Re: Need help on difference between statements being provable and being true.
- From: "T.H. Ray" <thray123@xxxxxxx>
- Date: Fri, 18 Dec 2009 09:03:58 EST
Darren wrote
Thank you for your response. You have given me quite
a few things to
chew on.
Thanks. Correcting an error above, the Popper twin
primes formulation is H: for every natural number x > 2,
there exists at least one natural number such that x + y
and (2 + x) + y are both prime.
The only difference between the Goldbach and twin primes
conjectures is the sign change, + y or - y in the second
pair of terms. Goldbach is falsifiable (in principle)
by iterated calculation; i.e., a program to test the
conjecture potentially halts when it finds a
counterexample. The twin primes conjecture could never
be tested in this manner, because it requires infinite
iterations (in principle). We already know (by Euclid's
classical proof), that prime integers are infinite.
Why does the verification of Goldbach, up to the huge
number of terms so far, not constitute an acceptable
proof? -- To say such and such is a true mathematical
statement, one must be able to explicitly verify not
only the result of calculation, but the terms of
existence in which the theorem coheres. Mathematicians
are not usually interested in conjectures that don't
promise a larger domain of knowledge. If the range
is limited to that consistent arithmetic system S,
then the meaning is also limited to S; we don't learn
anything new. For example, Euclidean geometry endured
as the only geometry imaginable for a couple of thousand
years because the consistency of Euclid's postulates
is rock-solid. Yet we can only really appreciate that
consistency now that we know how to limit Euclidean
geometry to its domain, by substituting another postulate
for Euclid's fifth and doing geometry another way.
Theorems can be proven in both systems, and both are
consistent.
Point is, the rise of non-Euclidean geometry showed us
that meaning supersedes language; i.e., we know more
about geometry only because we can speak another
language of geometry.
Would a proof of Goldbach or twin primes teach us
anything more about arithmetic? We don't know. Is
there an arithmetic as strong as ZFC that yields novel
theorems not true in ZFC, comparable to the results
of Euclidean vs non-Euclidean geometry?
I only meant to correct my error, and I apologize if
I ran on too long.
Tom
.
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