> > 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.
> >
> This is very helpful as it helps isolate the source of disagreement. That
> mathematics might be full of sound and fury but signifying nothing is of
> course not particularly comforting. Someone like Goedel, who thought we
had
> special intuitionistic access to the logical/mathematical structure of the
> universe, I am sure is rolling over in his grave at your suggestion that
> this might be the case.
Lol!
> > 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.
> >
> >
> In response let me quote from the great fideist Descartes, (who often goes
> around in rationalistic drag):
>
> "I would dare not even dare to say that God cannot arrange that a mountain
> should exist without a valley, or that one and two should not make three;
> but I only say that He has given me a mind of such a nature that I cannot
> conceive a mountain without a valley or a sum of one and two which would
not
> be three, and so on, and that such things imply contradictions in my
> conception." (Letter to Arnauld, 29 July 1648).
Oh, the second example would be easy:
1+2=0 (mod 3)
:-)
Thus, there already exists an axiomatic system in which the sum of one and
two is not three... :-)
> With respect to your example, Descartes would have to say he cannot
conceive
> of finding such an isomorphy in 3-space because God has given him such a
> mind, but he dare not say that god could not find such a solution. Of
course
> you will say that this would be to change the axiom system. Descartes will
> respond that this may not be so, to assume that it must be the case is to
> assume that your understanding extends as far as God's will along this
> vector. Of course what the transhumanist fideist must do here is
substitute
> a created God III here for God.
> You didn't warm to my last analogy but let me try another. Children
> sometimes have a difficult time grasping the concept of volume, the usual
> mistake is to think that a squat beaker which holds say 300 ml is smaller
> than a tall slender beaker of equal volume. So if you ask a child what
will
> happen if you pour the contents of the full tall beaker into the empty
squat
> beaker they say that it will overflow. Even when one demonstrates that the
> two beakers are of equal volume, by pouring the liquid from one to the
> other, some still think that the taller beaker holds more liquid. They
> sometimes think that there is some slight of hand going on. On this
analogy,
> then, what looks impossible to us, the 3-space solution may be like the
> child thinking that the two beakers are equivalent in volume. To say that
> any solution requires changing the axioms is like the child saying there
> must be some slight of hand--what the child believes is impossible is
indeed
> entirely possible. The fideist response then is whether you might believe
> the magnificently powerful being who claims to be a God III and says that
> there is a 3-space solution within the axioms given, it is just that
humans
> are too stupid to understand it. Could you imagine your trust going this
> far?
Oh yes, I could imagine the trust going this far -- IFF we wouldn't already
have proven -- within that system -- that it is impossible.
There is a different between improbable and impossible theorems within an
axiomatic system. When it comes to a theorem that is proven not to hold,
then I will not let my trust go 'that far'. If it is about an hypothesis
that something may hold, for which neither positive nor negative proof
exists, I readily believe already that it might hold or might not even
though we have not yet seen it.
// Mikael Johansson
This archive was generated by hypermail 2b30 : Mon May 28 2001 - 09:59:44 MDT