Re: Robot Evolution
- From: "John Edser" <edser@xxxxxxxxxxxxxx>
- Date: Sat, 23 Dec 2006 14:36:00 -0500 (EST)
aatu.koskensilta@xxxxxxxxx wrote:-
Phil Roberts, Jr. wrote:
Godel demonstrated (1931) that for any formal (logical, mathematical,
rule driven, etc.) system capable of simple arithmetic, there is at
least one well-formed sentence or theorem, usually referred to as the
Godel or G sentence, that cannot be proven in the system.
Interestingly enough, because of our ability to attach meaning to the
symbols employed in such a proof, the G sentence is one that we humans
can quite easily "see" to be true in that its semantic interpretation
is simply: 'This sentence can not be proven in this system'.
An elementary, if widespread, confusion. Gödel's incompleteness
theorem (or, rather, Rosser's strenghtening of it), which is a
mathematical theorem no more and no less doubtful than, say, the
infinitude of primes, only tells us that given a formal theory T, there
exists, constructible by a mechanical procedure, a sentence G_T, such
that G_T is true and unprovable in T just in case T is consistent. The
only cases we can "see" that G_T is true are those in which we are
able, for whatever reasons, to see that T is consistent. For some
theories T we can indeed "see" this, in the sense of having perfectly
good reasons to conclude that T is consistent, and for some other T we
just don't have the slightest idea whether T is consistent. Consider,
for example, the theory K ) Peano airthmetic + Goldbach's conjecture.
JE:-
Since this is a science group and not a mathematics group explaining this
example in more detail would help. What appears to be your most important
statement: " ..there exists, constructible by a mechanical procedure, a
sentence G_T, such that G_T is true and unprovable in T just in case T is
consistent" does not make sense to me. I cannot fathom: "...just in case T
is consistent".
It seems to me that what you are stating is that mathematics is not real
(remains NON empirically based). Unfortunately, mathematics has to get real
(become empirically based) because human thinking remains entirely based on
_testable_ theories of nature. Mathematics is required to import assumed
constants of nature just to be able to break out of any of the enormous
number of abstract tautological assumptions (circular arguments such as
axioms, definitions of equality etc) which dominate mathematics.
Mathematical tautologies (which are not valid theories) do not exist in
isolation to non tautologous assumptions (which alone can be regarded as
valid theories). This is because without exception, all tautologous
assumptions have to be deduced from non tautologies. Mathematical circular
arguments can only exist as a deduction from NON tautologous assumptions all
of which are EMPIRICAL, i.e. all non tautologies remain outside of
mathematics. Only an identified empirical non tautology can possibly remove
mathematical incompleteness by introducing into mathematics an empirically
testable frame of reference from which all the circular arguments assumed
within mathematics can be deduced.
Regards,
John Edser
Independent Researcher
edser@xxxxxxxxxxxxxx
Regards,
John Edser
Independent Researcher
edser@xxxxxxxxxxxxxx
.
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