Re: [metatuning] Digest Number 1018

Hi Paul,

> Umm . . . Robert, that's overstating the case a bit.
> For example, have you seen Penrose's book _The Large, The Small, and
> the Human Mind_? Stephen Hawking and the other thinkers Penrose
> invited to contribute have certainly found flaws . . . or so they
> think . . . I've purchased other books on this too, wish I could
> remember . . . but lots of fine thinkers think they've found
> flaws . . .

Yes probably got a bit carried away. Sorry about that.
Just meant really that you won't find a one liner
type easy knock down argument of it, at least, he
will almost certainly have met it before and given his response
to it. He has the sort of integrity of a mathematician
that if he did find a flaw that he was convinced himself
was a flaw then he would withdraw the argument, and
tell everyone that he had made a mistake - I'm sure of that.
And he is an honest clear thinker from what I've
seen of him. So if there are perceived flaws
then they are ones that can be replied to.

> Aren't you forgetting model theory here? There is a model of
> arithmetic in which G is true, and also a model in which ~G is true.
> So this "truth" you speak of is, of course, determined independently
> of the axioms (assuming they're consistent), since the axioms can't
> select between one model and the other.

Yes of course, the sentence and its negation are both
consistent with the axioms, otherwise it wouldn't
be an extension of them. But the sentence can
be seen by a human to have to be true if the
original theory is consistent. The negation
of the sentence can only be true if it is an
inconsistent set of axioms.

So therefore in that sense the sentence can
be seen to be true, and if you are going to
add it in to extend the theory, you would add in the
sentence rather than its negation, even though
both are consistent with the original axioms.

Yes of course the truth of the axioms are determined
independently of the model. The model only establishes
consistency, not truth. Establishes truth if you
interpret everything in a particular way, so
true of that interpretation of the axioms, but that
re-interpretation needed for the model isn't the
same as the original intended interpretation
of the axioms. So the model doesn't tell
you whether the axiom modeled is true of the
original interpretation, only that it is
consistent with it which isn't qutie the
same thing.

Well it is the same thing if you are a formalist,
then you can say, okay, its consistent
isn't it, so of course I can add in the
negation of the godel sentence.
It won't convince a formalist. But
I think people do have an innate
idea of truth, even formalists.
If you do then it is clear which one is
the axiom to add.

Robert

----- Original Message -----
From: <metatuning@yahoogroups.com>
To: <metatuning@yahoogroups.com>
Sent: Thursday, July 08, 2004 7:55 PM
Subject: [metatuning] Digest Number 1018

