[FOM] FOM] Directions for Computability Theory Beyond the PureMathematical

[Peter Gerdes]

On 5/17/06, joeshipman at aol.com <joeshipman at aol.com> wrote:
[snip]
>   The evidence for such "knowledge" would not be mathematical but
>   physical, we would believe those mathematical facts for the same reaosn
>   we believe physical theories to be correct. Godel envisioned this
>   possibility.

Do you mean that our theory might (for example) stipulate that some
physical constant is calculable from the halting problem via some
specified computable algorithm.  Then we could test small fragments of
this theory (for values ZFC settles) and gain confidence in it.  Then
simplicity considerations would give us confidence that constant was
truly non-computable rather than merely some very complex computable
real that our experiments had yet to reject.

If so I agree and think this is a much better and less roundabout way
to establish the point I was trying to make.  I just wanted to clarify
that these sort of simplicity arguments where what you meant by the
knowledge would be physical.

No need to reply if this is a correct characterization.

Peter


On 5/17/06, joeshipman at aol.com <joeshipman at aol.com> wrote:
>
> There is nothing to prevent a more general physical theory than we
> currently have entailing that certain physically measurable
> dimensionless constants (such as the fine-structure constant, or lepton
> mass ratios) are DEFINABLE but NONRECURSIVE real numbers. If sich a
> theory were correct, one form of Church's thesis would be false and we
>  could, by an effective procedure, obtain a nonrecursive sequence (and
> thereby gain physical knowledge of mathematical facts which ZFC cannot
> prove, since any nonrecursive definable sequence has elements whose
> value ZFC does not settle).
>
> The evidence for such "knowledge" would not be mathematical but
> physical, we would believe those mathematical facts for the same reaosn
> we believe physical theories to be correct. Godel envisioned this
> possibility.
>
> -- JS
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Three passions, simple but overwhelmingly strong, have governed my
life: the longing for love, the search for knowledge, and unbearable
pity for the suffering of mankind.

--Bertrand Russell

-- 
Three passions, simple but overwhelmingly strong, have governed my
life: the longing for love, the search for knowledge, and unbearable
pity for the suffering of mankind.

--Bertrand Russell