Re: Robot Evolution
- From: "John Edser" <edser@xxxxxxxxxxxxxx>
- Date: Tue, 26 Dec 2006 00:18:55 -0500 (EST)
Aatu Koskensilta aatu.koskensilta@xxxxxxxxx wrote:-
JE:-this
Since this is a science group and not a mathematics group explaining
example in more detail would help. What appears to be your mostimportant
statement: " ..there exists, constructible by a mechanical procedure, ais
sentence G_T, such that G_T is true and unprovable in T just in case T
consistent" does not make sense to me. I cannot fathom: "...just incase T
is consistent".
A formal theory T is said to be consistent if there is no sentence A
such that both A and the negation of A are formally provable in T.
JE:-
Hello Astu,
I hope you bear with us for a little longer because sbe badly needs the
council of a logician who is also interested in evolutionary theory.
To express what you wrote in just a simple way: contradictions are not
allowed within any consistent theory. Is this correct?
Gödel's first incompleteness theorem shows that, for formal theories T
meeting certain criteria, it is possible to find a formula G_T with the
property that G_T is true just in case T is consistent, i.e. G_T is true
if T is consistent, and false if T is inconsistent.
JE:-
It appears G is deducible from T but not the reverse. G remains predicated
on T which cannot contain a contradiction (must remain consistent). If this
is not the case then G-T only remains a tautology. In the non tautologous
sense: if the T predicate does contain a contradiction (remains
inconsistent) then it is false so that any deduction from it, in this case
G, must also be false. I can't see what is new here because one of the
basics of reasoning is that if the predicate is false then any deduction
from it must also be false. However, if the subject is true it is entirely
possible to induce a false predicate from it (please refer to my post on
material implication).
Your comments beyond the above were not in any way related to anything I
said.
JE:-
I disagree. Evolutionary theory is a SCIENCE yet most gene centric Neo
Darwinistic reasoning remains predicated on just tautologous propositions of
mathematics. In my opinion these Neo Darwinist models have been misused.
Importing tautologies from mathematics into evolutionary science renders any
mathematical model of evolutionary science, oversimplified. This allows a
reverse of cause and effect within the model compared to the theory it was
oversimplified from. My detailed example here (there are many others) is
Hamilton's rule: rb>c. This rule does not define a single constant term
providing no frame of reference. Yet, it has been employed for over 50 years
to allow the evolution of organism fitness altruism which Darwinism
prohibits as a refutation of Darwinism. Without a testable frame of
reference Hamilton's Rule has no basis in science, only mathematics.
I claim that Hamilton's Rule (or any other mathematical tautology imported
into evolutionary science) can only be broken using a constant term
_necessarily_ imported from _outside_ of mathematics. IOW what Gödel was
driving at as far as the epistemology of science was concerned is that
mathematics remains entirely deducible from non mathematics and not the
reverse. This is what connects Gödel to most evolutionary theory arguments.
Logicians may not be very interested but evolutionary theorists have to be
because offering oversimplified mathematical models in place of their parent
theory constitutes an enormous error of science no matter that these models
remain mathematically valid.
I just wished to make the rather trivial observation that the
incompleteness theorem establishes an implication, "if T is consistent,
then the Gödel sentence of T is true but unprovable in T", ..
JE:-
As I understand this: If T contains no contradiction then it remains a
valid induction, the truth of which cannot be proven. This is because T on
its own remains just one isolated inductive assumption. Karl Popper argued
that because the sciences are empirically based the truth of T can be tested
because T is absolutely required to remain refutable against nature via
every possible valid deduction from T. As an example, Newton's assumption of
m and t as constants (providing a necessary frame of reference for Newtonian
Mechanics) was empirically refuted via the Michelson-Morley experiment which
proved that the velocity of light was not just a variable.
http://en.wikipedia.org/wiki/Michelson-Morley_experiment
...and unless we
happen to know of some particular T that it is consistent, no way of
"seeing" the truth of the Gödel sentence is to be found in the proof of
the first incompleteness theorem.
JE:-
Gödel appears to have outlined the mystery of the inductive process. Nobody
knows how the brain makes an inspired (or just any) guess, i.e. produces a
inductive inference. Within the sciences any induction is allowed (the more
the better), if and only if, they remain refutable against nature allowing
some to be discarded in favor of others. The problem is, most people (even
scientists) are easily seduced by claims of irrefutability because this
seems to provide an illusion of certainty. Many of these certainties are
just tautologous assumptions of mathematics.
Regards,
John Edser
Independent Researcher
edser@xxxxxxxxxxxxxx
.
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