Mark Walker wrote:
> I have a hard time with this stuff no matter what the time of day--and
about
> half my thesis is on this, i.e., on the philosophical foundations of
logic.
> To my mind, the question is not whether we can understand violations of
our
> most cherished logical laws, but whether we could verify this is the case.
> Here is an analogy: A child of 4 is told by her loving physicist mother
that
> E=Mcc. The child has good evidence that this is true--mum wouldn't lie
about
> this--but she cannot comprehend the meaning of this statement. Does the
> child know that E=Mcc even though she cannot understand what the statement
> means?
To begin with, I have a feeling that this is a flawed analogy...
That E=mc^2 in this kind of universe is again something that even the II can
manipulate at will. What I'm against in the scenario is that the III would
be able to say that
'Given that p and not-p implies falsum for all p, me and not-me implies
not-falsum'.
A threefoil knot is not isomorphic to the ring in 3-space. When embedded in
4 or higher spaces, it is trivially isomorphic. This follows from axioms in
set theory, in topology and in knot theory -- and again my problem with the
III God is that he is supposed to be able to break against axiomatic laws
without stepping out of that axiomatic system.
If we 'solve' the threefoil by embedding it in 4-space, we step out of the
original system. In 3-space, the problem of finding such an isomorphy is ex
definitio unsolvable.
'Violations of our most cherished logical laws' cannot be understood within
their axiomatic system -- but these violations form _ANOTHER_ axiomatic
system, with its own validity, its own pecularities and its own results. The
study of it will with any luck even be rather interesting; but it doesn't
have any impact whatsoever on the axiomatic systems already in study, since
they build on different axioms.
> If you answer yes, the question then is could we be in the same
> relation to a God III. A rationalistic response is to say no, because we
> cannot verify statements that we cannot understand. Fideism allows the
> possibility that we are in exactly the same position as the child, that we
> could trust a God III who said: "hey you humans, your logic is wrong, but
> then again, that is understandable, you were built that way. The fact that
> you cannot understand this is no more telling than the fact that a child
> can't understand relativity theory." To the rationalist, the fideist looks
> irrational placing so much evidence on interpersonal trust. To the
fideist,
> the rationalist looks inconsistent because asserting the supremacy of
human
> reason can itself only be a matter of trust. Could we build a God III that
> we could not understand but who we could trust? I leave that as a homework
> assignment. Mark and Not-Mark.
I wouldn't place myself in any of these slots. It would be one thing if he
came down unto us and spake of our misunderstanding -- i.e. "Hey, you
humans! Your logic is _incompatible_with_the_universe_ you live in, but then
again, that is understandable, you were built that way..."; that's a
statement of the relationship between the mathematics and the real world,
but it still wouldn't influence the validity of the pure mathematics -- our
axiomatic systems still would be valid, they just wouldn't say anything
useful about the world around us.
Axiom 1)
Implication follows the laws we're used to.
Axiom 2)
p ^ -p --/-> F
Axiom 3)
Mark is a possible p
Theorem 1)
Mark and Not-Mark
// Mikael Johansson
This archive was generated by hypermail 2b30 : Mon May 28 2001 - 09:59:44 MDT