ROG TO RIFF ON GODEL
RIFF:
Now back to my(?) statement:
"Any formal system of logic MAY be either complete OR consistent
but NOT BOTH."
I now must admit that I am completely unprepared to support this,
as well. As I said, I had thought that I was simply paraphrasing
from Hofstadter's "GEB" ("Godel, Escher, Bach:...", Basic Books,
1979.), but now I can find no such reference. If I had, I could,
at worst, claim that supporting the statement was not my job, and
simply point to Hofstadter (pretty BAD, I'll grant you, but such is the
state of my skills at present.). Now I have to ask if I have been
misrepresenting both Godel AND Hofstadter.
When I offered the above "quote" to my Ethics class, albeit in a
somewhat different context, the instructor acted as if he had been
waiting years for a student to make such an observation. He then
proceeded to attribute the statement to Heisenberg! Oh, well, at
least I'm not ALONE in my confusion about this.
ROG:
Below are two different credible references on Godel. Both writers are
considered leaders in the field of Quantum physics:
Quantum Gravitational Physicist Lee Smolin: "From [Godel's] theorem we learn
that any mathematical system complicated enough to include arithmatic can be
either consistent -- meaning without contradiction -- or complete -- meaning
that everything that is true can be proven -- but not both.". [From The Life
of the Cosmos.]
Quantum Computational Physicist David Deutsch: "Kurt Godel revolutionized
proof theory.... Godel proved first that any set of rules of inference that
is capable of correctly validating even the proofs of ordinary arithmetic
could never validate a proof of its own consistency.....This is called
'Godel's incompleteness theorem'.....Thanks to Godel, we know that there will
never be a fixed method of determining whether a mathematical proposition is
true, any more than there is a fixed way of determining whether a scientific
theory is true....Therefore, progress in mathematics will always depend on
the exercise of creativity." [From The Fabric of Reality]
Three additional points,
1) From what I have read, Godel's theorem is based on the assumption that a
proof can have only a finite number of steps (note the word 'assumption')
2) In a handful of books referencing Godel, nobody quotes him. This guy must
be really tough to read or something.....
3) Heisenberg's uncertainty principle is certainly similar to Godel's
theorem, but it is not the same thing. Ken Wilber considers Godel's to be a
"logical analogue" of Heisenberg.
Hope this helps,
Rog
"....all the old religions try to explain the same contents, the same
relations, and all of these hinge around questions of values......We ought to
make every effort to grasp their meaning, since it quite obviously refers to
a crucial aspect of reality; or perhaps we ought to try putting it into
modern language, if it can no longer be contained in the old."
[W. Heisenberg from Truth Lies in The Deep]
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