Message from discussion
Goedel - interesting problem?
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From: "Acme Diagnostics" <LFinezapt...@partpostmark.net>
Newsgroups: sci.logic
Subject: Re: Goedel - interesting problem?
Date: 20 Jun 2004 20:54:11 -0500
Lines: 281
Message-ID: <40d63f29$0$7275$45beb828@newscene.com>
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I wish to make a record concerning certain criticism of the variously-
named "Dolan Goedel description" posted earlier in this thread, also
to make a statement of intention regarding it.
The "description" is in fact my own compilation of excerpts taken
out of context from old routine posts without Dolan's knowledge or
ever his characterization of any of it. The original context was
limited to math or ordinary arithmetic, not any extensions into other
"axiom systems," etc. That is only one way my compilation is divorced
from the author. Thus all promotion and characterizations of "the
article" are mine alone, and Dolan must be held harmless of any
resulting criticism.
In criticizing my compilation and characterization of it, posters in
this group have variously asserted or implied:
*. There are no controlling and overarching editing criteria for a
short explanation of a complex theory.
*. There can be no approximate understanding of a complex theory.
*. Educated laypersons will not interpret words grammatically. *1
*. They will not inference what they do not understand.
*. They will think that precise theoretical accuracy is intended.
*. There is no legitimate use of a short explanation of a complex
theory. *2
*. The only useful description of a program is its code listing.
*. Opinions in sci.logic supersede "Goedel on the web." *3
*. Educated laypersons are unable to usefully compare technical info. *4
*. I am too ignorant of Goedel's Theorem to judge accuracy. *5
*. My characterization of "masterpiece of explanatory text" implies
precise theoretical accuracy. *6
None of the above are tenable assertions. The absurdity of
several, by themselves, refute a purely theoretical analysis. All
deflate or invalidate the criticism in some way. Not that my critics
are in any way deficient - this sort of thing goes on all the time
in newsgroups, but that's not my topic today.
I don't expect correspondents to be accomplished debaters.
But everyone learns in Logic 101 that privileged positions
are not allowed in logical argumentation. Most, if not all, critics
attempted to assert privileged positions at some point. My ears
are unfortunately programmed to close immediately on same.
There is nothing I can do about that.
Those are the reasons I reject these poster's criticism of the article.
However I have received some off-line criticism that has been
persuasive, mainly having to do with playing a song that some
of the audience is known to hate and necessary academic goals.
Thus in good faith I will footnote the article as follows:
- - - - -
Without prejudice to the author: I recommend this explanation
only for dialog, not general publication. This explanation favors
reader's goals over the legitimate academic goals of experts by
approximating theoretical accuracy in favor of explanatory value
and usefulness outside of theoretical context. Upon objection of
experts, or in company of experts, you should expend some
of the explanatory excellence in their favor by replacing the 2rd
and 5th occurrences of "set of axioms" with "axiom system."
- - - - -
Google count for ["axiom system" incompleteness]: 968
Uses in this sci.logic thread: 2
Chris:
>In some cases, yes, but not by reasoning in the axiom system being shown
>to be incomplete.
Aatu:
>.,..if the strange notion about truth "within" or "without" an axiom system
>was dropped...
- - - - -
My compilation of Kent Paul Dolan's posts, with two unauthorized
revisions to substitute "axiom system" for "set of axioms:"
[Goedel] proved that any set of axioms at least as rich as the
axioms of arithmetic has statements which are true in that axiom
system, but cannot be proved by using that set of axioms.
That does not prevent that those true things can be proved
with a more powerful set of axioms. It only conveys that the
stronger axiom system will in turn contain new truths which
cannot be proved using only those axioms.
Goedel's incompleteness theorem only shows that some true
math facts cannot be proved within math, not that none of them
can.
It isn't all that complicated to follow the proof, either, since it
uses only the axioms of arithmetic to achieve its goal.
- - - - -
Here are some editing aspects I noticed in my first reading, plus
a few I've noticed since. Having much experience with technical
writing for laypersons myself, each impressed me in some way.
*. I interpreted the last paragraph as motivation to learn more.
*. Separation of short paragraphs.
*. Avoidance of contractions with "cannot" and "does not" but not
with "isn't."
*. The constant repetition of the one technical and most important
Goedel term.
*. I don't think most readers would remember any of six technical
terms used once.
*. I noticed that certain phrases are repeated exactly, such as
"cannot be proved" being repeated 3 times.
*. I noticed "by using" and "using only" rather than "with."
*. I noticed the various terms "true statements," "things," "truths,"
and "true math facts" all to describe the same thing.
*. I noticed that "statements" became "things" and "truths" in the
second paragraph.
*. I noticed "true in that set" rather than "true within that set."
*. When reading Jeffrey Ketland's Tarski post, I noticed the
equivalence of "truth" and "fact," and that that equivalence is
repeated in the article.
*. I *especially* noticed the clear logic, devoid of such things as
"some false things but not some true things."
*. I noticed the use of "which" and "that" throughout.
*. I liked the superficially redundant ", either," in the last
paragraph.
