On 2/9/2012 3:40 PM, acw wrote:
I think the idea of Platonia is closer to the fact that if a sentence
has a truth-value, it will have that truth value, regardless if you
know it or not.
Sure, but it is not just you to whom a given sentence may have the
same
exact truth value. This is like Einstein arguing with Bohr with the
quip: "The moon is still there when I do not see it." My reply to
Einstein would be: Sir, you are not the only observer of the moon! We
have to look at the situation from the point of view of many observers
or, in this case, truth detectors, that can interact and communicate
consistently with each other. We cannot think is just solipsistic
terms.
Sure, but what if nobody is looking at the moon? Or instead of moon,
pick something even less likely to be observed. To put it differently,
Riemann hypothesis or Goldbach's conjecture truth-value should not
depend on the observers thinking of it - they may eventually discover
it, and such a discovery would depend on many computational
consequences, of which the observers may not be aware of yet, but
doesn't mean that those consequences don't exist - when the
computation is locally performed, it will always give the same result
which could be said to exist timelessly.
[SPK]
My point is that any one or thing that could be affected by the truth
value of "the moon has X, Y, Z properties" will, in effect, be an
observer of the moon since it is has a definite set of properties as
"knowledge". The key here is causal efficacy, if a different state of
affairs would result if some part of the world is changed then the
conditions of that part of the world are "observed". The same thing
holds for the truth value Riemann hypothesis or Goldbach's conjecture,
since there would be different worlds for each of their truth values. My
point is that while the truth value or reality of the moon does not
depend on the observation by any _one_ observer, it does depend for its
definiteness on the possibility that it could be observed by some
observer. It is the possibility that makes the difference. A object that
cannot be observer by any means, including these arcane versions that I
just laid out, cannot be said to have a definite set of properties or
truth value, to say the opposite is equivalent to making a truth claim
about a mathematical object for whom no set of equations or
representation can be made.
You're conjecturing here that there were worlds where Riemann
hypothesis or Goldbach's conjecture have different truth values. I
don't think arithmetical truths which happen to have proofs have
indexical truth values, this is due to CTT. Although most physical
truths are indexical (or depend on the axioms chosen).
We could limit ourselves to decidable arithmetical truths only, but
you'd bump into the problem of consistency of arithmetic or the
halting problem. It makes no sense to me that a machine which is
defined to either halt or not halt would not do either. We might not
know if a machine halts or not, but that doesn't mean that if when ran
in any possible world it would behave differently. Arithmetical truth
should be the same in all possible worlds. An observer can find out a
truth value, but it cannot "alter" it, unless it is an indexical
(context-dependent truth, such as "what time it is now" or "where do
you live").
Of course, we cannot talk about the truth value of undefined stuff,
that would be non-sense. However, we can talk about the truth value of
what cannot be observed - "this machine never halts" is only true if
no observation of the machine halting can ever be made, in virtue of
how the machine is defined, yet someone could use various
meta-reasoning to reach the conclusion that the machine will never
halt (consistency of arithmetic is very much similar to the halting
problem - it's only consistent if a machine which enumerates proofs
never finds a proof of "0=1"; of course, this is not provable within
arithmetic itself, thus it's a provably unprovable statement for any
consistent machine, thus can only be a matter of "theology" as Bruno
calls it).
Hi ACW,
I am considering that the truth value is a function of the theory
with which a proposition is evaluated. In other words, meaningfulness,
including truth value, is contextual while existence is absolute.
Onward!
Stephen
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