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: 19 Jun 2004 01:49:23 -0500
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"Jeffrey Ketland" <ketl...@ketland.fsnet.co.uk> wrote:
>Spike wrote in message <40d2c63b$0$92045$45beb...@newscene.com>...
>>
>>"Jeffrey Ketland" <ketl...@ketland.fsnet.co.uk> wrote in message
>[snip]
>> I assume that you posted the above only as a helpful answer to
>> Larry's question, so that I must exclude you from my remarks.
>> Larry's question, and thus the above definition, fits into the
>> larger debate over Dolan's piece by becoming an expansion of the
>> word "true" in the phrase:
>>
>> "...statements which are true in that set of axioms..."
>
>Saying that a statement is "true in that set of axioms" is merely evidence
>of confusion and/or ignorance.
Beginning a post with such an assertion typically flags that it will be
the assertion to be argued, also that the argument will be in
reasonably rigorous form. I.e. it is an attention getter, and you have
my attention.
>A statement is not "true in a set of axioms",
>because truth has nothing to do with *axioms*. For example, the set Tr of
>arithmetic truths, which I gave a definition of in the previous post, has
>nothing to do with axioms. In fact, the set Tr of arithmetic truths is not
>axiomatizable. This follows from Tarski's Indefinability Theorem.
Understood.
>In general, whether a statement is true depends upon whether the relevant
>state of affairs that it expresses holds or not.
>
> A statement is true iff it says that such-and-such is the
> case, and such-and-such is the case.
Understood.
>As Aristotle put it, "To say of what is, that it is, is true". Of course,
>whether a linguistic item like a statement is true depends upon its
>interpretation.
That's a good point and very true. I believe that to reasonably argue
your main assertion, you will need to answer questions related to this
truth. For example:
Who is the resident expert of linguistic interpretation, i.e. who
is most qualified to write a short approximate explanation of Goedel's
theorem in such a way that, in practice, a reader will come away with
the best approximation? And what is the "best approximation?"
Theoretical understanding? Use of the result of the theorem for various
purposes? Given that all concepts cannot be included, which theoretical
and/or useful concepts should be included? Should the complexity of
concepts affect which concepts are included? Is there such a thing as
one element of the complete, simplest theoretical description being
more important than another, or would a writer choose the elements
to be included on some other basis?
Please excuse this diversion from your line of reasoning. Continuing
from the Aristotle quote:
>But it should be stressed that in usual cases, this is left
>implicit. If someone asks "Was GW Bush's opening statement of his speech
>yesterday true?", nobody wonders whether he is using the word "Iraq" to
>refer to the planet Venus. He is speaking English, an interpreted language,
>in which "Iraq" is a singular term which refers to Iraq.
I get that understanding is extremely reliable when using well-known
labels for well-known physical objects.
>The language of arithmetic, for example, is an interpreted language---by
>definition. So, "0" refers to the number 0, and "+" denotes the addition
>operation and the quantifiers range over N, and so on. So, "2+2 = 4" is true
>if and only if 2+2=4. One could take the constant "0" to refer to Paul
>McCartney if one liked, but this would be a very weird re-interpretation of
>the language of arithmetic. Admissible, but weird and pointless.
Isn't this a restatement of part of Tarski's definition? "p" is true if
and only if p is true? "He means Iraq" is true only if he means Iraq.
>The statement "John Lennon was born in Liverpool" is true (in its standard
>English interpretation), because John Lennon was, in fact, born in
>Liverpool. This has nothing to do with "axioms".
I get that you have made a definition of "true," which you assert is
the proper definition in context of a Goedel explanation, and have
applied it to the phrase "true in that set of axioms," deciding that
"true" cannot apply under your definition.
I see that as a powerful argument to any mathematical logician. But
given that intended readers will not know any of that, and will not
understand "true in a set of axioms," what inference(s) will they make?