*. I noticed that the full title "Goedel's incompleteness theorem"
awaits the third paragraph.
*. I noticed "It isn't all that complicated" instead of just "It isn't
that complicated."
*. I noticed the two spaces separating sentences.
*. I liked "will in turn" in the second paragraph in place of just
"will."
*. I noticed "to achieve its goal" in the last paragraph instead of 20
other lesser alternatives.
*. I noticed the use of most common, least pretentious, least
problematic sufficient words throughout, which happens to be
my language dogma for interdisciplinary publishing, also to
laypeople and others, posted many times.
*. I noticed that the whole explanation uses less than 650 characters.
*. I compared that to my judgment of the typical attention span, in
number of characters, of a typical newsgroup reader when reading
typical Usenet posts, also for material attendant to other topics.
I thought this length convenient for various venues besides Newsgroup
posting, non-spam length, for instance, some discussion formats, edit
boxes, etc.
*. I thought to notice that all the important concepts that an educated
layperson could reasonably be expected to superficially grasp at
this high level of description in a such a short reading are included
and, most important, no more than that.
I could not find anything that even came close to meeting this criteria
in, by now, several hours of googling.
Beyond those types of observations, there is the artistic component.
I don't believe that the art in Art, as opposed to technique, can be
explained.
- - - - -
Notes:
*. Educated laypersons will not interpret words grammatically.*`1
(1) But if they don't know math or logic beyond a distant minimal
general math education, how else can they interpret them?
*. There is no legitimate use of a short explanation of a theory. *2
(2) Course descriptions, speech synopses, reviews, physics and AI
discussions, voters, politicians, students, parents, employers, etc., etc.
to decide allocation of resources, other examples posted. Analogies:
"Goedel, Esther, Bach" (Pulitzer Prize winning book).
*. Opinions in sci.logic supersede "Goedel on the web." *3
(3) When anyone googles academic information, they place most faith in
name-brand colleges and highly credentialed or acclaimed authors,
moving down a hierarchy of less and less credible types of sites, with
discussion groups being at the very bottom of the credibility scale.
*. Educated laypersons are unable to usefully compare technical info. *4
(4) Except for the one phrase now able to be substituted, it doesn't
take a rocket scientist to notice the terminology in simple sentence
construction at this high level of description being essentially
repeated on authoritative web site after site with obvious redundancy.
I've pasted excerpts here and criticism has been dropped or adjusted. I
have more, including an acclaimed CS/math author/Phd and several
university sites for "association" of "true theorems" with "set of
axioms" so that I wasn't quite as poor a googler as supposed.
Generally, it's well-known that educated laypersons correct physicians
using information on the web, and that physicians encourage them to
use the web for this purpose and to learn about their diseases.
Medicine is at least as technical as mathematical logic, and the
academic credentials of medical doctors compared to their googling
patients are typically quite higher than those of newsgroup academics
compared to their googling critics. The assertion is obviously
self-serving to certain 20th century academics.
*. I am too ignorant of Goedel's Theorem to judge accuracy. *5
(5) While this is true, it is irrelevant. Sometimes the most useful
description of a complex process contains untrue things. The only
goal is the best approximation of the process in the mind of the
reader reliable for a defined purpose in a given limit of characters.
Everyone in the business of writing short copy knows this elementary
and routine fact. There's an excellent example earlier in this thread,
but too long-winded to include here (ask). Thus, the phrase
"arithmetic has statements which are true in that set of axioms" does
not need to be theoretically correct, nor logically "sufficient," but only
reliable in practice for its intended use. I still think it is.
But, given that, how can I justify presenting something about Goedel's
theorem with a phrase that is nonsensical (if so) at the precise theory
level in a group mainly devoted to mathematical logic? The answer is
simple: My presentation was made in a peculiar context which I have
redundantly stated to deaf ears. Maybe it turned out to be a terrific
troll piece - but it was certainly never intended as such.
Further, I only ever represented it as a masterful piece of short copy
which described the *effect* of Goedel's theorem for *useful* purposes
as best it could in a certain *limit* of characters. That's not my
regular line of work, but I have much experience with it. So, it turns
out, my critics are the ignorant ones about the topic of discussion
which was never the Goedel proof, but an excellent example of
explanatory short-copy. Of course *some* knowledge of the theory is
required to write the piece, more than I had at the time. As I have
explained many times, I invested a great deal of time learning about
the author. I trusted that his knowledge was sufficient, and verified that
somewhat with an hour+ of googling for Goedel. That was more than
enough then for the goals of the article, and now after more hours of
Goedel googling, it still is.
*. My characterization of "masterpiece of explanatory text" implies
precise theoretical accuracy. *6
(6) Besides being qualified to the editing component, I chose
"masterpiece" for it's artistic connotation. I always assumed the
theoretical component to be approximate and still do, and doubt that
many would think "masterpiece" would apply to a dry, precise,
explication of a theory. There exists objective editing criteria as
basis for criticizing my characterization. However, there is an
artistic component to that as well. Judgment of art is entirely a
matter of taste, and "masterpiece" is simply one person's opinion of
the artistic merit of the piece.
- - - - -
Larry
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