>
> There are 7 messages in this issue.
>
> Topics in this digest:
>
> 1. Re: Re: The limits of science and human thought
> From: "Aaron K. Johnson" <akjmicro@...>
> 2. Re: The limits of science and human thought
> From: "Gene Ward Smith" <gwsmith@...>
> 3. Goodstein's theorem
> From: "Gene Ward Smith" <gwsmith@...>
> 4. Re: The limits of science and human thought
> From: "Gene Ward Smith" <gwsmith@...>
> 5. Re: Goodstein's theorem
> From: "Gene Ward Smith" <gwsmith@...>
> 6. Re: 1st Defs (was: Is Science a Religion?)
> From: "Jon Szanto" <JSZANTO@...>
> 7. Re: Is Science a Religion?
> From: "Carl Lumma" <clumma@...>
>
>
> ________________________________________________________________________
> ________________________________________________________________________
>
> Message: 1
> Date: Thu, 8 Jul 2004 12:14:45 -0500
> From: "Aaron K. Johnson" <akjmicro@...>
> Subject: Re: Re: The limits of science and human thought
>
> On Thursday 08 July 2004 10:05 am, Carl Lumma wrote:
> > > > > http://members.shaw.ca/tfrisen/Buckman.htm
> > > >
> > > > This is incredible:
> > > >
> > > > "(4) Also thoughts from the right side of the brain that
> > > > cross to the left side are not recognized as our own, but
> > > > coming from outside ourselves."
> > > >
> > > > Have you seen the book Johnny Reinhard frequently refers to:
> > > > THE ORIGIN OF CONSCIOUSNESS IN THE BREAKDOWN OF THE
> > > > BICAMERAL MIND (1976) by Julian Jaynes?
> > > >
> > > > The central thesis of this book, which seemed very far-fetched
> > > > to me, is made A WHOLE LOT more plausible by the above.
> > > >
> > > > If it ((4)) is true.
> >
> > I would tend to doubt anything on the above page is true.
>
>
> Carl,
> why not? what's the basis of your positiion?
>
> Best,
> Aaron Krister Johnson
> http://www.dividebypi.com
> http://www.akjmusic.com
>
>
> ________________________________________________________________________
> ________________________________________________________________________
>
> Message: 2
> Date: Thu, 08 Jul 2004 17:41:16 -0000
> From: "Gene Ward Smith" <gwsmith@...>
> Subject: Re: The limits of science and human thought
>
> --- In metatuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > i have absolutely no basis for estimating Bush's math
> > knowledge ... but based on the evidence of what he's
> > spoken during his "presidency" (dictatorship), it's
> > *extremely* hard for me to believe that his English
> > scores were above average.
>
> They were below average for a Yale undergraduate. Most people can't
> manage to reach the level of a dumb Yale undergraduate, so Bush gets
> three digits and not two.
>
>
>
>
> ________________________________________________________________________
> ________________________________________________________________________
>
> Message: 3
> Date: Thu, 08 Jul 2004 18:00:07 -0000
> From: "Gene Ward Smith" <gwsmith@...>
> Subject: Goodstein's theorem
>
> This gives a nice example of a theorem which can't be proven in Peano
> arithmetic:
>
> http://en.wikipedia.org/wiki/Goodstein's_theorem
>
>
>
> ________________________________________________________________________
> ________________________________________________________________________
>
> Message: 4
> Date: Thu, 08 Jul 2004 18:03:44 -0000
> From: "Gene Ward Smith" <gwsmith@...>
> Subject: Re: The limits of science and human thought
>
> --- In metatuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > As I think Kraig was hinting, it's possible that he does
> > a lot of that stuff on purpose. Like Bogey let his lip
> > get caught on his fillings on purpose.
>
> Eisenhower did that; during press conferences he would produce
> baffling verbiage which almost made him seem brain-damaged, but in
> private he spoke clearly and concisely, like a general. Nixon hired
> Zigler to do the same for him.
>
>
>
> ________________________________________________________________________
> ________________________________________________________________________
>
> Message: 5
> Date: Thu, 08 Jul 2004 18:08:10 -0000
> From: "Gene Ward Smith" <gwsmith@...>
> Subject: Re: Goodstein's theorem
>
> --- In metatuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > This gives a nice example of a theorem which can't be proven in
> Peano
> > arithmetic:
> >
> > http://en.wikipedia.org/wiki/Goodstein's_theorem
>
> For some reason the Sprint ad seems to destroy the link.
>
> http://tinyurl.com/2y2fb
>
>
>
> ________________________________________________________________________
> ________________________________________________________________________
>
> Message: 6
> Date: Thu, 08 Jul 2004 18:08:50 -0000
> From: "Jon Szanto" <JSZANTO@...>
> Subject: Re: 1st Defs (was: Is Science a Religion?)
>
> --- In metatuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> > There's something kind of tragic about the later Elgar, with his
> > strong reaction to the war and then the death of his wife--sort of
> > like Robin Williams in Good Will Hunting. Here was the epitome of
> > Edwardian optimism--people were basically good, the world was clearly
> > getting better, Germans are some of his good musical friends, and so
> > forth. It all comes crashing down, and you hear the echos in his
> > later music.
>
> For this reason the Cello Concerto is heart-breaking: an old man in
> the twilight of his career, mourning the loss of loved ones and
> watching his country slide from the hey-days of empire.
>
> Anyhow, you've pinpointed the poignancy of his later years, and this
> is all recounted (based on a lot of factual material) as he ruminates
> over his life on this boat journey. If you're interested:
>
> Gerontius
> by James Hamilton-Paterson
> http://tinyurl.com/2nsnj
>
> Cheers,
> Jon
>
>
>
> ________________________________________________________________________
> ________________________________________________________________________
>
> Message: 7
> Date: Thu, 08 Jul 2004 18:43:01 -0000
> From: "Carl Lumma" <clumma@...>
> Subject: Re: Is Science a Religion?
>
> > > > Of course, if you're enough of a relativist, you'll
> > > > have your head up your ass so far that none of this
> > > > will matter. Instead of seeing science as the great
> > > > political *leveller* and *equalizer* (witness the
> > > > genetic proof that whites and blacks are inherently
> > > > equal, and that there is no genetic basis for any
> > > > of the arguments for racism),
> > >
> > > I'm not familiar with that proof, but given our current
> > > understanding of the workings of genes it sounds dubious.
> >
> > We do know that genetically speaking there isn't any such
> > thing as a white race and certainly no such thing as a
> > black race,
>
> You're referring perhaps to the recent article by
> Michael J. Bamshad and Steve E. Olson in Scientific
> American? Totally unconvincing.
>
> > Humans are not, genetically speaking, widely varied,
> > being of recent vintage.
>
> And being generally inbred.
>
> > The superficial characteristics which are so
> > striking, at least to us, don't seem to amount to much
> > more than cosmetics.
>
> They ammount to preferential breeding, which is
> probably significant.
>
> -Carl
>
>
>
> ________________________________________________________________________
> ________________________________________________________________________
>
>
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--- In metatuning@yahoogroups.com, "Robert Walker"
<robertwalker@n...> wrote:

> Yes of course, the sentence and its negation are both
> consistent with the axioms, otherwise it wouldn't
> be an extension of them. But the sentence can
> be seen by a human to have to be true if the
> original theory is consistent. The negation
> of the sentence can only be true if it is an
> inconsistent set of axioms.