Have you heard the phrase "Judge a tree by its fruit?" Do you think it
is nonsense? Do you think this demonstrates that the author is
confused? I see a definition of pragmatism in there. I also see that
the author prefers empirical tests of facts. In other words, I
inference that statement into many paragraphs of explanatory text. But
why did the writer use a short statement which is literally
nonsensical? I believe you must answer that question before your
assertion will be accepted by me or those not heavily invested in
the result.
In writing a short approximation, one sometimes uses nonsensical
phrases purposely to force the reader to draw expected inferences
instead of preventing that with a trivial literal statement. This is
only one writing device among a hundred, and I am not saying Dolan used
this device. I am instead challenging you to prove he did not use this
device as one less obstacle to my acceptance of your assertion. I'm
helping you to persuade me and others.
>Similarly, the arithmetic
>statement "There are infinitely many primes" is true (in the standard
>arithmetic interpretation), because there are, in fact, infinitely many
>primes.
I see the correspondence, but it seems you are wandering regarding the
tests of facts. You equated "truth" with "fact" in your earlier post.
No matter, I take it that is not your point.
>These ideas were made precise by the Polish logician Alfred Tarski in a book
>published in 1933.
Agreed.
>Simplifying somewhat, if we are discussing (in English) the notion of truth
>for an *interpreted* object language L, then we always assert instances of
>the Tarski T-scheme,
>
> (T) S is true in L if and only if p
>
>where S is a sentence of L and "p" is replaced by the translation of S into
>English.
>For example,
>
> "Schnee ist weiss" is true in German if and only if snow is white,
Understood. Ve vill NOT dreenk ze yellow schnee!
(That is intended to say I don't speak German in a way that doesn't
put readers to sleep.)
>If the meta-language contains the object language, then the need for a
>translation lapses, and we have the "disquotational T-scheme"
>
> "p" is true if and only if p
Understood.
>To sum up: People who write things like "true in a set of axioms" are merely
>displaying their confusion,
Did you understand that the article being debated was intended for
educated laypersons, not logicians or mathematicians or having any of
that background other than a distant general education?
How will such people interpret "statements which are true in that set
of axioms?" I.e., since they won't know anything about your "'p' is
true if and only p" or Tarski's definition, and won't know what a "set
of axioms" is, or any math definition of "in", but will try to
inference the meaning as best they can, what is your guess as to how
they will inference that phrase? I've made a couple of guesses.
What are yours? What are your credentials for making such guesses?
Also, what definition are you using for "in" in that phrase? Given that
the reader can only understand common language, what common definition
will the reader use for "in?" What are your credentials for judging
that?
>like someone who doesn't know what a
>differentiable function is.
This seems to contradict your assertion, since the author has a
mathematics background and is highly likely to be very familiar with
that term. I say "seems" because I'm well aware of the fallacy.
Those serious readers who are not invested in the results of this
debate will no doubt find it most significant that you completely
snipped the argument you are replying to, ignoring same, and
only continuing your own (as one now infers that your Tarski post was
in fact joining the larger debate). In logical argumentation that is
called "changing the subject" (and other things). To an experienced
arguer it is a clear announcement of a resignation, though done and
usually accepted in this way not to force the loser to eat humble pie
(because we are all the loser on occasion).
I will ignore this usual announcement because it doesn't seem to me
that you are an experienced debater. In fact I found that "definition
of 'in'" argument rather weak. Why not refute it quickly instead of
giving the appearance (to some) of announcing a lost position?
OTOH, I give your argument it's moment of fame as a demonstration that
I am not the least intimidated by my opponent's arguments. I rather
seek the facts even if that means I will be corrected. I usually doubt
my opinions even if I am convinced of them. I like to call them
probabilities because that helps me to acknowledge they are wrong, i.e.
it is trivial that one can lose a game of chance.
Your argument is not in the least persuasive to me. In case you have
interest in persuading me or the audience not yet invested, I included
a likely roadmap for same in this reply.
Larry
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