Re: Meyer's Argument against Gödel's Theorem

I'm pleased that my novel about Gödel's ontological proof got
mentioned on this list. I was not very surprised that LauLuna had
reservations about my approach to Gödel. _God Proof_ is a venture
into hard SF at the boundary of math, religion & philosophy.

Combining math & philosophy is dicey. Throw in religion and things
get really treacherous. Add to that a writer who has decided to
deal with the whole business in the form of a novel, and even a
generous person might wonder if he's not dealing with a crank or
madman.

I took it from LauLuna's remarks that he'd not actually read _The
God Proof_, but that he had done me the favor of looking it over
to see if, despite everything, there might be something there. I
hope I don't presume on sci.logic's patience if I explain a bit why
I dealt with this material in the way I did.

Let me say that my research into modal logic
and Gödel was quite serious. I became proficient enough in the
math to assist two of the professionals then studying Gödel's
ontological proof. Melvin Fitting
(http://comet.lehman.cuny.edu/fitting/errata/book_errors/
godelbookerrors/godelerrata.pdf)
and Jordan Howard Sobel (by letter) were kind enough to acknowledge
my minor assistance. Assistance to another researcher at the
boundaries of philosophy and mathematics landed me a mention in
_Mind_ (V115, N459, p. 692). An acknowledgment does not compare
with a publication, but just the same seeing my name in the same
pages which have carried articles by Turing, Freud and James gives
me a shiver. On my own, I'm a published mathematician of minor note.
(Communications of the ACM, V29 #6, June 1986, pp. 556-558).

Far more serious qualifications than mine would be no guarantee
against error. LauLuna presents some paraphrases as evidence that
I've got Gödel wrong on the Incompleteness Theorems. For example,
LauLuna says,

[ a proof that the world cannot be proved consistent ], Kegler
writes, is not that bad, for if the world cannot be proven
consistent, well, that's a proof that it is indeed consistent.

While my original langauge was carefully chosen and I prefer it,
the paraphrase above is close enough for this purpose. LauLuna seems
to be saying that statements like this clearly demonstrate that I've
gone off the rails. I can't for the life of me see where.

Let's leaving aside my use of the philosophically-loaded term "world"
for the moment, the math is not only correct, but downright boringly
orthodox testbook stuff. Not very formally,
the argument goes like this:

1. An inconsistent system is, by definition, one with a logical
contradiction.

2. From a logical contradiction, you can deduce any statement
whatsoever. (This is the principle of explosion, very well-
established
in classical logic.)

3. Conversely, if there is any statement at all which cannot be
deduced in a system, the system must be consistent.

4. Gödel noticed that arithmetic cannot prove its own consistency.
This (the fact that there is something arithmetic cannot prove) is
a meta-proof that arithmetic is consistent. That's because an
inconsistent system proves everything, including both its own
consistency and its inconsistency.

I expect the above will be very familiar to a lot of you on sci.logic.
Now, OK, what's the point of my talking about the consistency of
worlds instead of logical systems?

If a novel is centered on math, it needs to be made real. Usually,
in math courses all the philosophical baggage is stripped off and
ignored. How good an idea this is I won't address, but it does
free up the syllabus for a march through the formal systems and
their consequences.

But Gödel felt, as I do, and as a novelist of math must, that math
is about real things of real concern to real people. That means
my narrator (Josh Bryant) is fated to tackle those philosopical
issues head-on.

Josh is taking the position that logic underlies the world of the
senses. This is not beyond debate, but it is very mainstream.
You're very hard put to justify why it's even worth an attempt to
do science unless the world of appearances is logically coherent.
And how do you do this without treating the basic laws and results
of logic as facts basic to the world?

I don't say that there aren't other approaches. But Josh's position
is very mainstream. Josh just states it a lot more clearly than is
usual. He's a character in a novel. You'd expect that.

In academia, math and philosophy are separated. Gödel's
Incompleteness Theorem is discussed as an exercise with formalisms
in the math literature. The relationship of logic to ontology is
dealt with in the philosophical literature. But if either mean
anything in reality (and in a novel, things must be meaningful
to the characters) the twain must meet.

Even in SF, writers are often just plain indifferent to accuracy.
I took great trouble to make _God Proof_ not just a book
that stirs the imagination, but one that is as accurate as a book
which avoids equations and technical language can be.
It's easy to check out how well I've succeeded: _The God Proof_ is
available as a free download (http://www.lulu.com/content/933192).
Those who want a print edition can order one from either Lulu or
Amazon.

I don't claim _The God Proof_ is inerrant. (In another of his
interpretations of Gödel's work, Josh claims that the
Second Incompleteness Theorem in fact
shows that inerrancy is possible only if inerrancy is not claimed.)
Readers might find statements which are not just informal, but plain
ol' incorrect. I'm grateful to have those pointed out.

thanks,

Jeffrey Kegler
.

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