I see a significant difference between Godel's theorem and
Godel-like principles/analogies. I would have to argue that both Tipler and
Penrose are using Godel-like analogies and are not using Godel's theorem. I
don't think this invalidates some of the ideas but it removes the "proven
theorem" authority for the claim.
A legitimate reaction to Godel's theorem is that it does not allow
one to reduce everything to an algorithm. Some things are "undecidable",
that is, neither they nor their contradiction is provable.
However, it is usually quite difficult to prove that a given
proposition is undecidable. And it is even harder to identify a "real
world" proposition as undecidable. So a claim that "Godel's theorem shows
one cannot prove X" is to be regarded with suspicion. In order to prove
such a claim, one needs a mathematical model which includes a proposition Y,
a proof that Y is not provable within that system, and a correspondence
between the model and the "real world" which maps the real world statement X
to the proposition Y.
(And someone claimed last week this list is too technical!)
I believe the issue in this debate is not Godel. It should focus on
the word "know". ("Epistemology", right? A friend has been teaching me
enough philosophy to make me dangerous.) What does one mean when they say
they "know" a signal was caused by intelligent life?
I certainly do not think a sequence of prime numbers would qualify
as proof of intelligent life. Evidence, maybe, but not proof. If you saw
01101010001 at the beginning of a signal, would you conclude the signal was
caused by intelligent life? I think we would have to discuss some
statistical models and probabilities... I think Harvey's "epsilon" will
appear here.
>Thus it is possible to know that a statement is true (a SETI signal which
>plays pictures on my TV proves that there is a message) but be unable to prove
>mathematically that the sequence contains a message. This is what I suggested
>earlier today.
>
>SETI is different from DNA. We can hope to design a device which decodes the
>message in SETI (i.e. plays the TV picture). But proving that DNA is the
>product of deliberate design, is something else. What TV set do I use to see
>(prove) the design?
I don't understand this argument. It has nothing to do with Godel,
IMO. Both signals seem to carry patterns and information. In one case
you've found a handy and common tool which *appears* to decode it. In the
other case you haven't. So?
>"Let us recall the arguments given in Chapter 4 establishing Godel's theorem
>and its relation to computability. It was shown there that whatever
>(sufficiently extensive) algorithm a mathematician might use to establish
>mathematical truth-or, what amounts to the same thing, whatever formal system
>he might adopt as providing his criterion of truth-there will always be
>mathematical propositions, such as the explicit Godel proposition Pk(k) of the
>system, that his algorithm cannot provide an answer for. If the workings of
>the mathematician's mind are entirely algorithmic, then the algorithm (or
>formal system) that he actually uses to form his judgements is not capable of
>dealing with the proposition Pk(k) constructed from his personal algorithm.
>Nevertheless, *we* can (in principle) see that Pk(k) is actually *true!* This
>would seem to provide *him* with a contradiction, since *he* ought to be able
>to see that also. Perhaps this indicates that the mathematician was *not*
>using an algorithm at all!" Roger Penrose, The Emporers New Mind, p. 417
There are theorems within the system and "meta-theorems" about the system.
Godel's theorem is not a proposition within the given smaller system, it is
a "meta-theorem" about the given system. However, in a larger system,
Godel's theorem has been proven. (By Godel.)
>What Yockey said about a sequence is that it is fundamentally undecidable if a
>sequence was produced by random or highly organized processes. This applies
>to DNA and to SETI sequences because both are "information". Was life
>designed or produced by random processes. Information theory and Godel's
>theorem tell us that it is an undecidable question from the direction of
>mathematics. It is NOT as Dr. Harvey suggested today that it is merely epsilon
>factor away from certainty.
Hmm, I guess I need to read the article by Yockey. But at the moment I have
to side with Harvey and epsilon. I am unaware of ANY "real world"
application of the term "undecidable" which has been satisfactory to
mathematicians. There are, frankly, very few mathematical propositions
which have been shown to be undecidable and none of them have been
particularly applicable.
>I would also suggest looking at Kolata, "Does Godel's Theorem Matter to
>Mathematics," Science 218, Nov. 19, 1982, p. 779-780
I'll look this up.
>Now, does anyone have a TV set that plays DNA?
Not yet....
Closing comments:
I am very suspicious of the paper by Yockey -- I've apparently
missed the reference; I would like to read the paper.
The impact of Godel's theorem has not been understood by most
scientists. Godel deflates "scientism", his theorem destroys the prevailing
belief that all we need is more knowledge, that it is possible to understand
*everything*. As Christians, it might be a tool we could use to challenge
those who worship science...
In Christ,
Ken
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Ken W. Smith, Professor of Mathematics
Interim Director, Office of Institutional Research "In the future
Central Michigan University, Mt. Pleasant, MI 48859 computers may weigh
Work phone: 517-774-7222, fax: 517-774-4250 as little as 1.5 tons."
Home phone & FAX: 517-772-5042 Popular Mechanics, 1949