Actually, it can merely be omega-inconsistent, which is far easier to
live with. And work from.

> So therefore in that sense the sentence can
> be seen to be true, and if you are going to
> add it in to extend the theory, you would add in the
> sentence rather than its negation, even though
> both are consistent with the original axioms.

Or, on a different model of arithmetic, you could add in its
negation. This is essentially what John Robinson did to create a
framework for Nonstandard Analysis. Which allows easier proofs of
calculus theorems.

Hi Paul,

> Actually, it can merely be omega-inconsistent, which is far easier to
> live with. And work from.

Sorry, yes shows how rusty I am. But not denying you can
work with those theories. You can even re-interpret them
to mean something that you can understand to be both interesting
and true. But that then is no longer the original thing that
you were studying.

> Or, on a different model of arithmetic, you could add in its
> negation. This is essentially what John Robinson did to create a
> framework for Nonstandard Analysis. Which allows easier proofs of
> calculus theorems.

Actually that is very much my home territory in Logic
as that is what my research was basically about, a form of
Nonstandard Analysis, but one using intutionistic
and classical modes of reasoning in the same theory,
combining both. There there was an intended model
too, philsophically based, on my ideas of
seeming infinity. Related to the work of Vopenka's
group in prague, who gave as their interesting
interpretation the idea of a very large finite
number with a seemingly endless sub _class_ of a finite set-
taking that as ones axiom, together with a
way of transferring results from the endless
subclass to a superset - if you had induction over
an endless subclass (ordinary set theoretic mathematical
induction) then you could deduce that the result
also held for at least one number beyond it.

There depending on ones philsophical sympatheis
one can think of it as a large finite number
that seems infinite, or as an infinite number
finite in form beyond all ordinary finite
numbers. Just like non standard analysis
but not based on a star transform at all,
rather axiomatised from the ground up
as a new theory which had infinitesimals in it.

I then had ideas of using intuitionistic
logic for the endlessness there, and classical
logic for the very large finite number -
and the thing there is that
in intuitionistic calculus normally
you don't have the trichotomy rule that
x < y || = y || x > y
for reals x, y and z.

You only have
x!=y -> x < y or x > y - which is sufficient
to prove the theorems of calculus but everything
gets much more complicated. It works but isn't
elegant.

But with my ideas you had
x <= y or x >= y which is almost
as good as trichotomy for proving calculus
theorems. Add in the infinitesimals
as well, which came with the theory
and you ended up with a hybrid
classical / intuitionistic theory
with particularly elegant proofs
of the theorems of calculus using
infinitesimals. But unlike non standard
analysis you can't prove equivalence
of the results with those derived classically
in this type of approach based on Vopenka's
idea.

Instead you need to prove them all again
from scratch. Which if you want just
another way of looking at existing maths
is a drag. But if you want to explore
what might possibly be new maths, it
is interesting. Well, I don't know whether
or not you can get new classical type
maths that way - I think that was a
hope in Vopenka's work somewhat -
but anyway you certainly did get something new if you
think of it as a kind of constructive maths
got easier proofs, and it seemed also possible
that you might do more than you normally think possible
constructively.

I even got as far as the beginnings of an intuitionistic
/ classical version of Lebesgue integration
I remember. But never publisehd any of it
being rather discouraged when the examiners
failed my thesis. But I was so poor
at presenting my ideas in those days.
They failed it mainly because they
didn't understand how I used a particular
axiom in my proofs, and didn't understand
how it could ever be used in the theory
at all. So unusually for a maths
thesis, they wanted me to remove one of
the axioms which then left nothing much
of interest in teh theory. but if I'd
been able to explian better then
I might have been able to explain what
it was doing in my theory. It was
a very unusual form of axiom but later
I saw the identical axiom used
in a published paper but didn't know
about that at teh time.

Some day I may get back to it though
if I had the time I'd also be
very inclined to spend that time
composing instead, I'd really
like to develop that side of things
more and find out more about how
chords and things work in various
tunings. Hard to say which is the
best to do really if I get the
opportunity when things are a bit
less hectic in the way of programming.

Robert