throopw@rtp47.UUCP (Wayne Throop) (10/28/85)
[John is able to read net.philosophy, but cannot post. Thus, he mailed
me his comments on this subject, and asked me to post them (since he is
following up to one of my articles). Comments in square brackets are
mine, the rest of the article is from John.]
From Wayne Throop:
>...assuming that physical laws are consistant (in about the same
>way that formal systems are consistant), there is reason to think that
>Humans are "simulated" or "implemented" using consistant, formal,
>physical laws. Thus, Humans are arguably *already* implemented in
>a "consistant formal system", so I see no reason that they couldn't
>be simulated to any degree of accuracy you please in another formal
>system.
>I also note in passing (as I understand things) that nobody has ever
>actually found a Godel sentence for the formal system of "mathematics"
>or "logic". It has simply been proven that such sentences exist.
The first claim that consistent physical laws "simulate" humans so Turing
machines should be able to is hard for me to understand. In any event,
you need the added premise that the set of physical laws is recursive,
something I see no good reason to believe. The second claim that
nobody has ever constructed a Godel sentence is false. Godel
constructed one in his original paper. The construction is tedious, but
not beyond the ability of anyone who has had a one semester course in
mathematical logic.
John
[I take John to mean that there is no reason to think that the set of
physical laws is *only* general recursive (that is, it might have more
power than a general recursive formal system). I agree with that,
which is why I said "arguably" in my posting.
Also, I was aware that Goedel sentences have been constructed. My point
was that they have been constructed for "toy" formal systems. The
"formal system" that is "all of mathematics in common use" is much
larger than the stripped down version used to construct incompleteness
proofs, and might not even be consistant (or so I think... I'm open to
correction). Thus, the claim that a human could construct a Godel
sencence for *any* formal system is, I think, still suspect.]
[about my second posting of this series, John comments: ]
You and Ted Tedrick had the following exchange:
Ted:
The issue is that humans seem to recognize that certain formal systems
are consistent, but that this consistency cannot be proved within the
system. This mysterious ability to recognize such things being something
lacking in deterministic machines, I claim there is a distinction
between the human mind and any Turing machine.
You:
First, Godel's theory didn't have anything at all to do with recognizing
consistent formal systems. It just states some properties that
consistant formal systems of "sufficent power" must have, in particular,
incompleteness. This "mysterious ability" is something of your own
invention.
Read Goedel's paper and you will see that he proves two theorems,
known as Goedel's 1st and 2nd Incompleteness Theorem, respectively. So
far, you have considered only his 1st theorem (any recursive set of
axioms that contains number theory is incomplete). Ted seems to be
talking about his 2nd (roughly, any formal system that can prove its own
consistency is inconsistent). His 2nd theorem is a bit harder to state
precisely since consistency statements can be disguised, but it is often
considered the more important theorem for refuting Hilbert's program. It
has also been used as the basis for an argument similar to yours based on
the claim that we can see that a particular Turing machine is consistent,
but the machine itself can't see it; therefore, we are not Turing machines.
Another version is that we know we are consistent, and since no Turing
machine can know its consistent, we are not Turing machines.
John
[Well, you got me there. I was indeed forgetting about the second form
of incompleteness. However, I think the "mysterious ability" of humans
to detect consistancy is still a suspect notion. Mathematicians, for
example, don't seem to beleive in it, since Goedel's work was a
culmination of centuries of effort to *prove* mathematics consistant.
If any human could *see* it to be consistant, the proof would be
superfluous. In short, Goedel didn't prove that humans are "more
powerful" than anything, let alone Turing machines. He merely
(merely, ha... *I* should be so mere... :-) showed some (perhaps
surprising) properties that formal systems *must have*. There are (at
least) two ways that humans could be "equivalent to" general recursive
formal systems, even given Goedel's results:
- Humans could be inconsistant
- Humans could indeed be limited in what they can "know" ]
--
Wayne Throop at Data General, RTP, NC
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