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


Tim Peters wrote:
...

[Tim Peters]
I expect it's a mistake to assume that James knows what mathematicians
mean when they say two elements of a ring are coprime, except (and only
except) when the ring is Z.

[Gene Ward Smith]
I think this has been the problem, or one of them, all along. James
concludes two algebraic integers are coprime, when if you look at his
reasoning that isn't what he ought to be saying. If he'd said of the
three roots of y^3-12*y^2+65=0, one of the roots is a 5-adic unit, then
he would have said something which made sense in terms of the arguments
he was bringing. Instead he says it is coprime to 5, which sends the
whole discussion off in the wrong direction. Moreover, James comes to
this conclusion indirectly, looking at the factorization of the norm
term, and doesn't understand what it is he's really got an example of.

That's right, but he (thinks he) knows what he wants to /conclude/, and that
really drives everything. 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,

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 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 - 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.


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. It is a consequence of the theory of ideals that none
of the roots are coprime to f in the algebraic integers. He
accepts this as a fact - if you accuse him of NOT accepting
it, he will shriek that you are lying about what he says
- 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!
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.

[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.]

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. [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.


but underlying many of his arguments I
sense anger at rings that don't behave like a unique factorization domain
(or sometimes just an integral domain) when he wants them to. They "block"
him, they have "problems" and "flaws", and mathematicians "lie" when they
say such rings are perfectly fine the way they are.

Certainly there is anger. It's not at all clear that "the"
object ring gives him what he needs - 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.
There is enormous anger because mathematicians haved mercilessly
proved him wrong every step of the way. There is HUGE anger at
what happened with the electronic journal SWJPAM. Harris thinks
getting a paper published in a Peer-Reviewed Journal is the
final absolute stamp of approval, the one act that proves your
proof, and that it cannot be reversed. But it was! It was like
a starving man having a delicious sandwich placed in his mouth,
and then having it yanked out and thrown away before he can even
swallow a bite! His righteous outrage at this horribly unjust, almost
unprecedented event constantly recurs to his consciousness. Prominent
mathematicians - the Powers That Be - politely (or nervously ...)
listen to his ravings and then give him the brush-off. There
is deep confusion as well. And as you implied in another post
yesterday, there are signs of deterioration. At one time Harris
was able to talk intelligibly about the Barlow-Abel relations.
I think *** Winter has said that Harris once had a valid proof
of one nontrivial case of FLT for n = 5. Those days are long
gone. He makes more trivial errors, forgets the form of his own
beloved polynomial. Now he cannot bring himself to even check
his work with computations. The cause? Either prolonged mental
distress and frustration, or the now-ingrained habit of lying to
himself and everyone else to prop up his delusions and refusing
to face the truth, or alcohol. I don't think he is mentally ill
yet but he is getting there. He is too young for senility. I
think the most likely explanation is alcohol. Look at the recent
pattern. He submits a few posts early in the weekend, maybe on
Saturday morning. He gets angrier as the day goes on. Finally
he stops. I am guessing that is the alcohol taking hold.

Marcus.

.