of them are coprime. After all,
that one term is coprime and two aren't is his "factoring by eyeball"
/conclusion/, so you might rationally expect him to move in that direction
;-) But, as above, that's not what he was talking about at time he decided
ideal theory is wrong.

The state he is in now is VERY strange. He acknowledges
that in the ring of algebraic integers, none of the roots of
his polynomial are coprime to f. There are 'object' rings
in which one of the roots is coprime to f. For some reason
he thinks this means that the theory of ideals is wrong,
although he has shown no contradiction of the theory of ideals
at all.

Perhaps the above clarifies it -- LOL.

It is a consequence of the theory of ideals that none
of the roots are coprime to f in the algebraic integers.

Right, but that's not what the killer quote from Dedekind was about at the
time he decided Dedekind was wrong.

He accepts this as a fact -

I'm not sure about that. He does accept as a fact that Galois Theory (or an
"over interpretation" of GT, whatever that means) implies none of the roots
are coprime to f in the AI. I don't expect he has any idea what ideal
theory says about anything, apart from the one quote of Dedekind's he read
in that old thread.

if you accuse him of NOT accepting it, he will shriek that you are
lying about what he says -

He reliably shrieks if you accuse him of saying Galois Theory is wrong. But
as he as no idea what GT "as often taught" says either, who knows what he
means.

but at the same time, and as far as I can tell, for this
same exact reason, he thinks the theory of ideals is wrong!

Do note that his papers don't even mention ideal theory. I suspect that
means part of him knows his claims about that are pure bluff -- aka "Extreme
Mathematics".

It seems to give him a sense of importance to say he has shown
it is wrong - it destroys over a century of accepted mathematics,
etc. - so it is permanently in his archive of delusional
accomplishments.

Yup!

[Interestingly, even though Galois theory implies the same
conclusion here as ideal theory, he will not say that Galois
theory is wrong - only that it is "misinterpreted" or "taught
incorrectly" or "misused". I think he identifies with
Galois - the isolated and highly original mathematician whose
genius was not appreciated in his own short lifetime - a
dashing tragic romantic figure. Perhaps he views Dedekind
as an academic establishment plodder who overlooked his own
errors, and mathematicians after him just accepted it without
question and just keep gushing over how beautiful the theory
of ideals is. All JSH needs to do to perfect his Galois self-
image is take up dueling.]

Indeed, I've sometimes wondered whether "the math wars" are his way of
transplanting duels to cyberspace. It would be in character to take the
easy, risk-free way out ;-)

To complicate things: his "proof" that one of the roots is
coprime to f does not invoke any assumptions about the ring
in which said coprimeness occurs. Therefore if his proof
were valid, it would be valid in the ring of algebraic integers
and it would contradict the fact mentioned in the preceding
paragraph which he accepts.

His inability to see this extremely elementary point has baffled me for
years. At least once he waffled about it, saying that his "proof" actually
required "field operations" -- although he didn't specify which ones or
where.

[A good mathematician checks that his claimed proofs do not prove
too much!] But Harris cannot admit this bit of logic into his
consciousness. Maybe he has glanced it out the corner of his eye,
like an evil wraith come to rob him of his treasure and kill his
brainchild, and he cannot bear to look at it straight on.

Well, he has a different conclusion: since what he claims is false in the
AIs, the AIs are "flawed". Recently he says they "have a coverage problem"
more often, but the only "flaw" or "problem" he's ever demonstrated is that
their existence as a ring proves his argument is wrong.

....
It's not at all clear that "the" object ring gives him what he
needs -

WRT which, it's hilarious that his current FLT "proof" starts by saying it's
working in "an object ring" -- and then never mentions the ring again!

it of course lacks some of the nice properties of the ring of
algebraic numbers, and the next step in the FLT proof even if
he gets what he wants from "the" object ring is totally murky
as far as I can tell.

But the strategy is clear, right? The object ring magically sprouts
/whatever/ properties are needed in order for James to "rigorously" ignore
sane objections to the bulk of his FLT "proof". It's an infinitely
adaptable counterexample-wisher-awayer.

... [speculation about why JSH is, umm, JSH] ...

I expect you know I agree, but I'm keeping strictly to math here ;-)


.

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